text
stringlengths 59
500k
| subset
stringclasses 6
values |
|---|---|
Received June 15, 2017. Published online August 6, 2018.
Abstract: A $*$-ring $R$ is strongly 2-nil-$*$-clean if every element in $R$ is the sum of two projections and a nilpotent that commute. Fundamental properties of such $*$-rings are obtained. We prove that a $*$-ring $R$ is strongly 2-nil-$*$-clean if and only if for all $a\in R$, $a^2\in R$ is strongly nil-$*$-clean, if and only if for any $a\in R$ there exists a $*$-tripotent $e\in R$ such that $a-e\in R$ is nilpotent and $ea=ae$, if and only if $R$ is a strongly $*$-clean SN ring, if and only if $R$ is abelian, $J(R)$ is nil and $R/J(R)$ is $*$-tripotent. Furthermore, we explore the structure of such rings and prove that a $*$-ring $R$ is strongly 2-nil-$*$-clean if and only if $R$ is abelian and $R\cong R_1, R_2$ or $R_1\times R_2$, where $R_1/J(R_1)$ is a $*$-Boolean ring and $J(R_1)$ is nil, $R_2/J(R_2)$ is a $*$-Yaqub ring and $J(R_2)$ is nil. The uniqueness of projections of such rings are thereby investigated. | CommonCrawl |
How much information is stored in a genome and how much in the distribution of genomes in a species?
Does regularity of distributions have anything to do with definiteness of their product?
Complex Tensors in General Relativity
Resources for theory of distributions (generalized functions) for physicists
Legal values of spin-1/2 field can take: $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, .. (Grassmann)?
Decoupling of Holomorphic and Anti-holomorphic parts in 2D CFT
What really is a Dirac delta function?
Multiparticle Quantum Mechanics with independent phases
Principal value of 1/x and few questions about complex analysis in Peskin's QFT textbook
When I learn QFT, I am bothered by many problems in complex analysis.
1) $$\frac{1}{x-x_0+i\epsilon}=P\frac{1}{x-x_0}-i\pi\delta(x-x_0)$$
I can't understand why $1/x$ can have a principal value because it's not a multivalued function. I'm very confused. And when I learned the complex analysis, I've not watched this formula, can anybody tell me where I can find this formula's proof.
2) $$\frac{d}{dx}\ln(x+i\epsilon)=P\frac{1}{x}-i\pi\delta(x)$$
3) And I also find this formula. Seemingly $f(x)$ has a branch cut, then $$f(z)=\frac{1}{\pi}\int_Z^{\infty}dz^{\prime}\frac{{\rm Im} f(z^{\prime})}{z^{\prime}-z}$$ Can anyone can tell the whole theorem and its proof, and what it wants to express.
Now I am very confused by these formula, because I haven't read it in any complex analysis book and never been taught how to handle an integral with branch cut. Can anyone give me the whole proof and where I can consult.
complex-numbers
asked Mar 30, 2014 in Theoretical Physics by user34669 (205 points) [ no revision ]
Unfortunately, there are two completely unrelated meanings of the term "principal value". The kind referred to here is the Cauchy principal value, which assigns values to otherwise undefined improper integrals. This has nothing to do with the principal value you had in mind, which is for selecting single-valued branches of multi-valued functions. I know, it's stupid. You'd think someone would have fixed all these weird ambiguities by now, but alas math is not French.This post imported from StackExchange Physics at 2014-03-30 12:29 (UCT), posted by SE-user David H
answered Mar 30, 2014 by David H (90 points) [ no revision ]
@DavidH Thanks a lot! Then the last question, can you give me some clues?
commented Mar 30, 2014 by user34669 (205 points) [ no revision ]
@user34669 I think the last expression goes by the name of "Kramers-Kronig" relation, it is a way to express a complex function in its real or imaginary part. So with either the real or imaginary part, you can reconstruct the whole function. For a proof, see en.wikipedia.org/wiki/Kramers%E2%80%93Kronig_relations
This post imported from StackExchange Physics at 2014-03-30 12:29 (UCT), posted by SE-user Funzies
commented Mar 30, 2014 by Funzies (5 points) [ no revision ]
Identities like these are often used to perform calculations with Greens functions. In particular, given a Hamiltonian H, its resolvent
\((ω-H)^{-1} = \sum_n \frac1{ω-E_n} |ψ_n\rangle\langleψ_n|\)
is an object of key interest. It's essentially the Fourier transform of the corresponding Greens function.
Now, the thing about the resolvent is that is analytic for all complex values ω that are not in the spectrum of the Hamiltonian H. Hence, we can apply identities from complex analysis to perform computations.
One of the key observation of complex analysis is that complex analytic functions are very rigid: knowing the values of a function \(f\)on a countable set of points with a limit point is enough to reconstruct the whole function \(f\). The most prominent example is Cauchy's integral formula, which states that if a function is complex analytic on and inside a disk, then the values inside can be reconstructed entirely from the values on the boundary, via the following integral formula:
\(f(ω) = \frac1{2πi} \oint_{\text{circle}} dz\frac{f(z)}{z-ω}\)
The identities you mention in your question are used for the same purpose, except that in this case, we want to reconstruct a function not from its values on a circle, but from its values on the real axis.
Formula 1 is to be understood in the sense of distributions: multiply with a test function \(g\)and integrate over the whole real axis. Then, the results are equal:
\(\int^∞_{-∞} dx \frac{g(x)}{x-x_0+iε} = \mathcal{P} \int_{-∞}^∞dx \frac{g(x)}{x-x_0}- iπg(x_0)\)
(We assume that g is any continuous, complex-valued test function that decays sufficiently fast at infinity.)
Formula 2 is actually formula 1 after you perform the differentiation on the left-hand side.
Formula 3 can be derived from formula 1 for the special case where the function g is related to the values of a function f which is complex analytic in the upper and lower half plane. It is similar Cauchy's integral formula in the sense that you reconstruct the values of f in the complex plane from just the imaginary part of its values on the real axis.
My favorite example for these kinds of calculations is the function \(f(z) = \sqrt{z}\). It is complex analytic everywhere, except for the negative real numbers, where this function has a branch cut. Plug this into the formula and try to calculate the result, doing so will hopefully help you understand what is going on. (Note that the branch cut is from -∞ to 0 here, i.e. mirrored compared to formula 3.)
answered Apr 4, 2014 by Greg Graviton (775 points) [ no revision ] | CommonCrawl |
Numerical solution of bilateral obstacle optimal control problem, where the controls and the obstacles coincide
NACO Home
Complex and quaternionic optimization
September 2020, 10(3): 257-273. doi: 10.3934/naco.2020001
Resource allocation: A common set of weights model
Sedighe Asghariniya 1,, , Hamed Zhiani Rezai 2, and Saeid Mehrabian 1,
Department of Mathematics, Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran
Department of Mathematics, Islamic Azad University, Mashhad Branch, Mashhad, Iran
* Corresponding author: Sedighe Asghariniya
Received November 2018 Revised October 2019 Published February 2020
Figure(4) / Table(5)
Allocation problem is an important issue in management. Data envelopment analysis (DEA) is a non-parametric method for assessing a set of decision making units (DMUs). It has proven to be a useful technique to solve allocation problems. In recent years, many papers have been published in this regard and many researchers have tried to find a suitable allocation model based on DEA. Common set of weights (CSWs) is a DEA model which, in contrast with traditional DEA models, does not allow individual weights for each decision making unit. In this manner, all DMUs are assessed through choosing a same set of weights. In this article, we will use the weighted-sum method to solve the multi-objective CSW problem. Then, via introducing a set of special weights, we will connect the CSW model to a non-linear (fractional) CSW model. After linearization, the proposed model is used for allocating resources. To illustrate our model, some examples are also provided.
Keywords: Data Envelopment Analysis, Weighted-Sum Method, Common Set of Weights, Resource Allocation, Multi Objective Problem, Linear Problem.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Sedighe Asghariniya, Hamed Zhiani Rezai, Saeid Mehrabian. Resource allocation: A common set of weights model. Numerical Algebra, Control & Optimization, 2020, 10 (3) : 257-273. doi: 10.3934/naco.2020001
Y. Almogy and O. Levin, A class of fractional programming problems, Operations Research, 19 (1971), 57-67. doi: 10.1287/opre.19.1.57. Google Scholar
A. Amirteimoori and S. Kordrostami, Allocating fixed costs and target setting: A dea-based approach, Applied Mathematics and Computation, 171 (2005), 136-151. doi: 10.1016/j.amc.2005.01.064. Google Scholar
R. D. Banker, Estimating most productive scale size using data envelopment analysis, European Journal of Operational Research, 17 (1984), 35-44. Google Scholar
J. E. Beasley, Allocating fixed costs and resources via data envelopment analysis, European Journal of Operational Research, 147 (2003), 198-216. Google Scholar
G. Bi, J. Ding, Y. Luo and and L. Liang, Resource allocation and target setting for parallel production system based on dea, Applied Mathematical Modelling, 35 (2011), 4270-4280. doi: 10.1016/j.apm.2011.02.048. Google Scholar
A. Biswas and A.K. De, A priority based fuzzy programming approach for multiobjective probabilistic linear fractional programming, In 2013 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), (2013), 1–6. Google Scholar
M. Carrillo and J. M. Jorge, A multiobjective dea approach to ranking alternatives, Expert Systems with Applications, 50 (2016), 130-139. Google Scholar
M. Chakraborty and S. Gupta, Fuzzy mathematical programming for multi objective linear fractional programming problem, Fuzzy Sets and Systems, 125 (2002), 335-342. doi: 10.1016/S0165-0114(01)00060-4. Google Scholar
C. T. Chang, Fuzzy linearization strategy for multiple objective linear fractional programming with binary utility functions, Computers & Industrial Engineering, 112 (2017), 437-446. Google Scholar
A. Charnes and W. W. Cooper, Programming with linear fractional functionals, Naval Research Logistics Quarterly, 9 (1962), 181-186. doi: 10.1002/nav.3800090303. Google Scholar
A. Charnes, W. W. Cooper and E. Rhodes, Measuring the efficiency of decision making units, European Journal of Operational Research, 2 (1978), 429-444. doi: 10.1016/0377-2217(78)90138-8. Google Scholar
C. Chiang, M. J. Hwang and Y. H. Liu, etermining a common set of weights in a dea problem using a separation vector, Mathematical and Computer Modelling, 54 (2011), 2464-2470. doi: 10.1016/j.mcm.2011.06.002. Google Scholar
C. I. Chiang and G. H. Tzeng, A new efficiency measure for dea: efficiency achievement measure established on fuzzy multiple objectives programming, Journal of Management, 17 (2000), 369-388. Google Scholar
W. D. Cook and M. Kress, Characterizing an equitable allocation of shared costs: A dea approach, European Journal of Operational Research, 119 (1999), 652-661. Google Scholar
W. D. Cook, Y. Roll and A. Kazakov, A dea model for measuring the relative eeficiency of highway maintenance patrols, INFOR: Information Systems and Operational Research, 28 (1990), 113-124. Google Scholar
W. D. Cook and J. Zhu, Allocation of shared costs among decision making units: A dea approach, Computers & Operations Research, 32 (2005), 2171-2178. Google Scholar
W. W. Cooper, L. M. Seiford and K. Tone, Data Envelopment Analysis: A Comprehensive Text with Models, Applications, References and DEA-Solver Software, 2$^{nd}$ edition, Springer, New York, 2006. Google Scholar
J. P. Costa, Computing non-dominated solutions in molfp, European Journal of Operational Research, 181 (2007), 1464-1475. Google Scholar
J. P. Costa and M. J. Alves, A reference point technique to compute nondominated solutions in molfp, Journal of Mathematical Sciences, 161 (2009), 820-831. doi: 10.1007/s10958-009-9603-z. Google Scholar
Y. Dai, J. Shi and S. Wang, Conical partition algorithm for maximizing the sum of dc ratios, Journal of Global Optimization, 31 (2005), 253-270. doi: 10.1007/s10898-004-5699-3. Google Scholar
P. K. De and M. Deb, Solution of multi objective linear fractional programming problem by taylor series approach, In 2015 International Conference on Man and Machine Interfacing (MAMI), pages 1–5. IEEE, 2015. Google Scholar
W. Dinkelbach, On nonlinear fractional programming, Management Science, 13 (1967), 492-498. doi: 10.1287/mnsc.13.7.492. Google Scholar
J. Du, W. D. Cook, L. Liang and J. Zhu, Fixed cost and resource allocation based on dea cross-efficiency, European Journal of Operational Research, 235 (2014), 206-214. doi: 10.1016/j.ejor.2013.10.002. Google Scholar
M. Ehrgott, Multicriteria Optimization, 2$^{nd}$ edition, Springer, Berlin, 2005. Google Scholar
J. E. Falk and S. W. Palocsay, Image space analysis of generalized fractional programs, Journal of Global Optimization, 4 (1994), 63-88. doi: 10.1007/BF01096535. Google Scholar
N. Güzel, A proposal to the solution of multiobjective linear fractional programming problem, Abstract and Applied Analysis, 2013 (2013), Article ID 435030, 4 pages. doi: 10.1155/2013/435030. Google Scholar
G. R. Jahanshahloo, A. Memariani, F. Hosseinzadeh Lotfi and H. Z. Rezai, A note on some of dea models and finding efficiency and complete ranking using common set of weights, Applied Mathematics and Computation, 166 (2005), 265-281. doi: 10.1016/j.amc.2004.04.088. Google Scholar
G. R. Jahanshahloo, J. Sadeghi and M. Khodabakhshi, Proposing a method for fixed cost allocation using dea based on the efficiency invariance and common set of weights principles, Mathematical Methods of Operations Research, 85 (2017), 223-240. doi: 10.1007/s00186-016-0563-z. Google Scholar
G. R. Jahanshahloo, B. Talebian, F. Hosseinzadeh Lotfi and J. Sadeghi, Finding a solution for multi-objective linear fractional programming problem based on goal programming and data envelopment analysis, RAIRO-Operations Research, 51 (2017), 199-210. doi: 10.1051/ro/2016014. Google Scholar
C. Kao and H. T. Hung, Data envelopment analysis with common weights: the compromise solution approach, Journal of the Operational Research Society, 56 (2005), 1196-1203. Google Scholar
J. SH. Kornbluth and R. E. Steuer, Multiple objective linear fractional programming, Management Science, 27 (1981), 1024-1039. Google Scholar
T. Kuno, A branch-and-bound algorithm for maximizing the sum of several linear ratios, Journal of Global Optimization, 22 (2002), 155-174. doi: 10.1023/A:1013807129844. Google Scholar
F. Li, J. Song, A. Dolgui and L. Liang, Using common weights and efficiency invariance principles for resource allocation and target setting, International Journal of Production Research, 55 (2017), 4982-4997. Google Scholar
F. Li, Q. Zhu and L. Liang, Allocating a fixed cost based on a dea-game cross efficiency approach, Expert Systems with Applications, 96 (2018), 196-207. doi: 10.1007/s11424-015-4211-0. Google Scholar
Y. Li, F. Yang, L. Liang and Z. Hua, Allocating the fixed cost as a complement of other cost inputs: A dea approach, European Journal of Operational Research, 197 (2009), 389-401. doi: 10.1016/j.ejor.2008.06.017. Google Scholar
Y. Li, M. Yang, Y. Chen, Q. Dai and L. Liang, Allocating a fixed cost based on data envelopment analysis and satisfaction degree, Omega, 41 (2013), 55-60. Google Scholar
R. Y. Lin., Allocating fixed costs or resources and setting targets via data envelopment analysis, Applied Mathematics and Computation, 217 (2011), 6349-6358. doi: 10.1016/j.amc.2011.01.008. Google Scholar
R. Y. Lin and Z. Chen, Fixed input allocation methods based on super ccr efficiency invariance and practical feasibility, Applied Mathematical Modelling, 40 (2016), 5377-5392. doi: 10.1016/j.apm.2015.06.039. Google Scholar
F. H. F. Liu and H. H. Peng, Ranking of units on the dea frontier with common weights, Computers & Operations Research, 35 (2008), 1624-1637. Google Scholar
F. Hosseinzadeh Lotfi, A. Hatami-Marbini, P.J. Agrell, N. Aghayi and K. Gholami, Allocating fixed resources and setting targets using a common-weights dea approach, Computers & Industrial Engineering, 64 (2013), 631-640. Google Scholar
F. Hosseinzadeh Lotfi, A. A. Noora, G. R. Jahanshahloo, M. Khodabakhshi and A. Payan, A linear programming approach to test efficiency in multi-objective linear fractional programming problems, Applied Mathematical Modelling, 34 (2010), 4179-4183. doi: 10.1016/j.apm.2010.04.015. Google Scholar
S. Lozano and G. Villa, Centralized resource allocation using data envelopment analysis, Journal of Productivity Analysis, 22 (2004), 143-161. Google Scholar
OB. Olesen, Some unsolved problems in data envelopment analysis: A survey, International Journal of Production Economics, 39 (1995), 5-36. Google Scholar
S. Ramezani-Tarkhorani, M. Khodabakhshi, S. Mehrabian and F. Nuri-Bahmani, Ranking decision-making units using common weights in dea, Applied Mathematical Modelling, 38 (2014), 3890-3896. doi: 10.1016/j.apm.2013.08.029. Google Scholar
N. Ramón, J. Ruiz and I. Sirvent, Common sets of weights as summaries of dea profiles of weights: With an application to the ranking of professional tennis players, Expert Systems with Applications, 39 (2012), 4882-4889. Google Scholar
Y. Roll, W. D. Cook and B. Golany, Controlling factor weights in data envelopment analysis, IIE transactions, 23 (1991), 2-9. Google Scholar
S. Saati, A. Hatami-Marbini, P. J. Agrell and M. Tavana, A common set of weight approach using an ideal decision making unit in data envelopment analysis, Journal of Industrial and Management Optimization, 8 (2012), 623-637. doi: 10.3934/jimo.2012.8.623. Google Scholar
S. Sadia, N. Gupta, Q. M. Ali and A. Bari, Solving multi-objective linear plus linear fractional programming problem, Journal of Statistics Applications & Probability, 4 (2015), 253-258. Google Scholar
X. Si, L. Liang, G. Jia, L. Yang, H. Wu and Y. Li, Proportional sharing and dea in allocating the fixed cost, Applied Mathematics and Computation, 219 (2013), 6580-6590. doi: 10.1016/j.amc.2012.12.085. Google Scholar
R. E. Steuer, Multiple Criteria Optimization, Wiley, New York, 1986. Google Scholar
E. Valipour, M. A. Yaghoobi and M. Mashinchi, An iterative approach to solve multiobjective linear fractional programming problems, Applied Mathematical Modelling, 38 (2014), 38-49. doi: 10.1016/j.apm.2013.05.046. Google Scholar
H. Yan, Q. Wei and G. Hao, Dea models for resource reallocation and production input/output estimation, European Journal of Operational Research, 136 (2002), 19-31. doi: 10.1016/S0377-2217(01)00046-7. Google Scholar
Z. Yang and Q. Zhang, Resource allocation based on dea and modified shapley value, Applied Mathematics and Computation, 263 (2015), 280-286. doi: 10.1016/j.amc.2015.04.063. Google Scholar
M. Zohrehbandian, A. Makui and A. Alinezhad, A compromise solution approach for finding common weights in dea: An improvement to kao and hung's approach, Journal of the Operational Research Society, 61 (2010), 604-610. Google Scholar
Figure 1. The graph of the weighted-sum example
Figure Options
Download as PowerPoint slide
Figure 2. PPS in a two dimensions case
Figure 3. PPS$ _{CCR} $ for the peresentedDMUs in Table 1
Figure 4. virtual plane of the example
Table 1. Information related to example
DMU A B C D Agregated
input1 1 2 6 3 12
output 1 1 1 1 4
Table 2. Optimal solution of model (5) and virtual inputs and outputs connected with it for the example
$ u^* $ $ v^* $ $ ({v^ * }{x_1},{u^ * }{y_1}) $ $ ({v^ * }{x_2},{u^ * }{y_2}) $ $ ({v^ * }{x_3},{u^ * }{y_3}) $ $ ({v^ * }{x_4},{u^ * }{y_4}) $ $ ({v^ * }{\bar x},{u^ * }{\bar y}) $
$ \frac{8}{47} $ $ (\frac{8}{47},\frac{1}{47}) $ $ (\frac{8}{47},\frac{8}{47}) $ $ (\frac{8}{47},\frac{8}{47}) $ $ (\frac{10}{47},\frac{8}{47}) $ $ (\frac{12}{47},\frac{8}{47}) $ $ (\frac{32}{47},\frac{47}{47}) $
Table 3. Data set of [14]
DMU Input1 Input2 Input3 Output1 Output2 $ EFF_CCR $
1 350 39 9 67 751 0.75663
6 360 29 17 83 1070 0.96112
7 540 18 10 72 457 0.85863
9 323 25 5 75 1074 1.00000
10 444 64 6 74 1072 0.83102
11 323 25 5 25 350 0.33325
12 444 64 6 104 1199 1.00000
Table 4. Allocated cost to DMUs obtained by different methods
DMU Allocated resource Efficiency DMU Allocated resource Efficiency
1 8.47412 1.00000 7 4.84712 1.00000
3 6.45563 1.00000 9 12.32408 1.00000
4 7.56372 1.00000 10 12.13035 1.00000
5 4.96678 1.00000 11 3.76675 1.00000
6 12.22981 1.00000 12 13.65374 1.00000
Table 5. Different allocation in selected methods
Eff. invariance Output orientation Input orientation
no no no no no no yes no yes
our Besley Du et al. Li et al. Hossein Zadeh Si et al. Cook and Lin and Yang and
DMU approach [4] [23] [34] Lotfi et al. [40] [49] Kress [14] Chen [38] Zhang [53]
1 8.47 6.78 5.79 5.54 8.20 7.65 14.52 9.83 7.54
2 6.91 7.21 7.95 7.53 7.46 8.41 6.74 7.53 8.65
4 7.56 8.47 11.10 7.87 9.30 8.11 5.6 5.20 9.05
6 12.23 10.06 13.49 11.50 15.37 9.57 8.15 9.10 8.81
7 4.85 5.09 7.10 5.90 0 8.33 8.86 5.85 8.17
9 12.32 15.11 16.68 11.90 16.33 8.65 7.31 8.07 10.46
10 12.13 10.08 5.42 11.38 11.60 8.35 10.08 9.96 8.01
11 3.77 1.58 0 2.74 0 2.80 7.31 8.07 4.55
12 13.65 13.97 10.41 14.14 15.31 11.85 10.08 12.56 9.12
Gap 9.88 13.53 16.68 11.40 16.33 9.05 8.92 7.36 5.91
Cheng-Kai Hu, Fung-Bao Liu, Cheng-Feng Hu. Efficiency measures in fuzzy data envelopment analysis with common weights. Journal of Industrial & Management Optimization, 2017, 13 (1) : 237-249. doi: 10.3934/jimo.2016014
Saber Saati, Adel Hatami-Marbini, Per J. Agrell, Madjid Tavana. A common set of weight approach using an ideal decision making unit in data envelopment analysis. Journal of Industrial & Management Optimization, 2012, 8 (3) : 623-637. doi: 10.3934/jimo.2012.8.623
Henri Bonnel, Ngoc Sang Pham. Nonsmooth optimization over the (weakly or properly) Pareto set of a linear-quadratic multi-objective control problem: Explicit optimality conditions. Journal of Industrial & Management Optimization, 2011, 7 (4) : 789-809. doi: 10.3934/jimo.2011.7.789
Gholam Hassan Shirdel, Somayeh Ramezani-Tarkhorani. A new method for ranking decision making units using common set of weights: A developed criterion. Journal of Industrial & Management Optimization, 2020, 16 (2) : 633-651. doi: 10.3934/jimo.2018171
Habibe Zare Haghighi, Sajad Adeli, Farhad Hosseinzadeh Lotfi, Gholam Reza Jahanshahloo. Revenue congestion: An application of data envelopment analysis. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1311-1322. doi: 10.3934/jimo.2016.12.1311
Qinglan Xia, Shaofeng Xu. On the ramified optimal allocation problem. Networks & Heterogeneous Media, 2013, 8 (2) : 591-624. doi: 10.3934/nhm.2013.8.591
Thomas R. Cameron, Sebastian Charmot, Jonad Pulaj. On the linear ordering problem and the rankability of data. Foundations of Data Science, 2021, 3 (2) : 133-149. doi: 10.3934/fods.2021010
Pooja Bansal, Aparna Mehra. Integrated dynamic interval data envelopment analysis in the presence of integer and negative data. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021023
Mahdi Mahdiloo, Abdollah Noorizadeh, Reza Farzipoor Saen. Developing a new data envelopment analysis model for customer value analysis. Journal of Industrial & Management Optimization, 2011, 7 (3) : 531-558. doi: 10.3934/jimo.2011.7.531
Michael V. Klibanov, Dinh-Liem Nguyen, Loc H. Nguyen, Hui Liu. A globally convergent numerical method for a 3D coefficient inverse problem with a single measurement of multi-frequency data. Inverse Problems & Imaging, 2018, 12 (2) : 493-523. doi: 10.3934/ipi.2018021
Reza Alizadeh Foroutan, Javad Rezaeian, Milad Shafipour. Bi-objective unrelated parallel machines scheduling problem with worker allocation and sequence dependent setup times considering machine eligibility and precedence constraints. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021190
Mohammad Afzalinejad, Zahra Abbasi. A slacks-based model for dynamic data envelopment analysis. Journal of Industrial & Management Optimization, 2019, 15 (1) : 275-291. doi: 10.3934/jimo.2018043
Cheng-Kai Hu, Fung-Bao Liu, Hong-Ming Chen, Cheng-Feng Hu. Network data envelopment analysis with fuzzy non-discretionary factors. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1795-1807. doi: 10.3934/jimo.2020046
Wen-Hung Wu, Yunqiang Yin, Wen-Hsiang Wu, Chin-Chia Wu, Peng-Hsiang Hsu. A time-dependent scheduling problem to minimize the sum of the total weighted tardiness among two agents. Journal of Industrial & Management Optimization, 2014, 10 (2) : 591-611. doi: 10.3934/jimo.2014.10.591
Hsin-Min Sun, Yu-Juan Sun. Variable fixing method by weighted average for the continuous quadratic knapsack problem. Numerical Algebra, Control & Optimization, 2022, 12 (1) : 15-29. doi: 10.3934/naco.2021048
Namsu Ahn, Soochan Kim. Optimal and heuristic algorithms for the multi-objective vehicle routing problem with drones for military surveillance operations. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021037
Nguyen Thi Toan. Generalized Clarke epiderivatives of the extremum multifunction to a multi-objective parametric discrete optimal control problem. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021088
Jiangfeng Huang, Zhiliang Deng, Liwei Xu. A Bayesian level set method for an inverse medium scattering problem in acoustics. Inverse Problems & Imaging, 2021, 15 (5) : 1077-1097. doi: 10.3934/ipi.2021029
Qiang Du, M. D. Gunzburger, L. S. Hou, J. Lee. Analysis of a linear fluid-structure interaction problem. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 633-650. doi: 10.3934/dcds.2003.9.633
Xiaoni Chi, Zhongping Wan, Zijun Hao. A full-modified-Newton step $ O(n) $ infeasible interior-point method for the special weighted linear complementarity problem. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021082
Impact Factor:
Sedighe Asghariniya Hamed Zhiani Rezai Saeid Mehrabian | CommonCrawl |
\begin{document}
\title{The bead process for beta ensembles} \author{Joseph Najnudel, B\'alint Vir\'ag} \date{} \maketitle
\begin{abstract} The bead process introduced by Boutillier is a countable interlacing of the determinantal sine-kernel (i.e. $\operatorname{Sine}_2$) point processes. We construct the bead process for general $\operatorname{Sine}_{\beta}$ processes as an infinite dimensional Markov chain whose transition mechanism is explicitly described. We show that this process is the microscopic scaling limit in the bulk of the Hermite $\beta$ corner process introduced by Gorin and Shkolnikov, generalizing
the process of the minors of the Gaussian Unitary and Orthogonal Ensembles.
In order to prove our results, we use bounds on the variance of the point counting of the circular and the Gaussian beta ensembles, proven in a companion paper \cite{NV19}.
\end{abstract} \section{Introduction} In Boutillier \cite{Bou}, a remarkable family of point processes on $\mathbb Z \times \mathbb R$, called {\it bead processes}, and indexed by a parameter $\gamma \in (-1,1)$, has been defined. They enjoy the following properties: \begin{description} \item[Interlacing.] The points of two consecutive lines interlace with each other. \item[Invariance.] The distribution of the point process is invariant and ergodic under the natural action of $\mathbb Z \times \mathbb R$ by translation. \item[Parameters.] The expected number of points in any interval is proportional to its length.
Given that $(0,0)$ is in the process, the expected value of the first positive point on line $1$ is proportional to $\arccos \gamma$. \item[Gibbs property.] The distribution of any point $X$, given the other points, is uniform on the interval which is allowed by the interlacing property. \end{description} It is not known whether these properties determine the point process uniquely, as the closely related results of Sheffield \cite{She05} do not directly apply.
Existence was shown by Boutillier, who considers a deteminantal process with an explicit kernel. Its restriction to a line is the standard sine-kernel process. Thus the above description proposes to be the purest probabilistic definition of the Gaudin-Mehta sine kernel process limit of the bulk eigenvalues of the Gaussian Unitary Ensemble (GUE).
Boutillier's result relies on taking limits of tilings on the torus. Since then, works starting with Johansson and Nordenstam \cite{JN} showed that the consecutive minor eigenvalues of the Gaussian Unitary Ensemble also converge to the bead process, where the tilt depends on the global location within the Wigner semicircle. These results have been refined and generalized in Adler, Nordenstam and Moerbeke \cite{ANVM}. However, the corresponding questions remained open for other matrix ensembles, as the Gaussian Orthogonal Ensemble (GOE), and the Gaussian Symplectic Ensemble (GSE): \begin{itemize} \item Is there a limit of the eigenvalue minor process? \item Is there a simple characterization as for $\beta=2$? \item Can one derive formulas related to the distribution of beads? \end{itemize} One of the main goals of this paper is to answer positively to these questions. The limiting process is defined as an infinite-dimensional Markov chain, the transition from one line to the next being explicitly described. This transition can be viewed as a generalization of the limit, when the dimension $n$ goes to infinity, of the random reflection walk on the unitary group $U(n)$. This walk is the unitary analogue of the random transposition walk studied, for example, in Diaconis and Shahshahani \cite{DS81}, Berestycki and Durrett \cite{BD06} and Bormashenko \cite{Bor11}.
The natural generalization of the transpositions to the setting of the orthogonal group corresponds to the reflections. The orthogonal matrix corresponding to the reflection across the plane with normal unit vector $v$ is $I-2vv^*$. To further generalize to the unitary group, we proceed as follows: given a fixed unit complex number $\eta$ and a unit vector $v$, we define the {\bf complex reflection} across $v$ with angle $\arg(\eta)$ as the isometry whose matrix is given by $I+(\eta -1)vv^*$. The random reflection walk $(Y_k)_{k \geq 1}$ on the unitary group $U(n)$ is then defined by $Y_k=X_1\dots X_k$, where $(X_j)_{j \geq 1}$ are independent reflections for which $v$ is chosen according to uniform measure on the complex unit sphere, and $\eta$ is fixed.
Note that since the multiplicative increments of the walk are invariant under conjugation by any group element, it follows that $\bar Y_k$, the conjugacy class of $Y_k$, also follows a random walk. This, of course, is given by the eigenvalues of $Y_k$; the transition mechanism can be computed as follows. Assuming that the eigenvalues $u_j$ of $\bar Y_k$ are distinct, the eigenvalues of $Y_{k+1}$ are the solutions of \begin{equation} \sum_{j=0}^{n-1} i\frac{u_j+z}{u_j-z}\rho_j = i\frac{1+\eta}{1-\eta} \label{evolutionunitary} \end{equation}
where for $|z|=1$ the summands and the right-hand side are both real. The only randomness is contained in the values $\rho_j$, which have a Dirichlet joint distribution with all parameters equal to 1. To summarize, in order to get the evolution of $(\bar Y_k)_{k \geq 1}$, we pick $(\rho_j)_{1 \leq j \leq n}$ from Dirichlet distribution, form the rational function given by the left-hand side of \eqref{evolutionunitary}, and look at a particular level set to get the new eigenvalues.
This equation can be lifted to the real line. Let $(\lambda_j)_{j \in \mathbb{Z}}$ be the $(2 \pi n)$-periodic set of $\lambda \in \mathbb{R}$ such that $e^{i\lambda/n} \in \{u_1, \dots, u_n\}$, and extend the sequence $(\rho_j)_{1 \leq j \leq n}$ periodically (with period $n$) to all integer indices. With $z=e^{ix/n}$, the left-hand side of \eqref{evolutionunitary} can be written as $$ \lim_{\ell \to\infty} \sum_{j=-\ell}^{\ell} \frac{2n\rho_j}{\lambda_j-x}. $$ and the level set of this at $i(1+\eta)/(1-\eta)$ gives the lifting of the eigenvalues at the next step. Notice now that essentially the only role of $n$ in the above process is given by the joint distribution of the $\rho$-s. These are $n$-periodic and Dirichlet; clearly, as $n\to\infty$ they converge, after suitable renormalization, to independent exponential variables, giving naturally an infinite-dimensional Markov chain.
In the present article, we prove rigorously the existence of this Markov chain, and we deduce a new construction of the bead process. By replacing the exponential variables by gamma variables with general parameter, we construct a natural generalization of the bead process, indexed by a parameter $\beta > 0$. For $\beta = 2$, this process is the bead process itself, and then it is the scaling limit (at microscopic scale) of the eigenvalues of the GUE minors when the dimension goes to infinity. For $\beta= 1$, we show that we get the limit of the eigenvalues of the GOE minors, for $\beta = 4$, we get the limit of the eigenvalues of the GSE minors, and we generalize this result to all $\beta > 0$, by considering the Hermite $\beta$ corners, defined by Gorin and Shkolnikov \cite{GS}, which can be informally viewed as the "eigenvalues of G$\beta$E minors".
The sequel of the present paper is organized as follows.
In Section 2, we detail the above discussion on the random reflection walk, and we deduce a property of invariance for the law of the spectrum of a Haar-distributed unitary matrix, for the transition given by the equation \eqref{evolutionunitary}. We generalize this property to circular beta ensembles for any $\beta > 0$.
In Section 3, we generalize the notion of Stieltjes transform to a class of infinite point measures on the real line for which the series given by the usual definition is not absolutely convergent.
In Section 4, we construct a family of Markov chains on a space of point measures, for which the transition mechanism is obtained by taking a level set of the Stieltjes transform defined in Section 3.
In Section 5, we show how the lifting of the unit circle on the real line defined above connects the results of Section 2 to those of Section 4.
In Section 6, we use some bound on the variance of the number of points of the circular beta ensembles in an arc, in order to take the limit
of the results in Section 5, when the period of the point measure goes
to infinity. We show a property of invariance enjoyed by the determinantal sine-kernel process and
its generalizations for all $\beta > 0$, for the Markov chain defined in Section 4. From this Markov chain,
we deduce the construction of a stationnary point process on $\mathbb{R} \times \mathbb{Z}$, for which
the points of a given line follow the distribution of the $\operatorname{Sine}_{\beta}$ process introduced in
Valk\'o and Vir\'ag \cite{VV}.
In Section 7, we show, under some technical conditions, a property of continuity of the Markov chain
with respect to the initial point measure and the weights.
From this result, and from a bound, proven in a companion paper \cite{NV19} on the variance of the number of points of the Gaussian beta ensemble in intervals, we deduce in Section 8 that
the generalized bead process constructed in Section 6 appears as a limit for the eigenvalues of the minors
of Gaussian Ensembles for $\beta \in \{1,2,4\}$. The case $\beta = 2$ corresponds to the GUE, for which the
convergence to the bead process defined by Boutillier \cite{Bou} is already known from Adler, Nordenstam and Moerbeke \cite{ANVM}. Combining our result with \cite{ANVM} then implies that our Markov chain has necessarily the same distribution as the bead process given in \cite{Bou}. The case
$\beta = 1$ gives the convergence of the renormalized eigenvalues of the GOE minors, and the case $\beta = 4$ gives the convergence of the renormalized eigenvalues of the GSE minors. For other values of $\beta$, we
get a similar result of convergence for the renormalized points of the Hermite $\beta$ corner defined in \cite{GS}. \section{Random reflection chains on the unitary group}
We start with a brief review of how multiplication by complex reflections changes eigenvalues. Let $U \in U(n)$ be a unitary matrix with distinct eigenvalues
$u_1, \dots, u_n$, and let $v$ be a unit vector. Let $a_1, \dots, a_n$ be the coefficients of $v$ in a basis of unit eigenvectors of $U$, and let $\rho_j = |a_j|^2$ for $1 \leq j \leq n$: $\rho_1, \dots, \rho_n$ do not depend on the choice of the eigenvector basis and the sum of these numbers is equal to $1$.
If $\eta\not= 1$ is a complex number of modulus 1, the complex reflection with angle $\arg \eta$ and vector $v$ corresponds to the unitary matrix $I+(\eta-1)vv^*$. If we multiply $U$ by this reflection, we get a new matrix whose eigenvalues $u$ satisfy $$ 0=\det(U(I+(\eta-1)vv^*)-u), $$ which can be rewritten as $$ 0=\det(U-u)\det(I+(\eta-1)Uvv^*(U-u)^{^{-1}}) $$ when $u$ is not an eigenvalue of $U$. Now, the second argument is $I$ plus a rank-1 matrix, so its determinant equals 1 plus the trace of the rank-1 matrix. Thus the equation above reduces to $$ 0=1+ (\eta-1){\rm tr}(Uvv^*(U-u)^{-1}) = 1 + (\eta-1)v^*((U-u)^{-1}U)v. $$ Expanding $U$ in the basis of its eigenvectors and eigenvalues $u_j$, we get $$ 1=(1 - \eta) \sum_{j=1}^n \rho_j \frac{u_j}{u_j-u} $$ or, after a transformation, \begin{equation} \sum_{j=1}^n i \rho_j \frac{u_j+u}{u_j-u}=i\frac{1+\eta}{1-\eta}. \label{equationinterlacing} \end{equation} As $u$ moves counterclockwise on the unit circle, and on each arc between two consecutive poles, the left-hand side of \eqref{equationinterlacing} is continuous and strictly increasing from $-\infty$ to $\infty$. Hence, the matrix $U(I+(\eta-1)vv^*)$ has exactly one eigenvalue in each arc between
eigenvalues of $U$: in other words, the eigenvalues of $U(I+(\eta-1)vv^*)$ strictly interlace between those of
$U$, and are given by the solutions $u$ of the equation \eqref{equationinterlacing}.
Consider the product of the unit sphere in $\mathbb{C}^n$ and $\mathbb R$, and a distribution $\pi$ on this space which is invariant under permutations of the $n$ coordinates of the sphere, and by multiplication of each of these coordinates by complex numbers of modulus one. For such a distribution, we can associate a Markov chain on unitary matrices as follows. Given $U_0,\ldots, U_k$, we pick a sample $((a_1, \dots, a_n),h)$ from $\pi$ independently from the past. Then, $U_{k+1}$ is defined as the product of $U_{k}$ by the reflection with parameter $\eta$ so that $h=i\frac{\eta+1}{\eta-1}$, and vector $v=\sum a_j \varphi_j$, where $(\varphi_j)_{1 \leq j \leq n}$ are unit eigenvectors of $U$ (from the assumption made on $\pi$, the law of $v$ does not depend on the choice of the phases of the eigenvectors $(\varphi_j)_{1 \leq j \leq n}$).
From the discussion above, it is straightforward that if $V_k$ is the spectrum of $U_k$, then $(V_k)_{k \geq 0}$ forms a Markov process as well; its distribution depends on the coefficients $a_j$ only through $\rho_j$. The transition is given as follows: given $V_j$, $(\rho_j)_{1 \leq j \leq n}$ and $h$, $V_{j+1}$ is formed by the $n$ solutions of \eqref{equationinterlacing}.
When $a$ is uniform on the unit complex sphere of $\mathbb{C}^n$, and $h$ is independent of $a$, then
$(\rho_j)_{1 \leq j \leq n}$ has Dirichlet$(1,\ldots,1)$ distribution, and the corresponding reflection
is independent of $U_k$. Thus the Markov chain reduces to a random walk: $U_j=U_0R_1\ldots R_k$, where the reflections $(R_k)_{k \geq 1}$ are independent.
It is immediate that the Haar measure on $U(n)$ is invariant for this random walk. One deduces that if $(\rho_j)_{1 \leq j \leq n}$ follows a Dirichlet distribution with all parameters equal to $1$, if $h$ (and then $\eta$) is independent of $(\rho_j)_{1 \leq j \leq n}$, if the points of $V_0$ follow the distribution of the eigenvalues of the CUE in dimension $n$, and if $(V_k)_{k \geq 0}$ is the Markov chain described above, then the law of $V_k$ does not depends of $k$: the CUE distribution is invariant for this Markov chain.
This invariance property can be generalized to other distributions $\pi$.
Indeed, as in Simon \cite{Simon}, one can associate to the point measure $\sigma := \sum_{j=1}^n \rho_j \delta_{u_j}$ a so-called {\it Schur function}
$f_{\sigma}$, which is rational, and which can be
written, by Geronimus theorem, as
$$f_{\sigma}(u) = R_{\alpha_0} \circ M_u \circ R_{\alpha_1} \circ M_u \circ R_{\alpha_2} \circ \cdots \circ R_{\alpha_{n-2}} \circ M_u (\alpha_{n-1}),$$ where $M_u$ denotes the multiplication by $u$, the $(\alpha_j)_{0 \leq j \leq n-1}$ are the Verblunsky coefficients associated to the orthogonal polynomials with respect to the measure $\sigma$, and for all $\alpha \in \mathbb{D}$, $R_{\alpha}$ is the M\"obius transformation given by $$R_{\alpha} (z) = \frac{\alpha + z}{1 + \overline{\alpha} z}.$$ On the other hand, one has the equality of rational functions:
\begin{equation} \int_{\mathbb{U}}
i \frac{v+u}{v-u} d \sigma(v)
=i\frac{1+u f_{\sigma}(u)}{1-u f_{\sigma} (u)}. \end{equation}
Hence, the equation \eqref{equationinterlacing}
is satisfied if and only if $u f_{\sigma}(u) = \eta$, or equivalently, \begin{equation} M_{\eta^{-1}} \circ M_{u} \circ R_{\alpha_0} \circ M_u \circ R_{\alpha_1} \circ \cdots \circ M_u (\alpha_{n-1}) = 1. \label{MR} \end{equation} Now, $M_{\eta^{-1}}$ and $M_{u}$ commute and for $\alpha \in \mathbb{D}$, $M_{\eta^{-1}} \circ R_{\alpha} = R_{\alpha \eta^{-1}} \circ M_{\eta^{-1}}$. One deduces that \eqref{MR} is equivalent to $$M_u \circ R_{\alpha_0 \eta^{-1}} \circ M_u \circ R_{\alpha_1 \eta^{-1}} \circ \cdots \circ M_u (\alpha_{n-1} \eta^{-1}) = 1,$$
i.e. $u f_{\tau} (u) = 1$, where $\tau$ is the finitely supported probability measure whose Verblunsky coefficients are
$(\alpha_0 \eta^{-1}, \dots, \alpha_{n-1}\eta^{-1})$. Now, by the general construction of the Schur functions, the equation $u f_{\tau} (u) = 1$ is satisfied if and only if $u$ is a point of the support
of $\tau$: in other words, this support is the set of solutions of \eqref{equationinterlacing}.
We deduce that if the distribution $\pi$ and
the law of $\{u_1, \dots, u_n\}$ are chosen in such
a way that $(\alpha_0 \eta^{-1}, \dots, \alpha_{n-1}
\eta^{-1})$ has the same law as
$(\alpha_0, \dots, \alpha_{n-1})$, then
the law of $\{u_1, \dots, u_n\}$ is invariant for
the Markov chain described above. The precise statement is the following:
\begin{proposition} \label{circularinvariance}
Let $\pi$ be a probability
distribution on the product of the unit sphere
of $\mathbb{C}^n$ and $\mathbb{R}$, under which the
first component $(a_1, \dots, a_n)$ is independent of the second $h = i (1 + \eta)/(1- \eta)$.
We suppose that the law of
$(a_1, \dots, a_n)$ is invariant by permutation of the coordinates, and by their pointwise multiplication by
complex numbers of modulus $1$. Let
$\mathbb{P}$ be a probability measure of the
sets of $n$ points $\{u_1, \dots, u_n\}$,
such that under the product measure $\mathbb{P}
\otimes \pi$, the sequence $(\alpha_0, \dots,
\alpha_{n-1})$ of Verblunsky coefficients associated
to the measure $$\sigma = \sum_{1 \leq j \leq n}
\rho_j \delta_{u_j} = \sum_{1 \leq j \leq n}
|a_j|^2 \delta_{u_j}.$$
has a law which is invariant by multiplication by
complex numbers of modulus $1$.
Then, the measure $\mathbb{P}$ is invariant
for the Markov chain associated to $\pi$: more
precisely, under $\mathbb{P}
\otimes \pi$, the law of the set of solutions
of \eqref{equationinterlacing} is equal to
$\mathbb{P}$.
\end{proposition}
It is not obvious to find explicitly some measures $\mathbb{P}$ and $\pi$ under which the law of the Verblunsky coefficients is invariant by rotation. An important example is obtained by considering the so-called {\it circular beta ensembles}. These ensembles are constructed as follows: for some parameter $\beta > 0$, one defines a probability measure $\mathbb{P}_{n,\beta}$ on the sets of $n$ points on the unit circle, such that the corresponding $n$-point correlation function $r_{n,\beta}$ is given, for $z_1, \dots z_n \in \mathbb{U}$, by
$$r_{n, \beta} (z_1, \dots, z_N) = C_{n, \beta} \, \prod_{1 \leq j < k \leq N} |z_j - z_k|^{\beta},$$ where $C_{n, \beta} > 0$ is a normalization constant. Note that, for $\beta = 2$, one obtains the distribution of the spectrum of a random $n \times n$ unitary matrix following the Haar measure. Now, let $\pi_{n, \beta}$ be any distribution on the product of the unit sphere of $\mathbb{C}^n$ and $\mathbb{R}$, such that with the notation above, $h$ is independent of $(\rho_0, \dots, \rho_{n-1})$, which has a Dirichlet distribution with all parameters equal to $\beta/2$. Then, under $\mathbb{P}_{n, \beta}
\otimes \pi_{n, \beta}$, the distribution of the Verblunsky coefficients $(\alpha_0, \alpha_1, \dots, \alpha_{n-1})$ has been computed in Killip and Nenciu \cite{bib:KN04}. One obtains the following: \begin{itemize} \item The coefficients $\alpha_0, \alpha_1, \dots \alpha_{n-1}$ are independent random variables. \item The coefficient $\alpha_{n-1}$ is uniform on the unit circle.
\item For $j \in \{0, 1, \dots, n-2\}$, the law of $\alpha_j$ has density $(\beta/2)(n-j-1) (1-|\alpha_j|^2)^{(\beta/2)(n-j-1) - 1}$ with respect to the uniform probability measure on the unit disc: note that
$|\alpha_j|^2$ is then a beta variable of parameters $1$ and $\beta(n-j-1)/2$. \end{itemize} Therefore, the law of $(\alpha_0, \alpha_1, \dots, \alpha_{n-1})$ is invariant by rotation, and one deduces the following result: \begin{proposition} \label{CJEinvariance} The law of the circular beta ensemble is an invariant measure for the Markov chain associated to $\pi_{n, \beta}$. More precisely, under $\mathbb{P}_{n, \beta} \otimes \pi_{n, \beta}$, the set of solutions of \eqref{equationinterlacing} follows the distribution $\mathbb{P}_{n, \beta}$. \end{proposition} In the next sections, we will take a limit when $n$ goes to infinity. For this purpose, we need to consider point processes on the real line instead of the unit circle, and to find an equivalent of the equation \eqref{equationinterlacing} in this setting. \section{Stieltjes transform for point measures} Let $\Lambda$ be a $\sigma$-finite point measure on $\mathbb{R}$, which can be written as follows: $$\Lambda = \sum_{\lambda \in L} \gamma_{\lambda} \delta_{\lambda},$$ where $L$ is a discrete subset of the real line, $\gamma_{\lambda} > 0$ for all $\lambda \in S$, and $\delta_{\lambda}$ is the Dirac measure at $\lambda$. The usual definition of the Stieltjes transform applied to $\Lambda$ gives, for $z \in \mathbb{C} \backslash \{L\}$: \begin{equation} S_{\Lambda} (z) = \sum_{\lambda \in L} \frac{\gamma_{\lambda}}{\lambda - z}. \label{stieltjes} \end{equation} If the set $L$ is finite, then $S_{\Lambda}(z)$ is well-defined as a rational function. If $L$ is infinite and if the right-hand side of \eqref{stieltjes} is absolutely convergent, then this equation is still meaningful. The following result implies that under some technical assumptions, one can define $S_{\Lambda}$ even if \eqref{stieltjes} does not apply directly: \begin{theorem} \label{theoremstieltjes} Assume that the for all $a, b \in \mathbb{R}$, $\Lambda[0,x+a] - \Lambda[-x+b,0] = O(x/\log^2 x)$ as $x\to \infty$. Then, for all $z \in \mathbb{C} \backslash \{L\}$, there exists $S_{\Lambda} (z) \in \mathbb{C}$ such that $$ \sum_{\lambda \in L \cap [-c, c]} \frac{\gamma_{\lambda}}{\lambda - z} \underset{c \rightarrow \infty}{\longrightarrow} S_{\Lambda}(z).$$ The function $S_{\Lambda}$ defined in this way is meromorphic, with simple poles at the elements of $L$, and the residue at $\lambda \in L$ is equal to $- \gamma_{\lambda}$. The derivative of $S_{\Lambda}$ is given by \begin{equation} S'_{\Lambda}(z) = \sum_{\lambda \in L} \frac{\gamma_{\lambda}}{(\lambda - z)^2}, \label{derivative} \end{equation} where the convergence of the series is uniform on compact sets of $\mathbb{C} \backslash \{L\}$. For all pairs $\{\lambda_1, \lambda_2\}$ of consecutive points in $L$, with $\lambda_1 < \lambda_2$, the function $S_{\Lambda}$ is a strictly increasing bijection from $(\lambda_1, \lambda_2)$ to $\mathbb{R}$. Moreover, we have the following translation invariance: if $y\in \mathbb R$ and $\Lambda$ satisfies the conditions above, then so does its translation $\Lambda+y$, and one has $$S_{\Lambda + y}(z + y) = S_{\Lambda} (z)$$ for all $z \in \mathbb{C} \backslash \{L\}$. \end{theorem} \begin{remark} The bound $x/\log^2 x$ is somohow arbitrary and not optimal (any increasing function which is negligible with respect to $x$ and integrable against $dx/x^2$ at infinity would work). However, it will be sufficient for our purpose. \end{remark} \begin{proof}
Let $c_0 > 1$, and $z \in \mathbb{C}$ such that $|z| \leq c_0/2$. For $c > c_0$, we have: \begin{align*} \sum_{\lambda \in L \cap ([-c,-c_0] \cup [c_0,c] )} \frac{\gamma_{\lambda}}{\lambda - z } & = \sum_{\lambda \in L \cap [c_0,c]} \gamma_{\lambda} \int_{\lambda}^{\infty} \frac{d \mu}{(\mu -z)^2} - \sum_{\lambda \in L \cap [-c,-c_0]} \gamma_{\lambda} \int_{-\infty}^{\lambda} \frac{d \mu}{(\mu -z)^2} \\ & = \int_{c_0}^{\infty} \frac{ \Lambda([c_0, c \wedge \mu])}{(\mu -z)^2} \, d\mu - \int_{-\infty}^{-c_0} \frac{ \Lambda([(-c) \vee \mu, - c_0])}{(\mu -z)^2} \, d\mu \\ & = \int_{c_0}^{\infty} \left( \frac{ \Lambda([c_0, c \wedge \mu])}{(\mu -z)^2} - \frac{ \Lambda([-(c \wedge \mu), - c_0])}{(\mu +z)^2} \right)\, d \mu \\ & = \int_{c_0}^{\infty} \frac{ \Lambda([c_0, c \wedge \mu]) - \Lambda([-(c \wedge \mu), - c_0])}{\mu^2} \, d \mu \end{align*} $$ + \int_{c_0}^{\infty} \left( \frac{ (2 z \mu - z^2) (\Lambda([c_0, c \wedge \mu]) )}{\mu^2 (\mu-z)^2} + \frac{(2 z \mu + z^2) (\Lambda([-(c \wedge \mu), -c_0]) )}{\mu^2 (\mu+z)^2}
\right) \, d\mu.$$
Let $F$ be an increasing function from $\mathbb{R}_+$ to $\mathbb{R}_+^*$, such that $F(x)$ is equivalent to $x/\log^2 x$ when $x$ goes to infinity.
By assumption, there exists $C > 0$ such that for all $x \geq 0$,
$|\Lambda([0, x]) - \Lambda([-x, 0])| \leq C F(x) $, and then, for all $\mu \geq c_0$,
$$ |\Lambda([c_0, c \wedge \mu]) - \Lambda([-(c \wedge \mu), - c_0])| \leq C F(c \wedge \mu) + \Lambda([-c_0, c_0]) \leq \left( C + \frac{ \Lambda([-c_0, c_0]) }{F(0)} \right) \, F(\mu).$$ Since $\mu \mapsto F(\mu)/\mu^2$ is integrable at infinity, one obtains, by dominated convergence, $$ \int_{c_0}^{\infty} \frac{ \Lambda([c_0, c \wedge \mu]) - \Lambda([-(c \wedge \mu), - c_0])}{\mu^2} \, d \mu \underset{c \rightarrow \infty}{\longrightarrow} \int_{c_0}^{\infty} \frac{ \Lambda([c_0, \mu]) - \Lambda([-\mu, - c_0])}{\mu^2} \, d \mu,$$ where the limiting integral is absolutely convergent. Similarly, there exist $C', C'' > 0$ such that for all
$x \geq 0$, $|\Lambda([0,x+1]) - \Lambda([-x,0])| \leq C' F(x)$ and $|\Lambda([0,x]) - \Lambda([-x-1,0])| \leq C'' F(x)$, which implies that \begin{align*}
\Lambda((x,x+1]) + \Lambda([-x-1,-x)) & \leq |\Lambda([0,x+1]) - \Lambda([-x,0])| + |\Lambda([0,x]) - \Lambda([-x-1,0])| \\ & \leq (C'+C'') F(x). \end{align*} Hence, for all integers $n \geq 1$, \begin{align*} \Lambda([-n,n]) & = \Lambda(\{0\}) + \sum_{k=0}^{n-1} (\Lambda((k,k+1]) + \Lambda([-k-1,-k)) \\ & \leq \Lambda(\{0\}) + (C' + C'') \, \sum_{k=0}^{n-1} F(k) \leq K n F(n-1) \end{align*} where $K > 0$ is a constant, and then for all $x \geq 0$, $\Lambda([-x,x]) \leq K (1+x) F(x)$, which implies that for $\mu \geq c_0$,
$ \Lambda([-(c \wedge \mu), -c_0]) \leq K (1+ \mu) F(\mu)$ and $\Lambda([c_0, c \wedge \mu]) \leq K(1+ \mu) F(\mu)$. Moreover, since $|z| \leq c_0/2 \leq \mu/2$, one has $|\mu - z| \geq \mu/2$, $|\mu + z| \geq \mu/2$ and \begin{equation}
\left| \frac{ (2 z \mu - z^2)}{\mu^2 (\mu-z)^2} \right| + \left| \frac{ (2 z \mu + z^2)}{\mu^2 (\mu+z)^2} \right|
\leq 2 \, \frac{ 2.5 |z| \mu}{ \mu^2 (\mu/2)^2} = 20 |z| / \mu^3 \leq 10 \, c_0/\mu^3 \label{20z} \end{equation} Since $\mu \mapsto (1+ \mu) F(\mu) / \mu^3$ is integrable at infinity, one can again apply dominated convergence and obtain that $$\int_{c_0}^{\infty} \left( \frac{ (2 z \mu - z^2) (\Lambda([c_0, c \wedge \mu]) )}{\mu^2 (\mu-z)^2} + \frac{(2 z \mu + z^2) (\Lambda([-(c \wedge \mu), -c_0]) )}{\mu^2 (\mu+z)^2}
\right) \, d\mu$$
tends to $$\int_{c_0}^{\infty} \left( \frac{ (2 z \mu - z^2) (\Lambda([c_0, \mu]) )}{\mu^2 (\mu-z)^2} + \frac{(2 z \mu + z^2) (\Lambda([-\mu, -c_0]) )}{\mu^2 (\mu+z)^2}
\right) \, d\mu$$ when $c$ goes to infinity. Therefore,
\begin{align*}
\sum_{\lambda \in L \cap ([-c,-c_0] \cup [c_0,c] )} \frac{\gamma_{\lambda}}{\lambda - z } & \underset{c \rightarrow \infty}{\longrightarrow}
\int_{c_0}^{\infty} \frac{ \Lambda([c_0, \mu]) - \Lambda([-\mu, - c_0])}{\mu^2} \, d \mu
\\ & + \int_{c_0}^{\infty} \left( \frac{ (2 z \mu - z^2) (\Lambda([c_0, \mu]) )}{\mu^2 (\mu-z)^2} + \frac{(2 z \mu + z^2) (\Lambda([-\mu, -c_0]) )}{\mu^2 (\mu+z)^2}
\right) \, d\mu, \end{align*}
which proves the existence of the limit defining $S_{\Lambda} (z)$: explicitly, for $z \in \mathbb{C} \backslash \{L\}$ and for any $c_0 > 2|z| \vee 1$,
\begin{align}
S_{\Lambda}(z) & = \sum_{\lambda \in L \cap (-c_0,c_0)} \frac{\gamma_{\lambda}}{\lambda - z } + \int_{c_0}^{\infty} \frac{ \Lambda([c_0, \mu]) - \Lambda([-\mu, - c_0])}{\mu^2} \, d \mu \nonumber \\ & + \int_{c_0}^{\infty} \left( \frac{ (2 z \mu - z^2) (\Lambda([c_0, \mu]) )}{\mu^2 (\mu-z)^2} + \frac{(2 z \mu + z^2) (\Lambda([-\mu, -c_0]) )}{\mu^2 (\mu+z)^2}
\right) \, d\mu. \label{int}
\end{align}
For fixed $c_0 > 0$, the first term of \eqref{int} is a rational function of $z$, the second term of \eqref{int} does not depend on $z$, and by dominated convergence,
the third term can be differentiated in the integral if we restrict $z$ to the
set $\{|z| < c_0/2\}$. Hence, the restriction of $S_{\Lambda}$ to the set $\{z < c_0/2\}$ is meromorphic, with simple poles at points $\lambda \in L \cap (-c_0/2, c_0/2)$.
Since $c_0$ can be taken arbitrarily large, $S_{\Lambda}$ is in fact meromorphic on
$\mathbb{C}$, with poles $\lambda \in L$, the pole $\lambda$ having residue $- \gamma_{\lambda}$.
The derivative $S'_{\Lambda}(z)$ is given, for any $c_0 > 2 |z| \vee 1$, by:
\begin{align}
S'_{\Lambda} (z) & = \sum_{\lambda \in L \cap (-c_0,c_0)} \frac{\gamma_{\lambda}}{(\lambda - z)^2 } + 2 \, \int_{c_0}^{\infty} \left( \frac{ (\Lambda([c_0, \mu]) )}{ (\mu-z)^3} + \frac{ (\Lambda([-\mu, -c_0]) )}{ (\mu+z)^3}
\right) \, d\mu. \nonumber \\ &
= \sum_{\lambda \in L \cap (-c_0,c_0)} \frac{\gamma_{\lambda}}{(\lambda - z)^2 } +
\int_{c_0}^{\infty} \, \left( \sum_{\lambda \in L \cap [c_0, \mu]} \gamma_{\lambda} \right) \, \frac{2 \, d \mu}{(\mu-z)^3}
+ \int_{c_0}^{\infty} \, \left( \sum_{\lambda \in L \cap [-\mu, -c_0]} \gamma_{\lambda} \right) \, \frac{2 \, d \mu}{(\mu+z)^3}
\nonumber \\
& = \sum_{\lambda \in L \cap (-c_0,c_0)} \frac{\gamma_{\lambda}}{(\lambda - z)^2 } +
\sum_{\lambda \in L \cap [c_0, \infty)} \gamma_{\lambda} \int_{\lambda}^{\infty} \, \frac{2 \, d \mu}{(\mu-z)^3}
+ \sum_{\lambda \in L \cap (-\infty, c_0]} \gamma_{\lambda} \int_{-\lambda}^{\infty} \, \frac{2 \, d \mu}{(\mu+z)^3},
\nonumber
\end{align}
which implies \eqref{derivative}. Note that the implicit use of Fubini theorem is this computation is correct since all the sums and integral involved are
absolutely convergent.
Now, let $\mathcal{K}$ be a compact set of $\mathbb{C} \backslash L$, let $d > 0$ be the distance between $\mathcal{K}$ and $L$, and let
$A > 0$ be the maximal modulus of the elements of $\mathcal{K}$. For all $z \in \mathcal{K}$ and $\lambda \in L$, one has, for $|\lambda| \leq 2A+1$,
$$ \left| \frac{\gamma_{\lambda} }{(\lambda - z)^2} \right| \leq \frac{\gamma_{\lambda}}{d^2} \leq \frac{1+ (2A+1)^2}{d^2} \cdot \frac{\gamma_{\lambda}}{1 + \lambda^2} $$
and for $|\lambda| \geq 2A+1$,
$$\left| \frac{\gamma_{\lambda} }{(\lambda - z)^2} \right| \leq \frac{\gamma_{\lambda} }{ (|\lambda| - A)^2} \leq \frac{4 \gamma_{\lambda} }{ \lambda^2} \leq \frac{8 \gamma_{\lambda} }{1 + \lambda^2}.$$
Hence, in order to prove the uniform convergence of \eqref{derivative} on compact sets, it is sufficient
to check that $$\sum_{\lambda \in L} \frac{\gamma_{\lambda} }{1 + \lambda^2} < \infty,$$
but this convergence is directly implied by the absolute convergence of the right-hand side of \eqref{derivative} for any particular value
$z \in \mathbb{C} \backslash L$.
The formula \eqref{derivative} applied to $z \in \mathbb{R}$ implies immediately that for all pairs $\{\lambda_1, \lambda_2\}$ of consecutive points in $L$, with $\lambda_1 < \lambda_2$, the function $S_{\Lambda}$ is strictly increasing on the interval $(\lambda_1, \lambda_2)$. Moreover, one has for $\lambda \in \{\lambda_1, \lambda_2\}$ and $z \rightarrow \lambda$, $S_{\lambda}(z) \sim \gamma_{\lambda}/(\lambda - z)$, which implies that $S_{\Lambda}(z) \rightarrow -\infty$ for $z \rightarrow \lambda_1$ and $z >\lambda_1$, and $S_{\Lambda}(z) \rightarrow +\infty$ for $z \rightarrow \lambda_2$ and $z <\lambda_2$. We deduce that $S_{\Lambda}$ is a bijection from $(\lambda_1, \lambda_2)$ to $\mathbb{R}$.
It only remains to show the invariance by translation. If we fix $y \in \mathbb{R}$, then for all $a, b \in \mathbb{R}$, and for $x \geq 0$ large enough, \begin{align*} (\Lambda+y)([0,x+a]) & - (\Lambda+ y)([-x+b,0]) = \Lambda([-y, x+a-y])- \Lambda([-x + b-y, -y]) \\ &
= \Lambda([0, x+a-y])- \Lambda([-x + b-y, 0]) + O(\Lambda([-|y|,|y|])) \\ & =
O(x/\log^2 x) + O(1) = O(x/\log^2 x),
\end{align*}
and the assumptions of Theorem \ref{theoremstieltjes} are satisfied. One has
$$\Lambda + y = \sum_{\lambda \in L} \gamma_{\lambda} \delta_{\lambda + y},$$
and then for all $z \in \mathbb{C} \backslash L$, $$ S_{\Lambda + y} (z+y) = \lim_{c \rightarrow \infty} \, \sum_{\lambda \in (L+y) \cap [-c,c]} \, \frac{ \gamma_{\lambda-y} }{z + y- \lambda }
= \lim_{c \rightarrow \infty} \, \sum_{\lambda \in L \cap [-c-y,c-y]} \, \frac{ \gamma_{\lambda} }{z - \lambda }, $$ which is equal to $S_{\Lambda}(z)$, provided that we check that
$$\sum_{\lambda \in L \cap [-c-y,c-y]} \, \frac{ \gamma_{\lambda} }{z - \lambda } - \sum_{\lambda \in L \cap [-c,c]} \, \frac{ \gamma_{\lambda} }{z - \lambda }
\underset{c \rightarrow \infty}{\longrightarrow} 0,$$ which is implied by \begin{equation}
\sum_{\lambda \in L \cap [-c-|y|,-c + |y|] } \frac{ \gamma_{\lambda} }{|z - \lambda| } + \sum_{\lambda \in L \cap [c-|y|,c + |y|] } \frac{ \gamma_{\lambda} }{|z - \lambda| } \underset{c \rightarrow \infty}{\longrightarrow} 0. \label{conv} \end{equation}
Now, for $c > |y|+|z| + 1$, the left-hand side of \eqref{conv} is smaller than or equal to \begin{align*}
& \frac{\Lambda([-c-|y|, -c + |y|]) + \Lambda([c-|y|, c+ |y|])}{c -|z| - |y|} \\ \leq &
\frac{|\Lambda([0,c+|y|]) - \Lambda([-c+|y|+1, 0])| + |\Lambda([0,c-|y|-1]) - \Lambda([-c-|y|,0])|}{ c-|y|-|z|} = O(1/\log^2 c),
\end{align*} for $c$ tending to infinity. \end{proof} The assumption of Theorem \ref{theoremstieltjes} depends on the fact that the measure $\Lambda$ is not too far from being symmetric with respect to a given point on the real line. The next proposition expresses this assumption in terms of the support $L$ of $\Lambda$ and the weights $(\gamma_{\lambda})_{\lambda \in L}$. The following result gives a sufficient condition for Theorem \ref{theoremstieltjes}: \begin{proposition} \label{propositionstieltjes} Consider the measure $$ \Lambda= \sum_{j\in \mathbb Z} {\gamma_j} \delta_{\lambda_j} $$ where $(\lambda_j)_{j \in \mathbb{Z}}$ is strictly increasing and neither bounded from above nor from below, and $\gamma_j\in \mathbb{R}_+^*$. Let $L$ be the set $\{\lambda_j, j \in \mathbb{Z}\}$. Assume that for some $c > 0$, $$\sum_{j=0}^k \gamma_j = ck + O(k/\log^2 k) \; \; \mathrm{ and } \; \; \sum_{j=0}^k \gamma_{-j} = ck + O(k/\log^2 k),$$ when $k\to\infty$. If for $x\to\infty$ one has $ \operatorname{Card} (L \cap [0,x] ) = O(x)$ and for all $a, b \in \mathbb{R}$, $\operatorname{Card} (L \cap [0,x+a] ) - \operatorname{Card} (L \cap [-x+b,0] ) = O(x/\log^2 x)$, then the assumptions of Theorem \ref{theoremstieltjes} are satisfied. \end{proposition} \begin{proof} For $y \in \mathbb{R}$, let $N(y)$ (resp. $N(y-)$) be the largest index $j$ such that $\lambda_j \leq y$ (resp. $\lambda_j < y$). One has, for $a, b \in \mathbb{R}$ and for $x$ large enough, $$\Lambda([0,x+a]) = \sum_{j=N(0-) +1}^{N(x+a)} \gamma_j \; \; \mathrm{ and } \; \; \Lambda([-x+b,0]) = \sum_{j=N((-x+b)-)}^{N(0)} \gamma_j,$$ which implies that for $x \rightarrow \infty$ and then $ N(x+a) \rightarrow \infty$, $N((-x+b)-) \rightarrow - \infty$: \begin{align*}
\Lambda([0,x+a]) - \Lambda([-x+b,0]) & = c ( N(x+a) - |N((-x+b)-)|) \\ & + O\left(\frac{N(x+a)}{ \log^2 (N(x+a))} + \frac{|N((-x+b)-)|}{ \log^2 |N((-x+b)-)|} \right). \end{align*} Now, we have the following estimates: $$N(x+a) = \operatorname{Card} (L \cap [0,x+a] ) + O(1) = O(x+a) + O(1) = O(x),$$
$$\frac{N(x+a)}{ \log^2 (N(x+a))} = O(x/\log^2 x),$$
$$N(x+a) - |N((-x+b)-)| = \operatorname{Card} (L \cap [0,x+a] ) - \operatorname{Card} (L \cap [-x+b,0] ) + O(1) = O(x/\log^2 x),$$
$$ |N((-x+b)-)| \leq N(x+a) + |N(x+a) - |N((-x+b)-)| | \leq O(x) + O(x/\log^2 x) = O(x)$$ and
$$ \frac{|N((-x+b)-)|}{ \log^2 |N((-x+b)-)|}= O(x/\log^2 x).$$ Putting all together gives: $$\Lambda([0,x+a]) - \Lambda([-x+b,0]) = O(x/\log^2 x)$$ and then the assumptions of Theorem \ref{theoremstieltjes} are satisfied. \end{proof} As written in the statement of Theorem \ref{theoremstieltjes}, the function $S_{\Lambda}$ induces a bijection between each interval $(\lambda_1,\lambda_2)$, $\lambda_1$ and $\lambda_2$ being two consecutive points of $L$, and the real line. It is then natural to study the inverse of this bijection, which should map each element of $\mathbb{R}$ to a set of points interlacing with $L$. The precise statement we obtain is the following:
\begin{proposition} \label{L'} Let $\Lambda$ be a measure, whose support $L$ is neither bounded from above nor from below, and satisfying the assumptions of Theorem \ref{theoremstieltjes}. Then, for all $h \in \mathbb{R}$, the set $S^{-1}_{\Lambda} (h)$ of $z \in \mathbb{C} \backslash\{L\}$ such that $S_{\Lambda}(z) = h$ is included in $\mathbb{R}$, and interlaces with $L$, i.e. it contains exactly one point in each open interval between two consecutive points of $L$. Moreover, if $\Lambda$ satisfies the assumptions of Proposition \ref{propositionstieltjes}, then it is also the case for the set $L' := S^{-1}_{\Lambda} (h)$, i.e. for $x$ going to infinity, one has $\operatorname{Card} (L' \cap [0,x] ) = O(x)$ and for all $a, b \in \mathbb{R}$, $\operatorname{Card} (L' \cap [0,x+a] ) - \operatorname{Card} (L' \cap [-x+b,0] ) = O(x/\log^2 x)$. \end{proposition} \begin{proof} The interlacing property of points of $S^{-1}_{\Lambda}(h) \cap \mathbb{R}$ comes from the discussion above, so the first part of the proposition is proven if we check that $S_{\Lambda}(z) \notin \mathbb{R}$ if $z \notin \mathbb{R}$. Now, for all $z \in \mathbb{C} \backslash L$, \begin{align*} \Im \left( S_{\Lambda}(z) \right) = \lim_{c \rightarrow \infty} \sum_{\lambda \in L \cap [-c,c]} \Im \left( \frac{\gamma_{\lambda}}{\lambda -z} \right) & = \lim_{c \rightarrow \infty} \sum_{\lambda \in L \cap [-c,c]} \frac{ - \gamma_{\lambda} \, \Im (\lambda-z)}{ \Re^2(\lambda-z) + \Im^2(\lambda- z) }\\ & = \lim_{c \rightarrow \infty} \sum_{\lambda \in L \cap [-c,c]} \frac{ \gamma_{\lambda} \Im (z)}{ \Re^2(\lambda-z) + \Im^2(z) }. \end{align*} If $z \notin \mathbb{R}$, each term of the last sum is nonzero and has the same sign as $\Im(z)$. One deduces that
$\Im \left( S_{\Lambda}(z) \right)$ has the same properties, and then $S_{\Lambda}(z) \notin \mathbb{R}$.
Now, the interlacing property implies that for any finite interval $I$, $$| \operatorname{Card} (L' \cap I) - \operatorname{Card} (L \cap I) | \leq 1.$$
If $\Lambda$ satisfies the assumptions of Proposition \ref{propositionstieltjes}, then for $a, b \in \mathbb{R}$ and for $x$ going to infinity,
$$ \operatorname{Card} (L' \cap [0,x] ) = \operatorname{Card} (L \cap [0,x] ) + O(1) = O(x) + O(1) = O(x)$$
and
\begin{align*}
\operatorname{Card} (L' \cap [0,x+a] ) & - \operatorname{Card} (L' \cap [-x+b,0] ) = \operatorname{Card} (L \cap [0,x+a] ) \\ & - \operatorname{Card} (L \cap [-x+b,0] ) + O(1)
= O(x/\log^2x).
\end{align*}
\end{proof}
Proposition \ref{L'} shows that the Stieltjes transform gives a way to construct a discrete subset of $\mathbb{R}$ from another, provided that we get
a family $(\gamma_j)_{j \in \mathbb{Z}}$ of weights and a parameter $h \in \mathbb{R}$. In the next section, we use and randomize this procedure in order to define a family of Markov chains satisfying some remarkable properties.
\section{Stieltjes Markov chains}
In order to put some randomness in the construction above, we need to define precisely a measurable space in which the point processes will be contained.
The choice considered here is the following:
\begin{itemize}
\item We define $\mathcal{L}$ as the family of all the discrete subsets $L$ of $\mathbb{R}$, unbounded from above and from below, and satisfying the assumptions of
Proposition \ref{propositionstieltjes}, i.e. for $x$
going to infinity, $ \operatorname{Card} (L \cap [0,x] ) = O(x)$ and for all $a, b \in \mathbb{R}$, $\operatorname{Card} (L \cap [0,x+a] ) - \operatorname{Card} (L \cap [-x+b,0] ) = O(x/\log^2 x)$. \item We define, on $\mathcal{L}$, the $\sigma$-algebra $\mathcal{A}$ generated by the maps $L \mapsto \operatorname{Card} (L \cap I)$ for all open, bounded intervals $I \subset \mathbb{R}$, which is also the $\sigma$-algebra generated by the maps $L \mapsto \operatorname{Card} (L \cap B)$ for all Borel sets $B \subset \mathbb{R}$.
\end{itemize}
A similar choice of measurable space has to be made for the weights $(\gamma_j)_{j \in \mathbb{Z}}$:
\begin{itemize}
\item We define $\Gamma$ as the family of doubly infinite sequences $(\gamma_j)_{j \in \mathbb{Z}}$ satisfying the assumptions of Proposition \ref{propositionstieltjes}, i.e.
for $k$ going to infinity,
$$\sum_{j=0}^k \gamma_j = ck + O(k/\log^2 k) \; \; \mathrm{ and } \; \; \sum_{j=0}^k \gamma_{-j} = ck + O(k/\log^2 k),$$ where $c > 0$ is a constant. \item We define, on $\Gamma$, the $\sigma$-algebra $\mathcal{C}$ generated by the coordinate maps $\gamma_j$, $j \in \mathbb{Z}$. \end{itemize}
Let $\mathcal{D}$ be the map from $\mathcal{L} \times \Gamma \times \mathbb{R}$ to $\mathcal{L}$, defined by: $$\mathcal{D}(L, (\gamma_j)_{j \in \mathbb{Z}}, h) = S^{-1}_{ \sum_{j \in \mathbb{Z}} \gamma_j \delta_{\lambda_j}} (h),$$ where $\lambda_j$ is the unique increasing labeling of $L$ so that $\lambda_{-1}<0\le \lambda_0$. Proposition \ref{L'} shows that this is indeed a map to $\mathcal L$. It is easy to show that $\mathcal D$ is measurable. Now for any probability measure $\Pi$ on $\Gamma\times \mathbb R$, it naturally defines a Markov chain $(X_k)_{k \geq 0}$ on $\mathcal L$. To get $X_{k+1}$ from $X_k$, just take a fresh sample $G_k$ (independent of $X_k$ and its past) and set $$X_{k+1}=\mathcal D(X_k,G_k).$$ By construction, $X_k$ is then a time-homogeneous Markov chain.
Clearly, if the distribution of $X_0$ is invariant under translations of $\mathbb R$, and the distribution of the $((\gamma_j)_{j \in \mathbb{Z}},h)$ in $G_k$ is invariant under translations of the indices $j$, it follows that $X_1$ also has a translation-invariant distribution.
There are two important examples of probability measures $\Pi$ for which this construction applies: \begin{itemize} \item Under $\Pi$, $(\gamma_j)_{j \in \mathbb{Z}}$ is a family of i.i.d, square-integrable random variables, and $h$ is independent of $(\gamma_j)_{j \in \mathbb{Z}}$. \item Under $\Pi$, $(\gamma_j)_{j \in \mathbb{Z}}$ is a family of random variables, $n$-periodic for some $n \geq 1$, such that $(\gamma_0, \gamma_1, \dots, \gamma_{n-1}) = (\gamma_1, \dots, \gamma_{n-1}, \gamma_0)$ in law, and $h$ is independent of $(\gamma_j)_{j \in \mathbb{Z}}$. \end{itemize} The fact that $(\gamma_j)_{j \in \mathbb{Z}}$ is almost surely in $\Gamma$ comes from the law of the iterated logarithm in the first example, and directly from the periodicity in the second example.
\section{Periodic Stieltjes Markov chains}
Consider the case when $$\Lambda =\sum_{j\in \mathbb Z} \gamma_j \delta_{\lambda_j}$$ is invariant by translation of $2\pi n$, and when there are $n$ point masses in every interval of length $2\pi n$ with total weight $2n$. In this case, $\Lambda$ can be thought as $2n$ times the lifting of the measure $$ \sigma=\sum_{j=0}^{n-1} \frac{\gamma_j}{2n} \delta_{e^{i{\lambda_j}/n}} $$ on the unit circle $\mathbb U$ under a covering map. Moreover, with $u=e^{iz/n}$ the Stieltjes transform of $\Lambda$ can be expressed in terms of $\sigma$ by $$ S_\Lambda(z) = \sum_{j=0}^{n-1} i \frac{\gamma_j}{2n} \frac{e^{i\lambda_j/n} + u}{e^{i\lambda_j/n} - u}. $$ Indeed, periodicity implies that for $z\notin L$, we have \begin{align*} S_\Lambda (z) & = \underset{k \rightarrow \infty}{\lim} \sum_{j = - k n}^{ kn-1} \frac{\gamma_j}{\lambda_j - z} = \underset{k \rightarrow \infty}{\lim} \sum_{j=0}^{n-1} \gamma_j \, \left( \sum_{\ell=-k}^{k-1} \frac{1}{2 \pi n\ell+ \lambda_j - z} \right) \\ & = \sum_{j=0}^{n-1} \gamma_j \left( \underset{k \rightarrow \infty}{\lim} \sum_{\ell=-k}^{k-1} \frac{1}{2 \pi n \ell + \lambda_j - z} \right) = \frac{1}{2n} \, \sum_{j=0}^{n-1} \gamma_j \cot \left(\frac{\lambda_j - z}{2n} \right). \end{align*} Therefore, if we set $\rho_j := \gamma_j/2n$ and $u_j = e^{i \lambda_j/n}$, we can check that $\mathcal{D} (L, (\gamma_n)_{n \in \mathbb{Z}}, h)$ is the set of $z \in \mathbb{R}$, such that $e^{iz/n}$ satisfies \eqref{equationinterlacing}, for $h = i(1+\eta)/(1-\eta)$.
This property shows that the lifting $u \mapsto \{z \in \mathbb{R}, e^{iz/n} = u \}$ from $\mathbb{U}$ to $\mathbb{R}$ defined above transforms the Markov chain defined in Section 2 to the Markov chain defined in Section 4. In particular, from Propositions \ref{circularinvariance} and \ref{CJEinvariance}, we deduce the following results: \begin{theorem} Let $\Pi$ be a probability measure on the space $(\Gamma \times \mathbb{R}, \mathcal{C} \otimes \mathcal{B}(\mathbb{R}))$, under which the following holds, for some integer $n \geq 1$: \begin{itemize} \item Almost surely under $\Pi$, $(\gamma_n)_{n \in \mathbb{Z}}$ is $n$-periodic, and $\sum_{j=0}^{n-1} \gamma_j = 2n$. \item The law of $(\gamma_0, \dots, \gamma_{n-1})$ is invariant by permutation of the coordinates. \item The sequence $(\gamma_j)_{j \in \mathbb{Z}}$ is independent of $h$. \end{itemize} Let $\mathbb{Q}$ be a probability on $(\mathcal{L}, \mathcal{A})$ under which almost surely, the set $L$ is $(2n \pi)$-periodic and contains exactly $n$ points in the interval $[0, 2 \pi n)$: in this case, there exists a sequence $( u_1, \dots, u_{n})$ of elements of $\mathbb{U}$, with increasing argument in $[0, 2\pi)$, and such that $$L = \{ z \in \mathbb{R}, e^{iz/n} \in \{ u_1, \dots, u_{n}\} \}.$$ Under the probability $\mathbb{Q} \otimes \pi$, one can define a random probability measure $\sigma$ on the unit circle by: $$\sigma:= \frac{1}{2n} \sum_{j=1}^{n} \gamma_j \delta_{u_j}.$$ Let us assume that the joint law of the Verblunsky coefficients $(\alpha_0, \dots, \alpha_{n-1})$ of $\sigma$ is invariant by rotation, i.e. for all $u \in \mathbb{U}$, $$(\alpha_0 u, \dots \alpha_{n-1} u) = (\alpha_0, \dots, \alpha_{n-1})$$ in distribution. Then, the probability measure $\mathbb{Q}$ is an invariant measure for the Markov chain associated to $\pi$. \end{theorem}
\begin{theorem} \label{betafinite} Let $\beta > 0$, $n \geq 1$, and let $\Pi$ be a probability measure under which the following holds almost surely: \begin{itemize} \item The sequence $(\gamma_n)_{n \in \mathbb{Z}}$ is $n$-periodic. \item The tuple $(\gamma_0/2n, \dots, \gamma_{n-1}/2n)$ follows a Dirichlet distribution with all parameters equal to $\beta/2$. \item The sequence $(\gamma_j)_{j \in \mathbb{Z}}$ is independent of $h$. \end{itemize} Let $\mathbb{Q}_{n, \beta}$ be the distribution of the set $$\{z \in \mathbb{R} , e^{iz/n} \in V\},$$ where $V$ is a subset of $\mathbb{U}$ following $\mathbb{P}_{n,\beta}$, i.e. a circular beta ensemble with parameter $\beta$. Then, $\mathbb{Q}_{n, \beta}$ is an invariant measure for the Markov chain associated to $\Pi$. \end{theorem} In the next section, we will let $n \rightarrow \infty$ and we will obtain a similar result in which the variables $(\gamma_n)_{n \geq 1}$ will be independent and identically distributed.
\section{An invariant measure for independent gamma random variables}
In Theorem \ref{betafinite}, we have found an invariant measure on $\mathcal{L}$, corresponding to a measure $\Pi$ under which the sequence $(\gamma_j)_{j \in \mathbb{Z}}$ is periodic, each period forming a renormalized Dirichlet distribution. For $n \geq 1$ and $\beta > 0$ fixed, and under $\Pi$, the sequence $(\gamma_j)_{j \in \mathbb{Z}}$ can be written in function of a sequence $(g_j)_{j \in \mathbb{Z}}$ of i.i.d Gamma variables with parameter $\beta/2$, as follows: $$\gamma_{j} = \frac{2 n g_k}{\underset{-n/2 < \ell \leq n/2}{\sum} g_{\ell}},$$ where $-n/2 < k \leq n/2$ and $k \equiv j$ modulo $n$. For $\beta$ fixed, if we construct the sequence $(\gamma_j)_{j \in \mathbb{Z}}$ for all values of $n$, starting with the same sequence $(g_j)_{j \in \mathbb{Z}}$, we obtain, by the law of large numbers, that for all $j \in \mathbb{Z}$, $\gamma_j$ tends almost surely to $4 g_j/\beta$ when $n$ goes to infinity. Hence, if we want to make $n \rightarrow \infty$ in Theorem \ref{betafinite}, we should consider a measure $\Pi$ under which $(\beta \gamma_j/4)_{j \in \mathbb{Z}}$ is a sequence of i.i.d. Gamma random variables of parameter $\beta/2$.
On the other hand, for $n$ going to infinity, the probability $\mathbb{Q}_{n, \beta}$ converges to a limiting measure $\mathbb{Q}_{\beta}$, which is the distribution of the so-called $\mathrm{Sine}_{\beta}$ {\it point process}, constructed in \cite{bib:KSt09} and \cite{VV}.
Therefore, taking the limit $n \rightarrow \infty$ in Theorem \ref{betafinite} suggests the following result, whose proof is given below: \begin{theorem} \label{betainfinite} Let $\beta > 0$, and let $\Pi$ be a probability measure under which the random variables $h$ and $(\gamma_j)_{j
\in \mathbb{Z}}$ are all independent, $\gamma_j$ being
equal to $4/\beta$ times a gamma random variable of parameter $\beta/2$. Then, the law $\mathbb{Q}_{\beta}$ of the $\mathrm{Sine}_{\beta}$ point process is carried by the space $\mathcal{L}$ and it is an invariant measure for the Markov chain associated to $\Pi$. \end{theorem}
\begin{remark} Since the variables $(\gamma_j)_{j \in \mathbb{Z}}$ are i.i.d. and square-integrable, we have already checked that the Markov chain associated to $\Pi$ is well-defined. Recall what means the fact that $\mathbb{Q}_{\beta}$ is carried by $\mathcal{L}$: if $L$ is the set of points corresponding to a $\mathrm{Sine}_{\beta}$ process, then $L$ is unbounded from above and from below, for $x$ going to infinity, $ \operatorname{Card} (L \cap [0,x] ) = O(x)$ and for all $a, b \in \mathbb{R}$, $\operatorname{Card} (L \cap [0,x+a] ) - \operatorname{Card} (L \cap [-x+b,0] ) = O(x/\log^2 x)$. \end{remark} In order to show the theorem just above, we will use the following results, proven in \cite{NV19}: \begin{proposition} \label{estimatequadraticCbeta} Let $L$ be a random set of points in $\mathbb{R}$, whose distribution is $\mathbb{Q}_{n, \beta}$ or $\mathbb{Q}_{\beta}$. Then, there exists $C > 0$, depending on $\beta$ but not on $n$, such that for all $x > 0$, $$\mathbb{E} [ \left(\operatorname{Card} (L \cap [0,x]) - x/2 \pi \right)^2] \leq C \log(2+x)$$ and $$\mathbb{E} [ \left(\operatorname{Card} (L \cap [-x,0]) - x/2 \pi \right)^2] \leq C \log(2+x).$$ \end{proposition} \begin{proposition}\label{estimatealmostsureCbeta} Under the previous assumptions, for all $\alpha > 1/3$, there exists
a random variable $C > 0$, stochastically dominated by a finite random variable depending only on $\alpha$ and $\beta$, such that almost surely, for all $x \geq 0$,
$$|\operatorname{Card} (L \cap [0,x])
- x/2 \pi| \leq C (1+x)^{\alpha},$$ and
$$|\operatorname{Card} (L \cap [-x,0])
- x/2 \pi| \leq C (1+x)^{\alpha}.$$ \end{proposition}
\begin{remark} The periodicity of $L$ implies that
$|\operatorname{Card} (L \cap [0,x])
- x/2 \pi|$ is almost surely bounded when $x$ varies. Hence, the result above becomes trivial if one allows $C$ to depend on $n$. Moreover, we expect that it remains true for any $\alpha > 0$, and not only for $\alpha > 1/3$. \end{remark}
{\it Proof of Theorem \ref{betainfinite}. } Let $\Pi$ be a probability measure which satisfies the assumptions of Theorem \ref{betainfinite}, and for $n \geq 1$, let $\Pi_n$ be a measure satisfying the assumptions of Theorem \ref{betafinite}, for the same value of $\beta$. We also assume that the law of $h$ is the same under $\Pi_n$ and under $\Pi$ (note that $\Pi_n$ and $\Pi$ are uniquely determined by this law). By the discussion preceding the statement of Theorem \ref{betainfinite}, it is possible, by using a unique family $(g_j)_{j \in \mathbb{Z}}$ of i.i.d. gamma variables with parameter $\beta/2$, to construct some random sequences $(\gamma_j)_{j \in \mathbb{Z}}$ and $(\gamma^{n}_j)_{j \in \mathbb{Z}}$ (for all $n \geq 1$) and an independent real-valued random variable $h$, such that the following holds: \begin{itemize} \item $((\gamma_j)_{j \in \mathbb{Z}}, h)$ follows the law $\Pi$. \item For all $n \geq 1$, $((\gamma^n_j)_{j \in \mathbb{Z}}, h)$ follows the law $\Pi_n$. \item For all $j \in \mathbb{Z}$, $\gamma^n_j$ tends almost surely to $\gamma_j$ when $n$ goes to infinity. \end{itemize} Now, for all $n \geq 1$, let $L_n$ be a point process following the distribution $\mathbb{Q}_{n, \beta}$, and let $L$ be a point process following $\mathbb{Q}_{\beta}$. We already know that $L_n \in \mathcal{L}$ almost surely.
From Proposition \ref{estimatealmostsureCbeta} under $\mathbb{Q}_{\beta}$, we immediately deduce the weaker estimates $\operatorname{Card} (L \cap [0,x]) = x/2 \pi + O(x/\log^2 x)$ and
$\operatorname{Card} (L \cap [-x,0]) = x/2 \pi + O(x/\log^2 x)$ for $x$ going to infinity, which means that $L \in \mathcal{L}$ almost surely: $\mathbb{Q}_{\beta}$ is
carried by $\mathcal{L}$.
Moreover, by \cite{bib:KSt09}, the measure $\mathbb{Q}_{n,\beta}$ tends to $\mathbb{Q}_{\beta}$ when $n$ goes to infinity, in the following
sense: for all functions $f$ from $\mathbb{R}$ to $\mathbb{R}_+$, $C^{\infty}$ and compactly supported, one has
\begin{equation}
\sum_{x \in L_n} f(x) \underset{n \rightarrow \infty}{\longrightarrow} \sum_{x \in L} f(x) \label{weakconvergence}
\end{equation}
in distribution. By the Skorokhod representation theorem, one can assume that the convergence \eqref{weakconvergence} holds almost surely, and one can also suppose that $(L_n)_{n \geq 1}$ and $L$ are independent of $(\gamma^n_j)_{n \geq 1, j \in \mathbb{Z}}$, $(\gamma_j)_{j \in \mathbb{Z}}$ and $h$.
For $n \geq 1$, let $(\lambda^n_j)_{j \in \mathbb{Z}}$ be the strictly increasing sequence containing each point of $L_n$, $\lambda^n_0$ being the smallest nonnegative point, and let $(\lambda_j)_{j \in \mathbb{Z}}$ be the similar sequence associated to $L$. One can check that the convergence \eqref{weakconvergence} and the fact that $\mathbb{P} [0 \in L] = 0$ imply that for all $j \in \mathbb{Z}$, $\lambda^n_j$ converges almost surely to $\lambda_j$ when $n$ goes to infinity. Now, for all $c > 0$, $z \in \mathbb{C} \backslash \left(L \cup \left(\bigcup_{n \geq 1} L_n \right) \right)$, let us take the following notation:
$$S_{n,c}(z) := \sum_{j \in \mathbb{Z}} \, \frac{\gamma^n_j}{\lambda^n_j - z} \, \mathbf{1}_{|\lambda^n_j| \leq c}, \; \; S_{c}(z) := \sum_{j \in \mathbb{Z}} \, \frac{\gamma_j}{\lambda_j - z} \, \mathbf{1}_{|\lambda_j| \leq c},$$ $$S_N(z) := \lim_{c \rightarrow \infty} S_{N,c} (z), \; \; S(z) := \lim_{c \rightarrow \infty} S_c (z).$$ Almost surely, all the points of $L$ and $L_n$ ($n \geq 1$) are irrational. If this event occurs, then for all $c \in \mathbb{Q}^*_+$, there exists almost surely a finite interval (possibly empty)
$I_c$ such that $|\lambda_j| \leq c$ if and only if $j \in I_c$, and for all $n \geq 1$ large enough, $|\lambda^n_j| \leq c$ if and only if $j \in I_c$. Hence, for all $c \in \mathbb{Q}^*_+$, $z \in \mathbb{Q}$, one has almost surely $$S_{n,c}(z) = \sum_{j \in I_c} \frac{\gamma^n_j}{\lambda^n_j - z}, \; \; S_{c}(z) := \sum_{j \in I_c} \, \frac{\gamma_j}{\lambda_j - z},$$ if $n$ is large enough. Since $I_c$ is finite, $\gamma^n_j$ tends a.s. to $\gamma_j$, and $\lambda^n_j$ tends a.s. to $\lambda_j$ when $n$ goes to infinity, one deduces that almost surely, for all $c \in \mathbb{Q}^*_+$, $z \in \mathbb{Q}$, \begin{equation} S_{n,c}(z) \underset{n \rightarrow \infty}{\longrightarrow} S_c(z). \label{asconv} \end{equation} One the other hand, by \eqref{int}, and by the fact that $c$ and $-c$ are a.s. not in $L$ or in $L_n$, one deduces that almost surely, for all $c \in \mathbb{Q}^*_+$,
$z \in \mathbb{Q}$ such that $c > 2|z| \vee 1$, and for all $n \geq 1$, \begin{align*} S_n (z) - S_{n,c} (z) & = \int_{c}^{\infty} \frac{ \Lambda_n([c, \mu]) - \Lambda_n([-\mu, - c])}{\mu^2} \, d \mu \\ & + \int_{c}^{\infty} \left( \frac{ (2 z \mu - z^2) (\Lambda_n([c, \mu]) )}{\mu^2 (\mu-z)^2} + \frac{(2 z \mu + z^2) (\Lambda_n([-\mu, -c]) )}{\mu^2 (\mu+z)^2}
\right) \, d\mu
\end{align*}
and
\begin{align*} S (z) - S_{c} (z) & = \int_{c}^{\infty} \frac{ \Lambda([c, \mu]) - \Lambda([-\mu, - c])}{\mu^2} \, d \mu \\ & + \int_{c}^{\infty} \left( \frac{ (2 z \mu - z^2) (\Lambda([c, \mu]) )}{\mu^2 (\mu-z)^2} + \frac{(2 z \mu + z^2) (\Lambda([-\mu, -c]) )}{\mu^2 (\mu+z)^2}
\right) \, d\mu,
\end{align*} where $\Lambda_n:= \sum_{j \in \mathbb{Z}} \gamma^n_j \delta_{\lambda^n_j}$ and $\Lambda := \sum_{j \in \mathbb{Z}} \gamma_j \delta_{\lambda_j}$. If for any bounded interval $I$, one defines $\Lambda^{(0)}_n (I) := \Lambda_n(I) - \mathbb{E} [ \Lambda_n(I)] $ and $\Lambda^{(0)} (I) := \Lambda (I) - \mathbb{E} [ \Lambda(I)] $, one has by \eqref{20z}, the triangle inequality, and the fact that $ \mathbb{E} [ \Lambda_n(I)] $ is proportional to the Lebesgue measure on $I$: \begin{align*}
|S_n(z) - S_{n,c}(z)| & \leq \int_c^{\infty} \frac{ |\Lambda^{(0)}_n ([c, \mu])| + |\Lambda^{(0)}_n ([-\mu, -c])|}{\mu^2} \, d \mu
\\ & + \int_c^{\infty} \frac{20|z| \, d \mu}{\mu^3} \, \left[ C_1 \mu + |\Lambda^{(0)}_n ([c, \mu])| + |\Lambda^{(0)}_n ([-\mu, -c])|\right] \\
& \leq C_2 (1+|z|) \left( \frac{1}{c} + \int_c^{\infty} \frac{ |\Lambda^{(0)}_n ([c, \mu])| + |\Lambda^{(0)}_n ([-\mu, -c])|}{\mu^2} \, d \mu \right), \end{align*}
where $C_1, C_2 > 0$ are universal constants. Since the distribution of $L_n$ is invariant by translation (recall that its points are the rescaled arguments of the circular beta ensemble on the unit circle), one has $$\mathbb{E}[ |\Lambda^{(0)}_n ([c, \mu])|] = \mathbb{E}[ |\Lambda^{(0)}_n ([-\mu, -c])|]=
\mathbb{E}[ |\Lambda^{(0)}_n ([0, \mu-c])|]$$ and
$$\mathbb{E}[|S_n(z) - S_{n,c}(z)|] \leq C_3 (1+|z|) \left( \frac{1}{c} + \int_0^{\infty} \frac{ \mathbb{E}[ |\Lambda^{(0)}_n ([0, \nu])| ]}{(\nu + c)^2} \, d \nu \right),$$ where $C_3 > 0$ is a universal constant. Similarly,
$$\mathbb{E}[|S(z) - S_{c}(z)|] \leq C_3 (1+|z|) \left( \frac{1}{c} + \int_0^{\infty} \frac{ \mathbb{E}[ |\Lambda^{(0)} ([0, \nu])| ]}{(\nu + c)^2} \, d \nu \right).$$ Now, from Proposition \ref{estimatequadraticCbeta} under $\mathbb{Q}_{n, \beta}$ and $\mathbb{Q}_{\beta}$, one immediately deduces that \begin{equation}
\int_0^{\infty} \frac{ \mathbb{E}[ |\Lambda^{(0)} ([0, \nu])| ]+ \sup_{n \geq 1} \mathbb{E}[ |\Lambda_n^{(0)} ([0, \nu])| ]}{(1+\nu)^2} \, d\nu < \infty. \label{domination} \end{equation} Hence, by dominated convergence, there exists a function $\phi$ from $[1, \infty)$ to $\mathbb{R}^*_+$, tending to zero at infinity, such that
$$\mathbb{E}[|S_n(z) - S_{n,c}(z)|] \leq (1+|z|) \, \phi(c)$$ and
$$\mathbb{E}[|S(z) - S_{c}(z)|] \leq (1+|z|) \, \phi(c).$$ We deduce that for all $c \in \mathbb{Q}^*_+$,
$z \in \mathbb{Q}$ such that $c > 2|z| \vee 1$, $n \geq 1$ and $\epsilon> 0$,
\begin{align*}
\mathbb{P} [ |S(z) - S_{n} (z)| \geq \epsilon] & \leq
\mathbb{P} [ |S_c(z) - S_{n,c}(z) | \geq \epsilon/3 ]
+ \mathbb{P} [ |S(z) - S_{c}(z) | \geq \epsilon/3 ] \\ & + \mathbb{P} [ |S_n(z) - S_{n,c}(z)| \geq \epsilon/3 ]
\\ & \leq \mathbb{P} [ |S_c(z) - S_{n,c}(z) | \geq \epsilon/3 ] + \frac{6}{\epsilon} \, (1+|z|) \, \phi(c).
\end{align*} By the almost sure convergence \eqref{asconv}, which implies the corresponding convergence in probability, one deduces
$$\underset{n \rightarrow \infty}{\lim \sup} \; \mathbb{P} [ |S(z) - S_{n} (z)| \geq \epsilon]
\leq \frac{6}{\epsilon} \, (1+|z|) \, \phi(c).$$ Now, by taking $z \in \mathbb{Q}$ fixed, $c \in \mathbb{Q}$ going to infinity and then $\epsilon \rightarrow 0$, one deduces that for all $z \in \mathbb{Q}$, $$S_n(z) \underset{n \rightarrow \infty}{\longrightarrow} S(z)$$ in probability. By considering diagonal extraction of subsequences, one deduces that there exists a strictly increasing sequence $(n_k)_{k \geq 1}$ of integers, such that almost surely, \begin{equation} S_{n_k} (z) \underset{k \rightarrow \infty}{\longrightarrow} S(z) \label{asconv2} \end{equation} for all $z \in \mathbb{Q}$.
Now, for all $j \in \mathbb{Z}$, $n \geq 1$, let $\mu^n_j$ (resp. $\mu_j$) be the unique point of $\mathcal{D} (L_n, (\gamma^n_j)_{j \in \mathbb{Z}}, h)$ (resp. $\mathcal{D} (L, (\gamma_j)_{j \in \mathbb{Z}}, h)$) which lies in the interval $(\lambda^n_j, \lambda^n_{j+1})$ (resp. $(\lambda_j, \lambda_{j+1})$). Let us fix $j \in \mathbb{Z}$, $\epsilon > 0$, and let us consider two random rational numbers $q_1$ and $q_2$ such that almost surely, $$(\mu_j - \epsilon) \vee \lambda_j < q_1 < \mu_j < q_2 < (\mu_j + \epsilon) \wedge \lambda_{j+1},$$ which implies that $$S(q_1) < h < S(q_2).$$ By \eqref{asconv2}, one deduces that almost surely, for $k$ large enough, $$S_{n_k}(q_1) < h < S_{n_k}(q_2),$$ which implies that $\mathcal{D} (L_{n_k}, (\gamma^{n_k}_j)_{j \in \mathbb{Z}}, h)$ has at least one point in the interval $(q_1,q_2)$. On the other hand, since $\lambda^n_j$ (resp. $\lambda^n_{j+1}$) tends a.s. to $\lambda_j$ (resp. $\lambda_{j+1}$) when $n$ goes to infinity, one has almost surely, for $k$ large enough, $$\lambda^{n_k}_j < q_1 < q_2< \lambda^{n_k}_{j+1}.$$ Hence, $\mathcal{D} (L_{n_k}, (\gamma^{n_k}_j)_{j \in \mathbb{Z}}, h)$ has exactly one point in $(q_1, q_2)$, and this point is necessarily $\mu^{n_k}_j$. One deduces that almost surely,
$|\mu^{n_k}_j - \mu_j| \leq \epsilon$ for $k$ large enough, which implies, by taking $\epsilon \rightarrow 0$, that $\mu^{n_k}_j$ converges almost surely to $\mu_j$ when $k$ goes to infinity.
Now, let $f$ be a function from $\mathbb{R}$ to $\mathbb{R}_+$, $C^{\infty}$ and compactly supported. Since $L$ is locally finite, there exists a.s. an integer $j_0 \geq 1$ such that the support of $f$ is included in $(\lambda_{-j_0}, \lambda_{j_0})$, and then in $(\lambda^{n_k}_{-j_0}, \lambda^{n_k}_{j_0})$ for $k$ large enough, which implies that $f(\mu^{n_k}_j) = f(\mu_j) = 0$ for $|j| > j_0$. Hence, a.s., there exists $j_0, k_0 \geq 1$, such that for $k \geq k_0$,
$$\sum_{j \in \mathbb{Z}} f(\mu^{n_k}_j) = \sum_{|j| \leq j_0} f(\mu^{n_k}_j)$$ and
$$\sum_{j \in \mathbb{Z}} f(\mu_j) = \sum_{|j| \leq j_0} f(\mu_j),$$ which implies that \begin{equation} \sum_{j \in \mathbb{Z}} f(\mu^{n_k}_j) \underset{k \rightarrow \infty}{\longrightarrow} \sum_{j \in \mathbb{Z}} f(\mu_j), \label{asconv3} \end{equation} since $f(\mu_j^{n_k})$ tends to $f(\mu_j)$ for each $j \in \{-j_0, -j_0+1, \dots, j_0\}$.
The almost sure convergence \eqref{asconv3} holds a fortiori in distribution, which implies
that the law of $\mathcal{D} (L_{n_k}, (\gamma^{n_k}_j)_{j \in \mathbb{Z}}, h)$ tends to the law of $\mathcal{D} (L, (\gamma_j)_{j \in \mathbb{Z}}, h)$. On the other hand, by Theorem \ref{betafinite}, $\mathcal{D} (L_{n_k}, (\gamma^{n_k}_j)_{j \in \mathbb{Z}}, h)$ has distribution $\mathbb{Q}_{n_k, \beta}$, and then $\mathcal{D} (L, (\gamma_j)_{j \in \mathbb{Z}}, h)$ follows the limit of the distribution $\mathbb{Q}_{n_k, \beta}$ for $k$ tending to infinity, i.e. $\mathbb{Q}_{\beta}$. Theorem \ref{betainfinite} is then proven.
$\square$
\section{Properties of continuity for the Stieltjes Markov chain}
In the previous section, we have deduced the convergence of the Markov mechanism associated to $\mathbb{Q}_{n_k, \beta}$ towards the one corresponding to $\mathbb{Q}_{\beta}$ from the convergence of $\mathbb{Q}_{n_k, \beta}$ to $\mathbb{Q}_{\beta}$ itself, and the convergence of the associated weights. Later in the paper, we will prove similar results related to the Gaussian ensembles, for which the situation is more difficult to handle, in particular because of the lack of symmetry of the G$\beta$E at the macroscopic scale, when we rescale around a non-zero point of the bulk. Moreover, we will have to consider several steps of the Markov mechanism at the same time.
That is why we will need a more general result, giving a property of continuity of the Markov mechanism described above, with respect to its initial data.
The main results of the present paper concern convergence in distribution of point processes. In this section, we will assume properties of strong convergence, which can be done with the help of Skorokhod's representation theorem.
The notion of convergence of holomorphic functions usually considered is the uniform convergence on compact sets. This notion cannot be directly applied to the meromorphic functions involved here, because of the poles on the real line. That is why we will need an appropriate notion of uniform convergence of meromorphic functions.
More precisely, we say that a sequence $(f_n)_{n \geq 1}$ of meromorphic functions on an open set $U \subset \mathbb{C}$ converges uniformly
to a function $f$ from $U$ to the Riemann sphere $\mathbb{C} \cup \{\infty\}$ if and only if this convergence holds for the distance $d$ on $\mathbb{C} \cup \{\infty\}$, given by $$d(z_1, z_2) = \frac{|z_2 - z_1|}{ \sqrt{(1+ |z_1|^2)(1+|z_2|^2)}}$$ for $z_1, z_2 \neq \infty$, and extended by continuity at $\infty$ ($d$ corresponds to the distance of the points on the euclidian sphere, obtained via the inverse stereographic projection). It is a classical result that the limiting function $f$ should be meromorphic on $U$. One deduces the following: if a sequence $(f_n)_{n \geq 1}$ of meromorphic functions on $\mathbb{C}$ converges to a function $f$ from $\mathbb{C}$ to $\mathbb{C} \cup \{\infty\}$, uniformly on all bounded subsets of $\mathbb{C}$, then $f$ is meromorphic on $\mathbb{C}$. Morover, the following lemma will be useful: \begin{lemma} \label{lemmameromorphic}
Let $(f_n)_{n \geq 1}$ (resp. $(g_n)_{n \geq 1}$) be a sequence of meromorphic functions on an
open set $U$, uniformly convergent (for the distance $d$) to a function $f$ (resp. $g$),
necessarily meromorphic. We assume that none of these functions is identically $\infty$ on a connected component of $U$, and that $f$ and $g$ have no common pole. Then the sequence $(f_n + g_n)_{n \geq 1}$ of meromorphic functions tends uniformly to $f+g$ on all the compact sets of $U$. \end{lemma} \begin{proof} Let $K$ be a compact subset of $U$, let $z_1, z_2, \dots, z_p$ be the poles of $f$ in $K$, and $z'_1, z'_2, \dots, z'_q$ the poles of $g$ in $K$. There exists a neighborhood $V$ of $\{z_1,z_2, \dots, z_p\}$ containing no pole of $g$, and a neighborhood $W$ of $\{z'_1,z'_2, \dots, z'_p\}$ containing no pole of $f$. If $A > 0$ is fixed, one can assume the following (by restricting $V$ and $W$ if it is needed): \begin{itemize}
\item The infimum of $|f|$ on $V$ is larger than $2A+1$ and also larger than
the supremum of $2|g|+1$ on $V$.
\item The infimum of $|g|$ on $W$ is larger than $2A+1$ and also larger than
the supremum of $2|f|+1$ on $W$. \end{itemize} By the assumption of uniform convergence, we deduce, for $n$ large enough: \begin{itemize}
\item The infimum of $|f_n|$ on $V$ is larger than $2A$ and also larger than
the supremum of $2|g_n|$ on $V$.
\item The infimum of $|g_n|$ on $W$ is larger than $2A$ and also larger than
the supremum of $2|f_n|$ on $W$. \end{itemize} Now, for all $z \in V$ and $n$ large enough, one has
$$|f_n(z) + g_n(z)| \geq |f_n(z)| - |g_n(z)| \geq |f_n(z)| - \frac{ |f_n(z)|}{2}
= \frac{ |f_n(z)|}{2} \geq A.$$ and also
$$|f(z) + g(z)| \geq A,$$ which implies $$d(f_n(z) + g_n(z), f(z) + g(z)) \leq 2/A.$$ Similarly, this inequality is true for $z \in W$. Moreover, there exists a compact set $L \subset K$, containing no pole of $f$ or $g$, and such that $K$ is included in $L \cup V \cup W$. Since the meromorphic functions $f$ and $g$ have no pole on the compact set $L$, they are bounded on this set. Since $(f_n)_{n \geq 1}$ (resp. $(g_n)_{n \geq 1}$) converges to $f$ (resp. $g$) on $L$, uniformly for the distance $d$, and $(f_n)_{n \geq 1}$ (resp. $(g_n)_{n \geq 1}$) is uniformly bounded, the uniform convergence holds in fact for the usual distance. Hence, $(f_n + g_n)_{n \geq 1}$ tends uniformly to $f+g$ on $L$ for the usual distance, and a fortiori for $d$: by using the previous bounded obtained in $V$ and $W$, one deduces, since $L$, $V$ and $W$ cover $K$: $$\underset{n \rightarrow \infty} {\lim \sup} \, \underset{z \in K}{\sup} \, d(f_n(z) + g_n(z), f(z) + g(z)) \leq 2/A.$$ Since we can choose $A > 0$ arbitrarily, we are done. \end{proof} From this lemma, we deduce the following statement \begin{lemma} \label{lemmarational} Let $p \geq 1$, and let $(\lambda_k)_{1 \leq k \leq p}$, $(\lambda_{n,k})_{n \geq 1, 1 \leq k \leq p}$, $(\gamma_k)_{1 \leq k \leq p}$, $(\gamma_{n,k})_{n \geq 1, 1 \leq k \leq p}$ be some complex numbers such that all the $\lambda_k$'s are distincts, all the $\gamma_k$'s are nonzero, and for all $k \in \{1, \dots, p\}$, $$\lambda_{n,k} \underset{n \rightarrow \infty}{\longrightarrow} \lambda_k$$ and $$\gamma_{n,k} \underset{n \rightarrow \infty}{\longrightarrow} \gamma_k.$$ Then, one has, for $n$ going to infinity, the convergence of the rational function $$z \mapsto \sum_{k=1}^p \frac{\gamma_{n,k}}{\lambda_{n,k} - z}$$ towards the function $$z \mapsto \sum_{k=1}^p \frac{\gamma_{k}}{\lambda_{k} - z},$$ uniformly on all the compact sets, for the distance $d$. \end{lemma} \begin{proof} Let us first prove the result for $p=1$, which is implied by the following convergence $$\frac{\gamma_{n,1}}{\lambda_{n,1} - z } \underset{n \rightarrow \infty}{\longrightarrow} \frac{\gamma_{1}}{\lambda_{1} - z },$$ uniformly on $\mathbb{C}$ for the distance $d$. Let us fix $\epsilon > 0$. For $n$ large enough, we have
$|\lambda_{n,1} - \lambda_{1}| \leq \epsilon$ and
$|\gamma_{n,1} - \gamma_{1}| \leq |\gamma_1|/2$. If these conditions are satisfied and if $|\lambda_{1} - z| \leq 2 \epsilon$, then
$$\left|\frac{\gamma_{1}}{\lambda_{1} - z } \right| \geq \frac{|\gamma_1|}{2 \epsilon}$$ and
$$\left|\frac{\gamma_{n,1}}{\lambda_{n,1} - z } \right| \geq \frac{|\gamma_1|}{6 \epsilon},$$
since $|\gamma_{n,1}| \geq |\gamma_1|/2$ and
$$|\lambda_{n,1} - z| \leq |\lambda_{1} - z| + |\lambda_{n,1} - \lambda_{1}|\leq 3 \epsilon.$$ Hence, there exists $n_0 \geq 1$, independent of $z$ satisfying
$|\lambda_{1} - z| \leq 2 \epsilon$, such that for $n \geq n_0$, $$d\left( \frac{\gamma_{1}}{\lambda_{1} - z }, \frac{\gamma_{n,1}}{\lambda_{n,1} - z } \right) \leq d\left( \frac{\gamma_{1}}{\lambda_{1} - z }, \infty \right) + d\left( \infty, \frac{\gamma_{n,1}}{\lambda_{n,1} - z } \right)
\leq \frac{2 \epsilon}{|\gamma_1|} + \frac{6 \epsilon}{|\gamma_1|} = \frac{8 \epsilon}{|\gamma_1|}.$$
Similarly, there exists $n_1 \geq 1$ such that for all $n \geq n_1$ and for all $z$ satisfying $|\lambda_{1} - z| \geq 2 \epsilon$, one has:
$$|\lambda_{n,1} - z| \geq |\lambda_{1} - z| - |\lambda_{n,1} - \lambda_{1}| \geq \epsilon.$$ This implies: \begin{align*}
\left| \frac{\gamma_{1}}{\lambda_{1} - z } - \frac{\gamma_{n,1}}{\lambda_{n,1} - z } \right|
& \leq \left| \frac{\gamma_{1} - \gamma_{n,1} }{\lambda_{n,1} - z } \right|
+ |\gamma_1| \, \left| \frac{1}{\lambda_{1} - z} - \frac{1}{\lambda_{n,1} - z} \right|
\\ & \leq \frac{|\gamma_{1} - \gamma_{n,1} |}{\epsilon}
+ |\gamma_1| \, \frac{|\lambda_{1} - \lambda_{n,1}|}{(2\epsilon)(\epsilon)}. \end{align*} Since this quantity does not depend on $z$ and tends to zero at infinity, we deduce
$$\sup_{z \in \mathbb{C}, |\lambda_{1} - z| \geq 2 \epsilon} d\left( \frac{\gamma_{1}}{\lambda_{1} - z }, \frac{\gamma_{n,1}}{\lambda_{n,1} - z } \right) \underset{n \rightarrow \infty} {\longrightarrow} 0.$$ Since we know that $$\underset{n \rightarrow \infty}{\lim \, \sup}
\sup_{z \in \mathbb{C}, |\lambda_{1} - z| \leq 2 \epsilon} d\left( \frac{\gamma_{1}}{\lambda_{1} - z },
\frac{\gamma_{n,1}}{\lambda_{n,1} - z } \right)\leq \frac{8 \epsilon}{|\gamma_1|},$$ we get $$\underset{n \rightarrow \infty}{\lim \, \sup} \, \sup_{z \in \mathbb{C}} \, d\left( \frac{\gamma_{1}}{\lambda_{1} - z },
\frac{\gamma_{n,1}}{\lambda_{n,1} - z } \right)\leq \frac{8 \epsilon}{|\gamma_1|}.$$ Now, $\epsilon > 0$ can be arbitrarily chosen, and then the lemma is proven for $p=1$. For $p \geq 2$, let us deduce the result of the lemma, assuming that it is satisfied when $p$ is replaced by $p-1$. We define the meromorphic functions $(f_n)_{n \geq 1}$, $f$, $(g_n)_{n \geq 1}$, $g$ by the formulas: $$f_n(z) = \sum_{k=1}^{p-1} \frac{\gamma_{n,k}}{\lambda_{n,k} - z},$$ $$f(z) = \sum_{k=1}^{p-1} \frac{\gamma_{k}}{\lambda_{k} - z},$$ $$g_n(z) = \frac{\gamma_{n,p}}{\lambda_{n,p} - z},$$ $$g(z) = \frac{\gamma_{p}}{\lambda_{p} - z}.$$ Let $A > 0$. By the induction hypothesis, we know that $f_n$ converges to $f$ when
$n$ goes to infinity, uniformly on the set $\{z \in \mathbb{C}, |z| < 2A\}$ and for the distance $d$. Similarly, by the case $p=1$ proven above, $g_n$ converges to $g$, uniformly on the same set (in fact, uniformly on $\mathbb{C}$) and for the same distance. Moreover, the functions $f$ and $g$ have no common pole, since the numbers $(\lambda_k)_{1 \leq k \leq p}$ are all distinct. We can then apply Lemma \ref{lemmameromorphic} and deduce that $f_n+ g_n$ converges to $f + g$, uniformly on any compact set of
$\{z \in \mathbb{C}, |z| < 2A\}$, for example $\{z \in \mathbb{C}, |z| \leq A\}$,
and for the distance $d$. Since $A > 0$ can be arbitrarily chosen, we are done.
\end{proof}
We have now the ingredients needed to state the main result of this section. In this theorem, we deal with finite and infinite sequences together. So we will think of $k\mapsto \lambda_k$ as a function from $\mathbb Z\to \mathbb R\cup \{\emptyset\}$, with the convention that summation and other operations are only considered over the values that are different from $\emptyset$. We will also assume that the value $\emptyset$ is taken exactly on the complement of an interval of $\mathbb Z$.
The statement of the following result is long and technical, but as we will see in the next section, it will be adapted to the problem we are interested in.
\begin{theorem} \label{topologytheorem} Let $(\Xi_n)_{n \geq 1}$ be a sequence of discrete simple point measures on $\mathbb R$ (i.e. sums of Dirac masses at a locally finite set of points), converging to a simple point measure $\Xi$, locally weakly:
\begin{equation}\label{e:lambda1}
\Xi_n \longrightarrow \Xi. \end{equation} Let $L_n$ denote the support of $\Xi_n$, and $L$ the support of $\Xi$. We suppose that there exists $\alpha\in(0,1)$, a family $(\tau_\ell)_{\ell \geq 0}$ of elements of $\mathbb{R}_+^*$, with $\tau_\ell\to 0$ as $\ell\to\infty$, such that for all $n \geq 1$, $\ell \geq 1$, we have \begin{equation}\label{e:lambdasquared}
\int_{\mathbb R}\frac{\mathbf 1(|\lambda|>\ell)}{|\lambda|^{1+\alpha}}\,d\Xi_n(\lambda)
\le \tau_\ell \end{equation} Moreover, assume that the limits \begin{equation}\label{e:lambdasquared2}
h_{n,\ell} = \lim_{\ell'\to\infty} \int_{\mathbb R}\frac{\mathbf 1(\ell<|\lambda|<\ell')}{\lambda}\,d\Xi_n(\lambda) \end{equation} exist, and so does the similar limit $h_\ell$ defined in terms of $\Xi$. Assume further that for some $h\in \mathbb R$, the following equalities are well-defined and satisfied: \begin{equation}\label{e:hlim} \lim_{\ell\to \infty}\lim_{n\to \infty} h_{n,\ell} =h, \qquad \lim_{\ell\to \infty}h_{\ell} =0, \end{equation} when the limits are restricted to the condition: $\ell \notin L$ and $ -\ell \notin L$.
Further, let $(\gamma_{n,k})_{k\in \mathbb Z}$ be a strictly positive sequence.
Suppose it satisfies \begin{equation}
\gamma_{n,k} \to \gamma_k > 0 \label{e:gammalim} \end{equation} for each $k$, as $n\to\infty$. Also for some ${\bar \gamma},c>0$ and all $n,m \geq 1$, we assume \begin{eqnarray}\label{e:gammatight}\notag
\left|\sum_{k=0}^{m-1} \gamma_{n,k} - {\bar \gamma} m\right| &\le& c m^{\alpha'}\\
\left|\sum_{k=-m}^{-1} \gamma_{n,k} - {\bar \gamma} m\right| &\le& c m^{\alpha'} \end{eqnarray} with $0<(1+\alpha)\alpha'<1$. Let $\lambda^*$ be a point outside $L$, and consider the weighted version $\Lambda$ of $\Xi$ where the $k$-th point after $\lambda^*$ (for $k \leq 0$, the $(1-k)$-th point before $\lambda^*$) has weight $\gamma_k$. For $n$ large enough, one has also $\lambda^* \notin L_n$: define $\Lambda_n$ similarly. Then the limit $$ S_n(z)=\lim_{\ell\to\infty} \int_{[-\ell,\ell]} \frac{1}{\lambda-z}\,d\Lambda_n(\lambda) $$ exists for all $z\notin L_n$, is meromorphic with simple poles at $L_n$, and converges, uniformly on compacts with respect to the distance $d$ on the Riemann sphere $\mathbb{C} \cup \{\infty\}$, to $S(z)+\bar{\gamma}h$, where $S$ is a meromorphic function with simple poles at $L$, such that for all $z \notin L$, $$ S(z)=\lim_{\ell\to\infty} \int_{[-\ell,\ell]} \frac{1}{\lambda-z}\,d\Lambda(\lambda). $$
Moreover, for every $h'\in \mathbb R$, the sum of delta masses $\Xi'_n$ at $S_n^{-1}(h'+\bar{\gamma}h)$ converges locally weakly to the sum of delta masses $\Xi'$ at $S^{-1}(h')$, and $(\Xi'_{n})_{n \geq 1}$, $\Xi'$ satisfy the assumptions from \eqref{e:lambda1} to \eqref{e:hlim}. \end{theorem}
\begin{proof} Let $n \geq 1$, large enough in order to ensure that $\lambda^* \notin L_n$, $\ell > \ell_0 > 1$, and let $z$ be a complex number with modulus smaller than $\ell_0/2$. Let $k_{n, \ell_0}$ be the smallest index $k$ (if it exists) such that $\lambda_{n,k} > \ell_0$, where $\lambda_{n,k}$ is (if it exists) the $k$-th point of $L_n$ after $\lambda^*$. Similarly, let $K_{n, \ell}-1$ be the largest index $k$ (if it exists) such that $\lambda_{n,k} \leq \ell$. If for $m \in \mathbb{Z}$, $$\Delta_{n,m} := \mathbf{1}_{m \geq 0} \sum_{k=0}^{m-1} \gamma_{n,k} - \mathbf{1}_{m < 0} \sum_{k=m}^{-1} \gamma_{n,k} - \bar{\gamma} \,m,$$ then, in the case where $k_{n, \ell_0}$ and $K_{n, \ell}$ are well-defined and $k_{n, \ell_0} < K_{n, \ell}$: \begin{align*}
\int_{(\ell_0,\ell]} \frac{1}{\lambda-z}\,d\Lambda_n(\lambda) & = \sum_{k_{n,\ell_0} \leq k < K_{n,\ell} } \frac{ \gamma_{n,k}}{\lambda_{n,k} - z}
= \bar{\gamma} \sum_{k_{n,\ell_0} \leq k < K_{n,\ell}} \frac{1}{\lambda_{n,k} - z}
\\ & + \sum_{k_{n,\ell_0} \leq k < K_{n,\ell}} \frac{\Delta_{n,k+1} - \Delta_{n,k}}{\lambda_{n,k} - z}
= \bar{\gamma} \left( \sum_{k_{n,\ell_0} \leq k < K_{n,\ell}} \frac{1}{\lambda_{n,k} - z} \right) +
\frac{\Delta_{n,K_{n,\ell}}}{ \lambda_{n, K_{n,\ell}-1} - z } \\ & - \frac{\Delta_{n, k_{n,\ell_0}}}{ \lambda_{n, k_{n,\ell_0}} - z }
+ \sum_{k_{n,\ell_0} +1 \leq k < K_{n,\ell} } \Delta_{n,k} \left(\frac{1}{\lambda_{n,k-1} -z} - \frac{1}{\lambda_{n,k} -z} \right), \end{align*} which implies \begin{align*}
\int_{(\ell_0,\ell]} \frac{1}{\lambda-z}\,d\Lambda_n(\lambda) & - \bar{\gamma} \int_{(\ell_0, \ell]} \, \frac{d\Xi_n(\lambda)}{\lambda} = \bar{\gamma} z \left( \sum_{k_{n,\ell_0} \leq k < K_{n,\ell}} \frac{1}{\lambda_{n,k} (\lambda_{n,k} - z)} \right) + \frac{\Delta_{n,K_{n,\ell}}}{ \lambda_{n, K_{n,\ell} - 1} - z } \\ &
- \frac{\Delta_{n,k_{n,\ell_0}}}{ \lambda_{n, k_{n,\ell_0}} - z }
+ \sum_{k_{n,\ell_0} +1 \leq k < K_{n,\ell} } \Delta_{n,k} \left(\frac{\lambda_{n,k}
- \lambda_{n,k-1}}{(\lambda_{n,k-1} -z)
(\lambda_{n,k} -z)} \right). \end{align*} Note that in case where $k_{n,\ell_0}$ or $K_{n,\ell}$ is not well-defined, and in case where $k_{n,\ell_0} \geq K_{n,\ell}$, the left-hand side is zero, since $L_n$ has no point in the interval $(\ell_0, \ell]$. Let us now check that for $\ell$ going to infinity, this quantity converges,
uniformly in $\{z \in \mathbb{C}, |z| < \ell_0/2\}$, to the function $T_{n, \ell_0}$, holomorphic on this open set, and given by
\begin{align} T_{n, \ell_0}(z) & = \bar{\gamma} z \left( \sum_{k \geq k_{n,\ell_0} } \frac{1}{\lambda_{n,k} (\lambda_{n,k} - z)} \right)
+ \frac{\Delta_{n,K_{n,\infty}}}{ \lambda_{n, K_{n,\infty}-1} - z } - \frac{\Delta_{n,k_{n,\ell_0}}}{ \lambda_{n, k_{n,\ell_0}} - z } \nonumber \\ &
+ \sum_{ k \geq k_{n,\ell_0}+1} \Delta_{n,k} \left(\frac{\lambda_{n,k}
- \lambda_{n,k-1}}{(\lambda_{n,k-1} -z)
(\lambda_{n,k} -z)} \right), \label{Tnz}
\end{align}
if $L_n$ has at least one point in $(\ell_0, \infty)$, and $T_{n, \ell_0} (z) = 0$
otherwise. In the formula above, $K_{n, \infty}-1$ denotes the index of the largest point
of $L_n$ if this set is bounded from above, and otherwise,
$$\frac{\Delta_{n,K_{n,\infty}}}{ \lambda_{n, K_{n,\infty}-1} - z } := 0.$$
In order to prove this convergence, it is sufficient to check,
in case where
$L_n \cap (\ell_0, \infty) \neq \emptyset$,
the uniform convergence
$$
\frac{\Delta_{n,K_{n,\ell}}}{ \lambda_{n, K_{n,\ell} - 1} - z }
\underset{\ell \rightarrow \infty}{\longrightarrow}
\frac{\Delta_{n,K_{n,\infty}}}{ \lambda_{n, K_{n,\infty} - 1} - z },$$
for $|z| \leq \ell_0/2$, and the fact that
\begin{equation}
\sup_{z \in \mathbb{C}, |z| \leq \ell_0/2} \left( \sum_{k \geq k_{n,\ell_0} } \frac{1}{|\lambda_{n,k}||\lambda_{n,k}
- z|}+ \sum_{ k \geq k_{n,\ell_0}+1} |\Delta_{n,k}| \left(\frac{|\lambda_{n,k}
- \lambda_{n,k-1}|}{|\lambda_{n,k-1} -z| \,
|\lambda_{n,k} -z|} \right) \right)< \infty. \label{deltafinite}
\end{equation}
The first statement is immediate if $L_n$ is bounded from above. If $L_n$ is unbounded from
above,
let us remark that for $k \geq k_{n, \ell_0}$, $|z| \leq \ell_0/2$, one has $|\lambda_{n,k} - z| \geq \lambda_{n,k}/2$, and then it is sufficient to show: \begin{equation} \frac{\Delta_{n,K_{n,\ell}}}{ \lambda_{n, K_{n,\ell} - 1}} \underset{\ell \rightarrow \infty}{\longrightarrow} 0. \label{deltak1} \end{equation} Similarly, the statement \eqref{deltafinite} is implied by: \begin{equation} \sum_{k \geq k_{n,\ell_0} } \frac{1}{\lambda_{n,k}^2}+ \sum_{ k \geq k_{n,\ell_0}+1} \frac{
|\Delta_{n,k}| (\lambda_{n,k}
- \lambda_{n,k-1})}{\lambda_{n,k-1} \,
\lambda_{n,k}} < \infty. \label{deltak2} \end{equation} In order to prove \eqref{deltak1}, let us first use the majorization
\eqref{e:lambdasquared}, which implies, for all $\ell > 2$,
$$\Xi_n ([2, \ell]) \leq
\ell^{1 + \alpha} \, \int_{2}^{\ell} \frac{d\Xi_n(\lambda)}{\lambda^{1 + \alpha}}
\leq \ell^{1 + \alpha} \, \int_{\mathbb R}\frac{\mathbf 1(|\lambda|>1)}{|\lambda|^{1+\alpha}}\,d\Xi_n(\lambda) \leq \tau_1 \, \ell^{1 + \alpha},$$
and then
$$\Xi_n ([\lambda^*, \ell]) \leq \tau \, \ell^{1+ \alpha}$$
where
\begin{equation}
\tau := \tau_1 + \Xi_n ([\lambda^* \wedge 2, 2]). \label{tau}
\end{equation} We deduce, for $k \geq 1$ large enough in order to insure that $\lambda_{n,k} > 2$, $$k = \Xi_n ([\lambda^*, \lambda_{n,k}]) \leq \tau \, \lambda_{n,k}^{1+\alpha}$$ and then \begin{equation} \lambda_{n,k} \geq (k/ \tau)^{1/(1+\alpha)}. \label{minorizationlambda} \end{equation} By using \eqref{e:gammatight}, this inequality implies: $$\frac{\Delta_{n,k+1}}{ \lambda_{n,k}} \leq c (k+1)^{\alpha'} (k/\tau)^{-1/(1+\alpha)},$$ which tends to zero when $k$ goes to infinity, since $\alpha' < 1/(1+\alpha)$ by assumption. Therefore, we have \eqref{deltak1}.
Moreover, the left-hand side of \eqref{deltak2} is given by \begin{align*}
& \int_{\mathbb R} \frac{\mathbf 1(|\lambda|>\ell_0)}{|\lambda|^{2}}\,d\Xi_n(\lambda) + \sum_{ k \geq k_{n,\ell_0}+1}
|\Delta_{n,k}| \, \left( \frac{1}{\lambda_{n,k-1}} - \frac{1}{\lambda_{n,k}} \right)
\\ & \leq \int_{\mathbb R}\frac{\mathbf 1(|\lambda|>\ell_0)}{|\lambda|^{1+\alpha}}\,d\Xi_n(\lambda)
+ c \, \sum_{ k \geq k_{n,\ell_0}+1} |k|^{\alpha'} \, \left( \frac{1}{\lambda_{n,k-1}} - \frac{1}{\lambda_{n,k}} \right)
\\ & \leq \tau_{\ell_0} + c \left( \frac{|k_{n,\ell_0}+1|^{\alpha'}}{\lambda_{n,k_{n,\ell_0}}}
+ \sum_{ k \geq k_{n,\ell_0}+1} \frac{(|k+1|^{\alpha'} - |k|^{\alpha'})}{\lambda_{n,k}} \right). \end{align*}
If $L_n$ is bounded from above, the finiteness of this quantity is obvious. Otherwise, we know that for $k$ large enough, $(|k+1|^{\alpha'} - |k|^{\alpha'})$ is bounded by a constant times $k^{\alpha' - 1}$, and $\lambda_{k, n}$ dominates $k^{1/(1+\alpha)}$. Hence, it is sufficient to check the finiteness of the following expression: $$ \sum_{k = 1}^{\infty} k^{\alpha' -1} k^{-1/(1+\alpha)},$$ which is satisfied since by assumption, $$\alpha' - 1 - \frac{1}{1 + \alpha} < -1.$$ We have now proven: \begin{equation} \int_{(\ell_0,\ell]} \frac{1}{\lambda-z}\,d\Lambda_n(\lambda) - \bar{\gamma} \int_{(\ell_0, \ell]} \, \frac{d\Xi_n(\lambda)}{\lambda} \underset{\ell \rightarrow \infty}{\longrightarrow} T_{n, \ell_0}(z), \label{uniformTnz} \end{equation}
uniformly on the set $\{z \in \mathbb{C}, |z| < \ell_0/2\}$, where the holomorphic function $T_{n, \ell_0}$ is given by the formula \eqref{Tnz}.
Similarly, there exists an holomorphic function $U_{n, \ell_0}$ on
$\{z \in \mathbb{C}, |z| < \ell_0/2\}$, such that uniformly on this set,
\begin{equation} \int_{[-\ell, -\ell_0)} \frac{1}{\lambda-z}\,d\Lambda_n(\lambda) - \bar{\gamma} \int_{[-\ell, -\ell_0)} \, \frac{d\Xi_n(\lambda)}{\lambda} \underset{\ell \rightarrow \infty}{\longrightarrow} U_{n, \ell_0}(z). \label{uniformUnz} \end{equation}
The function $U_{n, \ell_0}$ can be explicitly described by a formula similar to \eqref{Tnz} (we omit the detail of this formula). By combining \eqref{e:lambdasquared2}, \eqref{uniformTnz} and \eqref{uniformUnz}, one deduces the following uniform convergence on $\{z \in \mathbb{C}, |z| < \ell_0/2\}$:
$$\int_{[-\ell, \ell] \backslash [-\ell_0, \ell_0]} \frac{1}{\lambda-z}\,d\Lambda_n(\lambda) \underset{\ell \rightarrow \infty}{\longrightarrow} T_{n, \ell_0}(z) + U_{n, \ell_0}(z) + \bar{\gamma} h_{n, \ell_0}.$$ One deduces, by using Lemma \ref{lemmameromorphic}, that $$\int_{[-\ell, \ell]} \frac{1}{\lambda-z}\,d\Lambda_n(\lambda) \underset{\ell \rightarrow \infty}{\longrightarrow} T_{n, \ell_0}(z) + U_{n, \ell_0}(z) + \bar{\gamma} h_{n, \ell_0} + \int_{[-\ell_0, \ell_0]} \frac{1}{\lambda-z}\,d\Lambda_n(\lambda) =: S_{n, \ell_0} (z),$$
uniformly on any compact subset of $\{z \in \mathbb{C}, |z| < \ell_0/2\}$, for the distance $d$ on the Riemann sphere. One checks immediately that the poles of $S_{n, \ell_0}$ with modulus smaller than or equal to $\ell_0/2$ are exactly the points of $L_n$ satisfying the same condition. Moreover, the convergence just above implies that for $\ell_1 > \ell_0 > 1$, the meromorphic functions $S_{n, \ell_0}$ and $S_{n, \ell_1}$ coincide on
$\{z \in \mathbb{C}, |z| < \ell_0/2\}$: hence, there exists a meromorphic function
$S_n$ on $\mathbb{C}$, such that for all $\ell_0 > 1$, the restriction of $S_n$ to $\{z \in \mathbb{C}, |z| < \ell_0/2\}$ is equal to $S_{n, \ell_0}$. The poles of $S_n$ are exactly the points of $L_n$, and one has, uniformly on all compact sets of $\mathbb{C}$ and for the distance $d$, $$\int_{[-\ell, \ell]} \frac{1}{\lambda-z}\,d\Lambda_n(\lambda) \underset{\ell \rightarrow \infty}{\longrightarrow} S_n(z).$$ In particular, the convergence holds pointwise for all $z \notin L_n$.
In an exactly similar way, one can prove that uniformly on compact sets of $\mathbb{C}$, for the distance $d$, $$\int_{[-\ell, \ell]} \frac{1}{\lambda-z}\,d\Lambda(\lambda) \underset{\ell \rightarrow \infty}{\longrightarrow} S(z)$$ where for all $\ell_0 > 1$, $$S(z) := T_{\ell_0}(z) + U_{\ell_0}(z) + \bar{\gamma} h_{\ell_0} + \int_{[-\ell_0, \ell_0]} \frac{1}{\lambda-z}\,d\Lambda(\lambda),$$
on the set $\{z \in \mathbb{C}, |z|< \ell_0/2\}$, $T_{\ell_0}$ and $U_{\ell_0}$ being defined by the same formulas as $T_{n,\ell_0}$ and $U_{n,\ell_0}$, except than one removes all the indices $n$. In order to show this convergence, it is sufficient to check that the assumptions \eqref{e:lambdasquared} and \eqref{e:gammatight} are satisfied if the indices $n$ are removed. For \eqref{e:gammatight}, it is an immediate consequence
of the convergence \eqref{e:gammalim}, since the constant $c$ does not depend on $n$. For \eqref{e:lambdasquared}, let us first observe that for all $\ell > 1$, and for any continuous function $\Phi$ with compact support, such that for all $\lambda \in \mathbb{R}$,
$$\Phi(\lambda) \leq \frac{\mathbf{1}(|\lambda| > \ell)}{|\lambda|^{1+\alpha}},$$ one has for all $n \geq 1$,
$$ \int_{\mathbb R} \Phi(\lambda) d\Xi_n(\lambda) \leq \int_{\mathbb R}\frac{\mathbf 1(|\lambda|>\ell)}{|\lambda|^{1+\alpha}}\,d\Xi_n(\lambda)
\le \tau_\ell.$$ Since $\Xi_n$ converges weakly to $\Xi$ when $n$ goes to infinity, one deduces: $$\int_{\mathbb R} \Phi(\lambda) d\Xi(\lambda) \leq \tau_{\ell}.$$ By taking $\Phi$ increasing to
$$\lambda \mapsto \mathbf{1}(|\lambda| > \ell)/ |\lambda|^{1+\alpha},$$ one obtains
$$\int_{\mathbb R}\frac{\mathbf 1(|\lambda|>\ell)}{|\lambda|^{1+\alpha}}\,d\Xi(\lambda)
\le \tau_\ell,$$
i.e. the equivalent of \eqref{e:lambdasquared} for the measure $\Xi$.
Once the existence of the functions $S_n$ and $S$ is ensured, it remains to
prove the convergence of $S_n$ towards $S+ \bar{\gamma} h$, uniformly on compact sets for the distance $d$.
In order to check this convergence, it is sufficient to prove that there exists
$\ell_0 > 1$ arbitrarily large, such that uniformly on any compact set
of $\{z \in \mathbb{C}, |z| < \ell_0/2\}$,
\begin{align*}
& T_{n, \ell_0}(z) + U_{n, \ell_0}(z) + \bar{\gamma} h_{n, \ell_0} + \int_{[-\ell_0, \ell_0]} \frac{1}{\lambda-z}\,d\Lambda_n(\lambda) \\ & \underset{n \rightarrow \infty}{\longrightarrow} T_{\ell_0}(z) + U_{\ell_0}(z) + \bar{\gamma} (h_{ \ell_0} + h ) + \int_{[-\ell_0, \ell_0]} \frac{1}{\lambda-z}\,d\Lambda(\lambda).
\end{align*}
In fact, we will prove this convergence for any $\ell_0 >2$ such that $\ell_0$ and
$-\ell_0$ are not in $L$, and then not in $L_n$ for $n$ large enough.
By Lemma \ref{lemmameromorphic}, it is sufficient to check for such an $\ell_0$:
\begin{equation}
h_{n, \ell_0} \underset{n \rightarrow \infty}{\longrightarrow} h_{\ell_0} + h, \label{convergenceh}
\end{equation}
\begin{equation}
\int_{[-\ell_0, \ell_0]} \frac{1}{\lambda-z}\,d\Lambda_n(\lambda) \underset{n \rightarrow \infty}{\longrightarrow}
\int_{[-\ell_0, \ell_0]} \frac{1}{\lambda-z}\,d\Lambda(\lambda), \label{convergencegamma} \end{equation}
uniformly on $\{z \in \mathbb{C}, |z| < \ell_0/2\}$ for the distance $d$, \begin{equation} T_{n, \ell_0}(z) \underset{n \rightarrow \infty}{\longrightarrow} T_{\ell_0}(z), \label{convergenceT} \end{equation}
uniformly on $\{z \in \mathbb{C}, |z| < \ell_0/2\}$, and \begin{equation} U_{n, \ell_0}(z) \underset{n \rightarrow \infty}{\longrightarrow} U_{\ell_0}(z), \label{convergenceU} \end{equation}
also uniformly on $\{z \in \mathbb{C}, |z| < \ell_0/2\}$. Since the proof of \eqref{convergenceU} is exactly similar to the proof of \eqref{convergenceT}, we will omit it and we will then show successively \eqref{convergenceh}, \eqref{convergencegamma} and \eqref{convergenceT}.
Let us first prove \eqref{convergenceh}. For all $\ell_1 > \ell_0 $ such that $-\ell_1$ and $\ell_1$ are not in $L$, one has
$$h_{n, \ell_0} - h_{n, \ell_1} = \int_{\mathbb R}\frac{\mathbf 1(\ell_0<|\lambda| \leq \ell_1)}{\lambda}\,d\Xi_n(\lambda)$$ and
$$h_{\ell_0} - h_{\ell_1} = \int_{\mathbb R}\frac{\mathbf 1(\ell_0<|\lambda| \leq \ell_1)}{\lambda}\,d\Xi(\lambda).$$ Now, $-\ell_1, -\ell_0, \ell_0, \ell_1$ are not in the support of $\Xi$, and since $\Xi$ is a discrete measure, there is a neighborhood of $\{-\ell_1, -\ell_0, \ell_0, \ell_1\}$ which does not charge $\Xi$. One deduces that there exist two functions $\Phi$ and $\Psi$ from $\mathbb{R}$ to $\mathbb{R}_+$, continuous with compact support, such that for all $\lambda \in \mathbb{R}$,
$$\Phi(\lambda) \leq \frac{\mathbf 1(\ell_0<|\lambda| \leq \ell_1)}{\lambda} \leq \Psi(\lambda)$$ and $$\int_{\mathbb R} \Phi(\lambda) \, d\Xi(\lambda) = h_{\ell_0} - h_{\ell_1} = \int_{\mathbb R} \Psi(\lambda) \, d\Xi(\lambda).$$ Since $\Xi_n$ tends weakly to $\Xi$ when $n$ goes to infinity, one deduces that $$\int_{\mathbb R} \Phi(\lambda) \, d\Xi_n(\lambda) \underset{n \rightarrow \infty} {\longrightarrow} \int_{\mathbb R} \Phi(\lambda) \, d\Xi(\lambda) = h_{\ell_0} - h_{\ell_1} $$ and similarly, $$\int_{\mathbb R} \Psi(\lambda) \, d\Xi_n(\lambda) \underset{n \rightarrow \infty} {\longrightarrow} h_{\ell_0} - h_{\ell_1}.$$ By the squeeze theorem, one deduces
$$ h_{n, \ell_0} - h_{n, \ell_1} = \int_{\mathbb R}\frac{\mathbf 1(\ell_0<|\lambda| \leq \ell_1)}{\lambda}\,d\Xi_n(\lambda)\underset{n \rightarrow \infty} {\longrightarrow} h_{\ell_0} - h_{\ell_1}.$$ Hence, $$\underset{n \rightarrow \infty}{\lim} h_{n, \ell_0} - \underset{n \rightarrow \infty}{\lim} h_{n, \ell_1} = h_{\ell_0} - h_{\ell_1}.$$ where, by assumption, the two limits in the left-hand side are well-defined. By \eqref{e:hlim}, one deduces, by taking $\ell_1 \rightarrow \infty$, $$\underset{n \rightarrow \infty}{\lim} h_{n, \ell_0} - h = h_{\ell_0},$$ which proves \eqref{convergenceh}. In order to show \eqref{convergencegamma}, let us first check the following properties, available for all $k \in \mathbb{Z}$: \begin{itemize} \item If $\lambda_k$ is well-defined, then $\lambda_{n,k}$ is well-defined for all $n$ large enough and tends to $\lambda_{k}$ when $n$ goes to infinity. \item If $\lambda_k$ is not well-defined, then for all $A > 0$, there are finitely many indices $n$ such that $\lambda_{n,k}$ is well-defined and in the interval $[-A,A]$. \end{itemize} By symmetry, we can assume that $k \geq 1$. We know that $\lambda^*$ is not in $L$, and then for $\epsilon > 0$ small enough, \begin{equation} L \cap [\lambda^* - 3 \epsilon, \lambda^* + 3 \epsilon] = \emptyset, \label{nolambda*1} \end{equation} Let us fix $\epsilon > 0$ satisfying this property. Since $\Xi_n$ tends locally weakly to $\Xi$, we deduce that for $n$ large enough, \begin{equation}
L_n \cap [\lambda^* - 2 \epsilon, \lambda^* + 2\epsilon] = \emptyset, \label{nolambda*2}
\end{equation} which implies that $\lambda_k \geq \lambda_1 > \lambda^* + 2\epsilon$. Now, let $\Phi$ and $\Psi$ be two continuous functions with compact support, such that: \begin{itemize} \item For $\lambda \leq \lambda^* - \epsilon$, $$\Phi(\lambda) = \Psi(\lambda) = 0.$$ \item For $\lambda^* - \epsilon \leq \lambda \leq \lambda^* + \epsilon$,
$$0 \leq \Phi(\lambda) = \Psi(\lambda) \leq 1.$$
\item For $\lambda^* + \epsilon \leq \lambda \leq \lambda_k - \epsilon$,
$$\Phi(\lambda) = \Psi(\lambda) = 1$$ (recall that
$\lambda^* + \epsilon < \lambda_k - \epsilon$). \item For $\lambda_k - \epsilon \leq \lambda \leq \lambda_k$, $$0 \leq \Phi(\lambda) \leq \Psi(\lambda) = 1.$$ \item For $\lambda_k \leq \lambda \leq \lambda_k+ \epsilon$, $$0 = \Phi(\lambda) \leq \Psi(\lambda) \leq 1.$$ \item For $ \lambda \geq \lambda_k+ \epsilon$, $$\Phi(\lambda) = \Psi(\lambda) = 0.$$ \end{itemize} By using \eqref{nolambda*1}, we deduce $$\int_{\mathbb{R}} \Phi(\lambda) d \Xi(\lambda) \leq \Xi ([\lambda^* - \epsilon, \lambda_k)) =
\Xi ([\lambda^*, \lambda_k)) = k-1$$
and
$$\int_{\mathbb{R}} \Psi(\lambda) d \Xi(\lambda) \geq \Xi ([\lambda^* + \epsilon, \lambda_k]) =
\Xi ([\lambda^*, \lambda_k]) = k.$$
Hence, for $n$ large enough,
$$\int_{\mathbb{R}} \Phi(\lambda) d \Xi_n(\lambda) \leq k - 1/2$$
and
$$\int_{\mathbb{R}} \Psi(\lambda) d \Xi_n(\lambda) \geq k - 1/2,$$
which implies
$$ \Xi_n ([\lambda^*, \lambda_k - \epsilon)) =
\Xi_n ([\lambda^*+ \epsilon, \lambda_k - \epsilon))
\leq \int_{\mathbb{R}} \Phi(\lambda) d \Xi_n(\lambda) \leq k - 1/2$$
and
$$\Xi_n ([\lambda^*, \lambda_k + \epsilon)) =
\Xi_n ([\lambda^*- \epsilon, \lambda_k + \epsilon])
\geq \int_{\mathbb{R}} \Psi(\lambda) d \Xi_n(\lambda) \geq k - 1/2.$$
Therefore, for $n$ large enough the point $\lambda_{n,k}$ is well-defined and between
$\lambda_{k} - \epsilon$ and $\lambda_{k} + \epsilon$. Since $\epsilon$ and
be taken arbitrarily small, we
have proven the convergence claimed above in the case where $\lambda_k$ is
well-defined. If $\lambda_k$ is not well-defined, let us choose $\epsilon > 0$
satisfying \eqref{nolambda*1}, and $A > |\lambda^*|$. Let $\Phi$ be
a continuous function with compact support, such that:
\begin{itemize}
\item For all $\lambda \in \mathbb{R}$, $\Phi(\lambda) \in [0,1]$.
\item For all $\lambda \in [\lambda^*, A]$, $\Phi(\lambda) = 1$.
\item For all $\lambda \notin (\lambda^*-\epsilon, A + \epsilon)$, $\Phi(\lambda) = 0$.
\end{itemize} Since $\lambda_k$ is not well-defined, $$\int_{\mathbb{R}} \Phi(\lambda) d \Xi(\lambda) \leq \Xi([\lambda^*- \epsilon, A+ \epsilon ]) = \Xi([\lambda^*, A +\epsilon]) \leq \Xi([\lambda^*, \infty)) \leq k-1,$$ and then for $n$ large enough, $$\Xi_n([\lambda^*, A]) \leq \int_{\mathbb{R}} \Phi(\lambda) d \Xi_n(\lambda) \leq k-1/2,$$ which implies that $\lambda_{n,k}$ cannot be well-defined and smaller than or equal to $A$. This proves the second claim. Let us now go back to the proof of \eqref{convergencegamma}. If $L \cap [-\ell_0, \ell_0] = \emptyset$, then $L \cap [-\ell_0- \epsilon, \ell_0+ \epsilon] = \emptyset$ for some $\epsilon > 0$. Hence, there exists a nonnegative, continuous function with compact support $\Phi$ such that $\Phi(\lambda) = 1$ for all $\lambda \in [-\ell_0, \ell_0]$,
and
$$\int_{\mathbb{R}} \Phi(\lambda) d \Xi(\lambda) = 0,$$
which implies, for $n$ large enough,
$$\Xi([-\ell_0, \ell_0]) \leq
\int_{\mathbb{R}} \Phi(\lambda) d \Xi_n(\lambda) \leq 1/2,$$ i.e. $L_n \cap [-\ell_0, \ell_0] = \emptyset$. Hence, for $n$ large enough, the two expressions involved in \eqref{convergencegamma} are identically zero. If $L \cap [-\ell_0, \ell_0] \neq \emptyset$, let $k_1$ and $k_2$ be the smallest and the largest indices $k$ such that $\lambda_k \in (-\ell_0, \ell_0)$. Since $\lambda_{n,k_1}$ and $\lambda_{n,k_2}$ converge respectively to $\lambda_{k_1}$ and $\lambda_{k_2}$ when $n$ goes to infinity, one has $\lambda_{n,k_1}$ and $\lambda_{n,k_2}$ in the interval $(-\ell_0, \ell_0)$ for $n$ large enough. On the other hand, $\lambda_{k_2+1}$ is either strictly larger than $\ell_0$ (strictly because by assumption, $\ell_0 \notin L$), or not well-defined. In both cases, there are only finitely many indices $n$ such that $\lambda_{n,k_2+1} \leq \ell_0$. Similarly, by using the fact that $-\ell_0 \notin L$, one checks that there are finitely many indices $n$ such that $\lambda_{n,k_1-1} \geq -\ell_0$. Hence, for $n$ large enough, the indices $k$ such that $\lambda_{n,k} \in [-\ell_0, \ell_0]$ are exactly the integers between $k_1$ and $k_2$, which implies $$ \int_{[-\ell_0, \ell_0]} \frac{1}{\lambda-z}\,d\Lambda_n(\lambda) = \sum_{k= k_1}^{k_2} \frac{\gamma_{n,k}}{\lambda_{n,k} - z},$$ whereas $$ \int_{[-\ell_0, \ell_0]} \frac{1}{\lambda-z}\,d\Lambda(\lambda) = \sum_{k= k_1}^{k_2} \frac{\gamma_{k}}{\lambda_{k} - z}.$$ We have shown that for all $k$ between $k_1$ and $k_2$, $\lambda_{n,k}$ tends to $\lambda_k$ when $n$ goes to infinity and by assumption, $\gamma_{n,k}$ tends to $\gamma_k$. Moreover, the numbers $\lambda_k$ are all distincts, and by assumption, $\gamma_k \neq 0$ for all $k$. Hence, one can apply Lemma \ref{lemmarational} to deduce \eqref{convergencegamma}.
Let us now prove \eqref{convergenceT}. If $L \cap (\ell_0, \infty) = \emptyset$, this statement can be deduced from the following convergences, uniformly on $\{z \in \mathbb{C}, |z| < \ell_0/2\}$: \begin{equation} \mathbf{1}_{L_n \cap (\ell_0, \infty) \neq \emptyset} \, \sum_{k \geq k_{n,\ell_0} } \frac{1}{\lambda_{n,k} (\lambda_{n,k} - z)} \underset{n \rightarrow \infty}{\longrightarrow} 0, \label{convergenceterm1} \end{equation} \begin{equation} \mathbf{1}_{L_n \cap (\ell_0, \infty) \neq \emptyset} \, \frac{\Delta_{n,K_{n,\infty}}}{ \lambda_{n, K_{n,\infty}-1} - z } \underset{n \rightarrow \infty}{\longrightarrow} 0, \label{convergenceterm2} \end{equation} \begin{equation} \mathbf{1}_{L_n \cap (\ell_0, \infty) \neq \emptyset} \, \frac{\Delta_{n,k_{n,\ell_0}}}{
\lambda_{n, k_{n,\ell_0}} - z } \underset{n \rightarrow \infty}{\longrightarrow} 0 \label{convergenceterm3} \end{equation} and \begin{equation} \mathbf{1}_{L_n \cap (\ell_0, \infty) \neq \emptyset} \, \sum_{ k \geq k_{n,\ell_0}+1} \Delta_{n,k} \left(\frac{\lambda_{n,k}
- \lambda_{n,k-1}}{(\lambda_{n,k-1} -z)
(\lambda_{n,k} -z)} \right) \underset{n \rightarrow \infty}{\longrightarrow} 0. \label{convergenceterm4} \end{equation} If $L \cap (\ell_0, \infty) \neq \emptyset$, then we have proven previously that $\lambda_{n,k_{\ell_0}}$ is well-defined for $n$ large enough and converges to $\lambda_{k_{\ell_0}} > \ell_0$ when $n$ goes to infinity: in particular, $\lambda_{n,k_{\ell_0}} > \ell_0$ for $n$ large enough. Moreover, one of the two following cases occurs: \begin{itemize} \item If $\lambda_{k_{\ell_0} - 1}$ is well-defined, then it is strictly smaller than $\ell_0$ (strictly because $\ell_0$ is, by assumption, not in $L$), and then $\lambda_{n,k_{\ell_0} - 1}$ is, for $n$ large enough, well-defined and strictly smaller than $\ell_0$. \item If $\lambda_{k_{\ell_0} - 1}$ is not well-defined, and if $A > 0$,
then for $n$ large enough, $\lambda_{n,k_{\ell_0} - 1}$ is not well-defined or has an absolute value strictly greater than $A$. By taking $A = \lambda_{k_{\ell_0}} + 1$, one deduces that for $n$ large enough, $\lambda_{n,k_{\ell_0} - 1}$ is not-well defined, strictly smaller than $- \lambda_{k_{\ell_0}} - 1$ or strictly larger than $ \lambda_{k_{\ell_0}} + 1$. This last case is impossible for $n$ large enough, since $\lambda_{n,k_{\ell_0} - 1}$ is smaller than
$\lambda_{n,k_{\ell_0}}$, which tends to $\lambda_{k_{\ell_0}}$. Hence, there are
finitely many indices $n$ such that $\lambda_{n,k_{\ell_0} - 1}$ is well-defined
and larger than $- \lambda_{k_{\ell_0}} - 1$, and a fortiori, larger than
or equal to $\ell_0$. \end{itemize} All this discussion implies easily that for $n$ large enough, $k_{n,\ell_0} =
k_{\ell_0}$, and then it is sufficient to prove the uniform convergences on $\{z \in \mathbb{C}, |z| < \ell_0/2\}$: \begin{equation}
\sum_{k \geq k_{\ell_0} } \frac{1}{\lambda_{n,k} (\lambda_{n,k} - z)} \underset{n \rightarrow \infty}{\longrightarrow} \sum_{k \geq k_{\ell_0} } \frac{1}{\lambda_{k} (\lambda_{k} - z)}, \label{convergenceterm5} \end{equation} \begin{equation}
\frac{\Delta_{n,k_{\ell_0}}}{ \lambda_{n, k_{\ell_0}} - z } \underset{n \rightarrow \infty}{\longrightarrow} \frac{\Delta_{k_{\ell_0}}}{ \lambda_{ k_{\ell_0}} - z }
\label{convergenceterm6} \end{equation} and \begin{align} & \frac{\Delta_{n,K_{n,\infty}}}{ \lambda_{n, K_{n,\infty}-1} - z } + \sum_{ k \geq k_{\ell_0}+1} \Delta_{n,k} \left(\frac{\lambda_{n,k}
- \lambda_{n,k-1}}{(\lambda_{n,k-1} -z)
(\lambda_{n,k} -z)} \right) \nonumber \\ & \underset{n \rightarrow \infty}{\longrightarrow}
\frac{\Delta_{K_{\infty}}}{
\lambda_{ K_{\infty}-1} - z } +\sum_{ k \geq k_{\ell_0}+1} \Delta_{k} \left(\frac{\lambda_{k}
- \lambda_{k-1}}{(\lambda_{k-1} -z)
(\lambda_{k} -z)} \right), \label{convergenceterm7} \end{align} with obvious notation.
Let us first prove \eqref{convergenceterm1}. If $L \cap (\ell_0, \infty) = \emptyset$, then $L \cap (\ell_0 - \epsilon, \infty) = \emptyset$
for some $\epsilon > 0$ (recall that $\ell_0 \notin L$). Hence, for all $A > \ell_0$, and $n$ large enough depending on $A$, $L_n \cap (\ell_0,A] = \emptyset$, which implies, for $|z| \leq \ell_0/2$, \begin{align*}
\left|\mathbf{1}_{L_n \cap (\ell_0, \infty) \neq \emptyset} \, \sum_{k \geq k_{n,\ell_0} } \frac{1}{\lambda_{n,k} (\lambda_{n,k} - z)} \right| & \leq 2 \int_{\mathbb{R}} \frac{\mathbf{1} (\lambda > \ell_0)}{\lambda^2} \, d \Xi_n(\lambda) \\ & = 2 \int_{\mathbb{R}} \frac{\mathbf{1} (\lambda > A)}{\lambda^2} \, d \Xi_n(\lambda) \\ & \leq 2\int_{\mathbb{R}} \frac{\mathbf{1} (\lambda > A)}{\lambda^{1+\alpha}} \, d \Xi_n(\lambda) \leq 2 \tau_A. \end{align*} By letting $n \rightarrow \infty$ and then $A \rightarrow \infty$, one deduces \eqref{convergenceterm1}.
Let us prove \eqref{convergenceterm2} and \eqref{convergenceterm3}. By using the estimates \eqref{tau} and \eqref{minorizationlambda} proven above, one deduces that for $$\tilde{\tau} := \tau_1 + \Xi([\lambda^* \wedge 2, 2]) + \sup_{n \geq 1} \Xi_n([\lambda^* \wedge 2, 2]),$$ one has, for any $k \geq 1$, \begin{equation} \lambda_{k} \geq (k/\tilde{\tau})^{1/(1+\alpha)} \label{minorizationlambda2}, \end{equation} if $\lambda_{k}> 2$, and uniformly in $n$, \begin{equation} \lambda_{k,n} \geq (k/\tilde{\tau})^{1/(1+\alpha)} \label{minorizationlambda3}, \end{equation} if $\lambda_{k,n} > 2$. Now, let us assume that $L \cap (\ell_0, \infty) = \emptyset$ and $L_n \cap (\ell_0, \infty) \neq \emptyset$. If $n$ is large enough, then for any index $k$ such that $\lambda_{n,k} > \ell_0$, one has also $\lambda_{n,k} > \lambda^* \vee 2$, since $L \cap (\ell_0-\epsilon, (\lambda^* \vee 2) + 1) = \emptyset$ for some $\epsilon > 0$, and $\Xi_{n} \rightarrow \Xi$. Hence,
$k \geq 1$ and \eqref{minorizationlambda3} is satisfied. By using this inequality and \eqref{e:gammatight}, one deduces, for $|z| \leq \ell_0/2$, \begin{align*}
& \left|\mathbf{1}_{L_n \cap (\ell_0, \infty) \neq \emptyset} \, \frac{\Delta_{n,K_{n,\infty}}}{ \lambda_{n, K_{n,\infty}-1} - z } \right| +
\left|\mathbf{1}_{L_n \cap (\ell_0, \infty) \neq \emptyset} \, \frac{\Delta_{n,k_{n,\ell_0}}}{
\lambda_{n, k_{n,\ell_0}} - z }\right| \\ &
\leq \sup_{k \geq 1} \frac{2 c (k+1)^{\alpha'}} {(k/\tilde{\tau})^{1/(1+\alpha)} \vee \lambda_{n, k_{n,\ell_0}} }
+\sup_{k \geq 1} \frac{2 c k^{\alpha'}} {(k/\tilde{\tau})^{1/(1+\alpha)} \vee \lambda_{n, k_{n,\ell_0}} } \\ & \leq 4c (1+2^{\alpha'}) (1+\tilde{\tau})^{1/(1+\alpha)} \sup_{k \geq 1} \frac{k^{\alpha'}} {k^{1/(1+\alpha)} \vee \lambda_{n, k_{n,\ell_0}} } \\ & = 4c (1+2^{\alpha'}) (1+\tilde{\tau})^{1/(1+\alpha)}\sup_{k \geq 1} \frac{(k^{1/(1+ \alpha)})^{\alpha'(1+\alpha)}} {k^{1/(1+\alpha)} \vee \lambda_{n, k_{n,\ell_0}} } \\ & \leq 4c (1+2^{\alpha'}) (1+\tilde{\tau})^{1/(1+\alpha)}\sup_{k \geq 1}
(k^{1/(1+\alpha)} \vee \lambda_{n, k_{n,\ell_0}})^{\alpha'(1+\alpha) - 1}
\\ & \leq 4c (1+2^{\alpha'}) (1+\tilde{\tau})^{1/(1+\alpha)}\lambda_{n, k_{n,\ell_0}}^{\alpha'(1+\alpha) - 1},
\end{align*}
where $\lambda_{n, k_{n,\ell_0}}$ is taken equal to $\infty$ for
$L_n \cap (\ell_0, \infty) = \emptyset$. Note that in the previous computation, the last inequality is a consequence of the inequality $\alpha'(1+\alpha) - 1 <0$. Now, $\lambda_{n, k_{n,\ell_0}}$ tends to infinity with
$n$, since for all $A > \ell_0$, one has $L_n \cap (\ell_0, A] = \emptyset$ for $n$ large enough. Hence,
we get \eqref{convergenceterm2} and \eqref{convergenceterm3}.
Moreover, in case where $L \cap (\ell_0, \infty) = \emptyset$,
$L_n \cap (\ell_0, \infty) \neq \emptyset$, $n$ is large enough, and $|z| < \ell_0/2$, the left-hand side of \eqref{convergenceterm4}
is smaller than or equal to:
\begin{align*}
4 c & \sum_{ k \geq k_{n,\ell_0}+1} |k|^{\alpha'} \left(\frac{\lambda_{n,k}
- \lambda_{n,k-1}}{\lambda_{n,k-1}
\lambda_{n,k} } \right) =
4 c \sum_{ k \geq k_{n,\ell_0}+1} |k|^{\alpha'} \left(\frac{1}{\lambda_{n,k-1}}
-\frac{1}{\lambda_{n,k}} \right)
\\ & = 4c \left(\frac{|k_{n, \ell_0}+1|^{\alpha'}}{\lambda_{n,k_{n, \ell_0}}}
+ \sum_{ k \geq k_{n,\ell_0}+1} \frac{|k+1|^{\alpha'} - |k|^{\alpha'}}{\lambda_{n,k}} \right) \end{align*}
$$ \leq 4c \left(\frac{|k_{n, \ell_0}+1|^{\alpha'}}{\lambda_{n,k_{n, \ell_0}} \vee
(|k_{n, \ell_0}|/\tilde{\tau})^{1/(1+\alpha)} }
+ \sum_{ k \geq 1} \frac{(k+1)^{\alpha'} - k^{\alpha'}}{\lambda_{n,k_{n, \ell_0}}
\vee (k/\tilde{\tau})^{1/(1+\alpha)}} \right),$$
when $k_{n, \ell_0} \geq 1$, which occurs for $n$ large enough. The first term of the last quantity is dominated by
$$(\lambda_{n,k_{n, \ell_0}} \vee (k_{n, \ell_0}/\tilde{\tau})^{1/(1+\alpha)})^{\alpha' (1+\alpha)-1}
\leq (\lambda_{n,k_{n, \ell_0}})^{\alpha'(1+\alpha)-1},$$
which tends to zero when $n$ goes to infinity, since $\lambda_{n,k_{n, \ell_0}}$ goes to infinity
and $\alpha'(1+\alpha)-1 < 0$. Similarly,
$$ \sum_{ k \geq 1} \frac{(k+1)^{\alpha'} - k^{\alpha'}}{\lambda_{n,k_{n, \ell_0}}
\vee (k/\tilde{\tau})^{1/(1+\alpha)}} \underset{n \rightarrow \infty}{\longrightarrow} 0,$$
by dominated convergence. Hence, we get \eqref{convergenceterm4}.
We can now assume $L \cap (\ell_0, \infty) \neq \emptyset$ and it remains to prove \eqref{convergenceterm5}, \eqref{convergenceterm6} and \eqref{convergenceterm7}.
For $k \geq k_{\ell_0}$, let us define $\lambda_{n,k}$ and $\lambda_{k}$ as $\infty$ if these numbers are not well-defined: this does not change the quantities involved in \eqref{convergenceterm5}. Moreover, for all $k \geq k_{\ell_0}$: \begin{itemize} \item If $\lambda_{k}$ is well-defined as a finite quantity, then $\lambda_{n,k}$ is also well-defined for $n$ large enough and tends to $\lambda_{k}$ when $n$ goes to infinity. \item If $\lambda_k = \infty$, then for all $A >0$, and for $n$ large enough, one has $\lambda_{n,k} \notin [-A,A]$. Since for $n$ large enough, $$\lambda_{n,k} \geq \lambda_{n, k_{\ell_0}} > \lambda_{k_{\ell_0}} -1 > \ell_0 - 1 > 0,$$ one has $\lambda_{n,k} > A$: in other words, $\lambda_{n,k}$ tends to infinity with $n$.
\end{itemize}
We have just checked that with the convention made here, one has always
$\lambda_{n,k}$ converging to $\lambda_{k}$ when $n$ goes to infinity, for all $k \geq k_{\ell_0}$.
Hence, \eqref{convergenceterm5} is a consequence of the dominated convergence theorem and
the majorization:
$$\sum_{k \geq k_{\ell_0}} (\lambda_k \wedge \inf_{n \geq n_0} \lambda_{n,k})^{-2} < \infty,$$
for some $n_0 \geq 1$.
Now, there exists $n_0 \geq 1$ such that for all $n \geq n_0$, one has $k_{n,\ell_0} = k_{\ell_0}$, and then for
all $k \geq 1 \vee k_{\ell_0}$, $\lambda_{k} > \ell_0 > 2$, $\lambda_{n,k} >2$ and $k \geq 1$, which
implies the minorizations \eqref{minorizationlambda2} and \eqref{minorizationlambda3}.
Hence one gets \eqref{convergenceterm5}, since
$$\sum_{k \geq 1} (k/\tilde{\tau})^{-2/(1+\alpha)} < \infty.$$
Since \eqref{convergenceterm6} is easy to check, it remains to show \eqref{convergenceterm7}, which can
be rewritten as follows:
$$ \sum_{ k \geq k_{\ell_0}+1} \Delta_{n,k} \left(\frac{1}{\lambda_{n,k-1} -z}
- \frac{1}{\lambda_{n,k} - z} \right)
\underset{n \rightarrow \infty}{\longrightarrow}
\sum_{ k \geq k_{\ell_0}+1} \Delta_{k} \left(\frac{1}{\lambda_{k-1} -z}
- \frac{1}{\lambda_{k} - z} \right),
$$
where for $k \geq K_{n,\infty}$ (resp. $k \geq K_{\infty}$), one defines $\lambda_{k,n} := \infty$
(resp. $\lambda_{k} := \infty$). Note that with this convention,
$\lambda_{n,k}$ tends to $\lambda_k$ when $n$ goes to infinity, for all $k \geq k_{\ell_0}$.
Note that each term of the left-hand side of this last convergence converges uniformly
on $\{z \in \mathbb{C}, |z| < \ell_0/2\}$ towards the corresponding term in the
right-hand side. Indeed, for $n$ large enough, for all $k \geq k_{\ell_0}$, for $|z| < \ell_0/2$,
and for $\lambda_k$, $\lambda_{n,k}$ finite,
\begin{align*}
\left|\frac{1}{\lambda_{n,k} - z} - \frac{1}{\lambda_{k} - z} \right|
& = \frac{|\lambda_{k} - \lambda_{n,k}|}{|\lambda_{n,k} - z||\lambda_{k} - z|}
\leq \frac{4 |\lambda_{k} - \lambda_{n,k}|}{\lambda_{n,k} \lambda_{k} }
\\ & \leq 4 \left| \frac{1}{\lambda_{n,k} } - \frac{1}{\lambda_{k}}\right|
\underset{n \rightarrow \infty}{\longrightarrow} 0,
\end{align*}
this convergence, uniform in $z$, being in fact also true if $\lambda_{n,k}$ or $\lambda_k$ is infinite.
Hence, one has, for all $k' > k_{\ell_0}+1$, the uniform convergence:
$$ \sum_{ k_{\ell_0}+1 \leq k \leq k'} \Delta_{n,k} \left(\frac{1}{\lambda_{n,k-1} -z}
- \frac{1}{\lambda_{n,k} - z} \right)
\underset{n \rightarrow \infty}{\longrightarrow}
\sum_{k_{\ell_0}+1 \leq k \leq k'} \Delta_{k} \left(\frac{1}{\lambda_{k-1} -z}
- \frac{1}{\lambda_{k} - z} \right),
$$
Hence, it is sufficient to check, for $n_0 \geq 1$ such that $k_{n,\ell_0} =k_{\ell_0}$ if
$n \geq n_0$, that \begin{equation}
\sup_{n \geq n_0, |z| < \ell_0/2}
\left| \sum_{ k > k'} \Delta_{n,k} \left(\frac{1}{\lambda_{n,k-1} -z}
- \frac{1}{\lambda_{n,k} - z} \right) \right| \underset{k' \rightarrow \infty}{\longrightarrow} 0
\label{convergencek'1} \end{equation} and \begin{equation}
\sup_{|z| < \ell_0/2}
\left| \sum_{ k > k'} \Delta_{k} \left(\frac{1}{\lambda_{k-1} -z}
- \frac{1}{\lambda_{k} - z} \right) \right| \underset{k' \rightarrow \infty}{\longrightarrow} 0.
\label{convergencek'2} \end{equation}
Now, for $k' \geq 1 \vee (k_{\ell_0} + 1)$, $n \geq n_0$ and $|z| < \ell_0/2$, one has \begin{align*}
\left| \sum_{ k > k'} \Delta_{n,k} \left(\frac{1}{\lambda_{n,k-1} -z}
- \frac{1}{\lambda_{n,k} - z} \right) \right| &
\leq \sum_{ k > k'} |\Delta_{n,k}| \, \left|\frac{1}{\lambda_{n,k-1} -z}
- \frac{1}{\lambda_{n,k} - z} \right|
\\ & \leq 4 c \sum_{ k > k'} k^{\alpha'} \, \left( \frac{1}{\lambda_{n,k-1}}
- \frac{1}{\lambda_{n,k}} \right)
\\ & = 4 c \left( \frac{(k'+1)^{\alpha'}}{\lambda_{n,k'}} +
\sum_{k > k'} \frac{(k+1)^{\alpha'} - k^{\alpha'}}{\lambda_{n,k}} \right)
\\ & \leq 4 c \left( \frac{(k'+1)^{\alpha'}}{(k'/\tilde{\tau})^{1/(1+\alpha)}} +
\sum_{k > k'} \frac{(k+1)^{\alpha'} - k^{\alpha'}}{(k/\tilde{\tau})^{1/(1+\alpha)}} \right)
\\ & \underset{k' \rightarrow \infty}{\longrightarrow} 0, \end{align*}
which proves \eqref{convergencek'1}. One shows \eqref{convergencek'2} in an exactly similar way,
which finishes the proof of the convergence of $S_n$ towards $S+\bar{\gamma}h$, uniformly on compact sets for the
distance $d$. It remains to prove that $\Xi'_{n}$ and $\Xi'$ satisfy the assumptions from \eqref{e:lambda1} to \eqref{e:hlim}. Note that the observation of the sign of the imaginary parts $\Im(S_n)$ and $\Im(S)$ implies
that the sets $\Xi'_n$ and $\Xi'$ are included in $\mathbb{R}$. Moreover, the derivatives $S_n'$ and $S'$
are strictly positive, respectively on $\mathbb{R} \backslash L_n$ and $\mathbb{R} \backslash L$, and
all the left (resp. right) limits of $S_n$ and $S$ at their poles are equal to $+ \infty$ (resp. $-\infty$).
We deduce that the support of $\Xi'_n$ (resp. $\Xi'$) strictly interlaces with the points
in $L_n$ (resp. $L$).
The convergence \eqref{e:lambda1} is a direct consequence of the convergence of $S_n$ towards $S+\bar{\gamma}h$, as written in the statement of Theorem \eqref{topologytheorem}. More precisely, for two points $a$ and $b$ ($a < b$) not in the support of $\Xi'$ and such that $\Xi'((a,b)) = k$, there exist real numbers $a = q_0 < q_1 < q_2 < \dots < q_{2k}< b = q_{2k+1}$ such that $-\infty < S(q_{2j-1}) < h' < S(q_{2j}) < \infty$ for $ j \in \{1, \dots, k\}$, which implies that these inequalities are also satisfied for $S_n - \bar{\gamma}h$ instead of $S$ if $n$ is large enough: one has $\Xi'_n((a,b)) \geq k$. On the other hand, since the support of $\Xi'$ has exactly one point on each interval $[q_{2j-1}, q_{2j}]$ ($1 \leq j \leq k$) and no point on the intervals $[q_{2j},q_{2j+1}]$ ($0 \leq j \leq k$), one deduces that $S$ is bounded on the intervals $[q_{2j-1}, q_{2j}]$ and bounded away from $h'$ on the intervals $[q_{2j},q_{2j+1}]$. These properties remain true for $S_n - \bar{\gamma}h$ if $n$ is large enough, and one easily deduces that $\Xi'_n((a,b)) \leq k$.
The properties \eqref{e:lambdasquared}, \eqref{e:lambdasquared2} and \eqref{e:hlim} can be deduced from the property of interlacing. More precisely, for $\ell \geq 1$, $$\int_{\mathbb{R}} \frac{\mathbf{1}(\lambda > \ell)}{\lambda^{1+\alpha}} \, d \Xi'_n(\lambda) = \sum_{\lambda \in L'_n \cap (\ell, \infty)} \frac{1}{\lambda^{1+\alpha}},$$ where $L'_n$ is the support of $\Xi'_n$. By the interlacing property, if $L'_n \cap (\ell, \infty)$ is not empty and if its smallest element is $\lambda' > \ell$, then it is possible to define an injection
between $(L'_n \cap (\ell, \infty) ) \backslash \{\lambda'\}$ and
$L_n \cap (\ell, \infty)$, such that the image of each point is smaller than this point.
One deduces
$$\int_{\mathbb{R}} \frac{\mathbf{1}(\lambda > \ell)}{\lambda^{1+\alpha}} \, d \Xi'_n(\lambda)
\leq \frac{1}{\lambda'} + \sum_{\lambda \in L_n \cap (\ell, \infty)} \frac{1}{\lambda^{1+\alpha}}
\leq \frac{1}{\ell} + \int_{\mathbb{R}} \frac{\mathbf{1}(\lambda > \ell)}{\lambda^{1+\alpha}} \, d \Xi_n(\lambda).$$
By looking similarly at the integral for $\lambda < -\ell$, one deduces
$$\int_{\mathbb{R}} \frac{\mathbf{1}(|\lambda| > \ell)}{|\lambda|^{1+\alpha}} \, d \Xi'_n(\lambda)
\leq \frac{2}{\ell} + \int_{\mathbb{R}} \frac{\mathbf{1}(|\lambda| > \ell)}{|\lambda|^{1+\alpha}} \, d \Xi_n(\lambda) \leq \frac{2}{\ell} +\tau_{\ell} \underset{\ell \rightarrow \infty}{\longrightarrow} 0,$$
which proves \eqref{e:lambdasquared} for the measure $\Xi'_n$.
By a similar argument, for $\ell'' > \ell' > \ell \geq 1$,
$$\int_{\mathbb{R}} \frac{\mathbf{1}(\ell' \leq \lambda< \ell'')}{\lambda} \, d \Xi'_n(\lambda)
\leq \int_{\mathbb{R}} \frac{\mathbf{1}(\ell' \leq \lambda< \ell'')}{\lambda} \, d \Xi_n(\lambda)
+ \frac{1}{\ell'},$$
and one has the similar inequalities obtained by exchanging $\Xi_n$ and $\Xi'_n$, and
by changing the sign of $\lambda$.
Hence,
$$ \left|\int_{\mathbb{R}} \frac{\mathbf{1}(\ell' \leq |\lambda|< \ell'')}{|\lambda|} \, d \Xi'_n(\lambda)
- \int_{\mathbb{R}} \frac{\mathbf{1}(\ell' \leq |\lambda|< \ell'')}{|\lambda|} \, d \Xi_n(\lambda)
\right| \leq \frac{4}{\ell'}. $$
This inequality and the existence of the limit $h_{n,\ell}$ for the measure
$\Xi_n$ implies that
$$\underset{\ell' \wedge \ell'' \rightarrow \infty}{\limsup}
\left|\int_{\mathbb{R}} \frac{\mathbf{1}(\ell < |\lambda|< \ell'')}{|\lambda|} \, d \Xi'_n(\lambda)
- \int_{\mathbb{R}} \frac{\mathbf{1}(\ell < |\lambda|< \ell')}{|\lambda|} \, d \Xi'_n(\lambda) \right|
= 0,$$ and then the limit given by \eqref{e:lambdasquared2} exists for the measure $\Xi'_n$. Moreover, for all $\ell \geq 1$, one gets the majorization:
\begin{equation}\left|\underset{\ell' \rightarrow \infty}{\lim}
\int_{\mathbb{R}} \frac{\mathbf{1}(\ell < |\lambda|< \ell')}{|\lambda|} \, d \Xi'_n(\lambda)
- \underset{\ell' \rightarrow \infty}{\lim}
\int_{\mathbb{R}} \frac{\mathbf{1}(\ell < |\lambda|< \ell')}{|\lambda|} \, d \Xi_n(\lambda)
\right| \leq \frac{4}{\ell},
\label{xxxx2017}
\end{equation}
and a similar inequality without the index $n$. This implies \eqref{e:hlim}, provided that we check the
existence of the limit: \begin{equation}
\lim_{n \rightarrow \infty} \underset{\ell' \rightarrow \infty}{\lim}
\int_{\mathbb{R}} \frac{\mathbf{1}(\ell < |\lambda|< \ell')}{|\lambda|} \, d \Xi'_n(\lambda) \label{existencelimit}
\end{equation}
for each $\ell$ such that $\ell$ and $-\ell$ are not in the support of $\Xi'$. Let us first assume that $\ell$ and
$-\ell$ are also not in the
support of $\Xi$.
We have, for all $\ell'' > \ell$, \begin{equation} \label{xxxx2018}\underset{\ell' \rightarrow \infty}{\lim}
\int_{\mathbb{R}} \frac{\mathbf{1}(\ell < |\lambda|< \ell')}{|\lambda|} \, d \Xi'_n(\lambda)
= \int_{\mathbb{R}} \frac{\mathbf{1}(\ell < |\lambda| \leq \ell'')}{|\lambda|} \, d \Xi'_n(\lambda)
+ \underset{\ell' \rightarrow \infty}{\lim}
\int_{\mathbb{R}} \frac{\mathbf{1}(\ell''< |\lambda|< \ell')}{|\lambda|} \, d \Xi'_n(\lambda),
\end{equation}
and a similar equality with $\Xi'_n$ replaced by $\Xi_n$.
Since $\ell$ and $-\ell$ are not in the support of $\Xi$ or $\Xi'$, the convergences of $\Xi_n$ towards $\Xi$ and of $\Xi'_n$ towards $\Xi'$ imply that the lower and upper limits (when $n$ goes to infinity) of the first term of \eqref{xxxx2018} (both with $\Xi'_n$ and with $\Xi_n$) differ by $O(1/\ell'')$. For the second term, the difference between the lower and upper limits should change only by $O(1/\ell'')$ when we replace \eqref{xxxx2018} by the same equation with $\Xi_n$, thanks to
\eqref{xxxx2017}. Hence, this observation is also true for the sum of the two terms.
On the other hand, the existence of the limit of $h_{n,\ell}$ when $n$ goes to infinity (for $\Xi_n$) implies that
in \eqref{xxxx2018} with $\Xi'_n$ replaced by $\Xi_n$, the difference between the upper and lower limit is zero.
Therefore, the difference is
$O(1/\ell'')$ without replacement of $\Xi'_n$ by $\Xi_n$ : letting $\ell'' \rightarrow 0$ gives the existence of the limit
\eqref{existencelimit} for $-\ell, \ell$ not in the support of $\Xi$ and $\Xi'$. If $-\ell$ or $\ell$
is in the support of $\Xi$ (but not in the support of $\Xi'$), we observe that for some $\epsilon > 0$ and
$n$ large enough,
there is no point in the supports of $\Xi'_n$ and $\Xi'$ in the intervals $\pm \ell + (-\epsilon, \epsilon)$, which implies
that the integral involved in \eqref{existencelimit} does not change if we change $\ell$ by less than $\epsilon$. By suitably moving $\ell$, we can then also avoid the support of $\Xi$.
\end{proof}
\section{Convergence of Hermite corners towards the bead process}
In this section, we consider, for all $\beta > 0$, the Gaussian $\beta$ Ensemble, defined
as a a set of $n$ points
$(\lambda_j)_{1 \leq j \leq n}$ whose joint density, with respect to the Lebesgue measure is proportional to $$e^{-\beta \sum_{k=1}^n \lambda_k/4}
\prod_{j < k} |\lambda_j - \lambda_k|^{\beta}.$$
We will use the following crucial estimate, proven in \cite{NV19}: \begin{theorem} \label{boundvariance2017} For $-\infty \leq \Lambda_1 < \Lambda_2 \leq \infty$, let $N(\Lambda_1, \Lambda_2)$ be the number of points, between $\Lambda_1$ and $\Lambda_2$, of a Gaussian beta ensemble with $n$ points, and let $N_{sc}(\Lambda_1, \Lambda_2)$ be $n$ times the measure of $(\Lambda_1, \Lambda_2)$ with respect to the semi-circle distribution on the interval $[- 2\sqrt{n}, 2 \sqrt{n}]$: $$N_{sc}(\Lambda_1, \Lambda_2) := \frac{n}{2\pi} \int_{\Lambda_1/\sqrt{n}}^{\Lambda_2/\sqrt{n}} \sqrt{(4 - x^2)_+} \, dx. $$ Then, $$\mathbb{E} [(N(\Lambda_1, \Lambda_2) - N_{sc}(\Lambda_1, \Lambda_2))^2] = O( \log (2 + (\sqrt{n}(\Lambda_2 - \Lambda_1) \wedge n))).$$ \end{theorem}
For $\beta \in \{1,2,4\}$, the Gaussian $\beta$ Ensemble can be represented by the
eigenvalues of real symmetric (for $\beta = 1$), complex Hermitian (for $\beta = 2$), or quaternionic Hermitian (for $\beta = 4$) Gaussian matrices. The law of the entries of these matrices, corresponding respectively to the Gaussian Orthogonal Ensemble, the Gaussian Unitary Ensemble and the Gaussian Symplectic Ensemble, are given as follows: \begin{itemize} \item The diagonal entries are real-valued, centered, Gaussian with variance $2/\beta$. \item The entries above the diagonal are real-valued for $\beta = 1$, complex-valued for $\beta = 2$, quaternion-valued for $\beta = 4$, with independent parts, centered, Gaussian with variance $1/\beta$. \item All the entries involved in the previous items are independent. \end{itemize} By considering the top-left minors $A_n$ of an infinite random matrix $A$ following the law described just above, and their eigenvalues, we get a family of sets of points, the $n$-th set following the G$\beta$E of order $n$. Conditionally on the matrix $A_n$, whose eigenvalues are denoted $(\lambda_1, \dots, \lambda_n)$, supposed to be distinct (this holds almost surely), the law of the eigenvalues of $A_{n+1}$ can be deduced by diagonalizing $A_n$ inside $A_{n+1}$, which gives a matrix of the form $$ \left( \begin{array}{ccccc} \lambda_1 & 0 & \cdots & 0 & g_1 \\ 0 & \lambda_2 & \cdots & 0 & g_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & \lambda_n & g_n \\
\overline{g_1} & \overline{g_2} & \cdots & \overline{g_n} & g \end{array} \right),$$
where $g_1, \dots g_n, g$ are independent, centered Gaussian, $g$ being real-valued with variance $2/\beta$, $g_1, \dots, g_n$ being real-valued of variance $1$ for $\beta = 1$, complex-valued with independent real and imaginary parts of variance $1/2$ for $\beta = 2$, quaternion-valued with independent parts of variance $1/4$ for $\beta = 4$.
Expanding the characteristic polynomial and dividing by the product of $\lambda_j - z$ for $1 \leq j \leq \lambda_n$, we see that the eigenvalues of $A_{n+1}$ are the solutions of the equation:
$$ g - z - \sum_{j=1}^n \frac{|g_j|^2}{ \lambda_j - z} = 0.$$
Hence, if for $n \geq 1$, we consider the eigenvalues of the matrices $(A_{n+k})_{k \geq 0}$, we
get an inhomogeneous Markov chain defined as follows:
\begin{itemize}
\item The first set corresponds to the $G \beta E$ with $n$ points.
\item Conditionally on the sets of points indexed by $0, 1, \dots, k$, the set indexed by $k$ containing the distinct points $\lambda_1, \dots, \lambda_{n+k}$, the set indexed by $k+1$ contains the zeros of
$$g - z - \sum_{j=1}^{n+k} \frac{(2/\beta) \gamma_{j}}{ \lambda_j - z},$$
$g$ being centered, Gaussian of variance $2/\beta$, $\gamma_j$ being a Gamma variable of parameter $\beta/2$, all these variables being independent.
\end{itemize}
This Markov chain can be generalized to all $\beta > 0$: this can be viewed as the "eigenvalues of the G$\beta$E minors".
In fact, what we obtain is equivalent (with suitable scaling) to the Hermite $\beta$ corners introduced by Gorin and Shkolnikov in \cite{GS}.
This fact is due to the following result, proven (up to scaling) in \cite{Forrester}, Proposition 4.3.2:
\begin{proposition}
The density of transition probability from the set $(\lambda_1, \dots, \lambda_n)$ to the set $(\mu_1, \dots, \mu_{n+1})$, subject to the interlacement property $$\mu_1 < \lambda_1 < \mu_2 < \dots < \mu_{n} < \lambda_n < \mu_{n+1},$$ is proportional to
$$\prod_{1 \leq p < q \leq n+1} (\mu_q - \mu_p) \prod_{1 \leq p < q \leq n} (\lambda_q - \lambda_p)^{1- \beta}
\prod_{1 \leq p \leq n, 1 \leq q \leq n+1} |\mu_q - \lambda_p|^{\beta/2 - 1}e^{- \frac{\beta}{4} \left( \sum_{1 \leq q \leq n+1} \mu_q^2 - \sum_{1 \leq p \leq n} \lambda_p^2 \right)}.$$
\end{proposition}
As explained in \cite{GS}, the marginals of the Hermite $\beta$ corner correspond to the Gaussian $\beta$ Ensemble, which implies the following: \begin{proposition}
For all $\beta > 0$, the set of $n+k$ points corresponding to the step $k$ of the Markov chain just above has the distribution of the Gaussian $\beta$ Ensemble of dimension $n+k$. In particular, if we take $n=1$, we get a coupling of the $G \beta E$ in all dimensions.
\end{proposition}
Now, we show that a suitable scaling limit of this Markov chain is the $\beta$-bead process introduced in the paper.
We choose $\alpha \in (-2,2)$ (this corresponds to the bulk of the spectrum), $n \geq 1$, and we center the spectrum around the level $\alpha \sqrt{n}$. The expected density of eigenvalues around this level is approximated by $\sqrt{n} \rho_{sc}(\alpha)$, where $\rho_{sc}$ is the density of the semi-circular distribution. In order to get an average spacing of $2\pi$, we should then scale the eigenvalues by a factor $2 \pi \sqrt{n} \rho_{sc}(\alpha) = \sqrt{n(4 - \alpha^2)}$. For $k \geq 0$, we then consider the simple point measure $\Xi_n^{(k)}$ given by putting Dirac masses at the points $( \lambda^{(n,k)}_j - \alpha \sqrt{n}) \sqrt{n (4 - \alpha^2)}$, where $(\lambda^{(n,k)}_j)_{1 \leq j \leq n+k}$ is the set of $n+k$ points obtained at the step $k$ of the Markov chain above. The sequence of measures $\Xi_n^{(k)}$ can be recovered as follows: \begin{itemize} \item For $k = 0$, $\Xi_n^{(0)}$ corresponds to the point measure associated with the suitably rescaled G$\beta$E point process, with $n$ points. \item Conditionally on $\Xi_n^{(k)}$, $\Xi_n^{(k+1)}$ is obtained by taking the zeros of $$ - \frac{\alpha}{ \sqrt{4 - \alpha^2}} + \frac{g^{(k)}}{\sqrt{n(4 - \alpha^2)}} - \frac{z}{n(4-\alpha^2)} - \int \frac{1}{ \lambda- z} d \Lambda_n^{(k)} (\lambda),$$ where $g^{(k)}$ is a centered Gaussian variable of variance $2/\beta$, and $\Lambda_n^{(k)}$ is the weighted version of $\Xi_n^{(k)}$, the weights being i.i.d. with distribution corresponding to $2/\beta$ times a Gamma variable of parameter $\beta/2$. \end{itemize} We are now able to prove the following result: \begin{theorem} The Markov chain $(\Xi_n^{(k)})_{k \geq 0}$ converges in law to the Markov chain defined in Theorem \ref{betainfinite}, for the topology of locally weak convergence of locally finite measures on $\mathbb{R} \times \mathbb{N}_0$, and for the level $$ h = - \frac{\alpha}{ \sqrt{4 - \alpha^2}}.$$ For $\beta = 2$ and $h \in \mathbb{R}$ fixed, the law of the Markov chain of Theorem \ref{betainfinite} corresponds (after dividing the points by $2$) to the bead process introduced by Boutillier, with parameter $$\gamma = - \frac{h}{\sqrt{1 + h^2}},$$ if we take the notation of \cite{Bou}. \end{theorem} \begin{proof} By the result of Valk\'o and Vir\'ag, $\Xi_n^{(0)}$ converges in distribution to the $\operatorname{Sine}_{\beta}$ point process.
Hence, the family, indexed by $n$, of the distributions of $(\Xi_n^{(0)})_{n \geq 1}$, is tight in the space of probability measures on $\mathcal{M}(\mathbb{R})$, $\mathcal{M}(\mathbb{R})$ being the space of locally finite measures on the Borel sets of $\mathbb{R}$, endowed with the topology of locally weak convergence. Hence, for $\epsilon > 0$, there exists $(C_K)_{K \in \mathbb{N}}$ such that with probability at least $1 - \epsilon$, the number of points in $[-K,K]$ of $\Xi_n^{(0)}$ is at most $C_K$ for all $K \in \mathbb{N}$, independently of $n$. Since the points of $\Xi^{(k)}_n$ interlace with those of
$\Xi^{(k-1)}_n$, the condition just above is satisfied with $\Xi^{(k)}_n$ instead of $\Xi^{(0)}_n$.
Hence, the family, indexed by $n$, of the laws of $(\Xi^{(k)}_n)_{k \geq 0}$ is tight in the space of probability measures
on $\mathcal{M}(\mathbb{R} \times \mathbb{N}_0)$, $\mathcal{M}(\mathbb{R} \times \mathbb{N}_0)$ being the space of locally finite measures on $\mathbb{R} \times \mathbb{N}_0$, again endowed with the topology of locally weak convergence.
From the tightness, it is enough to prove that the law of the Markov chain of Theorem \ref{betainfinite} is the only possible limit for a subsequence of the laws of $(\Xi^{(k)}_n)_{k \geq 1}$. Let us consider such a subsequence which converges in law. We define the following random variable $$Y_n := \sup_{\ell \geq 0}
( 1+ \ell)^{-3/4} (|\widetilde{\Xi}^{(0)}_n ([0, \ell]) |
+ |\widetilde{\Xi}^{(0)}_n ([-\ell, 0]) |)$$ where $$\widetilde{\Xi}^{(0)}_n ([a,b]) := \Xi^{(0)}_n([a,b]) - N_{sc} \left(\left[\alpha \sqrt{n} + \frac{a}{\sqrt{n(4-\alpha^2)}},\alpha \sqrt{n} + \frac{b}{\sqrt{n(4-\alpha^2)}} \right] \right),$$ for $$N_{sc} (\Lambda_1, \Lambda_2) := \frac{n}{2 \pi} \int_{\Lambda_1/\sqrt{n}}^{\Lambda_2/\sqrt{n}} \sqrt{(4-x^2)_+} dx.$$ The family $(Y_n)_{n \geq 0}$ is tight. Indeed, by Theorem \ref{boundvariance2017}, $$\mathbb{E} [(1+ \ell)^{-3/2}
(|\widetilde{\Xi}^{(0)}_n ([0, \ell]) |^2
+ |\widetilde{\Xi}^{(0)}_n ([-\ell, 0])|^2 )] = O \left( (1+ \ell)^{-3/2} \log \left( 2 + \sqrt{n} \frac{\ell}{\sqrt{n(4-\alpha^2)}} \right) \right),$$ which shows that $$\mathbb{E} \left[ \sum_{\ell=0}^{\infty} (1+ \ell)^{-3/2}
(|\widetilde{\Xi}^{(0)}_n ([0, \ell]) |
+ |\widetilde{\Xi}^{(0)}_n ([-\ell, 0])|)^2 \right] \leq C_{\alpha,\beta},$$ where $C_{\alpha,\beta} < \infty$ depends only on $\alpha$ and $\beta$ (in particular, not on $n$). This implies that $(Y_n)_{n \geq 1}$ is tight.
The point processes $\Xi_n^{(k)}$, $k \geq 0$, are constructed from $\Xi_n^{(0)}$, and families $\gamma_{n,k,k'}$ of weights, $\gamma_{n,k,k'}$ being
the weight, involved in the construction of $\Xi_n^{(k+1)}$,
of the $(k')$-th nonnegative point
of $\Xi_n^{(k)}$ if $k' > 0$, the $(1-k')$-th negative point of $\Xi_n^{(k)}$ if $k' \leq 0$.
All the variables $\gamma_{n,k,k'}$ are i.i.d., distributed like $2/\beta$ times a Gamma variable of parameter $\beta/2$, and independent of $\Xi_n^{(0)}$.
We can consider the variables
$$Z_{n,k} = \sup_{m \geq 1} m^{-0.51}
\left| m - \sum_{k' = 0}^{m-1} \gamma_{n,k,k'}\right| +
\sup_{m \geq 1} m^{-0.51}
\left| m - \sum_{k' = -m}^{-1} \gamma_{n,k,k'} \right|.$$
By classical tail estimates of the Gamma variables, $Z_{n,k} < \infty$ almost surely, and since its law does not depend on $n$ and $k$, $(Z_{n,k})_{n \geq 1, k \geq 0}$ is a tight family of random variables. Hence, $(Z_n := (Z_{n,k})_{k \geq 0} )_{n \geq 1}$ is a tight family of random variables on $\mathbb{R}^{\mathbb{N}_0}$, endowed with the $\sigma$-algebra generated by the sets
$\{(z_k)_{k \geq 0}, z_0 \in A_0, z_1 \in A_1, \dots, z_p \in A_p\}$ for $p \geq 0$ and $A_j \in \mathcal{B} (\mathbb{R})$.
Let us go back to our subsequence of $(\Xi_n^{(k)})_{k \geq 0}$ which converges in law. If we join
$(\gamma_{n,k, k'})_{k \geq 0, k' \in \mathbb{Z}}$, $Y_n$ and $Z_n$, we still get a tight family of probability measures on a suitable probability space. Hence, we can find a sub-subsequence for which the family of random variables $((\Xi_n^{(k)})_{k \geq 0}, (\gamma_{n,k, k'})_{k \geq 0, k' \in \mathbb{Z}}, Y_n, Z_n)$ converges in law, and a fortiori $(\Xi_n^{(0)}, (\gamma_{n,k, k'})_{k \geq 0, k' \in \mathbb{Z}}, Y_n, Z_n)$ converges in law. By Skorokhod representation theorem, this family has the same law as some family $(\Xi_n^{'(0)}, (\gamma'_{n,k, k'})_{k \geq 0, k' \in \mathbb{Z}}, Y'_n, Z'_n)$ which converges almost surely along a subsequence. Note that $Y'_n$ is function of $\Xi_n^{'(0)}$ and $Z'_n$ is function of the weights $\gamma'_{n,k, k'}$. Since we know that $\Xi_n^{'(0)}$ converges in law to a $\operatorname{Sine}_{\beta}$ process, its almost sure limit is a simple point measure.
From the boundedness of $Y'_n$ along our subsequence, and Proposition \ref{estimatealmostsureCbeta} (which implies a bound on the point distribution of the $\operatorname{Sine}_{\beta}$ process, and then the existence of $h_{\ell}$ and its vanishing limit when $\ell \rightarrow \infty$), we deduce that the part of Theorem \ref{topologytheorem} concerning $\Xi_n^{'(0)}$ is satisfied, with $$h = \underset{\ell \rightarrow \infty}{\lim} \underset{n \rightarrow \infty}{\lim} h^{sc}_{n, \ell},$$ for $$h^{sc}_{n, \ell} = \int_{(-\infty, -\ell] \cup [\ell, \infty)} \frac{1}{\lambda} d N_{sc} \left( \alpha \sqrt{n} + \frac{\lambda}{\sqrt{n (4 - \alpha^2)}} \right)$$ We have \begin{align*} d N_{sc} \left( \alpha \sqrt{n} + \frac{\lambda}{\sqrt{n (4 - \alpha^2)}} \right) & = \frac{n}{2 \pi} d \left(\int_{-\infty}^{\alpha +\left( \lambda/(n \sqrt{4 - \alpha^2})\right)} \sqrt{(4 - x^2)_+} dx \right) \\ & = \frac{1}{2 \pi \sqrt{4 - \alpha^2}} \sqrt{\left[4 - \left(\alpha +\left( \lambda/(n \sqrt{4 - \alpha^2})\right) \right)^2 \right]_+} d \lambda \end{align*} If we do a change of variable $\lambda = \mu n \sqrt{4 - \alpha^2}$, we get $$h_{n,\ell}^{sc} = \int_{(-\infty, - \ell/ (n \sqrt{4 - \alpha^2})] \cup [\ell/ (n \sqrt{4 - \alpha^2}), \infty)}
\frac{1} {2 \pi \sqrt{4 - \alpha^2}} \sqrt{[4 - (\alpha + \mu)^2]_+} \frac{d \mu}{ \mu}.$$ Taking $n \rightarrow \infty$, we get a quantity independent of $\ell$, given by $$h = \frac{1}{2 \pi \sqrt{4 - \alpha^2}} \int_{\mathbb{R}} \sqrt{(4 - y^2)_+} \frac{dy} {y - \alpha},$$ the integral in the neighborhood of $\alpha$ being understood as a principal value. From the value of the Stieltjes transform of the semi-circle law, we deduce $$h = - \frac{\alpha}{2 \sqrt{4 - \alpha^2}}.$$ From the boundedness of $Z'_{n,0}$, we deduce that the part of Theorem \ref{topologytheorem} concerning the weights is also satisfied. Finally, in this theorem, it is almost surely possible to take $\lambda^* = 0$, by the absolutely continuity of the densities of the ensembles which are considered.
All the assumptions of the theorem are satisfied. If we denote by $\Lambda_n^{'(0)}$ the measure constructed from $\Xi_n^{'(0)}$ and the weights
$\gamma'_{n,0,k'}$ ($k' \in \mathbb{Z}$),
and $\Lambda^{'(0)}$ the measure constructed from the a.s. limits of these points and weights, we deduce that for an independent standard Gaussian variable $g^{(0)}$,
the function
$$ z \mapsto - \frac{\alpha}{ \sqrt{4 - \alpha^2}} - \int \frac{1}{ \lambda- z} d \Lambda_n^{'(0)} (\lambda),$$ and then also the function
$$ z \mapsto - \frac{\alpha}{ \sqrt{4 - \alpha^2}} + \frac{g^{(0)}}{\sqrt{n(4 - \alpha^2)}} - \frac{z}{n(4-\alpha^2)} - \int \frac{1}{ \lambda- z} d \Lambda_n^{'(0)} (\lambda),$$ converges uniformly on compact sets, for the topology of the Riemann sphere given in Thorem \ref{topologytheorem}, to the function $$ z \mapsto -\frac{\alpha}{ \sqrt{4 - \alpha^2}} - \int \frac{1}{ \lambda- z} d \Lambda^{'(0)} (\lambda) - h = -\frac{\alpha}{2 \sqrt{4 - \alpha^2}} - \int \frac{1}{ \lambda- z} d \Lambda^{'(0)} (\lambda) .$$ As in the proof of Theorem \ref{topologytheorem}, we deduce that the point process $\Xi_n^{'(1)}$
given by $$ - \frac{\alpha}{ \sqrt{4 - \alpha^2}} + \frac{g^{(0)}}{\sqrt{n(4 - \alpha^2)}} - \frac{z}{n(4-\alpha^2)} - \int \frac{1}{ \lambda- z} d \Lambda_n^{'(0)} (\lambda) = 0,$$ locally weakly converges to the point process $\Xi^{'(1)}$ given by $$\underset{\ell \rightarrow \infty}{\lim} \int_{[-\ell, \ell]} \frac{1}{ \lambda- z} d \Lambda^{'(0)} (\lambda) = -\frac{\alpha}{2 \sqrt{4 - \alpha^2}}.$$
The points of $\Xi_n^{'(1)}$ and satisfy the assumptions of Theorem \ref{topologytheorem}, since they interlace with those of $\Xi_n^{'(0)}$. It is also the same for the weights $\gamma'_{n,1,k'}$ ($k' \in \mathbb{Z}$), by the boundedness of $Z'_n$. We then deduce that for an independent Gaussian variable $g^{(1)}$, the point process $\Xi_n^{'(2)}$ given by $$- \frac{\alpha}{ \sqrt{4 - \alpha^2}} + \frac{g^{(1)}}{\sqrt{n(4 - \alpha^2)}} - \frac{z}{n(4-\alpha^2)} - \int \frac{1}{ \lambda- z} d \Lambda_n^{'(1)} (\lambda) = 0$$ locally weakly converges to the process $\Xi^{'(2)}$ given by $$\underset{\ell \rightarrow \infty}{\lim} \int_{[-\ell, \ell]} \frac{1}{ \lambda- z} d \Lambda^{'(1)} (\lambda) = -\frac{\alpha}{2 \sqrt{4 - \alpha^2}},$$ where $\Lambda_n^{'(1)}$ is given by $\Xi_n^{'(0)}$ and the weights $\gamma'_{n,1,k'}$ and $\Lambda^{'(1)}$ are given by their limits. We can then iterate the construction, which gives a family of point processes $\Xi_n^{'(k)}$ ($k \geq 0$), converging to $\Xi^{'(k)}$. From the way we do this construction, we check that $(\Xi_n^{'(k)})_{k \geq 0}$ has the same law as $(\Xi_n^{(k)})_{k \geq 0}$, and that $\Xi^{'(k)}$ has the same law as the generalized bead process introduced in the present paper (with level lines at $-\alpha/2 \sqrt{4 - \alpha^2}$). Hence, any subsequence of $((\Xi_n^{(k)})_{k \geq 0})_{n \geq 1}$ converging in law has a sub-subsequence tending in law to the generalized bead process By tightness, we deduce the convergence of the whole sequence $((\Xi_n^{(k)})_{k \geq 0})_{n \geq 1}$.
This gives the first part of the theorem, after doubling the weights and the value of $h$.
The second part is deduced by using the convergence of the GUE minors towards the bead process introduced by
Boutillier, proven in \cite{ANVM}. The factor $2$ is due to the fact that the average density of points is $1/\pi$
in \cite{Bou} and $1/2 \pi$ here.
The value of the parameter $\gamma$ in \cite{Bou} ($a$ in \cite{ANVM}) corresponds to
$\alpha/2$ (the bulk corresponds to the interval $(-1,1)$ in \cite{ANVM} and to $(-2,2)$ in the present paper).
We then have
$$h = - \frac{\alpha}{\sqrt{4 - \alpha^2}} = - \frac{\gamma}{\sqrt{1 - \gamma^2}},$$
and finally
$$\gamma = - \frac{h}{\sqrt{1 + h^2}}.$$ \end{proof}
\noindent {\bf Acknowledgments.} B.V. was supported by the Canada Research Chair program, the NSERC Discovery Accelerator grant, the MTA Momentum Random Spectra research group, and the ERC consolidator grant 648017 (Abert).
\end{document} | arXiv |
\begin{document}
\renewcommand{(\roman{enumi})}{(\roman{enumi})}
\title {${\mathbf \F_4(2)}$ and its automorphism group}
\author{Chris Parker}
\author{Gernot Stroth}
\address{Chris Parker\\ School of Mathematics\\ University of Birmingham\\ Edgbaston\\ Birmingham B15 2TT\\ United Kingdom} \email{[email protected]}
\address{Gernot Stroth\\ Institut f\"ur Mathematik\\ Universit\"at Halle - Wittenberg\\ Theordor Lieser Str. 5\\ 06099 Halle\\ Germany} \email{[email protected]}
\email {}
\date{\today}
\begin{abstract} We present an identification theorem for the groups $\F_4(2)$ and $\mathrm{Aut}(\F_4(2))$ based on the structure of the centralizer of an element of order $3$. \end{abstract} \maketitle \pagestyle{myheadings}
\markright{{\sc }} \markleft{{\sc Chris Parker and Gernot Stroth}}
\section{Introduction}
In the classification of the finite simple groups a fundamental role was played by Timmesfeld's work on groups which contain a large extraspecial 2-subgroup \cite{Tim}. Timmesfeld determined the structure of the normalizer of such a subgroup and following this achievement several authors contributed to the classification of all the simple groups which contain a large extraspecial $2$-subgroup.
The notion of a large extraspecial $2$-subgroup of a group is generalized in the work of Meierfrankenfeld, Stellmacher and the second author \cite{MSS} to the concept of a large $p$-subgroup where $p$ is an arbitrary prime. The definition of a large $p$- subgroup is as follows: given a finite group $G$, a $p$-subgroup $Q$ of $G$ is \emph{large} if and only if
\begin{enumerate} \item[(L1)]\label{1} $Q = F^*(N_G(Q))$; and \item[(L2)]\label{2} for all non-trivial subgroups $U$ of $ Z(Q)$, $N_G(U)\le N_G(Q)$. \end{enumerate}
Recall that condition (L1) is equivalent to $Q=O_p(N_G(Q))$ and $C_G(Q)\le Q$. If $Q$ is extraspecial and $p = 2$ this definition coincides with Timmesfeld's definition of a large extraspecial 2-group. The classification of groups with a large $p$-subgroup is sometimes called the MSS-project. The first step of this project is \cite{MSS}, where in contrast to the work of Timmesfeld, it is not the normalizer of $Q$ which is determined but rather structural information about the maximal $p$-local subgroups of $G$ which are not contained in $N_G(Q)$ is provided.
Suppose now that $Q$ is a large subgroup of a group $G$ and let $S$ be a Sylow $p$-subgroup of $G$ containing $Q$. It is an elementary exercise to show that $F^\ast(N_G(U)) = O_p(N_G(U))$ for all non-trivial normal subgroups $ U$ of $ S$ (\cite[Lemma 2.1]{PS2}). Groups which satisfy this property are said to be of \emph{parabolic characteristic $p$}. If $F^\ast(N_G(U)) = O_p(N_G(U))$ for all $1 \not= U \le S$, then $G$ is of \emph{local characteristic $p$} (also called characteristic $p$-type). In \cite{MSS} it is assumed that $G$ has local characteristic $p$. However, there is work in progress which aims to remove this assumption, and so all the successor articles to \cite{MSS} will be produced under the weaker hypothesis that the group under investigation has a large $p$-subgroup. One reason for this is that, as mentioned above, a group with a large $p$-subgroup is of parabolic characteristic $p$, while demonstrating that a group has local characteristic $p$ may well be hard to verify in applications.
Nevertheless \cite{MSS} provides us with some $p$-local structure of the group $G$ and this is all what we require for the next step of the programme in which we aim to recognize $G$ up to isomorphism. For this recognition we typically build a geometry upon which a subgroup of $G$ acts. This means that we take some of the $p$-local subgroups of $G$ which contain $S$ and consider the subgroup $H$ of $G$ generated by them. The $p$-local subgroups are selected so that $O_p(H) = 1$. As the generic simple groups with a large $p$-subgroup are Lie type groups in characteristic $p$, in many cases we will be able to show that the coset geometry determined by the $p$-local subgroups in $H$ is a building. The recognition of $H$ is then achieved with help of the classification of buildings of spherical type \cite{Tits, local}. At this stage, as a third step of the programme, we would like to show that $G=H$. There is a general approach to achieve this goal. Since $H$ contains $S$ it also contains $Q$ and so we are able to identify $Q$ as a subgroup of $H$. Typically $Q=F^\ast(N_H(R))$ for some root group $R$ in $H$. We can then determine the structure of $N_G(Q)$. The aim is to show that $N_G(Q) = N_H(Q)$ and from this further show that $N_G(U) = N_H(U)$ for all $1 \not= U \unlhd S$. The final step is to show that, if $H$ is a proper subgroup of $G$, then $H$ is strongly $p$-embedded in $G$ and this contradicts the main results in \cite{Be} and \cite{PSStrong}.
However there are situations where it cannot be shown that $N_G(Q) = N_H(Q)$. This happens most frequently when $p = 2$ or $3$ and $N_H(Q)$ is soluble. For the final stage of the MSS-project one has to analyze exactly these more troublesome configurations; that is determine all the groups $G$ where $F^*(H)$ is a group of Lie type in characteristic $p$ containing a Sylow $p$-subgroup $S$ of $G$, $N_H(Q)$ is soluble and $N_H(Q) \not= N_G(Q)$. There are several configurations where this phenomenon arises. For example when $p=3$ we have $H \cong \Omega^-_6(3)$ contained in $G \cong \U_6(2)$. Similarly, there are containments $\Omega^+_6(3)$ in $\F_4(2)$, $\Omega_7(3)$ in ${}^2\E_6(2)$ and $\M(22)$, and $\Omega^+_8(3)$ in $\M(23)$ and $\F_2$. In all these cases $Q$ is an extraspecial 3-group and $N_H(Q)$ is soluble. In a series of papers \cite{PS1, PS3, PSS}, the larger groups in this list are determined from the approximate structure of the centralizer of an element of order 3, or equivalently from the structure of $N_G(Q)$. In this paper we identify $\F_4(2)$ from the approximate structure of the centralizer of a $3$-element. We are motivated by the embedding of $\Omega^+_6(3)$ in $\F_4(2)$, but we do not assume that $G$ contains this group. We just assume certain important structural information about the normalizer of $Q$ and, as a consequence, this present article is independent of the results in \cite{MSS}. This article should also be viewed as a companion to the authors' earlier work \cite{PS1} in which the groups $G$ with $\PSU_6(2)\le G \le \mathrm{Aut}(\PSU_6(2))$ are characterised by such information. Indeed in such groups, the centralizer of a $3$-element has a similar structure to that in $\F_4(2)$ or $\mathrm{Aut}(\F_4(2))$ but in these groups $Z(Q)$ is weakly closed in $Q$, while in $\F_4(2)$ and its automorphism group it is not. (Recall that, for subgroups $X \le Y \le L$, we say that $X$ is \emph{weakly closed} in $Y$ with respect to $L$ provided that if $ g \in L$ and $X^g \le Y$, then $X^ g = X$.) Unfortunately the arguments in these different situations are quite different. The theorems proved in \cite{PS1} and in this article are employed in \cite{PS2} to identify the corresponding groups.
We now make precise what we mean by the approximate structure of the centralizer of an element of order 3 in $\PSU_6(2)$ or $\F_4(2)$.
\begin{definition} We say that $X$ is similar to a $3$-centralizer in a group of type $\PSU_6(2)$ or $\F_4(2)$ provided the following conditions hold. \begin{enumerate} \item $Q=F^*(X)$ is extraspecial of order $3^5$ and $Z(F^*(X)) =Z(X)$; and \item $X/Q$ contains a normal subgroup isomorphic to $ \mathrm{Q}_8\times \mathrm{Q}_8$. \end{enumerate} \end{definition}
Our main theorem is as follows.
\begin{theorem}\label{MT} Suppose that $G$ is a group, $Z \le G$ has order $3$. If $C_G(Z)$ is similar to a $3$-centralizer in a group of type $\PSU_6(2)$ or $\F_4(2)$ and $Z$ is not weakly closed in $F^*(C_G(Z))$, then $G \cong \F_4(2)$ or $\mathrm{Aut}(\F_4(2))$. \end{theorem}
Combining Theorem~\ref{MT} and the main theorem from \cite{PS1} we obtain the following statement.
\begin{theorem}\label{combT} Suppose that $G$ is a group, $Z \le G$ has order $3$. If $C_G(Z)$ is similar to a $3$-centralizer in a group of type $\PSU_6(2)$ or $\F_4(2)$ and $Z$ is not weakly closed in a Sylow $3$-subgroup of $C_G(Z)$ with respect to $G$, then either $F^*(G) \cong \F_4(2)$ or $F^*(G) \cong \PSU_6(2)$. \end{theorem}
For groups $G$ with $C_G(Z)$ of type $\PSU_6(2)$ or $\F_4(2)$, the different $G$-fusion of $Z$ in $C_G(Z)$ manifests itself in the subgroup structure of $G$ very quickly. Indeed, if we let $S$ be a Sylow $3$-subgroup of $C_G(Z)$ and $Q= F^*(C_G(Z))$, then we easily determine that $S \in \syl_3(G)$ and the Thompson subgroup $J$ of $S$ has order $3^4$ or $3^5$ when $Z$ is weakly closed in $Q$, whereas, it has order $3^4$ if $Z$ is not weakly closed in $Q$. More strikingly, setting $L= N_G(J)$, we have $F^*(L/Q) \cong \Omega_4^-(3)$ in the first case and in the second case $L/Q \cong\Omega_4^+(3)$.
The paper is set out as follows. In Section~2 we gather pertinent information about that natural and spin modules for $\mathrm{Sp}_6(2)$ and the natural and orthogonal $\SU_4(2)$-module as well as collect together further identification theorems and results which we shall require for the proof of Theorem~\ref{MT}. In Section~3 we present Theorem~\ref{P=F4} which will be used to identify a subgroup $P$ of our target group which is isomorphic to $\F_4(2)$. The proof of Theorem~\ref{P=F4} involves the construction of a building of type $\F_4(2)$ on which $P$ acts faithfully. The proof of the main theorem commences in Section~4. Thus we assume that $G$ satisfies the hypothesis of Theorem~\ref{MT} and set $M= N_G(Z)$. We remark here that the information that is developed as the proof of Theorem~\ref{MT} unfolds becomes information about the groups $\F_4(2)$ and $\mathrm{Aut}(\F_4(2))$ once the theorem is proved. The initial objective of Section~4 is to determine more information about the structure of $M$. This is achieved by exploiting the fact that $Z$ is not weakly closed in $Q=O_3(M)$. The first significant result is presented in Lemma~\ref{structM} where it is shown that $$M/Q \approx (\mathrm{Q}_8\times \mathrm{Q}_8).\mathrm{Sym}(3) \text{ or } (\mathrm{Q}_8\times \mathrm{Q}_8).(2\times\mathrm{Sym}(3)).$$ In Section~4, we then move on, in Lemma~\ref{NJ}, to the determination of $L$ as described in the previous paragraph. At this stage we have shown that $L \approx 3^4:\GO_4^+(3)$ or $3^4:\mathrm{CO}_4^+(3)$. Thus $J$ supports a quadratic form and $G$-fusion of elements in $J$ is controlled by $L$. This allows us to parameterize the non-trivial cyclic subgroups of $J$ as singular, plus and minus (the latter two types are fused when $L\approx 3^4:\mathrm{CO}_4^+(3)$) and also the five types of subgroups of order $9$ which we label Type S, Type DP, Type DM, Type N+ and Type N- (the notation is chosen to indicate that the groups are singular, degenerate with three plus groups, degenerate with three minus groups, non-degenerate of plus-type and non-degenerate of minus-type).
We let $\rho_1$ and $\rho_2$ be elements of $Q \cap J$ each centralized by a $\mathrm{Q}_8$ (the quaternion group of order $8$) subgroup of $M$ and one generating a plus type and the other a minus type cyclic subgroup of $J$. In Section~6, we show that $C_G(\rho_1) \cong C_G(\rho_2) \cong 3 \times \SU_4(2)$ or $3 \times \mathrm{Sp}_6(2)$ see Lemmas~\ref{eitheror} and \ref{thesame}. It is the latter possibility that actually arises in our target groups. There is related work in \cite{FF} that we might refer to at this stage but they assume that $G$ is of characteristic $2$-type.
We let $r_1$ and $r_2$ be central involutions in the subgroup of $C_G(Z)$ isomorphic to $\mathrm{Q}_8\times \mathrm{Q}_8$ which do not invert $Q/Z$ and, for $i=1,2$, we set $K_i= C_G(r_i)$. Again when $L\approx \mathrm{CO}_4^+(3)$ these groups are conjugate. At this stage we know that $r_i$ centralizes the (simple) component of $C_G(\rho_i)$. The heart of the proof of Theorem~\ref{MT} is contained in Sections~7, 8, 9 and 10 where we determine the structure of $K_i$. Thus the aim is to show that $K_1$ and $K_2$ have shape $2^{1+6+8}.\mathrm{Sp}_6(2)$ where $O_2(K_1)$ and $O_2(K_2)$ are commuting products of an extraspecial group of order $2^9$ and an elementary abelian group of order $2^7$.
We begin our construction of $K_i$ by determining a large $2$-group $\Sigma_i$ which is normalized by $I_i= C_J(r_i)$. It turns out that $\Sigma_i$ is the extraspecial $2$-group of order $2^9$ and plus type we are seeking. In the case that $C_G(\rho_i) \cong 3 \times \SU_4(2)$, we are able to show that in fact $K_i = N_G(\Sigma_i) $ and $N_G(\Sigma_i)/\Sigma_i \cong \mathrm{Aut}(\SU_4(2))$ or $\mathrm{Sp}_6(2)$ and this leads to a contradiction as explained in Lemma~\ref{ItsSp62}. Thus we enter Section~9 knowing that $C_G(\rho_1) \cong C_G(\rho_2) \cong 3 \times \mathrm{Sp}_6(2)$. On the other hand $\Sigma_i$ is far from being a maximal signalizer for $I_i$. Thus is Section~9 we construct an even larger signalizer which in the end is a product $\Gamma_i=\Sigma_i\Upsilon_i$ where $\Upsilon_i$ is an elementary abelian group of order $2^7$. Thus $\Gamma_i$ has order $2^{15}$ and in fact $\Upsilon_i= Z(\Gamma_i)$ and this is proved in Lemma~\ref{Gammabasic}. We show that $N_{G}(\Gamma_i)/\Gamma_i \cong \mathrm{Sp}_6(2)$ in Lemma~\ref{itssp}. The final hurdle requires that we show that $K_i=N_G(\Gamma_i)$. This is proved in Lemma~\ref{H=K} and requires a sequence of lemmas which begins by showing that $\Upsilon_i$ is strongly closed in $\Gamma_i$ with respect to $K_i$ and culminates in the statement that $\Upsilon_i$ is strongly closed in a Sylow $2$-subgroup of $K_i$ with respect to $K_i$. At this stage we apply a Lemma~\ref{Gold} which is essentially Goldschmidt's Strongly Closed Abelian $2$-subgroup Theorem \cite{Goldschmidt} to conclude that $K_i= N_G(K_i) \approx 2^{1+6+8}.\mathrm{Sp}_6(2)$. Our final section exploits Theorem~\ref{P=F4} to produce a subgroup $P$ of $G$ with $P \cong \F_4(2)$. We show that a group closely related to $P$ is strongly $3$-embedded in $G$ and finally apply Holt's Theorem \cite{Ho} in the form presented in Lemma~\ref{Holt} to conclude the proof of the Theorem~\ref{MT}.
Throughout this article we follow the now standard Atlas \cite{Atlas} notation for group extensions. Thus $X\udot Y$ denotes a non-split extension of $X$ by $Y$, $X{:}Y$ is a split extension of $X$ by $Y$ and we reserve the notation $X.Y$ to denote an extension of undesignated type (so it is either unknown, or we don't care). Our notation follows that in \cite{AschbacherFG}, \cite{Gorenstein} and \cite{GLS2}. We use the definition of signalizers as given in \cite[Definition 23.1]{GLS2}. For odd primes $p$, the extraspecial groups of exponent $p$ and order $p^{2n+1}$ are denoted by $p^{1+2n}_+$. The extraspecial $2$-groups of order $2^{2n+1}$ are denoted by $2^{1+2n}_+$ if the maximal elementary abelian subgroups have order $2^{1+n}$ and otherwise we write $2^{1+2n}_-$. We expect our notation for specific groups is self-explanatory. For a subset $X$ of a group $G$, $X^G$ denotes the set of $G$-conjugates of $X$. If $x, y \in H \le G$, we write $x\sim _Hy$ to indicate that $x$ and $y$ are conjugate in $H$. Often we shall give suggestive descriptions of groups which indicate the isomorphism type of certain composition factors. We refer to such descriptions as the \emph{shape} of a group. Groups of the same shape have normal series with isomorphic sections. We use the symbol $\approx$ to indicate the shape of a group.
\noindent {\bf Acknowledgement.} The first author is grateful to the DFG for their support and thanks the mathematics department in Halle for their generous hospitality from January to August 2011.
\section{Preliminaries}
In this section we lay out certain facts about the groups $\mathrm{Sp}_6(2)$ and $\mathrm{Aut}(\U_4(2))$ which play a pivotal role in the proof of our main theorem. We also present other background results that are of key importance to our investigations.
\begin{lemma}\label{modfacts} Suppose that $X \cong \mathrm{Sp}_6(2)$ or $\mathrm{Aut}(\SU_4(2))$. Then there is a unique irreducible $\mathrm{GF}(2)X$-module of dimension $6$ and a unique irreducible $\mathrm{GF}(2)X$-module of dimension 8 all the other non-trivial irreducible $\mathrm{GF}(2)X$-modules have dimension at least $9$. \end{lemma}
\begin{proof} This is well known. See \cite{MOAT}. \end{proof}
In this section $U$ will denote the $\mathrm{Aut}(\SU_4(2))$ natural module and the $\mathrm{Sp}_6(2)$ spin module of dimension $8$ and $V$ will be the $\mathrm{Aut}(\SU_4(2))$ orthogonal module and the $\mathrm{Sp}_6(2)$ natural module of dimension $6$.
\begin{table} {\tiny
\begin{tabular}{|c|c|c|c|c|c|} \hline && Centralizer in $\mathrm{Aut}(\SU_4(2))$ & Centralizer in $\mathrm{Sp}_6(2)$&$\dim C_U(u_j)$&$\dim C_V(u_j)$\\ \hline $a_2$&$u_1$&$2^{1+4}_+.(\mathrm{SL}_2(2) \times \mathrm{SL}_2(2))$&$2^{1+2+4}.(\mathrm{SL}_2(2) \times \mathrm{SL}_2(2))$&6&4\\ $b_3$&$u_2$& $2 \times (\mathrm{Sym}(4) \times 2)$ &$2^7. 3$&4&3\\ $b_1$&$u_3$&$2 \times \mathrm{Sp}_4(2)$&$2^5.\mathrm{Sp}_4(2)$&4&5\\ $c_2$&$u_4$& $2^6. 3$ &$2^8. \mathrm{SL}_2(2)$&4&4\\\hline \end{tabular} \caption{Involutions in $\mathrm{Sp}_6(2)$ and $\mathrm{Aut}(\SU_4(2))$. The involutions in the first row are the \emph{unitary transvections.} The involutions labeled with ``$b$" those which are in $\mathrm{Aut}(\SU_4(2)) \setminus \SU_4(2)$.} \label{Table1}} \end{table}
For $X \cong \mathrm{Sp}_6(2)$, let $X_1, X_2 $ and $X_3$ be the minimal parabolic subgroups of $X$ containing a fixed Sylow $2$-subgroup $S$. Set $X_{ij}= \langle X_i, X_j\rangle$ where $1 \le i<j\le 3$ and fix notation so that $$X_{12}/O_2(X_{12})\cong \mathrm{SL}_3(2),$$ $$X_{23}/O_2(X_{23}) \cong \mathrm{Sp}_4(2) \text{ and} $$ $$X_{13}/O_2(X_{13}) \cong \mathrm{SL}_2(2)\times \mathrm{SL}_2(2).$$ There are three conjugacy classes of elements of order $3$ in $X$. Let $\tau_1$, $\tau_2$ and $\tau_3$ be representatives of these classes and choose so that on the natural $\mathrm{Sp}_6(2)$-module $V$, for $1 \le i \le 3$, $\dim [V,\tau_i]= 2i$.
\begin{lemma}\label{sp62facts} Suppose that $Y \cong \mathrm{Aut}(\SU_4(2))$ and that $X \cong \mathrm{Sp}_6(2)$ with $Y \le X$. Assume that $V$ and $U$ are the faithful $\mathrm{GF}(2)X$-modules of dimension $6$ and $8$ respectively. \begin{enumerate} \item $X$ and $Y$ each have four conjugacy classes of involutions and for each involution $u\in X$ we have $u^X\cap Y$ is a conjugacy class in $Y$. In column one of Table~\ref{Table1} we { provide the Suzuki names (see \cite[page 16]{AschSe}) for each class of involutions.}
\item The shape of the centralizers of involutions in $X$ and $Y$ is given in Table~\ref{Table1}. \item For each involution in $u\in X$, $\dim C_V(u)$ and $\dim C_U(u)$ is given in Table~\ref{Table1}. \item $X$ does not contain any subgroup of order $2^4$ in which all the involutions are conjugate. \item $X$ does not contain an extraspecial subgroup of order $2^7$.
\item If $x$ is an involution of type $b_1$, then a Sylow $3$-subgroup of $C_Y(u)$ contains two conjugates of $\langle \tau_1\rangle $ and two conjugates of $\langle \tau_2\rangle$.
\item $E=\langle \tau_1,\tau_2,\tau_3\rangle$ is the Thompson subgroup of a Sylow $3$-subgroup of $G$ and every element of order $3$ is $X$-conjugate ($Y$-conjugate) to an element of $E$. \end{enumerate} \end{lemma}
\begin{proof} {Parts (i)-(iii) follow from \cite[Proposition 2.12, and Table 1]{PS1}.}
Suppose that $A \le X$ has order $2^4$ and that all the non-trivial elements are conjugate in $X$. We use the character table of $X$ given in \cite[page 47]{Atlas}. Let $\chi$ be an irreducible character of $X$. Then, as $(\chi|_A,1_A) \ge 0$, we have $$(\chi|_A,1_A) = \frac{1}{|A|} \sum_{a\in A} \chi(a) \ge 0.$$ Taking $\chi$ to be the degree $7$ character we see that all the non-trivial elements in $A$ are in Suzuki class $c_2$ (Atlas \cite{Atlas} $2C$). Now considering the character of degree $35$ denoted $\chi_7$ in \cite{Atlas} we obtain a contradiction.
Let $E$ be extraspecial of order $2^7$. Since $X$ has a faithful $7$-dimensional representation in characteristic $0$ and the smallest such representation of $E$ is $8$-dimensional, $E$ is not isomorphic to a subgroup of $X$.
Part (vi) follows from the action of $\mathrm{Sp}_4(2)$ on the natural module for $\mathrm{Sp}_6(2)$ as $\mathrm{Sp}_4(2)$ contains no conjugates of $\tau_3$.
Part (vii) is also elementary to verify.
\end{proof}
\begin{lemma}\label{sp62natural} Let $X \cong \mathrm{Sp}_6(2)$, $S$ a Sylow $2$-subgroup of $X$ and $V$ be the $\mathrm{Sp}_6(2)$ natural module. Then the following hold. \begin{enumerate} \item $X$ acts transitively on the non-zero vectors in $V$. \item $V$ is uniserial as an $S$-module. \item Suppose that, for $1 \le i \le 3$, $V_i$ is an $S$-invariant subspace of $V$ of dimension $i$. Then $X_{23}=N_X(V_1)$ and $X_{23}$ acts naturally as $\mathrm{Sp}_4(2)$ on $V_1^\perp/V_1$, $X_{13}=N_X(V_2)$, $O^2(X_3)$ centralizes $V_2$ and $V/V_2^\perp$, and $O^2(X_1)$ centralizes $V_2^\perp/V_2$ and $X_{12}=N_X(V_3)$ and acts naturally on both $V_3$ and $V/V_3$. \end{enumerate} \end{lemma}
\begin{proof} These are all well known facts about the action of $X$ on $V$. See for example \cite[Lemma 14.37]{SymplecticAmalgams} for (i) and (ii). \end{proof}
\begin{lemma}\label{sp62spin}Let $X \cong \mathrm{Sp}_6(2)$, $S$ a Sylow $2$-subgroup of $X$ and $U$ be the $\mathrm{Sp}_6(2)$ spin module. \begin{enumerate} \item $X$ has exactly two orbits on the non-zero vectors of $U$ one of length $135$ and one of length $120$. \item $N_X(C_U(S)) = X_{12}$ and $C_U(S)= C_U(O_2(X_{12}))$. \item If $U_2\le U$ is $S$-invariant of dimension $2$, then $N_X(U_2)=X_{13}$ and $O^2(X_1)$ centralizes $U_2$. \end{enumerate} \end{lemma}
\begin{proof} See \cite[Proposition 2.12]{PS1}. \end{proof}
\begin{lemma}\label{sp62line} Suppose that $X \cong \mathrm{Sp}_6(2)$ and $V$ is the natural module for $X$. Let $P= X_{13}$, $T \in \syl_3(P)$ and $Q= O_2(P)$. \begin{enumerate} \item $P/Q \cong \mathrm{SL}_2(2) \times \mathrm{SL}_2(2)$. \item The subgroups of order $3$ in $T$ are as follows: there are two subgroups $Z_1$ and $Z_2$ which are $X$-conjugate to $\langle \tau_3\rangle$, one subgroup which is $X$-conjugate to $\langle \tau_1\rangle$ (which we suppose is $\langle \tau_1\rangle$) and one subgroup which is $X$-conjugate to $\langle \tau_2\rangle $. The two subgroups of $T$ which are conjugate to $\langle \tau_3\rangle$ are conjugate in $N_P(T)$. \item $C_Q(Z_1) \cong C_Q(Z_2) \cong \mathrm{Q}_8$ and $[C_Q(Z_1),C_Q(Z_2)]=1$. \item $C_T(Z(Q))=\langle \tau_1\rangle$ and $C_Q(\tau_1)= Z(Q)$. \item If $U\le Q$ has order $2^3$ and if $U$ is $T$-invariant, then either $U = C_Q(Z_1)$, $U= C_Q(Z_2)$ or $U= Z(Q)$.
\item Let $Q^\prime = \langle t \rangle$. Then $t^X \cap Q\not\subseteq Z(Q)$. \end{enumerate} \end{lemma}
\begin{proof} Let $Y$ be the $P$-invariant isotropic $2$-space in $V$. Then $P$ preserves $0<Y < Y^\perp <V$. Let $I$ be a hyperbolic line and $J= I^\perp$ be chosen so $Y \le J$. Then the decomposition $I \perp J$ is preserved by $\mathrm{Sp}_2(2) \times \mathrm{Sp}_4(2)$ and the subgroup $K$ of this group which leaves $Y$ invariant has shape $\mathrm{Sp}_2(2) \times (2\times 2^2).\mathrm{SL}_2(2) \cong \mathrm{SL}_2(2) \times 2 \times \mathrm{Sym}(4)$. In particular, we now have (i) holds. Furthermore, we may suppose the first factor of $K$ contains $\langle \tau_1\rangle$ while the second factor contains $\langle \tau_2^*\rangle$, an $X$-conjugate of $\langle \tau_2\rangle$, acting fixed point freely on $J$. Set $T =\langle \tau_1, \tau_2^*\rangle$. Since $\tau_1$ is inverted in the first factor of $K$, we see the two diagonal products $\tau_1\tau_2^*$ and $\tau_1^2\tau_2^*$ are conjugate in $N_P(T)$. Furthermore these elements act fixed point freely on $V$ and so are $X$-conjugate to $\tau_3$. This is (ii).
Now consider $Q$. We know this group has order $2^7$. We further have $Q \cap K = O_2(K)$ centralizes $Y+I= Y^\perp$. Consequently $Q\cap K$ is normal in $P$ and as $[V,Q,Q\cap K]= [V,Q\cap K , Q]$ we additionally have $K\cap Q \le Z(Q)$. Note that $\langle \tau_1\rangle$ centralizes $Q\cap K$. Now $C_P(\tau_2^*)$ is contained in $K$ and so we see $C_Q(\tau_2^*) = Z(K)$ has order $2$. Now the centralizer in $X$ of $\tau_3$ supports a $\mathrm{GF}(4)$ structure and is isomorphic to $\SU_3(2)$. It follows that $\tau_1\tau_2^*$ and $\tau_1^2\tau_2^*$ can centralize only quaternion subgroups of order $8$ in $Q$. Since $C_Q(\tau_1\tau_2^*)$ and
$C_Q(\tau_1^2\tau_2^*)$ both centralize $Z(K)$ and $|Q|=2^7$ we have $C_Q(\tau_1\tau_2^*) \cong C_Q(\tau_1^2\tau_2^*) \cong \mathrm{Q}_8$ and $C_Q(\tau_1\tau_2^*)'= Z(K)$. Putting $Q_1 = C_Q(\tau_1\tau_2^*)C_Q(\tau_1^2\tau_2^*)$ we have $Q_1$ is $T$-invariant. Now $Q= C_Q(\tau_1\tau_2^*)C_Q(\tau_1^2\tau_2^*)(Q\cap K)$, $$[Q,\tau_1]= [C_Q(\tau_1\tau_2^*),\tau_1][C_Q(\tau_1^2\tau_2^*), \tau_1] = Q_1$$ is a normal subgroup of $Q$ and $Q_1\cap (Q\cap K)\le Z(K)$. Thus $Q_1$ is extraspecial and $Q'=Z(K)$ which has order $2$. In addition, $Q= C_Q(\tau_1\tau_2^*) [Q,\tau_1\tau_2^*]$ with $C_Q(\tau_1\tau_2^*) \cap [Q,\tau_1\tau_2^*]= Z(K)$. Since $$[C_Q(\tau_1\tau_2^*) , Q,\tau_1\tau_2^*] \le [Z(K),\tau_1\tau_2^*]=1$$ and $[C_Q(\tau_1\tau_2^*) ,\tau_1\tau_2^*,Q] =1,$ we also have $[C_Q(\tau_1\tau_2^*) , [Q,\tau_1\tau_2^*]]=1$ by the Three Subgroup Lemma. In particular, as $[Q,\tau_1\tau_2^*]= C_Q(\tau_1^2\tau_2^*)(Q \cap K)$, we now have (iii) and (iv) hold. If
$U$ is of order $2^3$ and is $T$-invariant, then $C_T(U)> 1$ and so (v) also follows from the above discussion. To prove (vi), we start with a transvection $r \in Z(Q)$. By Table~\ref{Table1} we have $E = O_2(C_X(r))$ is elementary abelian of order $2^5$. Now $|E \cap Q| \geq 2^3$. If $E \cap Q \leq Z(Q)$, then, as $E \leq C_{N_X(Q)}(E \cap Q)$, we get $|E \cap Q| \geq 2^4$, a contradiction. Hence $E \cap Q \not\leq Z(Q)$. Now as $N_X(E)$ acts transitively on $E/\langle r \rangle$, we have any coset of $\langle r \rangle$ in $E$ contains a conjugate of $t$. In particular $t^X \cap E \cap Q \not\subseteq Z(Q)$. \end{proof}
\begin{lemma}\label{Noover} Let $Y = \mathrm{Aut}(\SU_4(2))$ and $V$ be the natural $\OO^-_6(2)$-module. Then there are no elementary abelian subgroup $E$ of order $8$ in $Y$ such that $|V : C_V(E)| \leq 4$. \end{lemma}
\begin{proof} Suppose false and let $E$ be such a subgroup of order $8$. From Table~\ref{Table1} we see $E$ cannot contain elements of type $b_3$. If $E \not\leq Y^\prime$, then $E$ contains exactly four elements of type $b_1$. As there are at most three hyperplanes in $V$
containing $C_V(E)$, two of these elements have to centralize the same hyperplane of $V$. But then their product, which is an involution in $E \cap Y$, also centralizes this hyperplane. As $\Omega_6^-(2)$ does not contain transvections, we have $E \leq Y^\prime$. Therefore $|V : C_V(E)| = 4$ and $C_V(E) = C_V(e)$ for all $e \in E^\#$. As $C_V(e) = [V,e]^\perp$ we also have $[V,e] = [V,E]$ for all $e \in E^\#$ which means all the involutions in $E$ are conjugate. Now we use the character table of $\SU_4(2)$ as in the proof of Lemma~\ref{sp62facts}(iv) to obtain a contradiction. \end{proof}
Recall that a faithful $\mathrm{GF}(p)G$-module is an \emph{$F$-module} provided there exists a non-trivial elementary abelian $p$-subgroup $A \le G$ such that $|V:C_V(A)|\le |A|$. The subgroups $A \le G$ with $|V:C_V(A)|\le |A|$ are called \emph{offenders}.
\begin{lemma} \label{NotF}Suppose that $X \cong \mathrm{Sp}_6(2)$ or $\mathrm{Aut}(\SU_4(2))$ and $W$ is a $\mathrm{GF}(2)X$-module of dimension $14$ which has exactly two composition factors one of dimension 6 and one of dimension $8$. Then $W$ is not an $F$-module. \end{lemma}
\begin{proof} Suppose that $A \le X$ is an offender on $W$. Then $|A| \ge |W:C_W(A)|$. From Table~\ref{Table1}, for $a \in A$, we read $|A| \ge |W:C_W(a)| \ge 2^4$. Since the $2$-rank of $X$ is at most $6$, we also have that $A$ does not contain any involutions of type $b_3$.
Suppose that $|A|=2^4$. Then all the involutions in $A$ must be of type $a_2$. This contradicts Lemma
\ref{sp62facts}(iv). Hence $|A| \ge 2^5$ and $X \cong \mathrm{Sp}_6(2)$ as the $2$-rank of $\mathrm{Aut}(\SU_4(2))$ is $4$ (see \cite[Proposition 2.12 (x)]{PS1}). We use the notation for involutions from Table~\ref{Table1}. We may as well suppose $A \le C_X(u_3)$. Then as the $2$-rank of
$\mathrm{Sp}_4(2)$ is $3$, we have $A\cap O_2(C_X(u_3))\not=1$. Since $|C_U(O_2(C_X(u_3)))|= 2^4$ and
$|C_V(O_2(C_X(u_3)))|=2$ certainly $A \not = O_2(C_X(u_3))$. Now $O_2(C_X(u_3))$ contains 15 elements from
$u_1^X$, 15 elements from $u_4^X$ and one element from $u_3^X$ and multiplication by $u_3$ maps $u_1^X \cap O_2(C_G(u_3))$ to $u_4^X \cap O_2(C_X(u_3))$. Thus, if $A$ contains a conjugate of $u_3$, then $A \cap u_i^X \not=\emptyset $ for $i=1,3,4$. As $|A|= 2^5$, $A$ does not consist purely of elements of elements from class $u_1^X$ by Lemma~\ref{sp62facts} (iv) and consequently we must have elements from $u_4^X$ in $X$. It follows now from Table~\ref{Table1} that $|A| = 2^6$. There is a unique such elementary abelian subgroup in a Sylow $2$-subgroup of $X$ and its normalizer is a plane stabiliser in the action of $X$ on $V$. But then
$|W:C_W(A)|\ge 2^{10}$ which is a contradiction. \end{proof}
\begin{lemma}\label{nonsplitmods} Suppose that $X \cong \mathrm{Sp}_6(2)$, $W$ is a 7-dimensional $\mathrm{GF}(2)X$-module with $W/C_W(X)$ the natural $\mathrm{Sp}_6(2)$-module. If $S \in \syl_2(X)$, then $C_W(S)> C_W(X)$. \end{lemma}
\begin{proof} Consider the subgroup $K= K_1\times K_2$ of $X$ which preserves the decomposition of $W/C_W(X)$ in to a perpendicular sum of a non-degenerate $2$-space $A/C_W(X)$ and a non-degenerate $4$-space $B/C_W(X)$ with $K_1\cong \mathrm{Sp}_2(2)$ and $K_2 \cong \mathrm{Sp}_4(2)$. Let $t$ be an involution in $K_1$. Since $\dim A=3$, we have $\dim [A,t]=1$. Furthermore $B/C_B(t)\cong [B,t]$ as $K_2$-modules and so we must have $[B,t]=0$. Thus $[W,t] = [A,t]+[B,t]= [A,t]$ has dimension 1 and so $t$ is a transvection on $W$. Let $P$ be the stabiliser in $X$ of the $2$-space $L=[A,t]+C_W(X)$. Then $P= C_X(t)$ and contains $K_2$ and a Sylow $2$-subgroup $S$ of $X$. Furthermore, $P' = O_2(P)K_2'$ has index $2$ in $P$. Since $P'$ is perfect it centralizes $L$. Let $s \in P\setminus P'$ be a transvection on $V$. Then $s$ is $X$-conjugate to $t$, and so $s$ is a transvection on $W$. Then $[L,s]\le C_W(X)$ and if $[L,s]= C_W(X)$ then certainly $[W,s]= C_W(X)$ which is nonsense. Thus $[L,s]=0$ and we conclude that $P$ and hence $S$ centralizes $L$. This proves the claim. \end{proof}
\begin{theorem}[Prince] \label{PrinceThm} Suppose that $Y$ is isomorphic to the centralizer of a $3$-central element of order $3$ in $\PSp_4(3)$ and that $X$ is a finite group with a non-trivial element $d$ such that $C_X(d)\cong Y$. Let $P \in \Syl_3(C_X(d))$ and $E$ be the elementary abelian subgroup of $P$ of order $27$. If $E$ does not normalize any non-trivial $3^\prime$-subgroup of $X$ and $d$ is $X$-conjugate to its inverse, then either \begin{enumerate} \item
$|X:C_X(d)| =2$; \item $X$ is isomorphic to $\mathrm{Aut}(\SU_4(2))$; or \item $X$ is isomorphic to $\mathrm{Sp}_6(2)$. \end{enumerate} \end{theorem}
\begin{proof} See \cite[Theorem 2]{prince1}. \end{proof}
\begin{lemma} \label{cen3psp43}Suppose that $X$ is a group of shape $3^{1+2}_+.\mathrm{SL}_2(3)$, $O_2(X)=1$
and a Sylow $3$-subgroup of $X$ contains an elementary abelian subgroup of order $3^3$. Then $X$ is isomorphic to the centralizer of a non-trivial $3$-central element in $\PSp_4(3)$. \end{lemma} \begin{proof} See \cite[Lemma~6]{Parker1}.\end{proof}
\begin{lemma}\label{quadratic form} Suppose that $F$ is a field, $V$ is an $n$-dimensional vector space over $F$ and $G= \mathrm{GL}(V)$. Assume that $q$ is quadratic form of Witt index at least $1$ and with non-degenerate associated bilinear form $f$, where, for $v,w \in V$, $f(v,w) = q(v+w)-q(v)- q(w)$. Let $\mathcal S$ be the set of singular 1-dimensional subspaces of $V$ with respect to $q$. Then the stabiliser in $G$ of $\mathcal S$ preserves $q$ up to similarity. \end{lemma}
\begin{proof} See \cite[Lemma 2.10]{ParkerRowley}. \end{proof}
\begin{lemma}\label{GO4} Suppose that $p$ is an odd prime, $X = \mathrm{GL}_4(p)$ and $V$ is the natural $\mathrm{GF}(p)G$-module. Let $A =\langle a, b\rangle\le X$ be elementary abelian of order $p^2$ and assume that $[V,a] = C_V(b)$ and $[V,b]= C_V(a)$ are distinct and of dimension $2$. Let $v \in V\setminus [V,A]$. Then $A$ leaves invariant a non-degenerate quadratic form with respect to which $v$ is a singular vector. In particular, $X$ contains exactly two conjugacy classes of subgroups such as $A$. One is conjugate to a Sylow $p$-subgroup of $\GO_4^+(p)$ and the other to a Sylow $p$-subgroup of $\GO_4^-(p)$. \end{lemma} \begin{proof} See \cite[Lemma 2.11]{ParkerRowley}. \end{proof}
The $4$-dimensional orthogonal module of $+$-type will play a prominent role in the proof of our main theorem. We next introduce some notation which will be used in the proof.
\begin{notation}\label{o4} Let $V$ be a $4$-dimensional non-degenerate orthogonal space of $+$-type over $\mathrm{GF}(3)$. Assume that $X$ is a non-zero subspace of $V$. Then $\mathcal S(X)$ is the set of singular 1-dimensional subspaces in $X$, $\mathcal P(X)$ the set of 1-dimensional subspaces of $+$-type in $X$ and $\mathcal M(X)$ the set of 1-dimensional subspaces of $-$-type in $X$. \end{notation}
\begin{lemma}\label{hyper} Let $X$ be a $3$-dimensional subspace in a non-degenerate 4-dimensional orthogonal space of $+$-type over $\mathrm{GF}(3)$. Then $\mathcal S(X) \not=\emptyset$. \end{lemma}
\begin{proof} See \cite[21.3]{AschbacherFG}. \end{proof}
We now introduce some additional notation:
\begin{notation}\label{type} Let $V$ be a $4$-dimensional non-degenerate orthogonal space of $+$-type over $\mathrm{GF}(3)$ and $E$ be a $2$-dimensional subspace of $V$. The type of $E$ is determined by the number of one spaces of a given type in $E$. Thus we have
\begin{tabular}{ll}
{\rm Type S}:&$|\mathcal S(E)|=4$.\\
{\rm Type DP}:& $|\mathcal S(E)|=1$ and $|\mathcal P(E)|=3$.\\
{\rm Type DM}:&$|\mathcal S(E)|=1$ and $|\mathcal M(E)|=3$.\\
{\rm Type N+}:&$|\mathcal S(E)|=2$ and $|\mathcal M(E)|=|\mathcal P(E)|=1$.\\
{\rm Type N-}:& $|\mathcal P(E)|=|\mathcal M(E)|=2$. \end{tabular} \end{notation}
\begin{lemma}\label{types} Let $V$ be a $4$-dimensional non-degenerate orthogonal space over $\mathrm{GF}(3)$ of $+$-type and $E$ be a $2$-dimensional subspace of $V$. Then $E$ is of one of the types in Notation~{\rm \ref{type}}. \end{lemma}
\begin{proof} The subspaces of $V$ of dimension $2$ are either totally singular (S), degenerate with three elements of $\mathcal P(V)$ (DP), degenerate with three elements from $\mathcal M(V)$ (DM) , non-degenerate of plus type (N+),
or non-degenerate of minus type (N-). \end{proof}
\begin{theorem}\label{closed} Suppose that $G$ is a finite group, $Q$ is a subgroup of $G$ and $H= N_G(Q)$. Assume that the following hold \begin{enumerate} \item $H/Q \cong \mathrm{Aut}(\SU_4(2))$ or $\mathrm{Sp}_6(2)$; \item $Q=C_G(Q)$ is a minimal normal subgroup of $H$ and is elementary abelian of order $2^8$; \item $H$ controls $G$-fusion of elements of $H$ of order $3$; and \item if $g \in G\setminus H$ and $d \in H\cap H^g$ has order $3$, then $C_Q(d)=1$. \end{enumerate} Then $G= HO_{2'}(G)$. \end{theorem}
\begin{proof} This is \cite[Theorem 3.1]{ParkerRowley}. \end{proof}
\begin{lemma}\label{involutionsonexspec} Suppose that $G$ is a group, $E$ is an extraspecial $2$-group which is normal in $G$ and $x \in G\setminus C_G(E)$ is an involution. If $x $ is not $E$-conjugate to $xe$ where $e \in Z(E)^\#$, then $C_E(x)\ge [E,x]$ and $[E,x]$ is elementary abelian. \end{lemma}
\begin{proof} Certainly $C_{E/Z(E)}(x)\ge [E/Z(E),x]$. Therefore, if $C_E(x) \not \ge [E,x]$, then $[f,x,x] = e$ for some $f \in E$. Setting $w = [f,x]$ we then have $x^w=xe$ which contradicts our hypothesis on $x$. Hence $C_E(x) \ge [E,x]$.
We now show that every element of $[E,x]$ has order $2$. Let $f \in [E,x]$. Then $fe$ has the same order as $f$. Thus we may suppose that $f=[h,x]$ for some $h \in E$. As $[E,x]\le C_E(x)$, $x[h,x]=[h,x]x$ and so \begin{eqnarray*}f^2&=&[h,x][h,x] = h^{-1}xhx[h,x]=h^{-1}xh[h,x]x\\&=& h^{-1}xhh^{-1}xhxx=1\\
\end{eqnarray*}as required. This proves the lemma. \end{proof}
For a group $X$ with subgroups $A \le Y \le X$, we say that $A$ is \emph{strongly closed in $Y$ with respect to $X$} provided $A^x \cap Y \le A$ for all $x \in X$.
\begin{lemma}\label{Gold} Suppose that $K$ is a group, $O_{2'}(K)=1$, $A$ is an abelian $2$-subgroup of $K$ and $A$ is strongly closed in $N_K(A)$ with respect to $K$. Assume that $F^*(N_K(A)/C_K(A))$ is a non-abelian simple group.
Then $K= N_K(A)$. \end{lemma}
\begin{proof} Set $ L= \langle A^{ K}\rangle$. Since $O_{2'}(K)=1$, we have $O_{2'}(L)=1$. By Goldschmidt \cite[Theorem A]{Goldschmidt}, $ L= O_2( L)E( L)$ and $ A= O_2( L)\Omega_1( T)$ where $ T \in \syl_2( L)$ contains $ A$. If $E( L)=1$, then $ A$ is normal in $ K$ and we are done. Thus $E( L) \neq 1$.
Goldschmidt additionally states that $E( L)$ is a direct product of simple groups of type $\PSL_2(q)$, $q \equiv 3,5 \pmod 8$, ${}^2\mathrm{G}_2(3^a)$, $\mathrm{SL}_2(2^a)$, $\PSU_3(2^a)$, ${}^2\B_2(2^a)$ for some natural number $a$, or the sporadic simple group $\J_1$. It follows from the structure of these groups that $N_{ L}( A)$ is a soluble group which is not a $2$-group. On the other hand, $N_{ L}( A)= L \cap {N_K(A)}$ is a normal subgroup of $ {N_K(A)}$. Since $F^*(N_K(A)/C_K(A))$ is a non-abelian simple group and $N_{ L}( A)$ is soluble we now have $N_L(A) \le C_K(A)$ and this contradicts the structure of $E(L)$. Thus $A$ is normal in $K$ as claimed. \end{proof}
We will also need the following statement of Holt's Theorem \cite{Ho}.
\begin{lemma}\label{Holt} Suppose that $K$ is a simple group, $P$ is a proper subgroup of $K$ and $r$ is a $2$-central element of $K$. If $r^K\cap P= r^P$ and $C_K(r)\le P$, then $K\cong \PSL_2(2^a)$ ($a \ge 2$), $\PSU_3(2^a)$ ($a \ge 2$), ${}^2\B_2(2^a)$ ($a\ge 3$ and odd) or $\mathrm{Alt}(n)$ ($n \ge 5$) where in the first three cases $P$ is a Borel subgroup of $K$ and in the last case $P \cong \mathrm{Alt}(n-1)$.
\end{lemma}
\begin{proof} Set $\Omega= K/P$ and assume that $P<K$. The conditions $C_K(r) \le P$ and $r^K \cap P= r^P$ together imply that $r$ fixes a unique point of $\Omega$. Let $J$ be the set of involutions of $K$ which fix exactly one point of $\Omega$. Since $r$ is a $2$-central element of $K$, any $2$-group which fixes at least $3$ points when it acts on $\Omega$ commutes with an element of $J$. Hence Holt's criteria (*) from \cite{Ho} is satisfied. In addition, the simplicity of $K$ yields $K= \langle r^K\rangle= \langle J\rangle$. Thus \cite[Theorem 1]{Ho} implies that $K$ is isomorphic to one of the following groups $\PSL_2(2^n)$, $\PSU_3(2^n)$, ${}^2\B_2(2^n)$ ($n\ge 3$ and odd) or $\mathrm{Alt}(\Omega)$ where in the first three classes of groups the stabiliser $P$ is a Borel subgroup and in the latter case it is $\mathrm{Alt}(\Omega\setminus\{P\})$.
\end{proof}
For the final steps in the identification of $\F_4(2)$ we need information about its involutions and their centralizers.
\begin{lemma}\label{F42Classes} The group $X=\F_4(2)$ has four conjugacy classes of involutions $x_1, x_2, x_3$ and $x_4$ three of which are $2$-central. Furthermore we may assume that notation is chosen so that \begin{enumerate}\item $C_X(x_1) \cong C_X(x_2) \approx 2^{1+6+8}.\mathrm{Sp}_6(2)$; \item $C_X(x_3) \approx 2^{1+1+4+1+4+4+1+4}.\mathrm{Sp}_4(2)$; and \item $C_X(x_3) \approx 2^{[9]}.(\mathrm{SL}_2(2)\times \mathrm{SL}_2(2))$.\end{enumerate} \end{lemma}
\begin{proof} These facts can be found in Guterman \cite[Section 3]{Guterman} (see also \cite[Page 45]{AschSe}) . \end{proof}
\section{Identifying $\F_4(2)$}
The final step in the proof of Theorem~\ref{MT} demands that we can identify $\F_4(2)$ or $\mathrm{Aut}(\F_4(2))$ from the structure of the centralizer of a certain $2$-central involution. In this section we give such an identification. The centralizers of interest are the centralizers of the involutions $x_1$, $x_2$ in $\F_4(2)$ as given in Lemma~\ref{F42Classes} (i). Of course, we do not want to specify the isomorphism type of such a centralizer, but only the approximate shape of the group.
\begin{definition}\label{F4cent} We say the group $U$ is similar to a $2$-centralizer in a group of type $\F_4(2)$ if $U$ has the following properties. \begin{enumerate} \item $U/O_2(U) \cong \mathrm{Sp}_6(2)$, \item $O_2(U) $ is an product of $Z(O_2(U))$ by an extraspecial group of order $2^9$, $Z(O_2(U))$ is elementary abelian of order $2^7$. \item $U/O_2(U)$ induces the natural module on $Z(O_2(U))/O_2(U)'$ and the spin module on $O_2(U)/Z(O_2(U))$. \end{enumerate} \end{definition}
\begin{definition}\label{F4setup} Suppose that $G$ is a group and assume that the following hold: \begin{enumerate} \item For $i=1,2$, there are involutions $x_i$ in $G$ such that $U_i = C_G(x_i)$ is similar to a $2$-centralizer in a group of type $\F_4(2)$. \item There is a Sylow $2$-subgroup $T$ of $U_1$ such that $Z(T) = \langle x_1,x_2 \rangle$. \end{enumerate} Then we say that $U_1$, $U_2$, $T$ is an \emph{$\F_4$ set-up in $G$.} \end{definition}
Our identification theorem in this section is as follows: \begin{theorem}\label{P=F4} If $U_1$, $U_2$, $T$ is an $\F_4$ set-up in $G$, then $\langle U_1, U_2 \rangle \cong \F_4(2)$. \end{theorem}
For the remainder of this section we assume that $U_1$, $U_2$ and $T$ is an $\F_4$ set-up in $G$. Notice that because of Definition~\ref{F4cent} (ii), for $i=1,2$, $O_2(U_i)'= \langle x_i\rangle$ has order $2$. The first lemma details the relationship of $U_1$ with $U_2$.
\begin{lemma} \label{p23} The following hold:
\begin{enumerate}
\item $U_1\cap U_2 $ contains $T$;\item $(U_1\cap U_2)/O_2(U_1\cap U_2)\cong \mathrm{Sp}_4(2)$; \item $O_2(U_1\cap U_2)=O_2(U_1)O_2(U_2)$; and \item $Z(T) = Z(O_2(U_1)) \cap Z(O_2(U_2))$.\end{enumerate} \end{lemma}
\begin{proof} From the definition of an $\F_4$ set-up in $G$, we have $T \le U_1 \cap U_2$. This proves (i).
Since $Z(U_i)/\langle x_i\rangle$ is a natural $U_i/O_2(U_i)$-module and $|Z(T)|=4$, Lemma~\ref{nonsplitmods} implies $Z(T) \le Z(U_1) \cap Z(U_2)$. Therefore, by Lemma~\ref{sp62natural} (iii), \begin{eqnarray*}(U_1\cap U_2)/O_2(U_1\cap U_2)&=&C_{U_1}(Z(T))/O_2(C_{U_1}Z(T))\\ &=& C_{U_2}(Z(T))/O_2(C_{U_1}Z(T)) \cong \mathrm{Sp}_4(2).\end{eqnarray*} Hence (ii) holds.
Since $$(O_2(U_1) \cap O_2(U_2))' \le O_2(U_1)' \cap O_2(U_2)'= \langle x_1\rangle \cap \langle x_2\rangle=1,$$ $O_2(U_1)
\cap O_2(U_2)$ is abelian. Therefore, as $O_2(U_1)$ contains an extraspecial subgroup of order $2^9$, we have $$|O_2(U_1): O_2(U_1)
\cap O_2(U_2)|\ge 2^4.$$ Furthermore, as $O_2(U_1) O_2(U_2)/O_2(U_1) $ is normal in $(U_1\cap U_2)/O_2(U_1)$, $O_2(U_1\cap U_2)=O_2(U_1)O_2(U_2)$ follows from Lemma~\ref{sp62natural} (iii). This is (iii).
Finally, since $O_2(U_1\cap U_2)$ centralizes $Z(O_2(U_1)) \cap Z(O_2(U_2))$, we deduce $Z(T)= Z(O_2(U_1)) \cap Z(O_2(U_2))$ and this proves (iv).\end{proof}
Our method to prove Theorem~\ref{P=F4} is to use the $\F_4$ set-up $U_1$, $U_2$, $T$ in $G$ to construct a chamber system of type $\F_4(2)$ using the subgroup $P= \langle U_1, U_2\rangle$ of $G$. To accomplish this we first define $P_1, P_2, P_3$ to be subgroups of $U_1$ containing $T$ such that $P_j/O_2(U_1)$, $j= 1,2,3$, are the minimal parabolic subgroups of $U_1/O_2(U_1)$ containing $T/O_2(U_1)$. We additionally let $P_4$ be such that $U_2\ge P_4 \ge T$, $P_4 \not \le U_1$ and $P_4/O_2(U_2)$ is a minimal parabolic subgroup of $U_2/O_2(U_2)$. For $\emptyset \neq \sigma \subseteq \{1,2,3,4\}$ we set $P_\sigma= \langle P_ j \mid j \in \sigma\rangle$.
We may assume that notation has been chosen so that \begin{eqnarray*}P_{12}/O_2(P_{12})&\cong &\mathrm{SL}_3(2);\\
P_{13}/O_2(P_{13})&\cong &\mathrm{SL}_2(2)\times \mathrm{SL}_2(2); \text{ and} \\
P_{23}/O_2(P_{23})&\cong &\mathrm{Sp}_4(2).\\
\end{eqnarray*}
Note also that $P_j /O_2(P_j) \cong \mathrm{SL}_2(2)$ for $1 \le j \le 4$.
By Lemma~\ref{p23} (ii), $P_{23}= U_1\cap U_2$ and $P= \langle P_1,P_2,P_3,P_4\rangle$.
Set $\mathcal I= \{1,2,3,4\}$, and let $${\mathcal{C}} = (P/T, (P/P_k), k \in \mathcal I)$$ \noindent be the corresponding chamber system. Thus $\mathcal C$ is an edge coloured graph with colours from $\mathcal I=\{1,2,3,4\}$ and vertex set the right cosets $P/T$. Furthermore, two cosets $Tg_1$ and $Tg_2$ form a $k$-coloured edge if and only if $Tg_2g_1^{-1} \subseteq P_k$. Obviously $P$ acts on $\mathcal C$ by multiplication of cosets on the right and this action preserves the coloured edges. For $\mathcal J \subseteq \mathcal I$, set $P_{\mathcal J} = \langle P_k \mid k \in \mathcal J\rangle$ and $\mathcal C_\mathcal J = (P_{\mathcal J}/T, (P_{\mathcal J}/P_k), k \in \mathcal J)$. Then $\mathcal C_\mathcal J$ is the $\mathcal J$-connected component of $\mathcal C$ containing the vertex $T$.
We will show $\mathcal C$ locally resembles the corresponding chamber system in $\F_4(2)$. This means that for $\sigma \subset \mathcal I$ with $|\sigma|=2$ we will show $P_{\sigma}/O_2(P_{\sigma})$ is isomorphic to the corresponding group in $\F_4(2)$. Since $U_1/O_2(U_1) \cong \mathrm{Sp}_6(2)$ this is true if $\sigma \subseteq \{1,2,3\}$. Hence we may assume that $4 \in \sigma$. There are two possibilities for the relationship between $P_2$ and $P_4$ (they are both contained in $U_2$), but we may have $P_{24}/O_2(P_{24}) \cong \mathrm{SL}_3(2)$ or $P_{24} = P_2P_4$. We shall show that the latter is in fact the case. We will also prove $P_{14} = P_1P_4$. This is the purpose of the next lemma.
\begin{lemma} \label{buildingF4} The subgroup $Z_2(T)$ is normalized by $P_{14}$, $P_{14} = P_1P_4$ and $P_{24}= P_2P_4$. \end{lemma}
\begin{proof} Let $V= Z_2(T)$. Then, by Lemma~\ref{p23} (iv), $V\cap Z(O_2(U_2)) \not \le Z(O_2(U_1))$.
As $C_{O_2(U_1)/Z(O_2(U_1))}(T)$ has order $2$ by Lemma~\ref{sp62spin} and $|V \cap Z(O_2(U_2))|=2^3$ by Lemma~\ref{sp62natural}, we deduce $V= (V \cap Z(O_2(U_1)))(V\cap Z(O_2(U_2)))$ has order $2^4$ as $Z(T) =Z(O_2(U_1))\cap Z(O_2(U_2))$.
Using Lemmas~\ref{sp62natural} and \ref{sp62spin}, $V\cap Z(O_2(U_1))$ and $VZ(O_2(U_1))$ are both normalized by $P_1$. Set $$W= \langle V{^{P_{1}}}\rangle.$$ Then, as the set $V^{P_1}$ has size at most $3$, $W /(V\cap Z(O_2(U_1)))$ has order at most $2^3$ and $W=V(W\cap Z(O_2(U_1)))$. Since $(W\cap Z(O_2(U_1)))/(V\cap Z(O_2(U_1)))$ has order at most $2^2$, Lemma~\ref{sp62natural} implies $O^2(P_1)$ centralizes $(W\cap Z(O_2(U_1)))/(V\cap Z(O_2(U_1)))$. But then $W/(V\cap Z(O_2(U_1)))$ is centralized by $O^2(P_1)$. Thus $W=V$. We may apply the same argument to $U_2$ to see that $P_4$ also normalizes $V$ and so deduce that $P_{14}$ acts on $V$ which has order $2^4$.
We have $[V,O_2(P_1)] \le Z(O_2(U_1)) \cap Z(O_2(U_2))= Z(T)$. Hence, as $[V,O_2(P_1)]$ is normalized by $P_1$, $[V,O_2(P_1)]= \langle x_1\rangle$. Similarly $ \langle x_2\rangle = [V,O_2(P_4)]$. Therefore $O_2(P_1) \cap O_2(P_4)$ centralizes $V$ and has index $4$ in $T$. Thus $C_T(V)= O_2(P_1) \cap O_2(P_4)$. In particular, $O_2(P_1)$ acts as a transvection on $V$. Hence $C_V(O_2(P_1))$ has order $2^3$ and so $C_V(O_2(P_1))= V \cap Z(U_1)$ and $C_V(O_2(P_4))= V \cap Z(O_2(U_2))$. Because $C_G(V) \le U_1$, we have also shown $C_G(V)= O_2(P_1) \cap O_2(P_4)$.
Set $$D=\langle O_2(P_1)^{N_G(V)}, O_2(P_4)^{N_G(V)}\rangle C_G(V) /C_G(V).$$
Then $D \cap U_1 = P_1$ and, as $x_1$ has at most 15 conjugates under the action of $D$, $|D| \leq 12\cdot 15$. The structure of $\mathrm{Alt}(8) \cong\mathrm{GL}_4(2)$ therefore shows $D \cong \mathrm{SL}_2(2)\times \mathrm{SL}_2( 2),$ or $\mathrm O_4^-(2)\cong \mathrm{Sym}(5)$.
Let $Q_{12}=O_2(P_{12})$, $W_1$ be the preimage of $C_{Z(O_2(U_1))/\langle x_1\rangle}(Q_{12})$ and define $W= W_1V$. Then $W$ is elementary abelian of order $2^5$. Since $V=(V \cap Z(O_2(U_1)))(V\cap Z(O_2(U_2)))$, \begin{eqnarray*} [W,Q_{12}]&=&[W_1(V \cap Z(O_2(U_1)))(V\cap Z(O_2(U_2))),Q_{12}]\\& \le &\langle x_1\rangle [(V \cap Z(O_2(U_1)))(V\cap Z(O_2(U_2))),Q_{12}]\\&=&\langle x_1\rangle [(V\cap Z(O_2(U_2))),Q_{12}]\\&\le &\langle x_1\rangle [(V\cap Z(O_2(U_2))),T] \\&=&\langle x_1\rangle [(V\cap Z(O_2(U_2))),O_2(U_1)O_2(P_4)]\\ &=&\langle x_1\rangle [(V\cap Z(O_2(U_2))),O_2(U_1)]=\langle x_1\rangle.\\ \end{eqnarray*} As $O_2(U_1)/Z(O_2(U_1))$ is a spin module for $\mathrm{Sp}_6(2)$, $$C_{O_2(U_1))/Z(O_2(U_1))}(Q_{12})= WZ(O_2(U_1))/Z(O_2(U_1))$$ by Lemma~\ref{sp62spin}. We deduce that $W$ is the preimage of $C_{O_2(U_1)/\langle x_1\rangle}(Q_{12})$ and thus $W$ is normalized by $P_{12}$. Since $Z(O_2(U_1)) \cap Z(O_2(U_2)) = Z(T)$, we have $WZ(O_2(U_2))/Z(O_2(U_2))$ has order $2^2$. It follows from Lemma~\ref{sp62spin} that $O^2(P_{4})$ centralizes $WZ(O_2(U_2))/Z(O_2(U_2))$. Let $W_2= \langle W^{P_4}\rangle$. Then $W_2= W(W_2 \cap Z(O_2(U_2)))$. Since $W/V$ has order $2$, we infer that $W_2/V$ has order at most $2^3$. Thus $(W_2 \cap Z(O_2(U_2)))/(V\cap Z(O_2(U_2)))$ has order at most $2^2$. It follows from Lemma~\ref{sp62natural} that $(W_2 \cap Z(O_2(U_2)))/(V\cap Z(O_2(U_2)))$ is centralized by $O^2(P_4)$. Therefore $W/V$ is normalized by $TO^2(P_4)= P_4$. This shows that $W$ is normalized by $P_{124}$. Notice that along the way we have shown that $P_{24}=P_2P_4$.
Suppose that $P_{14}/O_2(P_{14})\cong \mathrm O_4^-(2)$. Then $P_{14}$ acts irreducibly on $V$ and so, as $P_{12}$ does not normalize $V$, $W$ is an irreducible $P_{124}$-module. As $P_{14}$ has orbits of length $10$ and $5$ on $V$ and $Z(T) \leq V$, we have that $P_{14}$ does not centralize any element in $W \setminus V$ and so $P_{14}$ acts transitively on the 16 elements of $W \setminus V$. This means the orbits of $P_{14}$ on the involutions of $W$ have lengths 5, 10 and 16. Since $5$ divides the order of $D$, we get that the number of conjugates of $x_1$ under $P_{124}$ is divisible by $5$ and, as $|x_1^{P_{12}}|=10$, we conclude $|x_1^{P_{124}}|=10$ or 15. But then $V = \langle x_1^{P_{124}} \rangle$, contradicting the fact that $P_{124}$ acts irreducibly on $W$. Hence $P_{14}/O_2(P_{14})\cong \mathrm{SL}_2(2)\times \mathrm{SL}_2(2)$ with $P_{14}=P_1P_4$ and this concludes the proof of the lemma. \end{proof}
\begin{proof}[Proof of Theorem~\ref{P=F4}] Using Lemma~\ref{buildingF4} and the observations before the lemma yields that the chamber systems $\mathcal C_{1,2}$, $\mathcal C_{3,4}$ are projective planes, $\mathcal C_{2,3}$ is a generalized quadrangle and in both cases the parameters are $3,3$ and the remaining $\mathcal C_J$ with $|J| = 2$ are all complete bipartite graphs again with parameters $3,3$. Thus $\mathcal C$ is a chamber system of type $\F_4$ (see \cite{local}) in which all panels have $3$ chambers. Since $U_1/O_2(U_1)\cong \mathrm{Sp}_6(2) \cong U_2/O_2(U_2)$, we have $\mathcal C_{1,2,3}$ and $\mathcal C_{2,3,4}$ are the $\mathrm{Sp}_6(2)$-building. Hence, as each connected rank $3$ residue of $\mathcal C$ is a building of type $\mathrm C_3$
and all the rank $2$ residues of $\mathcal C$ are Moufang polygons, applying \cite[Corollary 3]{local} yields
that the universal covering $\pi : {\mathcal{C}}^\prime \longrightarrow {\mathcal{C}}$ has $\mathcal{C}^\prime$ a building of type $\F_4$ which also has three chambers on each panel.
By \cite[Proof of Theorem 10.2 on page 214]{Tits} this building is uniquely determined by the two
residues of rank three with connected diagram. Thus $\mathcal C^\prime$ is isomorphic to the $\F_4(2)$ building and the type preserving automorphism group $F$ of $\mathcal C'$ is isomorphic to $\F_4(2)$. Since $\mathcal C'$ is a $2$-cover of $\mathcal C$, there is a subgroup $U$ of $F$ such that $U$ contains $U_1$ and $U/D \cong P$ for a suitable normal subgroup
$D$ of $U$. As $U_1$ is isomorphic to a maximal parabolic subgroup of $F$, we deduce that $U = F$ and $D=1$. Thus $P \cong F$.
\end{proof}
\section{The structure of $M$}
From now on we suppose that $G$ is a group which satisfies the assumptions of Theorem ~\ref{MT}. We set $M = N_G(Z)$. So $C_G(Z)$ has index at most $2$ in $M$. Let $S \in \syl_3(M)$ and $Q= F^*(M)= O_3(M)$.
\begin{lemma}\label{basic} We have $Z=Z(S)=Z(Q)$, $N_G(S)\le M$ and $S \in \syl_3(G)$. \end{lemma}
\begin{proof} Since $C_M(Q) \le F^*(Q)= Q$, we have that $Z= Z(Q)=Z(S)$. Therefore $N_G(S) \le N_G(Z)= M$ and, in particular, $S \in\Syl_3(N_G(S))\subseteq \Syl_3(G)$. \end{proof}
Let $R^*$ be a normal subgroup of $C_H(Z)$ such that $R^*/Q\cong \mathrm{Q}_8 \times \mathrm{Q}_8$ and let $R \in \Syl_2(R^*)$. We have that $M/Q$ embeds into $\mathrm{Out}(Q)$ and $\mathrm{Out}(Q)$ is isomorphic to $\GSp_4(3)$ by \cite[III(13.7)]{Hu}. We now locate $M/Q$ in $\mathrm{Out}(Q)$. We will show that $M/QR$ is isomorphic to $\mathrm{Sym}(3)$ or $2 \times \mathrm{Sym}(3)$, more precise information will be presented in Lemma~\ref{structM}. The next lemma provides our initial restriction on the structure of $M$.
\begin{lemma}\label{U6F4} We have that $M/Q$ normalizes $R^*/Q$ and is isomorphic to a subgroup of the subgroup $\mathbf M$ of $\GSp_4(3)$ which preserves a decomposition of the natural $4$-dimensional symplectic space over $\mathrm{GF}(3)$ into a perpendicular sum of two non-degenerate $2$-spaces. Furthermore, $R/Q$ maps to $O_2(\mathbf M)$. \end{lemma}
\begin{proof} See \cite[Lemma 3.1]{PS1}. \end{proof}
We next introduce a substantial amount of notation. We will use this for the remainder of the paper. We note now that the subgroups $Q_1$ and $Q_2$ defined below will be shown to have order $3^3$ in Lemma~\ref{Qaction}.
\begin{notation}\label{nota}
\begin{enumerate} \item Define $R_1$ and $R_2$ to be the two subgroups of $R$ isomorphic to $\mathrm{Q}_8$ which map to normal subgroups of $C_{\mathbf M}(Z(R)Q/Q)$.
\item For $i=1, 2$, let $r_i \in Z(R_i)^\#$ and $K_i = C_G(r_i)$. \item For $i=1,2$, define $$Q_i= [Q,R_i].$$ \item For $i=1,2$, let $A_i \le Q_i$ be a fixed $S$-invariant subgroup of $Q_i$ of order $3^2$ and set $A=A_1A_2$. \item For $i=1, 2$, we let $$\langle \rho_i \rangle \le A_i$$ be such that $\langle \rho_i\rangle$ is inverted by $r_i$. \item Set $J = C_S(A)$ and $L = N_G(J)$. \end{enumerate} \end{notation}
Most of this paper is devoted to the determination of $K_1$ and $K_2$. We will show that $K_i$ is similar to a $2$-centralizer in a group of type $\F_4(2)$ as defined in Definition~\ref{F4cent} and, for $T \in \syl_2(K_1)$, show that $K_1,K_2$ and $T$ is an $\F_4$ set-up. We then use Theorem~\ref{P=F4} to obtain a subgroup $P \cong \F_4(2)$ of $G$. Our interim goal to achieve this objective is to show that $C_G(\rho_i)$ is isomorphic to the corresponding centralizer in $\F_4(2)$ or $\mathrm{Aut}(\F_4(2))$. We eventually do this in Lemma~\ref{ItsSp62}. However we begin more modestly by determining the precise structure of $M$.
\begin{lemma}\label{Qaction} The following hold. \begin{enumerate}
\item $|S/Q|\le 3^2$. \item $Q_1=C_Q(r_2)$ and $Q_2= C_Q(r_1)$ and both are normal in $S$; and \item $Q_1 \cong Q_2 \cong 3^{1+2}_+$, $[Q_1,Q_2]=1$ and $Q=Q_1Q_2$; \item $A$ is elementary abelian of order $3^3$. \end{enumerate} In particular, $Q$ has exponent $3$. \end{lemma}
\begin{proof} Part (i) follows from Lemma~\ref{U6F4}.
That $Q_1$ and $Q_2$ are normalized by $S$ follows from the action of $M$ on $Q$ as $R_1 Q/Q$ and $R_2 Q/Q$ are normalized by $S/Q$.
For $i=1,2$, we have that $C_Q(r_i)$ and $Q_i=[Q,r_i]$ commute by the Three Subgroup Lemma. Since $Q_i$ has order $3^3$ it follows that $Q_i\cong 3^{1+2}_+$. As $r_1r_2$ inverts $Q/Z$, $r_2 $ inverts $C_{Q/Z}(r_1)$ and so $C_Q(r_1)= Q_2$ and $C_Q(r_2)=Q_1$. In particular, $Q_1$ and $Q_2$ commute and $Q=Q_1Q_2$. This proves (ii) and (iii). Finally (iv) follows from (ii) and (iii). \end{proof}
\begin{lemma}\label{ActionQ} Every element of $Q$ is $M$-conjugate to an element of $A$. \end{lemma}
\begin{proof} It suffices to prove that every element of $Q/Z$ is conjugate to an element of $A/Z$. Let $w\in Q/Z$. Then $w=x_1x_2$ where $x_i \in Q_i/Z$ by Lemma~\ref{Qaction} (iii). Since, from the definition of $A$, for $i=1, 2$, $(A\cap Q_i)/Z= A_i/Z$ has order $3$ and $R_i$ acts transitively on $Q_i/Z$, there exists $s_i\in R_i$ such that $w^{s_1s_2} = x_1^{s_1}x_2^{s_2} \in A/Z$. This proves the claim. \end{proof}
Recall that by hypothesis $Z$ is not weakly closed in $Q$. Hence there is a $g\in G$ such that $Y= Z^g \le Q$ and $Y \neq Z$. We set \begin{eqnarray*} V&=&ZY;\\
H&=&\langle Q, Q^g\rangle; \text{ and } \\ W &=& C_{Q^g}(Z)C_Q(Y).\\ \end{eqnarray*} Notice that $C_Q(Y)$ normalizes $C_{Q^g}(Z)$ and so $W$ is indeed a subgroup of $G$. Because of Lemma~\ref{ActionQ} we may and do suppose that $V \le A$. In particular, $V$ is normalized by $S$. Before we continue our study of $M$, we investigate $H$.
\begin{lemma}\label{Z weak Q} The following statements hold. \begin{enumerate} \item $S> Q$; \item $Q \cap Q^g$ is elementary abelian of order $3^3$ and is a normal subgroup of $S$; \item $W=C_Q(Y)C_{Q^g}(Y)$ is a normal subgroup of $H$, $H/W\cong \mathrm{SL}_2(3)$, $WQ \in \syl_3(H)$ and $W/(Q\cap Q^g)$ is a natural $H/W$ module; \item for $i=1,2$, $V \cap Q_i=Z$ and $A \neq Q\cap Q^g$; \item $A=[Q,W]\le W$, $A/Z= C_{Q/Z}(S)= C_{Q/Z}(W)$ and $A$ is normal in $N_G(S)$; and \item for $i=1,2$, $[WQ/Q,R_iQ/Q]\neq 1$. \end{enumerate} \end{lemma}
\begin{proof} As $Q$ is extraspecial, $C_Q(Y)$ is non-abelian of order $3^4$. By Lemma~\ref{basic}, $M^g/Q^g$ has Sylow $3$-subgroups of order at most $9$ and $C_Q(Y)\le M^g$ so we have $Z=C_Q(Y)' \le Q^g$. In particular we now have $S> Q$ for else $C_Q(Y) \le Q^g$ and then $Z= C_Q(Y)' \le (Q^g)'= Y$ which is a contradiction. In particular, (i) holds.
Since $\Phi(Q\cap Q^g) \le Z \cap Y= 1$, $Q \cap Q^g$ is elementary abelian.
Because $V \le Q\cap Q^g$, we have $[V,Q]= Z$ and $[V,Q^g]=Y$ and so $H$ normalizes and acts non-trivially on $V$ with $H/C_H(V) \cong \mathrm{SL}_2(3)$.
Turning our attention to $W$, we have $$[W,Q] = [C_Q(Y)C_{Q^g}(Z),Q] = Z[C_{Q^g}(Z),Q].$$ Since $[[C_{Q^g}(Z),Y],Q]=1=[Q,Y,C_{Q^g}(Z)]$, the Three Subgroup Lemma implies that $[C_{Q^g}(Z),Q]\le C_{Q}(Y) \le W$. Therefore $$[Q,W]\le C_Q(Y)\le W$$ and, similarly, $[W,Q^g]\le C_{Q^g}(Z)\le W$. Hence $H$ normalizes $W$ and of course $W \le C_G(V)$.
As $[C_H(V),Q]\le C_Q(V)=C_Q(Y)\le W$, $H/W$ is a central extension of $\mathrm{SL}_2(3)$. Since $H$ acts transitively on the four subgroups of order $3$ in $V$, and each such subgroup determines uniquely a subgroup of $H$ we have that $Q^H$ has exactly $4$ members. Now $O^{3}(H)W/W$ is a central extension of a nilpotent group and is thus nilpotent. Let $T$ be a Sylow $2$-subgroup of $O^3(H)$. Then as $O^3(H)/W$ is nilpotent, $Q$ normalizes and does not centralize $T$. It follows that $H= WTQ$ and then the action of $Q$ on $T$ and the fact that $T/C_T(V)\cong\mathrm{Q}_8$ implies that $T \cong \mathrm{Q}_8$ and that $H/W \cong \mathrm{SL}_2(3)$, as by \cite[Satz V.25.3]{Hu} the Schur multiplier of a quaternion group is trivial.
Using that $O^3(H)$ acts transitively on $V^\#$, we see that $O^3(H)$ does not normalize any non-trivial subgroup of $(W\cap Q)/(Q\cap Q^g)$.
Assume $Q \cap Q^g = V$. Then $|W|=3^6$. As $W'\le V$, $W$ is generated by groups of exponent $3$ and $W$ is non-abelian, we have $\Phi(W) = V$. Let $f \in H$ be an involution. Then $fW \in Z(H/W)$ and, by Burnside's Lemma, $f$ does not centralize $W/\Phi(W)$ and neither does it invert $W/\Phi(W)$, for then, as $f$ inverts $V$, $W$ would be abelian. Therefore, setting $W_0= C_W(f)V$, we have $W_0>V$. Then, as the faithful representations of $\mathrm{SL}_2(3)$ in characteristic $3$ have even dimension and the minimal faithful representation for $\PSL_2(3)$
is $3$, $|W_0/V|=3^2$ and $W_0$ is centralized by $O^3(H)$ and normalized by $Q$; in particular, $Q\cap W_0 \le V$ by the comments at the end of the last paragraph. But then $(W\cap Q)W_0= W_0(W\cap Q^g)= W$ which means that $$[W, Q] =[W_0,Q][W\cap Q,Q]\le W_0.$$ Consequently $O^3(H)$ centralizes $W/V$ which is a contradiction as we have already remarked that $f$ does not centralize $W/V$. Therefore $Q\cap Q^g > V$.
Since $Q\cap Q^g$ is abelian and $Q$ is extraspecial of order $3^5$, we now have that $|Q\cap Q^g|= 3^3$ and $W/(Q\cap Q^g)$ is a natural $\mathrm{SL}_2(3)$-module. This completes the proof of the first two statements in (ii) and all of (iii).
Since $H$ acts two transitively on the non-trivial cyclic subgroups of $V$, $N_G(V)= (N_M(V)\cap N_{M^g}(V)) H$ and therefore $N_G(V)$ normalizes $Q\cap Q^g$. From the choice of $V \le A$, we have $S \le N_G(V)$. This is the last statement in (ii).
Suppose that $V \le Q_i$ for some $i\in \{1,2\}$. Then $C_M(V) \ge R_{3-i}$ and so $R_{3-i}$ acts on $Q\cap Q^g$. Since $|Q\cap Q^g:V|= 3$, we obtain $Q\cap Q^g \le C_Q(r_{3-i})= Q_i$ contrary to $Q\cap Q^g$ being elementary abelian of order $3^3$. Hence $V$ is not contained in $Q_i$ for $i=1,2$. If $A=Q\cap Q^g $, then $$Y=[A,C_{Q^g}(Z)]\le[A,S]= Z,$$ which is impossible. Hence we also know that $A \neq Q\cap Q^g$. Thus (iv) holds.
If $[Q_1,W]\le Z$, then $[Q,W]=[Q_1,W][Q_2,W]\le A_2$. Therefore using (iv), $$[C_Q(V),W]= [C_Q(V),C_{Q^g}(V)]Z
\le Q\cap Q^g \cap A_2= Z.$$ Since $|Q\cap Q^g|= 3^3$ by (ii), $Y=[Q\cap Q^g, C_{Q^g}(V)]\le [Q,W]=Z$ which is impossible. Thus $[Q_1,W] = A_1$ and similarly $[Q_2,W] = A_2$. Now $[Q,W]= A$ and consequently $[Q,S]= A$. This proves (v).
Finally, suppose that $[WQ,R_1Q] \le Q$. Then $[Q_1,W] \le A_1$ and is $R_1$-invariant. Hence $[Q_1,W]\le Z$ and this contradicts (v). Thus $[WQ,R_1Q] \not \le Q$ and (vi) holds. \end{proof}
Now we are in a position to determine $M$. For this set
$$M_0 = RQ$$ and let $f$ be an involution in $H$. Then $f$ inverts $V$ and thus $f \in M$. We refine our choice of $R$ so that $R\langle f \rangle$ is a Sylow $2$-subgroup of $M_0S\langle f \rangle$.
\begin{lemma}\label{structM} The following hold. \begin{enumerate}
\item $S= WQ$ and $|S/Q|=3$; and \item One of the following holds: \begin{enumerate} \item $M= M_0S\langle f \rangle$, $C_M(Z)= M_0S$ and $M/M_0 \cong \mathrm{Sym}(3)$; or
\item $|M: M_0S\langle f\rangle|=2$, $C_M(Z) = M_0S\langle t \rangle$ where $t$ is an involution which exchanges $R_1$ and $R_2$, centralizes $V$ and inverts $SM_0/M_0$ and $M/M_0 = \langle t,f\rangle SM_0/M_0 \cong 2 \times \mathrm{Sym}(3)$ with centre $\langle tf\rangle M_0/M_0$. \end{enumerate} \end{enumerate} \end{lemma}
\begin{proof} We have seen in Lemma~\ref{Z weak Q} (i) and (v) that $|S/Q| \ge 3$ and $A/Z=C_{Q/Z}(S)= C_{Q/Z}(W)$.
Suppose that $|S/Q|= 3^2$ and assume that $B$ is an abelian subgroup of $Q$ which is normal in $S$ of order $3^3$ with $B\neq A$. For $i=1,2$, let $s_i \in S$ be such that $[s_i,R_{3-i}]\le Q$. Then $[B,s_i]\le B \cap A\cap Q_i \le A_i$. Thus if $s_i$ does not centralizes $B/Z$, then $A_i \le B$. Since $S= \langle s_1,s_2\rangle$ and $B\neq A$, without loss of generality we may suppose that $A_1 \le B$ and $[B,s_2]\le Z$. In particular, $B \le Q_1 A$ as $C_{Q/Z}(s_2)= Q_1A/Z$. But then $A_1$ is centralized by $AB= Q_1A$ and we have a contradiction as $Z(Q_1A)= A_2$. Thus, if $B\le Q$ is a normal abelian subgroup of $S$ of order $3^3$, then $B=A$. Taking
$B = Q\cap Q^g$, we now have that $Q\cap Q^g= A$ a possibility which is eliminated by Lemma~\ref{Z weak Q} (iv). Thus $|S/Q|=3$. This proves (i).
We know that $f$ inverts $W/(Q\cap Q^g)$ and so $WQ/Q$ is inverted by $f$. In particular, $M_0S\langle f\rangle/M_0 \cong \mathrm{Sym}(3)$. If $M=M_0S\langle f\rangle$, then (ii)(a) holds. So assume that $M>M_0S\langle f\rangle$. As $M$ inverts $Z$, we have $M= C_M(Z)\langle f\rangle$. Since, by Lemma~\ref{U6F4}, $C_M(Z)/Q$ is isomorphic to a subgroup of $\mathrm{Sp}_2(3)\wr 2$ and since $S/Q$ has order $3$, Lemma~\ref{Z weak Q} (vi) implies that $C_M(Z)/M_0 \cong 3 \times 2$ or $\mathrm{Sym}(3)$. Especially, there is a $2$-element $t\in C_M(Z)\setminus M_0$ which normalizes $R\langle f\rangle$ and swaps $R_1$ and $R_2$. Because $R\langle t \rangle$ is isomorphic to a Sylow $2$-subgroup of $\mathrm{Sp}_2(3)\wr 2$, we may as well assume that $t$ is an involution and that $t$ normalizes $S$.
Since $t$ normalizes $S$ and swaps $R_1$ and $R_2$, $t$ also interchanges $Q_1$ and $Q_2$ and normalizes $A$. It follows that $t$ normalizes $V$. Without loss of generality we may now additionally assume that $t$ normalizes $Y$. Thus $t$ normalizes $Q\cap Q^g$ as well as $A$. Since $t$ centralizes $Z$, $[Q,t]$ is extraspecial of order $3^{1+2}$. Hence either $t$ centralizes $V$ and $Q/C_Q(V)$ or $t$ inverts $V/Z$ and $Q/C_Q(V)$. Multiplying $t$ by $r_1r_2$, we may assume that $t$ centralizes $V$. If $S/Q$ is centralized by $t$, we now have $S/C_Q(V)$ is centralized by $t$. However, as $[Q,S](Q\cap Q^g)= C_Q(V)/(Q\cap Q^g)$, we see that $S/(Q\cap Q^g)$ is extraspecial and since $t$ centralizes $S/C_Q(V)$, Burnside's Lemma implies that $t$ centralizes $S/(Q\cap Q^g)$. Then $t$ also centralizes $Q$ which is a contradiction. Hence $t$ inverts $S/Q$ and therefore $C_M(Z)/M_0$ has the structure described in (ii)(b). \end{proof}
\section{The structure of $L = N_G(J)$}
In this section we continue to use the notation introduced in \ref{nota}. We also recall $H = \langle Q , Q^g\rangle$ and $f$ is an involution in $H\cap M$ which inverts $Z$.
We will show that $J$ is the Thompson subgroup of $S$ and determine $L = N_G(J)$.
Set $$H_1 = H^{r_1}, W_1= W^{r_1} \mbox{ and }V_1= V^{r_1}.$$
\begin{lemma}\label{HnotH1} We have $W \not= W_1$ and $H \not= H_1$. \end{lemma}
\begin{proof} Notice that $r_1$ inverts $A_1/Z$ and centralizes $A_2/Z$. Therefore, $V^{r_1} \neq V$. Since $$W' = [C_Q(V),C_{Q^g}(V)]V \le Q\cap Q^g \cap [Q,W]= Q\cap Q^g\cap A= V,$$ we see $W'= V$ and $W_1'= V_1$. Thus $W$ and $W_1$ are not equal and so also $H \neq H_1$. \end{proof}
\begin{lemma}\label{3classes} For $i=1,2$, we have $\rho_i$ is not $G$-conjugate to an element of $Z$. In particular, $A$ contains exactly seven $G$-conjugates of $Z$. \end{lemma}
\begin{proof} Assume that $i\in \{1,2\}$ and set $U= \langle \rho_i \rangle Z$. If $\rho_i$ was $G$-conjugate to an element of $Z$, then Lemma~\ref{Z weak Q}(iv) applied with $U$ in place of $V$ yields $\langle \rho_i\rangle = U\cap Q_i \le Z$ which is absurd. Thus $\rho_i$ is not $G$-conjugate to an element $Z$. Since $V\cup V_1 \subset A$, we now see $A$ contains exactly seven $G$-conjugates of $Z$ three $Q$-conjugates of $\langle\rho_1\rangle$ and three $Q$-conjugates of $\langle \rho_2\rangle$. \end{proof}
We can now describe the structure of $L$.
\begin{lemma}\label{NJ} The following hold. \begin{enumerate} \item $J= J(S)$ is elementary abelian of order $3^4$. \item $L$ controls $G$-fusion of elements of $J$. \item $J= C_G(J)$. \item $L$ preserves a quadratic form $\mathrm q$ of $+$-type on $J$ up to similarity. \item Set $L_* = \langle H,H_1, r_1,r_2\rangle$. Then $L_*/J \cong \GO_4^+(3)$ and either \begin{enumerate} \item if $M= M_0S\langle f\rangle$, then $L=L_*$; or \item if $M> M_0S\langle f\rangle$, then $L/J\cong \mathrm {CO}_4^+(3)$. (Here $\mathrm {CO}_4^+(3)$ is the group which preserves $\mathrm q$ up to similarity.) \end{enumerate} \end{enumerate} \end{lemma}
\begin{proof} By construction $A$ is elementary abelian and so $A \le C_Q(V) \le W$ and $A \le C_Q(V_1) \le W_1$. Since $S$ centralizes $A/Z$ and since in $\mathrm{GL}_3(3)$ such a centralizer has order $18$, we infer that $J=C_S(A)$ has order $3^4$. Since $A$ has index $3$ in $J$, $J$ is abelian. Suppose that $B$ is an abelian subgroup of $S$ of order at least $3^4$. We may assume that $B \ge Z$. Thus by Lemma~\ref{structM}, $B \cap Q$ is an abelian subgroup of $Q$ of order at least $3^3$ and hence of order exactly $3^3$. Using that $(B\cap Q)/Z$ is centralized by $QB=S$, Lemma~\ref{Z weak Q} (iii) yields $B\cap Q= A$. But then $B \le C_S(A)=J$ and we have $B=J$. Hence $J= J(S)$ is the Thompson subgroup of $S$. Since $J$ centralizes $V$, $J \le S\cap C_G(V)= W$. Thus $J= J(W) $ and similarly $J= J(W_1)$. In particular, $L \ge \langle H, H_1\rangle N_G(S)$. Since $J$ contains $A$, if $J$ is not elementary abelian, then $\Phi(J)= Z$. But then $Z$ is normalized by $H$, which is a contradiction as $H$ acts irreducibly on $V$. Thus $J$ is elementary abelian. This proves (i). Part (ii) follows from \cite[37.6]{AschbacherFG} as $J$ is abelian.
We have that $C_G(J) \le C_G(Z)<M$. Since $J$ acts non-trivially on both $R_1Q/Q$ and $R_2Q/Q$, and $JM_0/M_0$ is inverted by $t$ when $M> M_0S\langle f\rangle$ (see Lemma~\ref{structM} (ii)), we have $C_M(J)\le S\langle r_1,r_2\rangle$. Since $r_1Q$ and $r_2Q$ act non-trivially on $A/Z$, we have $C_G(J) \le S$. Hence $J\le C_G(J)= C_S(J)\le C_S(A)\le J$ and this proves (iii).
Define $$\mathcal S (J) = \{j \in J^\#\mid j^{l}\in Z \text{ for some } l \in L\}.$$
Consider $S/J = Q_1Q_2J/J$. Then $S /J \in \syl_3(L_*/J) \subseteq \syl_3(L/J)$. We have $[J,Q_1]= A_1= C_J(Q_2)$ and $[J,Q_2]= A_2= C_J(Q_1)$. In addition, $[J,S]= [J,Q]= [W,Q]=A$ and $C_J(S)= Z$.
Now $\langle Z^{L_*}\rangle \ge \langle Z^H\rangle \langle Z^{H_1}\rangle = VV_1= A$ and, as $A \not \le Q\cap Q^g$, $A$ is not normalized by $H$. Hence $\langle Z^{L_*}\rangle=J$ and, in particular, $L_*$ and, consequently, $L$ acts irreducibly on $J$. Thus there are members of $\mathcal S(J)$ in $J\setminus A$. By Lemma~\ref{3classes} there are exactly $14$ elements of $\mathcal S(J)$ in $A$ and in $J\setminus A$ there are a multiple of 18 such elements. Thence $|\mathcal S(J)| = 14+ n\cdot 18$ for some integer $n\ge 1$. Since $|J|= 3^4$, using the fact that $|\mathcal S(J)| $ divides $|\mathrm{GL}_4(3)|$ we infer that $|\mathcal S(J)|= 32$.
Using Lemma~\ref{GO4} with $\langle a \rangle = Q_1J/J$ and $\langle b \rangle = Q_2J/J$, yields that $S$ preserves a quadratic form with any element of $\mathcal S(J)$ as a singular vector. Since $S/J$ contains $W_1/J$ and $W_2/J$ which both act quadratically on $J$ with $[J,W]= [J,J(Q\cap Q^g)]= [J,(Q\cap Q^g)]=V$ and $[J,W]= [J,W]^{r_1}= V_1$ we see that for any such form $V$ and $V_1$ would consist of singular vectors. It follows that $\mathcal S(J)$ is the set of singular vector of a $+$-type quadratic form on $J$. Since this set is by design invariant under the action of $L$, we have $L/J $ is isomorphic to a subgroup of $\mathrm {CO}_4^+(3)$ by Lemma~\ref{quadratic form}. Thus (iv) is true. Now $HH_1$ contains $S=WW_1$ which is a Sylow $3$-subgroup of $G$, $H$ acts irreducibly on $V$ and $H_1$ acts irreducibly on $V_1$, it follows that $HH_1/J \cong \Omega_4^+(3)$. Conjugation by $r_1$ exchanges $H$ and $H_1$, $\langle r_1r_2\rangle H_1/W_1 \cong \mathrm{GL}_2(3)$ and so we infer that $L_*/J \cong \mathrm {GO}_4^+(3)$ and $L_*$ is normal in $L$. By the Frattini Argument, $L= N_L(S) L_* = N_M(S)L_*$ and so (v) holds. \end{proof}
\begin{lemma}\label{fusion3elts} We have $\rho_1$ is $G$-conjugate to $\rho_2$ if and only if $SR\langle f \rangle$ has index $2$ in $M$. \end{lemma}
\begin{proof} This is a consequence of Lemma~\ref{NJ}(ii) and (v). \end{proof}
Recall the notation introduced in ~\ref{o4} and ~\ref{type}.
\begin{lemma} The sets $\mathcal P(J)$ and $\mathcal M(J)$ are fused in $L$ if $L > L_*$ and we have $|\mathcal S(J)|= 16$, $|\mathcal P(J)|=|\mathcal M(J)|= 12$. \end{lemma}
\begin{proof} This follows directly from Lemma~\ref{NJ}. \end{proof}
\begin{lemma}\label{I1} For $i=1,2$, $C_L(r_i) = C_{L_*}(r_i)$, $[J,r_i]= \langle \rho_i\rangle$, $|C_J(r_i)|=3^3$ and $C_L(r_i)/C_J(r_i) \langle r_i\rangle \cong \GO_3(3) \cong 2 \times \mathrm{Sym}(4)$. \end{lemma}
\begin{proof} We have that $|C_S(r_i)| = 3^4$ and $r_i$ inverts $Q_iJ/J$. Hence $|C_J(r_i)|= 3^3$. It follows that both $r_1$ and $r_2$ are reflections on $J$. If $L> L_*$, then $r_1^t= r_2$ and so $C_L(r_i) = C_{L_*}(r_i)$. Since $r_1$ and $r_2$ are reflections and since $L_*/J \cong \GO_4^+(3)$ by Lemma~\ref{NJ}, we have $C_L(r_i)/C_J(r_i) \langle r_i\rangle \cong \GO_3(3) \cong 2 \times \mathrm{Sym}(4)$. \end{proof}
From Lemma~\ref{I1} we have $[J,r_1] = \langle \rho_1\rangle$ and $[J,r_2] =\langle \rho_2\rangle$ are non-singular 1-dimensional spaces in $J$. We fix notation so that $\langle \rho_1\rangle \in \mathcal P(J)$ and $\langle \rho_2\rangle \in \mathcal M(J)$.
\begin{lemma}\label{type1} The following hold: \begin{enumerate} \item $V$ and $V_1$ are of Type S; \item $A_1$ is of Type DP; \item $A_2$ is of Type DM; \item $\langle \rho_1,\rho_2 \rangle$ is of type $N+$;
\item $|\mathcal S(C_J(r_1))|=4$, $|\mathcal M(C_J(r_1))|=6$ and $|\mathcal P(C_J(r_1))|=3$; and
\item $|\mathcal S(C_J(r_2))|=4$, $|\mathcal M(C_J(r_2))|=3$ and $|\mathcal P(C_J(r_2))|=6$. \end{enumerate} \end{lemma}
\begin{proof} Parts (i)--(iv) are obvious. By Lemma~\ref{I1} we have that $|C_J(r_i)| = 3^3$ for $i = 1,2$. Since $J$ is a quadratic space of plus type, it follows that $C_J(r_1)$ has an orthonormal basis consisting of members of $\mathcal P(J)$ and $C_J(r_2)$ has an orthonormal basis consisting of elements of $\mathcal M(J)$. Thus (v) and (vi) hold. \end{proof}
\begin{lemma}\label{I2} If $\wt {\rho_{i}} \in C_J(r_i)$ is $L_*$-conjugate to $\rho_i$, then $\langle \rho_i, \wt \rho_i\rangle$
has Type N-. In particular, $|\mathcal P (\langle \rho_i, \wt \rho_i\rangle)|=|\mathcal M(\langle \rho_i, \wt
\rho_i\rangle)|=2 $. \end{lemma}
\begin{proof} Suppose that $\wt {\rho_{i}} \in C_J(r_i)$ is $L_*$-conjugate to $\langle \rho_i\rangle$. Then, as $\langle \rho_i\rangle= [J,r_i]$, $\rho_i$ is perpendicular to $C_J(r_i)$. It follows that $\wt {\rho_{i}} $ is perpendicular to $\rho_i$ and this means that $ \langle \rho_i, \wt \rho_i\rangle$ is of Type N-. \end{proof}
\section{Two $3$-centralizers}
In this section we determine the structure of $C_G(\rho_1)$ and $C_G(\rho_2)$. We first show that these centralisers do not have non trivial normal $3^\prime$-subgroups. Recall the notation of \ref{nota} and that $f\in M$ is an involution inverting $Z$.
\begin{lemma}\label{JSig} $J$ does not normalize any non-trivial $3'$-subgroups. \end{lemma}
\begin{proof} Suppose that $Y$ is a non-trivial $3'$-subgroup normalized by $J$. Then, as every subgroup of $J$ of order $27$ contains a conjugate of $Z$ by Lemma~\ref{hyper}, we may assume that $X= C_{Y}(Z)\neq 1$. As $X$ is normalized by $A = J \cap Q$ and $X$ normalizes $Q$, $[A,X] \le Q \cap X=1$ and hence $X \le C_M(A) = J $ as $A$ is a maximal abelian subgroup of $Q$. But then $X=1$ which is a contradiction. This proves the lemma. \end{proof}
\begin{lemma}\label{princeprep} For $i=1,2$, $C_M(\rho_i) = Q_{3-i}R_{3-i}J\langle fr_i \rangle$ and $C_{C_M(Z)}(\rho_i)/\langle \rho_i \rangle$ is isomorphic to the centralizer of a non-trivial $3$-central element in $\PSp_4(3)$ and $Z$ is inverted in $C_M(\rho_i)$. \end{lemma}
\begin{proof} Since $\rho_i \in A_i\le J$ and since $[Q_1,Q_2]=1$ and $[Q_i,R_{3-i}]=1$, we certainly have $C_M(\rho_i) \ge Q_{3-i}R_{3-i}J$. Furthermore, $f$ inverts $J$ and so $f$ inverts $\rho_i$ and as $r_i$ also inverts $\rho_i$, we have $C_M(\rho_i) \ge Q_{3-i}R_{3-i}J\langle fr_i \rangle$ which has index either $24$ or $48$ in $M$ dependent upon whether or not $M = RS\langle f \rangle$ respectively. Since $Q_i$ contains twelve $Q$-conjugates of $\langle \rho_i\rangle$, Lemma~\ref{fusion3elts} implies $C_M(\rho_i) = Q_{3-i}R_{3-i}J\langle fr_i \rangle$.
Because $r_if$ inverts $Z$, we have $C_{C_M(Z)}(\rho_i)/\langle \rho_i\rangle= Q_{3-i}R_{3-i}J/\langle \rho_i\rangle$ with $R_{3-i}$ acting faithfully on $Q_{3-i}$. Thus the final statement also is valid by Lemma~\ref{cen3psp43}. \end{proof}
In the next two lemmas we pin down two possible structures of $C_G(\rho_1)$ and $C_G(\rho_2)$. In fact in $\F_4(2)$ we have that both are isomorphic to $3 \times \mathrm{Sp}_6(2)$. That this is the case in our group will be proved later in Lemma~\ref{ItsSp62}.
\begin{lemma}\label{eitheror} For $i=1,2 $ either $C_G(\rho_i) \cong 3 \times \mathrm{Aut}(\SU_4(2))$ or $C_G(\rho_i) \cong 3 \times \mathrm{Sp}_6(2)$. Furthermore, $r_i$ inverts $\rho_i$ and centralizes $C_G(\rho_i)/\langle \rho_i\rangle$. \end{lemma}
\begin{proof} We consider $C_G(\rho_i)/\langle \rho_i\rangle$. By Lemma~\ref{princeprep}, $C_{C_M(Z)}(\rho_i)/\langle \rho_i\rangle$ is isomorphic to
a $3$-centralizer in $\PSp_4(3)$. Since $J/\langle \rho_i\rangle$ normalizes
no non-trivial $3'$-subgroup of $C_G(\rho_i)$ by Lemma~\ref{JSig} and $Z$ is inverted by $fr_i$, we may apply Theorem~\ref{PrinceThm} to obtain $C_G(\rho_i)/\langle \rho_i\rangle \cong \mathrm{Aut}(\SU_4(2))$ or $\mathrm{Sp}_6(2)$ or that $C_G(\rho_i) = C_M(\rho_i)$. The latter possibility is dismissed as $C_{L}(\rho_i)$ has index $2$ in $\langle \rho_i\rangle C_{L_*}(r_i) $ and so, by Lemma~\ref{I1}, $$C_{L}(\rho_i)\cong 3 \times 3^3:(2 \times \mathrm{Sym}(4))$$ does not normalize $Z$.
The Sylow $3$-subgroup of $C_G(\rho_i) $ is $\langle \rho_i\rangle \times Q_{3-i}C_J(r_i)$ and hence the extension $C_G(\rho_i)/\langle \rho_i\rangle$ splits by Gasch\"utz Theorem. Finally we have that $r_i$ centralizes $Q_{3-i}J/\langle \rho_i\rangle$ and, as no automorphism of either $\mathrm{Aut}(\SU_4(2))$ or $\mathrm{Sp}_6(2)$ of order $2$ centralizes such a subgroup, we infer that $r_i$ centralizes $C_G(\rho_i)/\langle \rho_i\rangle$ and of course we also know that $\rho_i$ is inverted by $r_i$. \end{proof}
\begin{lemma}\label{thesame} We have $C_G(\rho_1) \cong C_G(\rho_2)$. \end{lemma}
\begin{proof} By Lemma \ref{eitheror}, $C_G(\rho_1)/\langle \rho_1 \rangle \cong \mathrm{Sp}_6(2)$ or $\mathrm{Aut}(\SU_4(2))$.
Assume that $C_G(\rho_1)/\langle \rho_1 \rangle \cong \mathrm{Sp}_6(2)$. Using Lemma~\ref{type1} (v), we have some $\wt {\rho_1} \in \mathcal P(C_J(\rho_1))$ and as $|\mathcal P(C_J(\rho_1))| = 3$, $C_{E(C_G(\rho_1))}(\wt {\rho_1}) \cong 3 \times \mathrm{Sp}_4(2)$ from the structure of $\mathrm{Sp}_6(2)$. Therefore $E(C_G(\langle \rho_1, \wt{\rho_1} \rangle)) \cong \mathrm{Sp}_4(2)'$. Lemma~\ref{I2}, yields that $\mathrm{Sp}_4(2)'$ is involved in the centralizer of a $3$-element in $C_G(\rho_2)$. As there are no such $3$-elements in $\SU_4(2)$ \cite{Atlas}, Lemma \ref{eitheror} implies $E(C_G(\rho_2))/\langle \rho_2 \rangle \cong \mathrm{Sp}_6(2)$. Hence Lemma~\ref{thesame} holds. \end{proof}
\section{Building a signalizer in the centralizers of $r_1$ and $r_2$}
In this section we begin the construction $K_i= C_G(r_i)$ for $i=1,2$. We give a brief overview of our plans for $i = 1$ to guide the reader through the technicalities involved. Our final aim is to show that $K_1$ is similar to a $2$-centralizer in a group of type $\F_4(2)$ (see Definition~\ref{F4cent}). Hence we aim to show that $K_1$ is an extension of a 2-group by $\mathrm{Sp}_6(2)$. Furthermore this $2$-group is a product of an extraspecial group of order $2^9$ by an elementary abelian group. Our first aim is to construct the extraspecial group $\Sigma_1$, and show that it is normalized by $C_L(r_1)$. Note that $C_J(r_1)\le C_L(r_1)$ and the former group is elementary abelian of order $3^3$.
We briefly consider the situation in our target group. In $\F_4(2)$ there are exactly four maximal subgroups of $C_J(r_1)$ with centralizers in $\Sigma_1$ which properly contain $\langle r_1\rangle$ and these maximal subgroups centralize a quaternion group of order eight in $\Sigma_1$. In our group $G$, the first problem is to find these quaternion groups. For this we pick a set of four maximal subgroups of $C_J(r_1)$, which are conjugate to $A_2$. They all contain a conjugate of $\rho_2$. By Lemma~\ref{eitheror} there are exactly two possibilities for the structure of $C_G(\rho_2)$. Examining these structures shows $C_{C_G(\rho_2)}(A_2)/\langle \rho_2 \rangle \cong 3^{1+2}_+{:}\mathrm{SL}_2(3)$. Hence $C_{C_G(\rho_2) \cap C_G(r_1)}(A_2)/\langle \rho_2 \rangle \cong \mathrm{SL}_2(3)$. This shows that $O_2(C_{C_G(\rho_2) \cap C_G(r_1)}(A_2)) \cong \mathrm{Q}_8$, and this is one of the quaternion groups we are looking for. As $A_2$ has four conjugates under $C_L(r_1)$, we now get a set of four quaternion groups. The problem is to show these four quaternion groups generate a 2-group $ \Sigma_1$ which is extraspecial of order $2^9$. This will be done in Lemma~\ref{sigmai}. Furthermore, the very construction guarantees us that $C_L(r_1)$ acts on $\Sigma_1$.
We continue to use the notation from \ref{o4}, \ref{type} and \ref{nota}. Additionally we introduce \begin{notation}\label{notar} For $i=1,2$, $I_i = C_J(r_i)$ and $F_i = C_L(r_i)$. \end{notation}
Notice that by Lemma~\ref{I1}, $F_i$ acts on $I_i$ and $F_i/I_i \langle r_i\rangle\cong 2 \times \mathrm{Sym}(4)$. As explained above we intend to determine a large signalizer for $I_i$, that is a $3^\prime$-group which is normalized by $I_i$. We begin with two easy observations.
\begin{lemma}\label{CSr} For $i=1,2$, $C_{C_M(Z)}(r_i) = Q_{3-i}R_1R_2I_i$ and $C_S(r_i) = Q_{3-i}I_i\in \Syl_3(C_M(r_i)) \subseteq \Syl_3(K_i)$. \end{lemma}
\begin{proof} Obviously $C_{C_M(Z)}(r_i) \ge Q_{3-i}R_1R_2C_J(r_i)$ and then Lemma~\ref{structM} (ii) implies the equality. Therefore, $C_S(r_i) = Q_{3-i}I_i \in \syl_3(C_M(r_i))$ and $Z(C_S(r_i))= Z$. Thus $N_{K_i}(C_S(r_i)) \le N_G(Z)=M$. In particular, $C_S(r_i) \in \syl_3(K_i)$. \end{proof}
\begin{lemma}\label{fusionr1r2} We have $r_1$ is $G$-conjugate to $r_2$ if and only if $r_1$ is $M$-conjugate to $r_2$. \end{lemma}
\begin{proof} Obviously if $r_1$ and $r_2$ are conjugate in $M$ then they are conjugate in $G$. Suppose then that $r_1=r_2^g$ for some $g\in G$. By Lemma~\ref{CSr}, for $i=1,2$, $C_S(r_i) \in \Syl_3(C_G(r_i))$ and $Z = Z(C_S(r_i))$. Since $r_1= r_2^g$, $C_S(r_2)^g \in \syl_3(C_G(r_1))$. Thus there is $h \in C_G(r_1)$ such that $C_S(r_2)^{gh} = C_S(r_1)$. But then $$Z^{gh}= Z(C_S(r_2))^{gh} = Z(C_S(r_1))=Z$$ which means that $gh \in M$. Hence $r_1$ and $r_2$ are $M$-conjugate. \end{proof}
Recall, for $i=1,2$, $$I_i=C_J(r_i)=J \cap E(C_G(\rho_i))$$ as, by Lemma \ref{eitheror}, $ E(C_G(\rho_i)) = C_{C_G(\rho_i)}(r_i)$.
\begin{lemma}\label{crossover} Suppose that $\wt \rho_1 \in \mathcal P(I_1)$ and $\wt \rho_2 \in \mathcal M(I_2)$. Then, for $i=1,2$, in $E(C_G(\wt \rho_i))\langle r_i\rangle$, $r_i$ is an involution which has $\mathrm{Sp}_4(2)'$ as a composition factor of its centralizer. Moreover, $I_i \cap E(C_G(\wt \rho_i))$ is of Type N-. \end{lemma}
\begin{proof} For $i=1,2$, the definition of $I_i$, yields $r_i \in C_G(\wt \rho_i)$. Now $r_i$ normalizes $E(C_G(\wt \rho_i))$ and centralizes $I_i \cap E(C_G(\wt \rho_i))$ which has order $9$.
On the other hand, in $C_G(\rho_i)$, as there are only three conjugates of $\langle \wt{\rho_i}\rangle$ in $I_i$ by Lemma~\ref{type1}(v) and (vi), we have that $$C_{E(C_G(\rho_i))}(\wt \rho_i)\approx 3 \times 3^2.\mathrm{Dih}(8)$$ if $E(C_G(\rho_i)) \cong \SU_4(2)$ and $$C_{E(C_G(\rho_i))}(\wt \rho_i)\approx 3 \times \mathrm{Sp}_4(2) $$ if $E(C_G(\rho_i)) \cong \mathrm{Sp}_6(2)$. As $I_i \le E(C_G(\rho_i))$, it follows that $$I_i \cap [I_i,C_{E(C_G(\rho_i))}(\wt \rho_i)]$$ is of Type N-. Now deploying Lemmas~\ref{sp62facts} and \ref{sp62line} (ii), $C_{E(C_G(\wt \rho_i))}(r_i) \cong \mathrm{Sp}_4(2)$ if $E(C_G(\wt \rho_i)) \cong \SU_4(2)$ and has shape $2^5.\mathrm{Sp}_4(2)$ when $E(C_G(\wt \rho_i))\cong \mathrm{Sp}_6(2)$. In particular, the main claim in the lemma is true. We have already observed that $I_i \cap [I_i,C_{E(C_G(\rho_i))}(\wt \rho_i)]$ has Type N- and as this group is $I_i \cap E(C_G(\wt \rho_i))$ we have the last part of the lemma. \end{proof}
We can now locate the four maximal subgroups of $I_i$, whose centralizers contain the quaternion groups we are looking for.
Recall that, for $i=1,2$, $A_{3-i}= A \cap Q_{3-i}$ is a hyperplane of $I_i$ which with respect to the quadratic form on $J$ is a degenerate $2$-dimensional subspace which contains one conjugate of $Z$ and three conjugates of $\langle \rho_{i}\rangle$. Therefore $A_{1}$ has Type DP and has $A_2$ Type DM in the sense of Notation~\ref{type}. Consequently the set $A_{3-i}^{F_i}$ has order $4$. We let the four $F_i$-conjugates of $A_{3-i}$ be $I_i^1=A_{3-i}$, $I_i^2$, $I_i^3$ and $I_i^4$. Then, for $1\le j < k \le 4$, we have $I_i^j \cap I_i^k $ is an $M$-conjugate of $\langle \rho_{3-i}\rangle$. We further select notation so that $$I_i ^1 \cap I_i^2 = \langle \rho_{3-i}\rangle.$$ The next lemma follows immediately from the 2-transitive action of $F_i$ on the set $\{I_i^1,I_i^2,I_i^3,I_i^4\}$.
\begin{lemma}\label{IijIik} For $1 \le l \le 4$ and $1 \le j < k \le 4$ we have \begin{enumerate} \item $I_1^l$ has Type DM and $I_1^j\cap I_1^k\in \mathcal M(I_1)$; and \item $I_2^l $ has Type DP and $I_2^j\cap I_2^k\in \mathcal P(I_2)$.\end{enumerate}\qed \end{lemma}
With these comments we have the following lemma directly from Lemmas~\ref{eitheror} and \ref{thesame}.
\begin{lemma}\label{Cij} For $i=1,2$ and for $1\le j< k\le 4$, we have $$C_G(I_i^k\cap I_i^j) \cong 3 \times \mathrm{Aut}(\SU_4(2)) \text{ or } 3 \times \mathrm{Sp}_6(2).$$ Furthermore, the isomorphism type of $C_G(I_i^k\cap I_i^j)$ does not depend on $i$, $j$ or $k$.\end{lemma}
Recall the Type N+ subgroups of order $9$ are just the non-degenerate subgroups of $J$ of plus type. \begin{lemma}\label{I1capI2} $I_1 \cap I_2$ is of Type N+. \end{lemma}
\begin{proof} We know that $I_1 \cap I_2 = C_J(\langle r_1,r_2\rangle)$ and is consequently non-degenerate. Since $Z \le I_1 \cap I_2$, it has Type N+. \end{proof}
The next lemma is an adaption of Lemma~\ref{NJ}(ii) to $K_i$.
\begin{lemma}\label{fusion1} $F_i = N_{K_i}(I_i)$ controls $K_i$-fusion of elements in $I_i$. \end{lemma}
\begin{proof} By Lemma~\ref{CSr}, $C_S(r_i)\in \syl_3(K_i)$ and thus $I_i $ is the Thompson subgroup of $C_S(r_i)$ and is elementary abelian. It follows from \cite[37.6]{AschbacherFG} that $N_{K_i}(I_i)$ controls fusion in $I_i$. As $C_G(I_i) \le M$, we calculate that $C_G(I_i)= J\langle r_i\rangle$. Hence $C_{K_i}(I_i)= I_i\langle r_i\rangle$ and $N_{K_i}(I_i) = L \cap K_i = F_i$. \end{proof}
For $i \in \{1,2\}$ and $1\le j < k\le 4$, $$E_i^{j,k} = E(C_G(I_i^j\cap I_i^k)).$$ So $E_i^{j,k} \cong \SU_4(2)$ or $\mathrm{Sp}_6(2)$ and we note again that the isomorphism type of this group does not depend on $i,j$ or $k$. At least one potential avenue for confusion is caused by this notation so please note that $E_i^{j,k}$ does not centralize $r_i$. Rather it centralizes a conjugate of $r_{3-i}$. Indeed $E_1^{1,2}= E(C_G(\rho_2))$ centralizes $r_2$ and $E_2^{1,2}= E(C_G(\rho_1))$ centralizes $r_1$ by Lemma~\ref{eitheror}.
Notice that $I_i$ is centralized by $r_i$ and so $r_i$ is contained in $C_G(I_i^j\cap I_i^k)$ and it centralizes $I_i \cap E_i^{j,k}$ and this contains $Z$. It follows that $I_i \cap E_i^{j,k}$ is of Type N+ as it must also be non-degenerate. This means that $r_i$ acts as an involution of type $a_2$ on $E_i^{j,k}$ in the sense of Table 1. Therefore, Lemma~\ref{sp62facts}(ii) gives the following result: \begin{lemma}\label{cri} We have \begin{eqnarray*}C_{K_i}(I_{i}^j \cap I_i^k) &=& C_{C_G(I_i^j \cap I_i^k)}(r_i)\\& \approx& \begin{cases}3 \times 2^{1+4}_+.(\mathrm{Sym}(3) \times \mathrm{Sym}(3))&E_i^{j,k} \cong \SU_4(2) \\3\times 2^{1+2+4}.(\mathrm{Sym}(3) \times \mathrm{Sym}(3))&E_i^{j,k} \cong \mathrm{Sp}_6(2)\end{cases}.\end{eqnarray*}\qed \end{lemma}
The next lemma now is the key. It shows that the groups $O_2(C_{K_i}(I_j^{i}))$ are quaternion groups of order eight which pairwise commute and so generate an extraspecial group of order $2^9$.
\begin{lemma}\label{Sigmapieces} Assume that $i=1,2$ and $1\le j< k\le 4$. \begin{enumerate} \item For $m \in \{j,k\}$, $I_i^m \cap E_i^{j,k}$ is a $3$-central element of $G$ and of $E_{i}^{j,k}$; \item $C_{G}(I_i^k) = (I_i^k\cap I_i^j) \times C_{E_i^{j,k}}(I_k \cap E_i^{j,k}) \approx 3 \times 3^{1+2}_+.\mathrm{SL}_2(3)$; \item \begin{enumerate}\item $O_2(C_{K_i}(I_i^j)) \cong O_2(C_{K_i}(I_i^k)) \cong \mathrm{Q}_8$; \item $O_2(C_{K_i}(I_i^j))O_2(C_{K_i}(I_i^k)) \le O_2(C_{K_i}(I_i^j\cap I_i^k))$ with equality if $E_i^{j,k} \cong \SU_4(2)$; and \item $[O_2(C_{K_i}(I_i^j)), O_2(C_{K_i}(I_i^k))]=1$; and\end{enumerate} \item $C_{I_i}(O_2(C_{K_i}(I_i^j))O_2(C_{K_i}(I_i^k)))= I_i ^j\cap I_i^k$. \end{enumerate} \end{lemma}
\begin{proof} It suffices to prove part (i) for $I_i^1 $ as then the result will follow by conjugating by $F_i$
So consider $I_i^1\cap I_i^2 = \langle \rho_{3-i}\rangle$. Then, by Lemma~\ref{princeprep}, $C_S(\rho_{3-i})= Q_iJ$ and $C_S(\rho_{3-i})' \cap Z(C_S(\rho_{3-i}))= Z$. Thus $Z \le I_i^1 \cap E_{i}^{1,j}$ is $3$-central in $G$ and in $E_{i}^{1,j}$. Part (i) follows as $F_i$ acts 2-transitively on $\{I_i^j\mid 1 \le j \le 4\}$.
Part (ii) follows from (i) as the centralizer of a $3$-central element in $\mathrm{Sp}_6(2)$ and $\SU_4(2)$ has shape $3^{1+2}_+.\mathrm{SL}_2(3)$.
To deduce part (iii), we first note that $$O_2(C_{K_i}(I_i^k)) \cong O_2(C_{K_i}(I_i^{j})) \cong \mathrm{Q}_8$$ follows from (ii) as $r_i$ is an involution in $C_G(I_i^{k})$. We have $l \in \{j,k\}$, $O_2(C_{K_i}(I_i^l)) \le C_{K_i}(I_{i}^j \cap I_i^k) $ and is normalized by $I_i^jI_i^k= I_i$. Since \begin{eqnarray*}C_{K_i}(I_{i}^j \cap I_i^k) &=& C_{C_G(I_i^j \cap I_i^k)}(r_i)\\& \approx& \begin{cases}3 \times 2^{1+4}_+.(\mathrm{Sym}(3) \times \mathrm{Sym}(3))&E_i^{j,k} \cong \SU_4(2) \\3\times 2^{1+2+4}.(\mathrm{Sym}(3) \times \mathrm{Sym}(3))&E_i^{j,k} \cong \mathrm{Sp}_6(2)\end{cases}\end{eqnarray*}by Lemma~\ref{cri}, it follows that $O_2(C_{K_i}(I_i^l)) \le O_2(C_{K_i}(I_{i}^j \cap I_i^k))$. Now we apply Lemma~\ref{sp62line}(iii) to see that $[O_2(C_{K_i}(I_i^k)),O_2(C_{K_i}(I_i^k))]=1$. (Recall that $O_2(C_{\SU_4(2)}(r_i)) \le O_2(C_{\mathrm{Sp}_6(2)}(r_i))$.)
Part (iv) follows as $I_i \cap E_{i}^{j,k}$ acts faithfully on $O_2(C_{K_i}(I_i^j))O_2(C_{K_i}(I_i^k))$.
\end{proof} We now introduce some further notation
\begin{notation}\label{sigma} For $i = 1,2$, $1\le k \le 4$, $$\Sigma_i^k= O_2(C_{K_i}(I_i^k))\cong \mathrm{Q}_8$$ and $$\Sigma_i= \langle \Sigma_i^k\mid 1\le k \le 4\rangle=\langle O_2(C_{K_i}(I_i^k)) \mid 1 \le k \le 4\rangle.$$ \end{notation} Note that $\Sigma_1^1= O_2(C_{K_1}(A_2)) = R_1$ and $\Sigma_2^1= O_2(C_{K_2}(A_1)) = R_2$.
\begin{lemma}\label{sigmai} We have $\Sigma_i$ is extraspecial of order $2^9$ and plus type, $Z(\Sigma_i)= \langle r_i\rangle$ and $F_i/\langle r_i\rangle$ acts faithfully on $\Sigma_i$. \end{lemma} \begin{proof} The structure of $\Sigma_i$ follows from Lemma~\ref{Sigmapieces} (iii) as the generating subgroups commute pairwise. To see the last part is suffices to show that $I_i$ acts faithfully on $\Sigma_i$ as every normal subgroup of $F_i$ which strictly contains $\langle r_i\rangle$ contains $I_i$. Using Lemma~\ref{Sigmapieces} (iv) we see that $C_{I_i}(\Sigma_i) = \bigcap_ {j=1}^4I_i^j= 1$.
\end{proof}
At this stage we have constructed the extraspecial group of order $2^9$ on which $F_i$ acts.
\begin{lemma} \label{celts} The following hold: \begin{enumerate} \item $C_{\Sigma_1}(Z)=R_1$, $C_{\Sigma_1}(I_1^j \cap I_1^k) =\Sigma_1^j\Sigma_1^k$ and, if $\langle x \rangle \in \mathcal P(I_1)$, then $C_{\Sigma_1}(x) =\langle r_1\rangle$. \item $C_{\Sigma_2}(Z)=R_2$, $C_{\Sigma_2}(I_2^j \cap I_2^k) =\Sigma_2^j\Sigma_2^k$ and, if $\langle x \rangle \in \mathcal M(I_2)$, then $C_{\Sigma_2}(x) = \langle r_2\rangle$. \end{enumerate} \end{lemma}
\begin{proof} We prove (i) the proof of (ii) being the same. Let $1 \le j \le 4$. We know that $\Sigma_1= \Sigma_1^1\Sigma_1^2\Sigma_1^3\Sigma_1^4$. Since $I_1$ acts faithfully on $\Sigma_1$, we have that $C_{I_1}(\Sigma_1^j) = I_1^j$. Thus the elements of $\mathcal P(I_1)$ act non-trivially on each $\Sigma_1^j$ and so $C_{\Sigma_1}(x) =\langle r_1\rangle$ for $\langle x \rangle \in \mathcal P(I_1)$. Since we know that $Z$ centralizes exactly $R_1= \Sigma_1^1$ on $\Sigma_1$ we now have that (i) holds. \end{proof}
\section{The structure of $C_G(\rho_1)$}
We continue to use our standard notation. In this section we are going to show that $C_G(\rho_1)$ is isomorphic to the corresponding centralizer in $\F_4(2)$. So our aim is to show that $C_{G}(\rho_1) \cong 3 \times \mathrm{Sp}_6(2)$. By Lemma~\ref{eitheror} we have that $C_G(\rho_1)$ either is as in $\F_4(2)$ or is isomorphic to $3 \times \mathrm{Aut}(\SU_4(2))$. We will show the latter case yields a contradiction.
\begin{lemma}\label{uniqueAisig1} Suppose that $C_G(\rho_i) \cong 3 \times \mathrm{Aut}(\SU_4(2))$. Then $\Sigma_i$ is the unique maximal signalizer for $I_i^1$ in $K_i$. \end{lemma}
\begin{proof} We simplify our notation by assuming that $i=1$. The argument for $i=2$ is the same. Notice that $$\{I_1^1 \cap I_1^j\mid 2\le j\le 4\}= \mathcal M(I_i^1) .$$ The only other proper subgroup of $I_1^1$ is $Z$ by Lemma~\ref{IijIik}. Hence, as $E_1^{1,j} \cong \SU_4(2)$ by assumption, Lemma~\ref{Sigmapieces} (iii)(b) implies that $$\Sigma_1 \ge O_2(C_{K_1}(I_1^k\cap I_1^j))= O_{3'}(C_{K_1}(I_1^k\cap I_1^j)).$$
Suppose that $\Theta$ is a signalizer for $I_1^1$.
Then $$\Theta= \langle C_\Theta(a)\mid a\in I_1^{1\#}\rangle.$$ However, $$C_\Theta (Z)\le O_{3'}(M\cap K_1)= R_1\le \Sigma_1$$ and, for $1<j \le 4$, by Lemma~\ref{cri} and Lemma~\ref{celts} $$C_{\Theta}(I_1^1\cap I_1^j) \le O_{3'}(C_{K_i}(I_1^1\cap I_1^j))= \Sigma_1^1\Sigma_1^j\le \Sigma_1.$$ Hence $\Theta\le \Sigma_1$. \end{proof}
The next lemma puts us firmly on the track of $\F_4(2)$ and $\mathrm{Aut}(\F_4(2))$.
\begin{lemma}\label{ItsSp62} We have $C_G(\rho_1) \cong C_G(\rho_2) \cong 3 \times \mathrm{Sp}_6(2)$. \end{lemma}
\begin{proof} Suppose that the lemma is false. Then by Lemmas~\ref{eitheror} and \ref{thesame} $$C_G(\rho_1) \cong C_G(\rho_2) \cong 3 \times \mathrm{Aut}(\SU_4(2)).$$ We claim that, for $i=1,2$, $\Sigma_i$ is self-centralizing in $K_i$. Let $W_i= C_G(\Sigma_i)$. Then $W_i \le K_i$ and, as $C_S(r_i) \in \Syl_3(K_i)$ by Lemma~\ref{CSr} and since this group acts faithfully on $\Sigma_i$ by Lemma~\ref{sigmai}, we have that $W_i$ is a $3'$-group which is normalized by $I_i^1$. By Lemma~\ref{uniqueAisig1}, $\Sigma_i$ is the unique maximal signalizer for $I_i^1$ and hence $\Sigma_i \ge W_i$.
Since $\Sigma_i$ is the unique maximal signalizer for $I_i^1$ in $K_i$ it is also the unique maximal signalizer of $Q_{3-i}\ge I_i^1$ and $I_i\ge I_i^1$ in $K_i$. It follows that $N_G(\Sigma_i) \ge \langle F_i, C_M(r_i)\rangle $ as $Q_{3-i} $ is a normal subgroup of $C_M(r_i)$. Now $$C_M(r_i)\Sigma_i/\Sigma_i = I_iQ_{3-i}R_{3-i}\langle f \rangle \Sigma_i/\Sigma_i$$ as $R_i \le \Sigma_i$. We now deduce $C_{C_M(Z)}(r_i)\Sigma_i/\Sigma_i $ is isomorphic to a $3$-centralizer in $\PSp_4(3)$. Furthermore, as $\Sigma_i$ is the unique maximal signalizer for $I_i$ in $K_i$, we have that $I_i$ does not normalize any non-trivial $3'$-subgroup of $N_G(\Sigma_i)/\Sigma_i$ and $f$ inverts $Z$. Therefore, since $F_i \le N_G(\Sigma_i)$, Prince's Theorem~\ref{PrinceThm} yields $$N_G(\Sigma_i)/\Sigma_i \cong \mathrm{Aut}(\SU_4(2))\text{ or } \mathrm{Sp}_6(2).$$ Observe that $N_G(\Sigma_i) \ge \langle F_i,C_M(r_i)\rangle \ge E(C_G(\rho_{i}))$.
We claim $N_G(\Sigma_i) = K_i$. To prove this we intend to apply Theorem~\ref{closed} to $K_i/\langle r_i\rangle$. We have already verified hypotheses (i) and (ii) of that theorem.
As $N_G(\Sigma_i)/\Sigma_i \cong \mathrm{Aut}(\SU_4(2))\text{ or } \mathrm{Sp}_6(2)$, every element of $C_S(r_i)\Sigma_i/\Sigma_i$ is $N_G(\Sigma_i)/\Sigma_i$-conjugate to an element of $I_i \Sigma_i/\Sigma_i= J(C_S(r_i))\Sigma_i/\Sigma_i$ the Thompson subgroup of $C_S(r_i)\Sigma_i/\Sigma_i$. Since $F_i$ controls fusion in $I_i$ by Lemma~\ref{fusion1}, we also have hypothesis (iii) of Theorem~\ref{closed}.
Again to simplify notation, assume that $i=1$. Suppose that $d $ is an element of order $3$ with $ d\in N_G(\Sigma_1) \cap N_G(\Sigma_1)^h $ for some $h \in K_1$ is such that $C_{\Sigma_1}(d) \neq \langle r_1\rangle$. Then, by Lemma~\ref{celts} (i), we may suppose that $\langle d \rangle = Z$ or $\langle d \rangle = I_1^1\cap I_1^2= \langle \rho_2\rangle$. Then, as $N_{K_1}(Z)=C_{M}(r_1) \le N_G(\Sigma_1)$ and $C_{K_1}(\rho_2)=C_{C_G(\rho_{2})}(r_1) \le N_G(\Sigma_1)$, we deduce $$C_{K_1}(d) \le N_G(\Sigma_1).$$ On the other hand, $C_{{N_G(\Sigma_1)}^h}(d)$ contains a $K_1$-conjugate $X$ of $I_1$. Since $X \le C_{K_1}(d) \le N_G(\Sigma_1)$, we may suppose that ${N_G(\Sigma_1)} \cap {N_G(\Sigma_1)}^h \ge I_1$. But then $\Sigma_1 = \Sigma_1^h$ and $N_G(\Sigma_1)= {N_G(\Sigma_1)}^h$ as $\Sigma_1$ is the unique maximal signalizer for $I_1$ in $K_1$ by Lemma~\ref{uniqueAisig1}. Thus the hypothesis of Theorem~\ref{closed} fulfilled and therefore $K_1=N_G(\Sigma_1)$.
Suppose that $N_G(\Sigma_1)/\Sigma_1\cong \mathrm{Aut}(\SU_4(2))$. Let $\wt \rho_1 \in \mathcal P(I_1)$. Then, as $|\mathcal P(I_1)|=3$, $$C_{N_G(\Sigma_1)/\Sigma_1} (\wt {\rho_1}\Sigma_1) \cong 3^3.\mathrm{Dih}(8) $$ by Lemma~\ref{type1} (v). On the other hand, by Lemma~\ref{crossover} this group is non-soluble which is a contradiction. We conclude that $N_G(\Sigma_1)/\Sigma_1 \cong \mathrm{Sp}_6(2)$. Repeating the arguments for $N_G(\Sigma_2)$ yields $N_G(\Sigma_2)/\Sigma_2 \cong \mathrm{Sp}_6(2)$. Furthermore, the elements from $\mathcal P(I_1)$ act fixed point freely on $\Sigma_1/\langle r_1\rangle$ and the elements of $\mathcal M(I_2)$ act fixed point freely on $\Sigma_2/\langle r_2\rangle$, in both cases, $i=1,2$, $\Sigma_i/\langle r_i\rangle$ is the spin module for $N_G(\Sigma_i)/\Sigma_i$.
Since $r_2$ commutes with $I_1\cap I_2 \le N_G(\Sigma_1)$ which has Type N+ by Lemma~\ref{I1capI2}, Table~\ref{Table1} indicates that $r_2$ acts as a unitary transvection on $\Sigma_1/\langle r_1\rangle$. Therefore $|C_{\Sigma_1/\langle r_1\rangle }(r_2)|= 2^6$ and
$$2^6 \le |C_{\Sigma_1}(r_2)|\le 2^7.$$ Since $\langle r_1, r_2 \rangle$ is centralized by $I_1\cap I_2$, $C_{\Sigma_1}(r_2)$ is $(I_1\cap I_2)$-invariant. Because the elements of $\mathcal P(I_1\cap I_2)$ act fixed point freely on $\Sigma_1/\langle r_1\rangle$ (see Lemma~\ref{sp62spin}) we infer that
$|C_{\Sigma_1}(r_2)|=2^7$. Now, as $K_i=N_G(\Sigma_i)$ for $i=1,2$, $C_{\Sigma_1}(r_2)$ normalizes $C_{\Sigma_2}(r_1)$ and vice versa, and so $$[C_{\Sigma_1}(r_2), C_{\Sigma_2}(r_1)] \le \Sigma_1 \cap \Sigma_2.$$ Since $r_1 \not \in \Sigma_2$ and $r_2 \not \in \Sigma_1$, $\Sigma_1 \cap \Sigma_2$ is abelian and is centralized by $C_{\Sigma_1}(r_2)C_{\Sigma_2}(r_1)$. In particular, $\Sigma_1 \cap \Sigma_2 \le Z(C_{\Sigma_1}(r_2))$. Thus, as
$|C_{\Sigma_1}(r_2)|= 2^7$ and $\Sigma_1$ is extraspecial it follows that $\Sigma_1 \cap \Sigma_2$ has order at most $2^2$ as $r_1 \not \in \Sigma_2$. We have that $I_1 \cap I_2$ acts on $\Sigma_1 \cap \Sigma_2$. Since $|I_1 \cap I_2|= 3^2$, there is $w \in C_{I_1\cap I_2}(\Sigma_1\cap \Sigma_2)^\#$. Now $(\Sigma_1 \cap \Sigma_2)\langle r_1 \rangle$ is elementary abelian. Since for $a \in \mathcal S(I_1\cap I_2)$ have $C_{\Sigma_1}(a) \cong \mathrm{Q}_8$ and $a \in \mathcal P(I_1 \cap I_2)$ have $C_{\Sigma_1}(a)= \langle r_1 \rangle$, we must have $\langle w \rangle \in \mathcal M(I_1\cap I_2)$. But then $\Sigma_1\cap \Sigma_2 \le C_{\Sigma_2}(w)=1$ by Lemma~\ref{celts}. This means that $\Sigma_1 \cap \Sigma_2=1$ which then forces $[C_{\Sigma_1}(r_2), C_{\Sigma_2}(r_1)]=1 $ and Lemma~\ref{sp62facts} (iv) provides a contradiction.\end{proof}
\section{Some subgroups in the centralizer of the involutions $r_1$ and $r_2$}
In this section, we finally construct $O_2(K_i)$ where $K_i= C_G(r_i)$. Recall from Definition~\ref{F4cent}, we expect $O_2(K_i) $ to be a product of an elementary abelian group of order $2^7$ by an extraspecial group of order $2^9$. We have already located the extraspecial group $\Sigma_i$. In this section we uncover the elementary abelian group. We consider the situation for $K_1$. In the previous section we proved that $C_G(\rho_2) \cong 3 \times \mathrm{Sp}_6(2)$. With this additional information we study $C_{K_1}(\rho_2)$. This group has shape $3 \times 2^{1+2+4}.(\mathrm{Sym}(3) \times \mathrm{Sym}(3))$. For us it is important that $Z(O_2(C_{K_1}(\rho_2) ))$ is elementary abelian of order 8. Furthermore $I_1=C_J(r_1)$ normalizes this group. This time there are six conjugates of this group under the action $C_L(r_1)$ and we define a group $\Upsilon_1$ generated by these six conjugates. We show that $\Upsilon_1$ is elementary abelian of order $2^7$ and centralizes $\Sigma_1$, the extraspecial group found earlier. Hence the product of both gives a $2$-group $\Gamma_1$ of order $2^{15}$, which is in fact isomorphic to the corresponding group in $\F_4(2)$. Furthermore we show that $N_G(\Gamma_1)/\Gamma_1 \cong \mathrm{Sp}_6(2)$ and so $N_G(\Gamma_1)$ is similar to a $2$-centralizer in $\F_4(2)$. In the next section we show $K_1 = N_G(\Gamma_1)$.
We use our, by now, standard notation. In particular recall the definition of $\Sigma_i$ from \ref{sigma} and $I_i^j$ the conjugates of $A_{3-i}$ under $F_i=C_L(r_i)$. Our first goal is to construct a signalizer for $I_i^1$, $i=1,2$, which contains $\Sigma_i$ properly. So, for $1 \le j<k\le 4$, we define $$\Theta_i^{j,k}= Z(O_2(C_{K_i}(I_i^j \cap I_i^k)))$$ and put $$\Upsilon_i = \langle \Theta_{i}^{j,k}\mid 1 \le j<k\le 4\rangle.$$ We will shortly show that $\Upsilon_i$ is elementary abelian group of order $2^7$.
As $C_G(I_i^j\cap I_i^k) \cong 3 \times \mathrm{Sp}_6(2)$, Lemma~\ref{cri} yields $$C_{K_i}(I_i^j\cap I_i^k)\approx 2^{1+2+4}.(\mathrm{Sym}(3) \times \mathrm{Sym}(3)).$$ Hence, by Lemmas~\ref{sp62line} (iii) and (iv) and \ref{Sigmapieces}(iii), $\Theta_{i}^{j,k} $ is elementary abelian of order $2^3$ and $$O_2(C_{K_i}(I_i^j\cap I_i^k))= \Sigma_i^{j}\Sigma_i^k\Theta_{i}^{j,k}.$$ We record this latter equality.
\begin{lemma}\label{O2} For $i= 1, 2$ and $1 \le j < k \le 4$, $O_2(C_{K_i}(I_i^j\cap I_i^k))= \Sigma_i^{j}\Sigma_i^k\Theta_{i}^{j,k}.$ \end{lemma}
\begin{lemma} \label{UPSstruct}Suppose that $i=1,2$ and $ \{j, k,l,m\} =\{1,2,3,4\}$. Then \begin{enumerate} \item $\Theta_i^{j,k}$ is elementary abelian of order $2^3$, contains $r_i$ and a $G$-conjugate $s_{3-i}$ of $r_{3-i}$ with $s_{3-i}\neq r_i$. \item $\Theta_i^{j,k}= \Theta_i^{l,m}$. \item $\Upsilon_i$ centralizes $\Sigma_i$. \item $\Theta_i^{j,k} \Theta_i^{k,l} $ is elementary abelian of order $2^5$. \item $\Upsilon_i$ is elementary abelian of order $2^7$ and is normalized by $I_i$.\end{enumerate} \end{lemma}
\begin{proof} To reduce the notational complexity of our argument we present the proof for $i=1$ the proof when $i=2$ is the same but we have to be careful when following the members of $\mathcal M(J)$ and $\mathcal P(J)$ in the arguments.
By definition $$\Theta_1^{j,k}=Z(O_2(C_{K_1}(I_1^j \cap I_1^k))).$$ We know $I_1^j\cap I_1^k \in \mathcal M(J)$ from Lemma~\ref{IijIik} and we know $C_{K_1}(I_1^j\cap I_1^k)\cap E_1^{j,k}$ is a line stabiliser in the natural symplectic representation of $E_1^{j,k} \cong \mathrm{Sp}_6(2)$. Thus $\Theta_1^{j,k}$ is elementary abelian of order $2^3$ by Lemma~\ref{sp62line} and of course $\Theta_1^{j,k}$ contains $r_1$ and, by Lemma~\ref{crossover}, $r_2$ is a 2-central involution in $E_1^{jk}$ and so $\Theta_1^{j,k}$ also contains a conjugate of $r_2$. This proves (i).
Now $J \cap E_1^{j,k}$ centralizes a conjugate of $r_{2}$ and is thus
$G$-conjugate to $I_2$. It follows from Lemma~\ref{type1} that $|\mathcal S(J \cap E_1^{j,k})|=4$, $|\mathcal P(J \cap E_1^{j,k})|= 6$ and $|\mathcal M(J \cap E_1^{j,k})|=3$. Now using Lemma~\ref{sp62line} (iv), we have $$X_1^{j,k}=C_{I_1 \cap E_1^{j,k}}(\Theta_1^{j,k}) \in \mathcal M(I_1 \cap E_1^{j,k}).$$ Observe $X_1^{j,k} \le I_1$ and so $X_1^{j,k}$ normalizes $\Sigma_1$.
Since $X_1^{j,k} \in \mathcal M(I_1)$, $C_{\Sigma_1} (X_1^{j,k})$ has order $2^5$ by Lemma~\ref{celts}. As $[\Sigma_1^j\Sigma_1^k,X_1^{j,k} ]= \Sigma_1^j\Sigma_1^k$ and $\Sigma_1$ is extraspecial, we deduce $$C_{\Sigma_1}(X_1^{j,k}) = \Sigma_1^l\Sigma_1^m = C_{\Sigma_1}(\Sigma_1^j\Sigma_1^k).$$ In particular, we now have $X_1^{j,k} = I_1^l \cap I_1^m$ by Lemma~\ref{celts}. This implies $\Theta_1^{j,k} \le C_G( I_1^l \cap I_1^m)$ and $\Theta_1^{j,k}$ is normalized by $I_1$; therefore $$\langle\Theta_1^{j,k},\Sigma_1^l\Sigma_1^m\rangle = O_2(C_{K_1}(I_1^l\cap I_1^m)).$$ Since $\Theta_1^{j,k}$ is $I_1$-invariant and elementary abelian, we infer $\Theta_1^{j,k}= \Theta_1^{l,m}$ and that $\Theta_1^{j,k}$ commutes with $\Sigma_1^j\Sigma_1^k$ as well as with $\Sigma_1^l\Sigma_1^m$. Since $\Sigma_1=\Sigma_1^j\Sigma_1^k\Sigma_1^l\Sigma_1^m$, we have now proved claims (ii) and (iii).
Because $\Theta_1^{j,k} = \Theta_1^{l,m}$ we have that $\Theta_1^{j,k}$ is centralized by $\langle X_1^{j,k}, X_1^{l,m}\rangle = \langle I_1^i\cap I_1^j,I_1^l\cap I_1^m\rangle$ which has Type N- as $\Theta_1^{j,k}$ does not commute with a conjugate of $Z$. Hence $\langle \Theta_1^{j,k} , \Theta_1^{k,l}\rangle$ is centralized by $$Y=\langle X_1^{j,k}, X_1^{l,m}\rangle \cap \langle X_1^{k,l}, X_1^{j,m}\rangle \in \mathcal P(J).$$
Now $C_G(Y) \cong 3 \times \mathrm{Sp}_6(2)$ and $I_1 \cap E(C_G(Y))$ is of Type N- by Lemma~\ref{crossover}. Since $\langle \Theta_1^{j,k} , \Theta_1^{k,l}\rangle$ centralizes $r_1$ and is normalized by $I_1$ we infer that $r_1$ is an involution of $E(C_G(Y))$ with centralizer of shape $2^{5}.\mathrm{Sp}_4(2)$ and that $\langle \Theta_1^{j,k} , \Theta_1^{k,l}\rangle \le O_2(C_{E(C_G(Y))}(r_1))$ which is elementary abelian. But then $$\langle \Theta_1^{j,k} , \Theta_1^{k,l}\rangle = \Theta_1^{j,k} \Theta_1^{k,l}$$ is elementary abelian of order at most $2^5$. It now follows that $\Upsilon_1= \Theta_1^{1,2}\Theta_1^{2,3}\Theta_1^{2,4}$ has order at most $2^7$ and is $I_1$-invariant. We have seen that $C_{I_1}(\Theta_1^{j,k}
\Theta_1^{k,l})$ is $I_1^j\cap I_1^k$. Thus $C_{I_1}(\Upsilon_i) \le I_1^1\cap I_1^2\cap I_1^3\cap I_1^4=1$. Hence $I_1$ acts faithfully on $\Upsilon_1$ and so $|\Upsilon_1|=2^7$. This completes the proof of (iv) and (v) and the verification of the statements in the lemma. \end{proof}
For $i=1,2$, we now set $$\Gamma_i = \Sigma_i\Upsilon_i.$$
\begin{lemma}\label{Gammabasic} For $i=1,2$, we have that $\Gamma_i$ has order $2^{15}$ and is normalized by $F_i$. Furthermore the following hold. \begin{enumerate} \item $Z(\Gamma_i) = \Upsilon_i$; and \item $[\Gamma_i,\Gamma_i]= \langle r_i\rangle$. \end{enumerate} \end{lemma}
\begin{proof} By Lemmas~\ref{sigmai} and \ref{UPSstruct}, $\Sigma_i$ has order $2^9$ and is extraspecial and $|\Upsilon_i|= 2^7$ and centralizes $\Sigma_i$. This yields $\Upsilon_i \cap \Sigma_i = \langle r_i\rangle$ and $\Gamma_i$ has order $2^{15}$. Furthermore, as $\Sigma_i$ is extraspecial, $\Upsilon_i$ is elementary abelian and $\Upsilon_i$ commutes with $\Sigma_i$ we have that $\Upsilon_i= Z(\Gamma_i)$ and $[\Gamma_i,\Gamma_i]= \langle r_i\rangle$. Hence (i) holds.
By the construction of $\Sigma_i$ and $\Upsilon_i$, $F_i$ normalizes both groups and consequently also normalizes their product. This is (ii). \end{proof}
\begin{lemma}\label{MaxSig} For $i=1,2$, $\Gamma_i$ is the unique maximal signalizer for $I_i^1$ in $K_i$. \end{lemma} \begin{proof} Assume that $W$ is an $I_i^1$ signalizer in $K_i$. Then $$W= \langle C_W(x) \mid x \in (I_i^1)^\#\rangle.$$ If $\langle x \rangle =Z\in \mathcal S (I_i^1)$, then $$O_{3'}(C_{K_i}(Z))=R_i =\Sigma_i^1 \le \Sigma_i\le \Gamma_i$$ is the unique maximal $I_i^1$ signalizer in $C_{K_i}(Z)$. All the other subgroups of order $3$ in $I_i^1$ are conjugate to $\langle \rho_{3-i}\rangle$ by an element of $Q_{3-i} \le F_i$. Hence we only need to consider $I_i^1$ signalizers in $C_{K_i}(\rho_{3-i})$. By Lemma \ref{ItsSp62}, $C_{G}(\rho_{3-i}) = C_G(I_i^1\cap I_i^2) \cong 3 \times \mathrm{Sp}_6(2)$ and we know from Lemma~\ref{cri} that $$C_{K_i}(\rho_{3-i}) \approx 3 \times 2^{1+2+4}.(\mathrm{Sym}(3) \times \mathrm{Sym}(3)).$$ Set $D= C_{K_i}(\rho_{3-i})$. Then $$O_2(D)= \Sigma_i^1\Sigma_i^2\Theta_i^{1,2} \le \Gamma_i$$ and, Lemma~\ref{sp62line}(ii), implies $ZO_2(D)/O_2(D)$ is diagonal in $D/O_2(D)$. Since $C_W(\rho_{3-i})$ is normalized by $Z$ we infer that $C_W(\rho_{3-i}) \le \Gamma_i$ as claimed. \end{proof}
\begin{lemma}\label{fusionr_i} For $i=1,2$, there is a $G$-conjugate of $r_i$ in $\Gamma_i\setminus \Upsilon_i$. \end{lemma}
\begin{proof} This fusion can already be seen in $$C_{K_i}(\rho_{3-i})\approx 3 \times 2^{1+2+4}.(\mathrm{Sym}(3)\times\mathrm{Sym}(3))$$ as $r_i$ is not weakly closed in $O_2(C_{K_i}(\rho_{3-i}))$ with respect to $C_G(\rho_{3-i})$ by Lemma~\ref{sp62line} (vi). \end{proof}
We are now able to determine the structure of $N_G(\Gamma_i)$.
\begin{lemma}\label{itssp} For $i=1,2$, the following hold. \begin{enumerate} \item $N_G(\Gamma_i)/\Gamma_i \cong \mathrm{Sp}_6(2)$; \item as $N_G(\Gamma_i)/\Gamma_i$-modules, $\Gamma_i/\Upsilon_i$ is a spin module and $\Upsilon_i/\langle r_i \rangle$ is a natural module; \item $\syl_2(N_G(\Gamma_i)) \subseteq \syl_2(K_i)$; and \item if $T \in \syl_2(N_G(\Gamma_i))$, then $\Gamma_i/\langle r_i \rangle = J(T/\langle r_i \rangle)$, $Z(T) \le \Upsilon_i$ and $Z(T)$ has order $4$. \end{enumerate} In particular, $N_G(\Gamma_i)$ is similar to a $2$-centralizer in $\F_4(2)$. \end{lemma}
\begin{proof} We already know that $\Gamma_i$ is normalized by $F_i$ and we have that $\Gamma_i$ is the unique maximal $I_i^1$-signalizer in $K_i$ by Lemma~\ref{MaxSig}. It follows that $\Gamma_i$ is also the unique maximal signalizer for $Q_{3-i} \ge I_i^1$ in $K_i$. Therefore $N_{E (C_G(\rho_i))}(Q_{3-i})$ also normalizes $\Gamma_i$. It follows from \cite[page 46]{Atlas} that $$X=\langle F_i, N_{E (C_G(\rho_i))}(Q_{3-i})\rangle\cong \mathrm{Aut}(\SU_4(2))$$ and $X$ normalizes $\Gamma_i$.
Since $C_{K_i}(Z)\Gamma_i/\Gamma_i$ is a $3$-centralizer of type $\PSp_4(3)$, $\Gamma_i$ is a maximal signalizer for $I_i^1$ and $Z$ is inverted in $N_G(\Gamma_i)/\Gamma_i$, we deduce
$N_G(\Gamma_i)/\Gamma_i \cong \mathrm{Sp}_6(2)$ or $\mathrm{Aut}(\SU_4(2))$ by using Theorem~\ref{PrinceThm}.
We know that $I_i$ acts faithfully on both $\Gamma_i/\Upsilon_i$ and $\Upsilon_i/\langle r_i \rangle$. In particular, as $|\Upsilon_i/\langle r_i\rangle|= 2^6$, if $N_G(\Gamma_i)/\Gamma_i\cong \mathrm{Aut}(\SU_4(2))$ then $\Upsilon_i/\langle r_i\rangle$ is an orthogonal module and if $N_G(\Gamma_i)/\Gamma_i\cong \mathrm{Sp}_6(2)$ then $\Upsilon_i/\langle r_i\rangle$ is a natural module. Similarly since $C_{\Sigma_i}(Z)= \Sigma_i^1$ and since this subgroup is not normalized by $F_i$ and $|\Gamma_i/\Upsilon_i|=2^8$, if $N_G(\Gamma_i)/\Gamma_i\cong \mathrm{Aut}(\SU_4(2))$, then $\Gamma_i/\Upsilon_i$ is an natural module and, if $N_G(\Gamma_i)/\Gamma_i\cong \mathrm{Sp}_6(2)$, then $\Gamma_i/\Upsilon_i$ is a spin module (see Lemma~\ref{modfacts}). So once we have proved part (i), part (ii) will also be proved.
Next we prove (iii) and the first part of (iv). Let $T \in \syl_2(N_G(\Gamma_i))$. Since, by Lemma~\ref{NotF}, $\Gamma_i/\langle r_i\rangle $ is not an $F$-module for $N_G(\Gamma_i)/\Gamma_i$, \cite[Lemma~26.15]{GLS2} implies that $ \Gamma_i/\langle r_i\rangle$ is the Thompson subgroup of $T/\langle r_i\rangle$. It follows that $N_{K_i}(T) \le N_G(\Gamma_i)$ and, in particular, $T \in \Syl_2(K_i)$ and $N_{K_i}(T)=T$. Notice furthermore that $N_G(\Gamma_i)/\langle r_i\rangle$ controls $K_i/\langle r_i\rangle$-fusion in $\Gamma_i /\langle r_i\rangle$. The last two parts of (iv) follow from the fact that $\Sigma_i$ is extraspecial and Lemma~\ref{nonsplitmods}.
It remains to prove (i). Assume that $N_G(\Gamma_i)/\Gamma_i \cong\mathrm{Aut}(\SU_4(2))$. Using Lemma~\ref{fusionr_i}, there exists $g\in G$ and $s \in \Gamma_i\setminus \Upsilon_i$ such that $s=r_i^g$. Since $N_G(\Gamma_i^g)$ contains a Sylow $2$-subgroup of $C_G(s)$, there is a $h \in C_G(s)$ such that $C_{\Gamma_1}(s)^h \le N_G(\Gamma_i^g)$ and we have $s= r_i^{gh}$ so we may suppose $g$ was chosen so $C_{\Gamma_1}(s)\le N_G(\Gamma_i^g)$. Note that, as $s \in \Gamma_i\setminus \Upsilon_i$, $s$ is conjugate in $\Gamma_i$ to $sr_i$ and, as $N_G(\Gamma_i)/\langle r_i\rangle$ controls $K_i/\langle r_i\rangle$-fusion in $\Gamma_i /\langle r_i\rangle$, $s$ is not $K_i$-conjugate to an element of $\Upsilon_i$.
Since $C_{\Gamma_1}(s)$ contains an extraspecial group of order $2^7$ with derived group $\langle r_i\rangle$, and $\mathrm{Aut}(\SU_4(2))$ does not (by Lemma~\ref{sp62facts}), we have $r_i \in \Gamma_i^g$. It follows that $C_{\Gamma_i^g}(r_i)$, which has index at most $2$ in $\Gamma_i$, also contains an extraspecial group of order $2^7$. As $T \in \Syl_2(K_i)$, there is $f \in K_i$ such that $C_{\Gamma_i^g}(r_i)^f=C_{\Gamma_i^{gf}}(r_i)\le T$. It follows that $s^f \in \Gamma_i \setminus \Upsilon_i$
and we may as well suppose that $s=s^f$ (though we may no longer have $C_{\Gamma_1}(s)\le N_G(\Gamma_i^g)$). With this choice of $s$, $|\Gamma_i^g : \Gamma_i^g\cap N_G(\Gamma_i)| \le 2$. Now $$\Phi(\Gamma_i^g \cap \Gamma_i) \le \Phi(\Gamma_i^g) \cap \Phi(\Gamma_i)= \langle s\rangle \cap \langle r_i\rangle=1$$ which means $\Gamma_i^g \cap \Gamma_i$ is elementary abelian. As $\Gamma_i$ contains $\Sigma_i$ which is extraspecial of order $2^9$, this yields $|\Gamma_i^g\cap \Gamma_i| \le 2^{11}$ and so $$|({\Gamma_i^g}\cap N_G(\Gamma_i))\Gamma_i/\Gamma_i| \ge 2^3.$$ Further
$$[\Upsilon_i \cap \Gamma_i^g, N_G(\Gamma_i) \cap \Gamma_i^g] \leq [\Gamma_i^g, \Gamma_i^g]\cap \Upsilon_i =\langle s \rangle \cap \Upsilon_i = 1.$$
Hence, as $|({\Gamma_i^g}\cap N_G(\Gamma_i))\Gamma_i/\Gamma_i| \ge 2^3$, Lemma~\ref{sp62facts}(iii) (which says that $\mathrm{Aut}(\SU_4(2))$ contains no fours group of unitary transvections) implies $|\Upsilon_1 \cap \Gamma_i^g| \leq 2^5$. Therefore $|\Gamma_i \cap \Gamma_i^g| \leq 2^9$. We have now shown $|(\Gamma_i^g \cap N_G(\Gamma_i))\Gamma_i/\Gamma_i| \geq 2^5$ which, as this group is elementary abelian and the $2$-rank of $\mathrm{Aut}(\SU_4(2))$ is $4$, is contradiction. Therefore $N_G(\Gamma_i)/\Gamma_i \cong \mathrm{Sp}_6(2)$ and this completes the proof of part (i) and thereby also (ii).
\end{proof}
\section{The centralisers of $r_1$ and $r_2$}
In this section we finally determine the structure of $K_i=C_G(r_i)$. We will prove $K_i = N_G(\Gamma_i)$ and hence conclude that $K_i$ is similar to a $2$-centralizer in $\F_4(2)$. The plan is to show $\Upsilon_i$ is strongly closed in a Sylow $2$-subgroup of $K_i$ with respect to $K_i$ and then to quote Goldschmidt's Theorem in the form of Lemma~\ref{Gold} to show that $K_i=N_G(\Gamma_i)$. To achieve this we study $K_i$-fusion of involutions. As most of the centralizers of involutions in $N_G(\Gamma_i)$ have order divisible by three, this will be reduced to fusion of 3-elements. Hence the first lemma we prove in this section will be that $N_G(\Gamma_i)$ is strongly $3$-embedded in $K_i$, which means that we have control of fusion of elements of order $3$ in $K_i$.
We use all our previous notation and furthermore for this section we set $H_i = N_G(\Gamma_i)$.
\begin{lemma}\label{Strong3} For $i=1,2$,
$H_i$ is strongly $3$-embedded in $K_i$. In particular, $H_i$ controls fusion of elements of order $3$ in $H_i$. \end{lemma}
\begin{proof}
Suppose that $x \in H_i$ has order $3$. We will show $C_{K_i}(x)$ normalizes $\Gamma_i$. Recall $C_S(r_i) \in \Syl_3(K_i)$ and $C_S(r_i) \le F_i \le H_i$ so $C_S(r_i)$ normalizes $\Gamma_i$. Since every element of order $3$ in $C_S(r_i)$ is $H_i$-conjugate into $I_i$, we may suppose $x \in I_i$.
Again to simplify our notation slightly we consider the case when $i=1$. Thus $|\mathcal S(I_1)|= 4$, $|\mathcal M(I_1)|= 6$ and $|\mathcal P(I_1)|=3$ by Lemma~\ref{I1}. If $\langle x \rangle \in \mathcal S(I_1)$, then we may suppose that $\langle x \rangle = Z$. In this case, by Lemma~\ref{CSr} $$C_{K_1}(Z) = Q_2R_1R_2I_1 \le H_1.$$ So suppose that $\langle x\rangle = \langle \rho_2\rangle \in \mathcal M(I_1)$. Then, by Lemma~\ref{O2}, $$C_{K_1}(\rho_2) = \Sigma_1^1\Sigma_1^2\Theta_1^{1,2}N_{F_1}(I_1 \cap E_1^{12}) \le \Gamma_1 F_1 \le H_1.$$ Suppose $\langle x \rangle = \wt {\rho_1} \in\mathcal P(I_1)$. Then, by Lemma~\ref{crossover}, $C_{K_1}(\wt {\rho_1}) \approx 3 \times 2^5{:}\mathrm{Sp}_4(2)$ and this has the same order as $C_{H_1}(\wt {\rho_1})$. Thus $C_{K_1}(\wt {\rho_1}) \le H_1$. Finally, $N_{K_1}(C_S(r_1)) \le N_{K_1}(Z)$ and so $H_1$ is strongly $3$-embedded in $K_1$ by \cite[Lemma 17.11]{GLS2}.
\end{proof}
We next show $H_i = K_i$ for $i = 1,2$ the proof is accomplished through a series of lemmas. It suffices to prove this with $i=1$ as the proof for $i=2$ is the same. By Lemma~\ref{itssp} (ii), $Z(H_1) = \langle r_1\rangle$, $\Upsilon_1/Z(H_1)$ is the natural $\mathrm{Sp}_6(2)$-module and $\Gamma_1/\Upsilon_1$ is the spin module for $\mathrm{Sp}_6(2)$. Let $T$ be a Sylow 2-subgroup of $H_1$. From Lemma~\ref{itssp} (iv) we have $T \in \syl_2(K_1)$.
\begin{lemma}\label{sc} \begin{enumerate}\item If $x \in \Upsilon _1^\#$ and $s\in x^{K_1}$, then $s$ and $sr_1$ are not $K_1$-conjugate.\item $\Upsilon_1$ is strongly closed in $\Gamma_1$ with respect to $K_1$. \end{enumerate} \end{lemma}
\begin{proof} (i) Obviously, if $x= r_1$, the result is true. So we may suppose that $x \in \Upsilon_1 \setminus \langle r_1\rangle$. Since $H_1$ acts transitively on $(\Upsilon_1/\langle r_1\rangle)^\#$, we may additionally assume $ x\langle r_1\rangle \in C_{{\Upsilon_1/\langle r_1\rangle}}( T)$ which has order $2$ by Lemma~\ref{sp62natural}. As by Lemma~\ref{nonsplitmods} the preimage of $ C_{{\Upsilon_1/\langle r_1\rangle}}( T)$ is centralized by $T$ we have $x \in Z(T)$.
Suppose that $x$ is $K_1$-conjugate to $xr_1$. Then as $x$ and $xr_1 \in Z(T)$, this conjugation must happen in $N_{K_1}(T)$. Since $T \in \syl_2(K_1)$, this is impossible and it follows that $x$ is not $K_1$-conjugate to $xr_1$. This proves (i)
Now consider $y \in \Gamma_1\setminus \Upsilon_1$. Then $[y,\Gamma_1] =\langle r_1\rangle$ and so $y$ is conjugate to $r_1y$ in $\Gamma_1$. Therefore (i) implies (ii). \end{proof}
\begin{lemma}\label{snormsigma} Let $x \in \Upsilon_1$, $g \in K_1$ and assume that $s=x^g$ with $s \in T \setminus \Gamma_1$. Then $s$ normalizes a $H_1$-conjugate of $I_1\Gamma_1$ and $\Sigma_1$. \end{lemma}
\begin{proof} Since in $H_1/\Gamma_1\cong \mathrm{Sp}_6(2)$ every involution is conjugate in to $N_{H_1/\Gamma_1}(I_1 \Gamma_1/\Gamma_1)$, we may as well suppose that $s $ normalizes $I_1 \Gamma_1$. In particular by Lemma~\ref{sigmai} we may additionally assume $\Sigma_1^s = \Sigma_1$. \end{proof}
\begin{lemma}\label{3prime} Let $x \in \Upsilon_1$, $g \in K_1$ and assume that $s=x^g$ with $s \in T \setminus \Gamma_1$.
Then the following hold: \begin{enumerate} \item $C_{\Gamma_1/\Upsilon_1}(s) = C_{\Gamma_1}(s)\Upsilon_1/\Upsilon_1$; and \item$C_{H_1}(s)$ is a $3^\prime$-group. \end{enumerate} \end{lemma} \begin{proof} By Lemma~\ref{snormsigma} we may assume that $s$ normalizes both $I_1\Gamma_1$ and $\Sigma_1$. Let $w \Upsilon_1 \in C_{\Gamma_1/\Upsilon_1}(s)$ and write $w= w_*u$ where $w_* \in \Sigma_1$
and $u \in \Upsilon_1$. Then $$[w,s]= [w_*u,s] = [w_*,s][u,s] \in \Upsilon_1.$$
As $s$ normalizes $\Sigma_1$, this means that $[w_*,s]\in \Sigma_1\cap \Upsilon_1 = \langle r_1\rangle$. Since $x$ is not $K_1$-conjugate to $sr_1$, we deduce that $w_*$ is centralized by $s$ and this proves (i).
Suppose that $W \in \Syl_3(C_{H_1}(s))$ and let $U \in \Syl_3(C_{H_1}(x))$. Then, as $ \Upsilon_1/\langle r_1\rangle$ is the natural $\mathrm{Sp}_6(2)$-module, $U$ has order $3^2$ by Lemma~\ref{sp62natural}. Since by Lemma~\ref{Strong3} $H_1$ is strongly $3$-embedded in $K_1$ we know that $U\in \syl_3(C_{K_1}(x))$ and so $U^g \in \Syl_3(C_{K_1}(s))$. Thus there exists $h \in C_{K_1}(s)$ so that $U^{gh} \ge W$. Consequently $W \le H_1 \cap H_1^{gh}$ and Lemma~\ref{Strong3} yields $gh \in H_1$ which contradicts the fact that $s = x^{gh}$, $s \in T \setminus \Sigma_1\Upsilon_1$ and $x \in \Upsilon_1$. \end{proof}
Suppose that $s^* \in s\Gamma_1$ is an involution which is conjugate to $s$ in $K_1$.
Then $ws= s^*$ with $w \in \Gamma_1$. We claim that $w \in C_{\Gamma_1}(s)$. To see this we note that the other possibility is that $w^{s}= w^{-1}= wr_1$ and then we calculate $${s^*}^{s}= (ws)^{s}= w^{s}s= w^{-1}s=wr_1s= s^*r_1$$ which contradicts Lemma~\ref{sc}(i).
Let $q \in C_{\Gamma_1}(s)$ and assume that $[w,q]\neq 1$. Then, by Lemma~\ref{Gammabasic}, $[w,q]=r_1$ and $$s^*{^q}=(ws)^q= w^qs= w[w,q]s= wsr_1=s^*r_1$$, which is also impossible. Therefore $w \in Z(C_{\Gamma_1}(s))$. Since $s$ normalizes $\Sigma_1$ and $\Sigma_1$ is extraspecial, the Three Subgroup Lemma implies $Z(C_{\Sigma_1}(s))= [\Sigma_1,s]$. Thus
Lemma~ \ref{sc}(i) implies that
\begin{lemma}\label{newclaim} Let $x \in \Upsilon_1$, $g \in K_1$ and assume that $s=x^g$ with $s \in T \setminus \Gamma_1$. If $s$ is $H_1$-conjugate to $s^*=w s$ where $w \in \Gamma_1$, then $w \in Z(C_{\Gamma_1}(s))\le [\Gamma_1,s]\Upsilon_1$. In particular, $s\Upsilon_1$ is $\Gamma_1/\Upsilon_1$-conjugate to $s^*\Upsilon_1$ and $C_{H_1/\Gamma_1}(s\Upsilon_1)= C_{H_1/\Upsilon_1}(s)\Gamma_1/\Gamma_1$. \end{lemma}
Now we are going to identify the involution $s\Gamma_1$ in $H_1/\Gamma_1 \cong \mathrm{Sp}_6(2)$.
\begin{lemma}\label{c2} Let $x \in \Upsilon_1$, $g \in K_1$ and assume that $s=x^g$ with $s \in T \setminus \Gamma_1$. Then $s\Gamma_1$ is an involution of type $c_2$ and all $K_1$-conjugates of $x $ in $H_1\setminus \Gamma_1$ project to elements of this type. \end{lemma}
\begin{proof}
By Lemma~\ref{sp62facts} (i), $s \Gamma_1$ is an involution of type $a_2$, $b_1$, $b_3$ or $c_2$ in
$H_1/\Gamma_1\cong\mathrm{Sp}_6(2)$. If $s\Gamma_1$ is of type $b_3$, then Lemma~\ref{sp62facts} implies that $[\Gamma_1/\langle r_1\rangle,s] = C_{\Gamma_1/\langle r_1\rangle}(s)$ and consequently $3$ divides $|C_{H_1}(s)|$. Hence $s\Gamma_1$ is not of type $b_3$ by Lemma~\ref{3prime} (ii).
If $s\Gamma_1$ is of type $b_1$ or $a_2$, then, by Lemma~\ref{newclaim}, $|C_{H_1/\Upsilon_1}(s)|$ is divisible by $3^2$. If $s\Gamma_1$ is of type $a_2$, then Lemma~\ref{sp62facts} implies $$|C_{\Upsilon/\langle r_1 \rangle}(s)/[\Upsilon/\langle r_1 \rangle,s]| = 4$$ and so $s$ is centralized by an element of order 3 contrary to Lemma~\ref{3prime} (ii). Thus $s\Gamma_1$ is not of type $a_2$. If $s\Gamma_1$ is of type $b_1$, then Lemma~\ref{sp62facts} yields $C_{\Upsilon/\langle r_1 \rangle}(s)/[\Upsilon/\langle r_1 \rangle,s]$ is the natural $\mathrm{Sp}_4(2)$-module and, as $\mathrm{Sp}_4(2)$ acts transitively on the non-trivial elements of this module, we again see $s$ is centralized by a $3$-element, a contradiction. Thus $s\Gamma_1$ must be of type $c_2$. \end{proof}
\begin{lemma}\label{strongc} $\Upsilon_1$ is strongly $2$-closed in $T$ with respect to $K_1$. \end{lemma}
\begin{proof} Let $x \in \Upsilon_1$, $g \in K_1$ and assume that $s=x^g$ with $s \in T \setminus \Gamma_1$. By Lemma~\ref{c2}, $s$ acts as an element of type $c_2$ on the natural $\mathrm{Sp}_6(2)$-module.
Let $F=C_{\Sigma_1}(s)=[\Sigma_1,s]$. Then $F$ has order $2^5$ by Lemma~\ref{sp62facts}. Thus the coset $Fs$ consists solely of conjugates of $s$ and of $sr_1$ and $F \cap \Upsilon_1 = \langle r_1\rangle$.
Recall that we may suppose that $x \in Z(T)$. So $s$ is a $2$-central element of $K_1$. Hence, as $F$ is a $2$-group which centralizes $s$, $F$ is contained in a Sylow $2$-subgroup $T_0$ of $K_1$ which centralizes $s$. Let $\Gamma_1^*$ be the preimage of $J(T_0/\langle r_1\rangle)$, $\Upsilon_1^*=Z(\Gamma_1^*)$ and $H^*= N_G(\Gamma_1^*)$. By Lemma~\ref{itssp} we have that $\Gamma_1^*$ is conjugate to $\Gamma_1$ in $K_1$. Then also $H^*$ is $K_1$-conjugate to $H_1$ and $H^*/\Gamma_1^*\cong \mathrm{Sp}_6(2)$.
Assume that $y \in F\setminus \langle r_1 \rangle$. Then $ys$ is conjugate to either $s$ or $sr_1$. In particular any coset of $\langle r_1 \rangle$ in $F$ contains some $y$ such that $ys$ is conjugate to $s$ in $K_1$. If $y \in \Gamma_1^*$, then, as $y \in \Gamma_1\setminus \Upsilon_1$, Lemma~ \ref{sc} (ii) yields $y \not \in \Upsilon_1^*$ and consequently we also have $ys \in \Gamma_1^*\setminus \Upsilon_1^*$ which contradicts Lemma~\ref{sc}. Thus $y \not \in \Gamma_1^*$ and the coset $y\Gamma_1^*$ contains $ys$. We deduce with Lemma~\ref{c2} that $y\Gamma_1^*$ is of type $c_2$ in $N_{K_1}(\Gamma_1^*)/\Gamma_1^*$ and $F\Gamma_1^*/\Gamma_1^*$ is a subgroup of order $2^4$ in which all the non-trivial elements are in class $c_2$. Since $\mathrm{Sp}_6(2)$ has no such subgroups by Lemma~\ref{sp62facts}, we have a contradiction. Therefore $\Upsilon_1$ is strongly 2-closed in $T$ with respect to $K_1$. \end{proof}
Next we can prove the main result of this section:
\begin{lemma}\label{H=K} For $i=1,2$, we have $H_i= K_i$. In particular, $K_1$ and $K_2$ are similar to $2$-centralizers in $\F_4(2)$, \end{lemma}
\begin{proof} Again it is enough to prove the lemma for $i = 1$. By Lemma~\ref{strongc} we have that $\Upsilon_1$ is strongly 2-closed in $T$ with respect to $K_1$. Therefore Lemma~\ref{Gold} yields $K_1\le N_{G}(\Upsilon_1)$. Now $C_{K_1}(\Upsilon_1) \cap C_S(r_1)=1$ and so $C_{K_1}(\Upsilon_1)$ is a $3'$-group. Since, by Lemma~\ref{MaxSig}, $\Gamma_1$ is the unique maximal $I_1^1$-signalizer in $K_1$, we conclude $\Gamma_1 \ge C_{K_1}(\Upsilon_1)$ and thus $\Gamma_1 =C_{K_1}(\Upsilon_1)$. It follows that $K_1=N_{K_1}(\Upsilon_1) = N_{K_1}(\Gamma_1)$ as claimed. \end{proof}
\section{Proof of Theorem~\ref{MT}}
Having determined the shapes of the centralizers of the involutions $r_1$ and $r_2$ in this section we accomplish the final identification of $G$.
Let $T \in \Syl_2(K_1)$, where $K_1 = C_G(r_1)$, and recall that $\Gamma_1= \Sigma_1\Upsilon_1= O_2(K_1)$. The conclusion of the work of the previous sections is that $K_1$ is similar to a $2$-centralizer in $\F_4(2)$.
By Lemma~\ref{UPSstruct}, $\Upsilon_1$ contains a $G$-conjugate $s_2$ of $r_2$ with $s_2\neq r_1$. As $K_1$
acts transitively on the non-trivial elements of $\Upsilon_1/\langle r_1\rangle$, Lemma~\ref{nonsplitmods} shows that we may further suppose that $s_2 \in Z(T)$ and $Z(T)=\langle r_1,s_2\rangle$. Define $U_2 = C_G(s_2)$. We have $U_2$ is $G$-conjugate to $K_2= C_G(r_2)$ and thus, as $|K_1|=|K_2|$, we have $T \in \syl_2(U_2)$.
We will use the two groups to construct a subgroup $P = \langle K_1,U_2 \rangle \cong F_4(2)$ using Theorem~\ref{P=F4}. Recall Definition~\ref{F4setup}, and note that $K_1$, $U_2$, $T$ is an $\F_4$ set-up.
\begin{lemma}\label{P=F4new} $P = \langle K_1, U_2\rangle \cong \F_4(2)$. \end{lemma}
\begin{proof} This follows directly from Theorem~\ref{P=F4}. \end{proof}
In fact we have the following corollary:
\begin{corollary}\label{F42sub} If $X$ is any group which satisfies the assumptions of Theorem~{\rm \ref{MT}}, then $X$ contains a subgroup isomorphic to $\F_4(2)$. \end{corollary}
\begin{proof} This follows immediately from Lemma~\ref{P=F4new}. \end{proof}
Our aim is to show that $G$ is isomorphic to either $\F_4(2)$ or $\mathrm{Aut}(\F_4(2))$. For this we will show that $P$ is normal in $G$. As a first step we show that $P$ is normalized by $M$ and that $P_0=PM$ is either $\F_4(2)$ or $\mathrm{Aut}(\F_4(2))$. We then produce a normal subgroup $G_*$ of $G$ of index at most two such that $P_0 \cap G_* = P$. Our objective is then to show $G_* = P$. This will be done using Holt's Theorem (Lemma~\ref{Holt}). Hence we have to gain control of $G_*$-fusion of involutions in $P$. For this we show that $P_0$ is strongly $3$-embedded in $G_*$, which will imply that $P$ controls $G_*$-fusion in $P$. We start with the proof that $M$ normalizes $P$.
We have $C_P(\rho_1) \cong C_P(\rho_2) \cong 3 \times \mathrm{Sp}_6(2)$ and so, by Lemma~\ref{ItsSp62}, $C_G(\rho_i) = C_P(\rho_i)$, $i = 1,2$. As $\langle C_M(\rho_1), C_M(\rho_2)\rangle = M \cap P$, we see $ \langle C_G(\rho_1) ,C_G(\rho_2)\rangle$ satisfies the assumptions of Theorem~\ref{MT}. By Corollary~\ref{F42sub} we get that $ \langle C_G(\rho_1) ,C_G(\rho_2)\rangle$ contains a subgroup isomorphic to $\F_4(2)$. As $P \cong \F_4(2)$, we obtain
\begin{lemma}\label{subF42} $ \langle C_G(\rho_1) ,C_G(\rho_2)\rangle = P$. \end{lemma}
\begin{lemma}\label{PM} $M$ normalizes $P$.
\end{lemma}
\begin{proof} Since $P \cong \F_4(2)$ and $\rho_1$ and $\rho_2$ are not conjugate in $P$, we have that $M \cap P = RS\langle f \rangle$. If $M \le P$, we have nothing to do. If $M > M \cap P = RS\langle f \rangle$, then, by Lemma~\ref{structM}, there is an element $t$ of $M\setminus M \cap P$ such that $\rho_1^t = \rho_2$. This element normalizes $P$ by Lemma~\ref{subF42}. Thus $M$ normalizes $P$. \end{proof}
Define $P_0= PM$.
\begin{lemma} \label{strong3a} $P_0$ is strongly $3$-embedded in $G$. \end{lemma}
\begin{proof} Since $P \cong \F_4(2)$, there are three conjugacy classes of elements of order $3$ in $P$ and they are all witnessed in $J$. For $\langle x\rangle \in \mathcal S( J)$, we have $N_G(\langle x \rangle) = M \le P_0$ and for $\langle x \rangle \in \mathcal M(J) \cup \mathcal P(J)$ we have $C_G(x) = C_P(x)$ by Lemma~\ref{ItsSp62}.
Since also $N_G(S) \le M \le P_0$ we have $P_0$ is strongly $3$-embedded in $G$ by \cite[Lemma 17.11]{GLS2}. \end{proof}
We can now determine the structure of $P_0$.
\begin{lemma}\label{P0syl} We have $P_0$ contains a Sylow $2$-subgroup of $G$ and either $P_0=P$ or $P_0 \cong \mathrm{Aut}(\F_4(2))$. \end{lemma}
\begin{proof} Assume that $T \not \in \syl_2(G)$ and let $T_1 >T$ normalize $T$. Then $T_1$ normalizes $Z(T) = \langle r_1, s_2\rangle$. Since $K_1 \le P$ and $U_2\le P$, there exists $x \in T_1$ such that $r_1^x \neq r_1$ and $s_2^x \neq s_2$. Since $Z(T)$ has order $4$, we deduce that $r_1^x= s_2$ and thus that $K_1^x=U_2$. Hence $x$ normalizes $P = \langle K_1, U_2 \rangle$ and $P_0=P\langle x \rangle \cong \mathrm{Aut}(\F_4(2))$.
Now let $T_0 \in \syl_2(P_0)$ ($P_0 = P$ or $P_0 = \mathrm{Aut}(P)$) and assume that $w \in N_G(T_0)$. As $r_1 \in T^\prime \leq T_0^\prime \leq T$, we have $r_1^w \in T \leq P$. Employing Lemma~\ref{F42Classes} we see that all involutions of $P$ commute with elements of order $3$. By Lemma~\ref{strong3a} $C_{P_0}(r_1^w)$ contains a Sylow $3$-subgroup of $C_G(r_1^w)$. Hence it follows that $r_1^w \in r_1^{P_0}\cup s_2^{P_0}$. Then there is $x \in P_0$ such that $r_1=r_1^{wx}$ or $s_2= r_1^{wx}$. Since $\langle K_1, U_2 \rangle = P$, we have $wx \in P$. However this means $w \in P_0$ and we infer $T_0\in \Syl_2(G)$. \end{proof}
Now we produce the normal subgroup $G_*$ with $G_* \cap P_0 = P$.
\begin{lemma}\label{index2} If $P_0 > P$, then $G$ has a subgroup $G_*$ of index $2$ with $P = P_0 \cap G_*$. Furthermore $G_*$ satisfies the hypothesis of Theorem~{\rm \ref{MT}}. \end{lemma}
\begin{proof} We let $T_0 \in \syl_2(P_0)$ and $T \in \syl_2(P)$ with $T_0>T$. Suppose that $t \in T_0$ is an involution and $C_{P_0}(t)$ has a non-trivial Sylow $3$-subgroup $D$. Then as $P_0$ is strongly $3$-embedded by Lemma~\ref{strong3a} we have that $D \in \Syl_3(C_G(t))$. Now by Lemma~\ref{F42Classes} $P$ has four conjugacy classes of involutions and their centralizers have $3$-parts of their orders $3^4$, $3^4$, $3^2$ and $3^2$. On the other hand, if we let $x \in T_0\setminus T$ with $C_{P_0}(x) \cong 2 \times {}^2\F_4(2)$, then $C_P(x)$ has Sylow $3$-subgroups which are extraspecial of order $3^3$. It follows that $x$ is not conjugate to any element in $T$ and consequently $G$ has a subgroup $G_*$ of index $2$ by Thompson's Transfer Lemma \cite[Lemma 15.16]{GLS2}. Obviously then $P_0 \cap G_* = P$ and $G_*$ satisfies the hypothesis of Theorem~\ref{MT}. \end{proof}
We finally prove that $G \cong \F_4(2)$ or $\mathrm{Aut}(\F_4(2))$.
\begin{proof}[Proof of Theorem~{\rm \ref{MT}}] By Lemma~\ref{index2}, we may suppose that $P=P_0.$ Using Lemma~\ref{F42Classes}, $P$ has exactly four conjugacy classes of involutions and each such involution $t$ has $|C_P(t)|_3 \neq 1$. Since $P$ is strongly $3$-embedded in $G$, $C_P(t)$ contains a Sylow $3$-subgroup of $C_G(t)$. Thus, as $|C_P(r_1)|_3= 3^4$, we have $r_1^G\cap P \subseteq r_1^P \cup r_2^P$. Since $r_1$ and $r_2$ are not $G$-conjugate by Lemma~\ref{fusionr1r2} and \ref{index2}, we get that $r_1^G \cap P= r_1^P$. We note that if $N$ is a non-trivial normal subgroup of $G$, then, as $C_G(r_1) \le P$ and $r_1 \not \in Z(P)$, $1 \neq C_N(r_1) \le N \cap P$ which means that $P \le N$. Because $N_G(S) \le P$, the Frattini Argument implies $G = N_G(S)N \leq PN =N$. Hence $G$ is a simple group. Now an application of Lemma~\ref{Holt} and the observation that $P$ is neither soluble nor an alternating group yields $G=P$ and the proof is complete. \end{proof}
\end{document}
\end{document} | arXiv |
Toric manifold
In mathematics, a toric manifold is a topological analogue of toric variety in algebraic geometry. It is an even-dimensional manifold with an effective smooth action of an $n$-dimensional compact torus which is locally standard with the orbit space a simple convex polytope.[1][2]
The aim is to do combinatorics on the quotient polytope and obtain information on the manifold above. For example, the Euler characteristic and the cohomology ring of the manifold can be described in terms of the polytope.
The Atiyah and Guillemin-Sternberg theorem
This theorem states that the image of the moment map of a Hamiltonian toric action is the convex hull of the set of moments of the points fixed by the action. In particular, this image is a convex polytope.
References
1. Jeffrey, Lisa C. (1999), "Hamiltonian group actions and symplectic reduction", Symplectic geometry and topology (Park City, UT, 1997), IAS/Park City Math. Ser., vol. 7, Amer. Math. Soc., Providence, RI, pp. 295–333, MR 1702947.
2. Masuda, Mikiya; Suh, Dong Youp (2008), "Classification problems of toric manifolds via topology", Toric topology, Contemp. Math., vol. 460, Amer. Math. Soc., Providence, RI, pp. 273–286, arXiv:0709.4579, doi:10.1090/conm/460/09024, MR 2428362.
| Wikipedia |
\begin{document}
\title{\bf Surfaces with triple points} \author{Stephan Endra\ss\ \and Ulf Persson \and Jan Stevens
\thanks{Partially supported by the
Swedish Natural Science Research Council (NFR)}} \maketitle
\begin{abstract} \noindent In this paper we compute upper bounds for the number of ordinary triple points on a hypersurface in $\mathbb{P}^3$ and give a complete classification for degree six (degree four or less is trivial, and five is elementary). But the real purpose is to point out the intricate geometry of examples with many triple points, and how it fits with the general classification of surfaces. \end{abstract}
\section*{Introduction}
The problem of finding the maximal number of simple singularities on projective hypersurfaces has attracted a lot of attention in the last twenty years. In this paper we study surfaces with a simple kind of non-simple singularities, namely ordinary triple points.
In contrast to the case of simple surfaces singularities (classified by DuVal as those which `do not affect the conditions of adjunction' \cite{duval}) the invariants of the surface and its type in the classification of surfaces may change. Normal surfaces with higher singularities provide interesting examples of surfaces found, so to speak, in our back-yard.
We warm up by looking at quintics with isolated triple points. Their analysis is very elementary, but yields examples which nicely illustrate many aspects of the general theory of surfaces. Quintics with many triple points were to our knowledge first studied by Gallarati \cite{gallarati}.
For higher degree surfaces the location of the triple points will matter very much, and the problem of finding the maximal number becomes very difficult. We derive several bounds on the number of triple points. We found an example of a septic with a high number (16) of triple points, which is one short of our upper bound. For sextics we give a classification, which takes up the main part of this paper.
Our research started out as search for sextics with many triple points. In particular, a sextic with 11 triple points would be very interesting. In fact, given the right configuration lying on a quadric, it would furnish a birational Abelian surface. However, we could not find such a surface; the putative example turned out to be a triple cover of a quadric. Then it was easily remarked that 11 triple points are {\it a priori} impossible. Instead we found many examples with 9 triple points and general arguments allowing to rule out possibilities. The successful construction of an example raises the question how special the construction is: can it be generalised? This can be decided by infinitesimal methods. Let $\Sigma^d_\nu$ be the stratum of surfaces with $\nu$ ordinary triple points in the parameter space of all surfaces of degree $d$. A lower bound for the dimension of $\Sigma^d_\nu$ in the point representing an explicit example $X$ is the number of moduli in the construction plus $15$ from coordinate transformations --- the stabiliser of the point configurations being discrete for large $\nu$. An upper bound is given by the dimension of the Zariski tangent space. If these dimensions are equal we know that the stratum is smooth of the given dimension in that point, and the construction gives the general element of the component of $\Sigma^d_\nu$.
To compute the Zariski tangent space we have to determine which polynomials of degree $d$ induce equisingular deformations of the singular points, which for each specific example can easily be computed with {\it Macaulay} \cite{bayer}.
The clue to classifying sextics with many triple points is the study of exceptional curves of the first kind on the minimal resolution. It turns out that only a few different cases can occur. For nine triple points, we find three families of $K3$ surfaces (theorem \ref{theorem:K3}) and two families of properly elliptic surfaces (theorem \ref{theorem:elliptic}). In the $K3$ cases we find for each possible configuration of the nine points a pencil of sextics of the form $\alpha g+\beta q^3$, where $q$ defines the (unique) quadric through the nine points, and $g$ is a reducible sextic. In the other two cases we find even a net of sextics, again containing $q^3$ and reducible sextics. Constructing reducible sextics with triple points is not so difficult.
Regarding the existence of a sextic $\{g=0\}$ with 10 triple points we first observe that the pencil $\alpha g+\beta q^3$, where $\{q=0\}$ passes through nine of them, falls into one of our five families of sextics with nine triple points. Assuming that an element of such a family has a tenth triple point gives conditions on the coefficients. The resulting equations are much to difficult to solve. By imposing extra symmetry we have been able to reduce the number of variables and equations, while keeping at least a one-parameter family of solutions.
This paper is organised as follows. First we study quintics with triple points. The next section describes the birational invariants of our surfaces. In the third section several bounds for the number of triple points on a surface of degree $d$ are given. Then we identify the exceptional curves.
The next section is the main part of this paper. First we study the exceptional curves on a sextic with triple points and find only a few different cases, depending on the geometric genus (corollary \ref{proposition:curves}). Based on the geometry of the exceptional curves, we give an overview over sextic surfaces with $0$, \ldots, $10$ ordinary triple points. For convenience, a list of sextics with triple points and their invariants is given (theorem \ref{theorem:sextics}). In the last section a septic with 16 ordinary triple points is constructed.
\section{Quintics}
A smooth quintic in $\mathbb{P}^3$ is one of the simplest examples of a surface of general type. Its Chern invariants are given by $c_1^2=5$, $c_2=55$ and thus $\chi=4$. Furthermore it has the Hodge invariants $p_g=4$, $q=0$ and $h^{1,1}=45$ (we give general formulas in the next section). The adjoint system consists of planes, thus is $4$-dimensional (illustrating $p_g=4$).
If nodes are allowed, or more generally rational double points, nothing happens to the invariants, as the corresponding resolutions are smooth deformations of smooth quintics. It is however an interesting question to try and decide what 'bouquets' of rational double points can be imposed. The maximal number of nodes is 31 (see below), and one may impose five $E_8$ singularities ($w^5=F(x,y,z)$ where $F$ is a plane quintic with five cusps) which is somewhat short of the maximal number $44=h^{1,1}-1$ (counted with multiplicity according to Milnor number) theoretically possible.
If other types of singularities are considered, more interesting things happen. Invariants change, and also the type of the surfaces. There exist classifications of quintics with isolated singularities (at least for those of general type), and interesting such examples occur for $\tilde{E}_8$ singularities (normal form: $z^2=y^3+\lambda x^2y^2+x^6$). We will however restrict ourselves only to ordinary triple points (locally $f(x,y,z)=0$ with $f$ a smooth plane cubic). It is easy to see (cf.~Section \ref{section:invariants}), that a triple point decreases $c_1^2$ by three and $c_2$ by nine (and thus $\chi$ by one). Furthermore the adjoint system consists of planes passing through the triple points. We can thus establish the following table, where $\nu$ denotes the number of triple points. $$ \halign{\qquad\quad$#$\quad&\quad\hfil$#$\hfil\quad&\quad$#$\quad&\quad$#$ \quad&\quad$#$\quad&\quad$#$\quad&\quad#\quad\hfil\cr \nu&c_1^2&c_2&\chi&p_q&q&Type of surface\cr \noalign{\vskip 0.5em} 0&5&55&5&4&0&general type\cr 1&2&46&4&3&0&general type\cr 2&-1&37&3&2&0&elliptic blown up once\cr 3&-4&28&2&1&0&$K3$ blown up four times\cr 4&-7&19&1&0&0&rational\cr 5&-10&20&0&0&1&ruled over elliptic curve\cr} $$ The most interesting thing about this table is the geometry of the special examples, which nicely illustrates many aspects of the general theory of surfaces. We should also note that the constructions of surfaces are elementary, as we can impose the triple points generically, and then simply solve linear equations in the coefficients. We observe that the line joining any two triple points lies by Bezout on the surface, and furthermore is exceptional. The latter is true for any conic passing through three triple points {\it and} lying on the quintic.
Let us now comment upon our small menagerie of surfaces, found, so to speak in our back-yard. \begin{description} \item[$\nu=1\!:$] This is a double octic, the double cover of $\mathbb{P}^2$ effected by projection from the triple point. (In fact all the quintics with triple points, can be considered as double octics, with one less triple point.) Note that not all double octics are of this type. \item[$\nu=2\!:$] An elliptic surface blown up. The elliptic fibration is given by the planes through the line joining the two triple points. The intersection consists of the line and a quartic with two double points at its intersection with the line. The resolution of such plane quartics (a resolution effected by the desingularisation of the triple points) is clearly elliptic. The canonical divisor will coincide with the elliptic fibration (plus the exceptional divisor). The elliptic curves which arise from desingularisation, will be bi-sections of the elliptic fibration. \item[$\nu=3\!:$] A $K3$ surface blown up four times. The canonical divisor is given by the plane through the three triple points. The intersection of the quintic with that plane is given by the three lines of the corresponding triangle, and the residual conic, passing through them all. If you make the resolved surface minimal, you get a $K3$ surface with three elliptic curves $E_i$ all passing through a common point $p$ and any two $E_i$, $E_j$ ($1\leq i\neq j\leq3$) also intersecting in another point $p_k$ ($i$, $j$, $k$ distinct integers). Each of those elliptic curves will give rise to elliptic fibrations, which are amusing to identify. All the $K3$ surfaces will have Picard number at least three, and the configuration of the elliptic curves will give a common sublattice to them all. Conversely starting with a $K3$ surface with such a sublattice of its Picard group, we choose a point $p$ and elliptic curves
$E_i$ in each of the corresponding pencils, passing through $p$. Those will define intersection points $p_k=E_i\cap E_j$ ($i$, $j$, $k$ distinct). Blowing up the points to exceptional divisors $F_k$ and $G$ (the latter corresponding to the blow up of $p$) we can in fact write down the divisor $H$ of degree 5, effecting the birational embedding. Namely $$ H=E_1+E_2+E_3-F_1-F_2-F_3-2G \;. $$ It is straightforward to check that $H^2=5$, $H\cdot E_i=0$ and $H\cdot F_i=1$ while $H\cdot G=2$. Independent examples of such $K3$ surfaces are furnished by quartics with three lines, each of which gives rise to an elliptic fibration. It could be a mildly challenging exercise for the reader to see how such a quartic can be transformed into a quintic with three triple points. It is also amusing to count parameters: there are 56 quintic monomials, 10 conditions for a triple point at a fixed location, and 6-dimensional family of projective linear transformations, fixing three points. This makes 19. On the other hand $K3$ surfaces with Picard number at least three, make up a 17-dimensional family, and we add 2 for the position of the point.
\item[$\nu=4\!:$] A rational surface blown up many times (15 or 16 times). You expect an infinite number of exceptional curves of the first kind. Ten of those are obvious, given by the six edges of the tetrahedron spanned by the triple points, and the four residual conics corresponding to each face. It is a challenge to find others. \item[$\nu=5\!:$] A ruled surface over an elliptic curve, blown up ten times. The twenty exceptional divisors, which come in pairs, are obvious. Each pair of triple points determine an exceptional line, and dually the three remaining triple points a residual exceptional conic. This also gives a clue to the ruling. Through six points one may always find a unique twisted cubic. For each point on the surface, consider the twisted cubic through it and the five triple points. By Bezout this curve has to lie on the surface. The resolved elliptic curves will be sections. By blowing down the ten lines (or the ten conics) we get a minimal ruled surface. This will turn out to be the one coming from the stable rank-two bundle on an elliptic curve. \end{description}
Note that six or more triple points is an impossibility. Such a putative surface would necessarily have $\chi<0$ and hence be ruled over a curve of genus $g>1$. In such surfaces there is no space for elliptic curves, as they can neither surject on to the base, nor squeeze into the fibres.
Quintics with at least four triple points can be simplified using a birational transformation, known as {\em reciprocal transformation} \cite[VIII \S\ 4]{semple}. The ordinary plane Cremona transformation using the linear system of conics through three points can be described in suitable coordinates by the formula $(x \mathop{\rm :} y\mathop{\rm :} z) \mapsto (1/x \mathop{\rm :} 1/y\mathop{\rm :} 1/z)$. This formula generalises to higher dimensions. In particular, the space transformation
\begin{equation*}
(x\mathop{\rm :} y\mathop{\rm :} z \mathop{\rm :} w)
\mapsto \left(\frac{1}{x}\mathop{\rm :}\frac{1}{y}\mathop{\rm :}\frac{1}{z}
\mathop{\rm :}\frac{1}{w}\right)
\end{equation*}
simultaneously blows up the vertices and blows down the faces of the coordinate tetrahedron. The vertices are called {\em fundamental points} of the reciprocal transformation. Let $Y\subset\mathbb{P}^3$ by a surface of degree $d$ not containing any of the coordinate planes. Let $m_1$, \dots, $m_4$ be the multiplicities of $Y$ in the fundamental points. Then the image $Y'$ of $Y$ is a surface of degree $3d-m_1-\cdots-m_4$. In many cases $Y'$ will be singular in the fundamental points with singularities obtained from contracting the intersection curves of $Y$ with the coordinate planes.
For a quintic $X$ the following happens: taking four triple points as fundamental points we transform the surface into one of degree $3\cdot 5 - 4\cdot 3 = 3$. Conversely, given four points on a cubic surface we obtain a surface of degree $3\cdot 3 - 4 =5$ with four new singularities, which are ordinary triple points if the tetrahedron (spanned by the four points) cuts out smooth curves. This argument shows that five triple points are maximal, and realisable by starting with a cubic cone. The construction also allows you to find the blown up rational quintic with four triple points. As we all know the cubic can be thought of $\mathbb{P}^2$ blown up six times. The four vertices of the tetrahedron will provide four more blow ups, and finally the six edges of the tetrahedron, each intersect the cubic in a residual point, each of which is blown up. This makes a total of sixteen.
It is amusing to continue. Setting $H$ to be the hyperplane section of the cubic (thus $H^2=3$) and letting $E_i$ denote the exceptional curves associated to the four vertices, and $E_{ij}$ the residual intersections associated to the six edges, we can write down the linear system on the cubic, which gives the quintic, as $$ 3H-2\sum_iE_i-\sum_{ij}E_{ij}\;. $$ This linear system blows down the four elliptic curves $$ H-E_i-E_j-E_k-E_{ij}-E_{ik}-E_{jk} $$ each of which has self-intersection $3-6=-3$. Furthermore the six exceptional curves $E_{ij}$ are mapped onto lines, namely the edges of the tetrahedron spanned by the triple points, while the exceptional curves $E_i$ are mapped onto residual conics. Each of the 27 lines of the cubic, which does not pass through a vertex, maps onto a twisted cubic passing simply through all four triple points. As there are exceptional curves of arbitrary high degree on the cubic blow-up, we see that on a quintic there may be exceptional curves of arbitrary high degree.
The above analysis is very elementary, and parts of it has not too surprisingly already appeared in the literature \cite{gallarati}. It can be generalised in a number of different ways. One, which we have already mentioned, is to consider more general singularities (in this way interesting surfaces of general type can be constructed \cite{yang}), the other is to consider triple points on surfaces of higher degree.
\section{Invariants} \label{section:invariants}
A surface singularity $P\in X$ is an {\em ordinary triple point of $X$\/} if there exist local coordinates $x$, $y$ and $z$ centred at $P$ such that $X$ is given by the equation
\begin{equation*}
x^3+y^3+z^3+\lambda xyz=0
\end{equation*}
for a $\lambda\in\mathbb{C}$ with $\lambda^3\neq -27$. Such a triple point is also called a singularity of type $\tilde{E}_6$ (resp.~$P_8$, $T_{3,3,3}$). The minimal resolution of an ordinary triple point is given as follows: let $\pi_P\colon\tilde{X}\rightarrow X$ be the blowup of $X$ in $P$ and let $E_P$ be the exceptional divisor. This is a smooth elliptic curve, given in suitable homogeneous coordinates $(x\mathop{\rm :} y\mathop{\rm :} z)$ by the equation $x^3+y^3+z^3+\lambda xyz=0$. In particular $\tilde{X}$ is smooth in every point of $E_P$ and the self intersection of $E_P$ on $\tilde{X}$ is $-3$.
Now let $X\subset \mathbb{P}^3$ be a projective surface of degree $d$ with $\nu$ isolated triple points. Let $\mathscr{S}=\left\{P_1,\ldots,P_\nu\right\}=X_{\it sing}$ be the singular locus of $X$ and let $\tilde{X}$ be the blow-up of $X$ in $\mathscr{S}$; it is a smooth model of $X$, however not minimal in general. We will denote the minimal model of $X$ by $\overline{X}$. Moreover let $E=\sum_{i=1}^\nu E_i$ be the sum of all exceptional divisors.
There are basically two types of invariants of $\tilde{X}$. To start with, there are invariants of local nature which take into account the number, but not the position of the triple points. This are the Chern numbers $c_1\brtX^2$, $c_2\brtX$ and the holomorphic Euler characteristic $\chi\left(\cO_{\smash{\tX}}\right)$. Second, there are invariants which are also influenced by the position of the triple points. Amongst them are the geometrical genus $p_g\left(\smash{\tX}\right)$, the irregularity $q\left(\smash{\tX}\right)$, the Betti numbers $b_i\left(\smash{\tilde{X}}\right)$, the Hodge numbers $h^{p,q}\left(\smash{\tilde{X}}\right)$ and the Kodaira dimension $\kod{(\tX)}$. We are going to compare these invariants with the invariants of a smooth hypersurface $X_s$ of degree $d$.
\begin{description} \boldmath \item[The canonical class and $c_1\brtX^2$:] \unboldmath we have
$K_{\smtX}\sim_{lin}\pi^\astK_{X_s}-E$.
This follows from the adjunction formula:
we know that $E_i\cdot (K_{\smtX}+E_i)=0$, so if
$K_{\smtX}\sim_{lin}\pi^\ast K_X-\alpha E$, then $E_i\cdot (K_{\smtX}+E_i)=
E_i\cdot (\pi^\astK_{X_s}-\alpha E +E_i)= 3(\alpha-1)$.
As ${K_X}^2={K_{X_s}}^2$ we find
\begin{equation}
c_1\brtX^2 = {K_{\smtX}}^2 = {K_{X_s}}^2-3\nu.
\end{equation}
Every triple point diminishes $c_1\brtX^2$ by three.
\boldmath\item[The Euler number $e\brtX=c_2\brtX$:]\unboldmath the drop in the Euler number is purely local and can be computed by means of topological considerations from the Milnor number $\mu\left(\smash{\tilde{E}_6}\right)=8$ and the Euler numbers $e\left(E_i\right)=e\left(\mbox{\it torus\/}\right)=0$ and $e\left(\mbox{\it point\/}\right)=1$ as follows:
\begin{align*}
e\brtX &= e\left(X_s\right) + \nu\left(-e\left(\mbox{\it point\/}\right)
+e\left(E_i\right)
-\mu\left(\smash{\tilde{E}_6}\right)\right)\\
&= e\left(X_s\right) - 9\nu.
\end{align*}
So every triple point diminishes the Euler number by nine.
\boldmath\item[The holomorphic Euler characteristic $\chi\left(\cO_{\smash{\tX}}\right)$:]\unboldmath the Noether formula on $\tilde{X}$ says $\chi\left(\cO_{\smash{\tX}}\right)=\left(c_1\brtX^2+c_2\brtX\right)/12$, hence
\begin{equation*}
\chi\left(\cO_{\smash{\tX}}\right) = \chi\left(\cO_{X_s}\right) -\nu.
\end{equation*}
Every triple point diminishes the holomorphic Euler characteristic by one. \end{description}
A quick (cheating) way of computing the above `drop' in invariants exploits the local character of the Chern invariants. So consider a smooth cubic, whose invariants are $c_1^2=3$ and $c_2=9$. A cubic with a triple-point is a cone over an elliptic curve, the resolution is a ruled surface over an elliptic curve, and hence $c_1^2=c_2=0$ giving indeed the `drops' three and nine, deduced above.
Now we come to the other invariants. The adjoint linear system on $X$ is cut out by those surfaces of degree $d-4$ which pass through every point of $\mathscr{S}$. Intuitively speaking, every triple point in general position puts a linear condition on the adjoint linear system of $X$, so it will diminish the geometric genus $p_g\left(\smash{\tX}\right)$ by one (if not already zero). Let $\alpha\in\mathbb{N}_0$ be the discrepancy defined by
\begin{equation*}
p_g\left(\smash{\tX}\right) = p_g\left(X_s\right) -\nu +\alpha.
\end{equation*}
As a consequence
\begin{equation*}
q\left(\smash{\tX}\right) = q\left(X_s\right) + \alpha.
\end{equation*}
Both $\tilde{X}$ and $X_s$ are K\"ahler, so we have the Hodge decompositions of $H^i(\tilde{X},\mathbb{Z})\otimes\mathbb{C}\cong H^i(\tilde{X},\mathbb{C})$ and $H^i(X_s,\mathbb{Z})\otimes\mathbb{C}\cong H^i(X_s,\mathbb{C})$. From the equalities $h^{p,q}=h^{q,p}=h^{4-p,q}$ and $b_0\left(\smash{\tX}\right) = b_0\left(X_s\right) = 1$ one easily computes
\begin{align*}
b_2\left(\smash{\tX}\right) &= b_2\left(X_s\right) -9\nu +4\alpha, \\
h^{1,1}\left(\smash{\tX}\right) &= h^{1,1}\left(X_s\right) -7\nu +2\alpha.
\end{align*}
The other Betti numbers and Hodge numbers do not give more information: $b_3\left(\smash{\tX}\right)=b_1\left(\smash{\tX}\right)=2q\left(\smash{\tX}\right)$, $h^{1,0}\left(\smtX\right)=h^{0,1}\left(\smtX\right)=h^{2,1}\left(\smtX\right)=h^{1,2}\left(\smtX\right)=q\left(\smash{\tX}\right)$, $h^{2,0}\left(\smtX\right)=h^{0,2}\left(\smtX\right)=p_g\left(\smash{\tX}\right)$ and $h^{0,0}\left(\smtX\right)=h^{2,2}\left(\smtX\right)=1$.
We list the invariants in terms of $d$, $\nu$ and $\alpha$ in the following table.
\begin{table}[H]
\centering
\begin{math}
\renewcommand{1.2}{1.2}
\begin{array}{|c||c|c|}\hline
& X_s & \tilde{X} \\\hline\hline
c_1^2 & d(d-4)^2 & d(d-4)^2-3\nu \\\hline
c_2 & d(d^2-4d+6) & d(d^2-4d+6)-9\nu \\\hline
\chi & d(d^2-6d+11)/6 & d(d^2-6d+11)/6-\nu \\\hline
p_g & \binom{d-1}{3} &
\binom{d-1}{3}-\nu+\alpha\\\hline
q & 0 & \alpha \\\hline
b_2 & d^3-4d^2+6d-2 & d^3-4d^2+6d-2 -9\nu +4\alpha \\\hline
h^{1,1} & d(2d^2-6d+7)/3 & d(2d^2-6d+7)/3 -7\nu +2\alpha \\\hline
\end{array}
\end{math}
\caption{The invariants of $X_s$ and $\tilde{X}$}
\end{table}
The Kodaira dimension $\kod{(\tX)}$ measures the growth of the plurigenera $P_n(\tilde{X})=h^0(\tilde{X},{\cal O}_{\tilde{X}}(nK_{\smtX}))$ as $n$ grows. In general we have \cite[ch.~I, thm.~7.2]{barth}
\begin{equation*}
\kod{(\tX)}\left\{\begin{array}{c@{\quad}l}
\geq 0 & \text{if $p_g\left(\smash{\tX}\right)\geq 1$ and}\\
\geq 1 & \text{if $p_g\left(\smash{\tX}\right)\geq 2$.}
\end{array}\right.
\end{equation*}
For surfaces of general type $P_2(\tilde{X})={K_{\smtX}}^2+\chi\left(\cO_{\smash{\tX}}\right)+\epsilon$, where $\epsilon$ is the number of exceptional divisors. Generally we have $P_n(\tilde{X})=\frac12n(n-1) ({K_{\smtX}}^2+\epsilon)+\chi\left(\cO_{\smash{\tX}}\right)$.
\section{Bounds for the number of triple points}
The surface $X$ can have only a finite number of ordinary triple points, the maximal number depending on its degree $d$. Let $\mu_3\left(d\right)$ be the maximal number of ordinary triple points of a degree $d$ surface. We immediately find
\begin{equation*}
\mu_3(1)=\mu_3(2)=0,\quad
\mu_3(3)=\mu_3(4)=1 \quad
\text{and}\quad
\mu_3(5)=5.
\end{equation*}
The only cubic surface with an ordinary triple point is the cone over a plane smooth elliptic curve. A quartic surface with two triple points is necessarily singular along the line joining these two points. As we have seen in the preceding section, for quintics the result is due to Gallarati \cite{gallarati}.
The position of the triple points cannot be too special, as also the maximal number of triple points of $X$ on a given curve or surface is bounded. Let $C\subset\mathbb{P}^3$ be a curve of degree $c$ and $V\subset\mathbb{P}^3$ a surface of degree $v$. \begin{lemma}\label{lemma:position}\noindent
\begin{itemize}
\item[\rm 1)] $C$ contains at most $c(d-1)/2$ triple points
of $X$ \lh(with multiplicity\rh).
\item[\rm 2)] If $V$ and $X$ do not have a common component, then
$V$ contains at most $vd(d-1)/6$ triple points
of $X$ \lh(with multiplicity\rh).
\end{itemize}
\end{lemma}
{\noindent\it Proof:\enspace } Consider the linear system $\mathscr{L}_p$ of polar surfaces of $X$, i.e.~the linear system generated by the partial derivatives of the degree $d$ polynomial defining $X$. Then $\mathscr{S}$ is exactly the base locus of $\mathscr{L}_p$ and the general member $X_p\in\mathscr{L}_p$ is a degree $d-1$ surface which is smooth except ordinary double points in the triple points of $X$. So $X_p$ does not contain a component of $C$.
In every triple point $P\in\mathscr{S}$ the intersection multiplicity of $C$ and $X_p$ in $P$ is $\operatorname{mult}_P(C,X_p)\geq 2$ and thus $2\nu\leq C \cdot X_p = c(d-1)$. This proves 1).
The surfaces $V$, $X$ and $X_p$ intersect in a finite number of points. In every triple point $P\in\mathscr{S}$ the intersection multiplicity of $V$, $X$ and $X_p$ in $P$ is $\operatorname{mult}_P(V,X,X_p)\geq 6$. Hence $6\nu\leq V \cdot X\cdot X_p = vd(d-1)$ and 2) holds. $\quad\square$
\par
We will now discuss three bounds for $\mu_3\left(d\right)$ with $d\geq 6$: the polar bound, the Miyaoka bound and the spectrum bound.
{\bf The polar bound:} Suppose that $p_g\left(\smash{\tX}\right)\geq 1$. Taking a general adjoint surface $V=K_p$ we find using lemma \ref{lemma:position} 2)
\begin{equation*}
\nu\leq \frac{1}{6}\,d(d-1)(d-4).
\end{equation*} The condition $p_g\left(\smash{\tX}\right)\geq 1$ is satisfied for $d>6$. This can be seen as follows. Substitute $\alpha=p_g\left(\smash{\tX}\right)-\binom{d-1}{3}+\nu$, then the inequalities $1+\nu\leqh^{1,1}\left(\smash{\tX}\right)$ and $q\left(\smash{\tX}\right)\geq 0$ imply
\begin{equation*}
p_g\left(\smash{\tX}\right)\geq \frac{1}{24}\left(d-1\right)\left(2d^2-16d+21\right).
\end{equation*}
So for $d\geq 7$ we have even $p_g\left(\smash{\tX}\right)\geq 2$. The bound $\nu\leq \frac{1}{6}d(d-1)(d-4)$, which we call polar bound, even holds for $d=6$ in case $p_g\left(\smash{\tX}\right)=0$. Then $b_2\left(\smash{\tX}\right)=h^{1,1}\left(\smash{\tX}\right)$ and the equation $1+\nu\leqh^{1,1}\left(\smash{\tX}\right)$ gives
\begin{equation*}
\nu\leq\left\lfloor\frac{1}{18}(d-1)(d^2+d-3)\right\rfloor
= 10. \end{equation*}
We get the following table.
\begin{table}[H]
\centering
\begin{math}
\begin{array}{c|cccccccc}
d & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\\hline
\nu\leq{} & 6 & 10 & 21 & 37 & 60 & 90 & 128 & 176
\end{array}
\end{math}
\caption{The polar bound}
\end{table}
{\bf The Miyaoka bound:} Miyaoka's famous bound \cite{miyaoka} applies only to quotient singularities of surfaces with nonnegative Kodaira dimension. However there is a generalisation by Wall \cite[cor.~2]{wall} which also applies to log-canonical singularities on surfaces with nonnegative Kodaira dimension. Applied to triple points, we get the bound
\begin{equation*}
\nu \leq \frac{2}{27}d(d-1)^2.
\end{equation*}
As $p_g\left(\smash{\tX}\right)\geq 2$ for $d=\deg(\tilde{X})\geq 7$ this bound holds for every surface with triple points of degree $\geq 7$ and we get the following table.
\begin{table}[H]
\centering
\begin{math}
\begin{array}{c|cccccccc}
d & 7 & 8 & 9 & 10 & 11 & 12 \\\hline
\nu\leq{} & 18 & 29 & 42 & 60 & 81 & 107
\end{array}
\end{math}
\caption{The Miyaoka bound}
\end{table}
{\bf The spectrum bound} \cite[sect.~14.3.2]{arnold} uses the semicontinuity of the spectrum of a singularity. Let $f\colon(\mathbb{C}^{n+1},0)\rightarrow(\mathbb{C},0)$ be the germ of an isolated hypersurface singularity with Milnor number $\mu$. Then the characteristic polynomial of the monodromy has $\mu$ eigenvalues which are roots of unity, and the Mixed Hodge Structure on the cohomology of the Milnor fibre gives a way to take logarithms. The precise definitions are not important for us now. The spectrum is easy to compute for a function of the form $f= x_0^{a_0}+ \ldots + x_n^{a_n}$: then the spectrum is the set of rational numbers (with multiplicity) of the form $i_0/a_0+...+i_n/a_n$ with the $i_j$ running from 1 to $a_j-1$. Specifically we can take $a_i=d$ for all $i$, and as the spectrum is invariant under $\mu$-constant deformations, we have it now for any homogeneous isolated singularity. A projective hypersurface with isolated singularities has a smooth hyperplane section, and the affine complement of the section is a small deformation of the affine cone over the hyperplane section, so a homogeneous isolated singularity. The important property of the spectrum is its semicontinuity, in the sense that for every open interval of length 1 the number of spectral numbers in it of the singularity in the special fibre is at least the sum of the spectral numbers in the same interval of all singularities in the general fibre of a 1-parameter deformation (of negative degree).
We consider the spectrum as a divisor on $\mathbb{Q}$. With this notation the spectrum for $d=5$ is
\newcommand{\spn}[1]{\!\left({\textstyle\frac{#1}{5}}\right)} \begin{equation*}
1\spn{3}+
3\spn{4}+
6\spn{5}+
10\spn{6}+
12\spn{7}+
12\spn{8}+
10\spn{9}+
6\spn{10}+
3\spn{11}+
1\spn{12}
\end{equation*}
The spectrum of an ordinary double point is $\left({\textstyle\frac{3}{2}}\right)$. As the open interval $\left(\frac{3}{5},\frac{8}{5}\right)$ contains $31$ spectral numbers, a quintic surface can contain at most 31 nodes.
The spectrum for an $\tilde{E}_6$ is
\begin{equation*}
1\spn{3}+
3\spn{4}+
3\spn{5}+
1\spn{6}.
\end{equation*}
The open interval $\left(\frac{4}{5},\frac{9}{5}\right)$ contains 40 spectrum numbers of the quintic, and seven of $\tilde{E}_6$, so $\lfloor\frac{40}{7}\rfloor = 5 $ is the spectrum bound. Analogous computations for higher degree give the following table.
\begin{table}[H]
\centering
\begin{math}
\begin{array}{c|cccccccc}
d & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\\hline
\nu\leq{} & 5 & 11 & 17 & 29 & 45 & 60 & 84 & 114
\end{array}
\end{math}
\caption{The spectrum bound}
\end{table}
Putting all bounds together we arrive at the
\begin{proposition}\label{proposition:bound}
Let $X\subset\mathbb{P}^3$ be a surface of degree $d\geq 3$ with
$\nu$ ordinary triple points as its only singularities.
Then $\nu$ is bounded as given by the following table.
\begin{equation*}
\begin{array}{c|cccccccccc}
d & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline
\nu\leq{} & 1 & 1 & 5 & 10 & 17 & 29 & 42 & 60 & 81 & 107
\end{array}
\end{equation*}
\end{proposition}
We want to classify surfaces in $\mathbb{P}^3$ with only ordinary triple points. In contrast to surfaces with only ordinary double points, the class of a surface can change if more triple points come into play. The more triple points, the less nef $K_{\smtX}$ will become and this makes the Kodaira dimension eventually drop. The cases $d\leq 4$ being obvious, we can state the
\begin{proposition}\label{proposition:multiplicity}
If $d\geq 7$, then $\tilde{X}$ is minimal.
If $d=6$, then the smooth rational
$(-1)$-curves on $\tilde{X}$ come from
rational curves $C$ of degree $c\geq2$
on $X$ through $2c+1$
triple points \lh(with multiplicity\rh), while for
$d=5$ they come from curves of degree
$c\geq1$ on $X$ through $c+1$
triple points.
\end{proposition}
{\noindent\it Proof:\enspace } Let $C\subset X$ be a rational curve of degree $c$ such that $\tilde{C}\subset\tilde{X}$ is a smooth rational $(-1)$-curve. Then
$-2=\degK_{\smash{\tC}}={(K_{\smtX}+\tilde{C})}|_{\tilde{C}}= K_{\smtX}\cdot\tilde{C}+\tilde{C}^2 = c(d-4)-\operatorname{mult}\left(C,\mathscr{S}\right)-1$. Hence
\begin{equation*}
\operatorname{mult}\left(C,\mathscr{S}\right) = c(d-4)+1.
\end{equation*}
Applying lemma \ref{lemma:position} 1) we find that
\begin{equation*}
c(d-4)+1= \operatorname{mult}\left(C,\mathscr{S}\right) \leq \frac{1}{2}\,c(d-1)
\end{equation*}
and consequently $c(d-7)\leq -2$. This implies $d\leq 6$ and $c\geq 2$ for $d=6$. $\quad\square$
\par
\begin{corollary}\label{corollary:minimal}
If $d\geq 7$, then $X$ is a minimal surface of general
type.
\end{corollary}
{\noindent\it Proof:\enspace } For $d\geq 7$ $\tilde{X}$ is minimal by proposition \ref{proposition:multiplicity}. $K_{\smtX}$ is effective,
so by the Enriques-Kodaira classification we just have to show $c_1\brtX^2 >0$. But $c_1\brtX^2 = d(d-4)^2-3\nu$, so $c_1\brtX^2\leq 0$ iff
\begin{equation*}
\nu\geq\frac{1}{3}d(d-4)^2.
\end{equation*}
Playing this inequality against the Miyaoka bound gives a contradiction.
$\quad\square$
\par
\section{Sextics}
\subsection{Exceptional curves} We intend to study the $(-1)$-curves on $\tilde{X}$. The amazing thing is that there are severe restrictions. The consequences of lemma \ref{lemma:position} and proposition \ref{proposition:multiplicity} are as follows.
\begin{itemize}
\item At most two triple points lie on a line $L\subset\mathbb{P}^3$.
\item At most five triple points lie in a plane $H\subset\mathbb{P}^3$
and if so, then $H\cdot X=3C$ for a smooth conic $C$,
and $\tilde{C}\subset\tilde{X}$ is
a smooth rational $(-1)$-curve.
\end{itemize} The conic through five triple points gives the simplest example of a $(-1)$-curve (this really does occur, see section \ref{section:less} for explicit examples). Such a conic will be called a $(-1)$-conic; by abuse of notation we also call the curve $C\subset X$ a $(-1)$-curve, if $\tilde{C}\subset\tilde{X}$ is an exceptional curve of the first kind. The next possible candidates would be a twisted cubic curve through seven triple points and a rational quartic curve through nine triple points. Surprisingly the twisted cubic is impossible.
\begin{proposition}\label{proposition:cubic}
At most six triple points lie on a cubic curve
$D\subset X$.
\end{proposition}
{\noindent\it Proof:\enspace } By lemma \ref{lemma:position} 1), a cubic curve contains at most seven triple points. So assume that $\operatorname{mult}(D,\mathscr{S})=7$. Then $D$ cannot be a plane curve and $D$ cannot split in three lines. So either $D=C+L$ for a nondegenerate conic $C$ and a line $L$ with $\operatorname{mult}(C,\mathscr{S})=5$ and $\operatorname{mult}(L,\mathscr{S})=2$ or $D$ is a twisted cubic curve.
In the first case let $H$ be the plane containing $C$. Then $H\cdot X=3C$, so $L$ and $C$ meet in one point $P\in X\setminus\mathscr{S}$. But then $L\cdot X\geq 7$, so $L\subset X$. Hence $L\subset T_{P,X}=H$, contradiction.
In the second case let $N$ be the net of quadrics with the twisted cubic $D$ as its base locus. So the general (smooth) quadric $Q\in N$ intersects $X$ in $S$ and a residual curve $D_Q$ of type $(4,5)$ with double points in $D\cap\mathscr{S}$. But on $Q$ we have $D\cdot D_Q=(2,1).(4,5)=14$. Hence $D\cap D_Q=D\cap\mathscr{S}$ for the general $Q\in N$. But for every $P\in D\setminus\mathscr{S}$ there exists a pencil of quadrics $N_P\subset N$ having contact to $X$ at $P$. This implies that $P\in D_Q$ for all $Q\in N_P$. Now two things can happen. Either $N_P\subset N$ moves if we move $P$ on $D\setminus\mathscr{S}$ or $N_P$ is constant. If $N_P$ moves it will sweep out a Zariski open subset of $N\simeq\mathbb{P}^2$. Then $D\cap D_Q=D\cap\mathscr{S}$ cannot hold for the general element of $N$. So $N_P$ is constant and for every $Q\in N_P$, $X$ has contact to $Q$ along $D$. But then for two elements $Q\neq Q'\in N_P$ one has $Q\cdot Q'=2D+D'$ for some curve $D'$, contradiction. $\quad\square$
\par
\begin{corollary}\label{corollary:cubic}
There is no $(-1)$-curve of degree three on $X$.
\end{corollary}
In the case $p_g\left(\smash{\tX}\right)\geq1$ we find further restrictions for the number of $(-1)$-curves and their degrees. First we show that every $(-1)$-curve is irreducible. Whenever the canonical divisor $K_{\smtX}$ is effective, any exceptional divisor $E$ is automatically a component, as $K_{\smtX}\cdot E=-1<0$. Therefore $E$ comes from a rational curve on $X$ of degree $c$ (by Proposition \ref{proposition:multiplicity} through $2c+1$ triple points), which is contained in the base locus of the system of quadrics through the triple points. This is the adjoint system; we will call every quadric in it a {\em canonical surface}.
\begin{proposition}\label{proposition:disjoint}
Let $C$ be an irreducible $(-1)$-curve on $X$, let $K$ be a canonical
divisor \lh(of degree 12\rh) and $C'$ the residual curve of $C$ in $K$.
Then the strict transform $\tilde{C}\subset \tilde{X}$ of $C$ is disjoint from
the strict transform $\tilde{C}'$ of $C'$.
\end{proposition}
{\noindent\it Proof:\enspace } First suppose that $C$ is a conic. Then the residual curve $C'$ has degree $10$, and no component of it lies in the plane through $C$. Therefore the intersection multiplicity $C\cdot C'$ is at most $10$. As $C'$ has multiplicity $2$ in each of the five triple points on $C$, the intersection multiplicity is exactly 10 and $\tilde{C}$ is disjoint from $\tilde{C}'$.
If $\deg C \geq 4$, then $C$ lies on an irreducible quadric $Q$. We shall show that $\tilde{C}$ is disjoint from $\tilde{C}'$ on the blow up of $Q$ in the points $P\in\mathscr{S}$. We first suppose that $Q$ is smooth. Then $C$ is a curve of type $(a,b)$ with arithmetic genus $p_a=(a-1)(b-1)$, and $C'$ has type $(6-a, 6-b)$, so $C\cdot C'=6(a+b)-2ab$. Suppose that $C$ has multiplicity 3 in $\tau$ points $P\in\mathscr{S}$, multiplicity $2$ in $\delta$ points, and passes simply though $\sigma$ points. Then $3\tau +2\delta +\sigma= 2(a+b)+1$. As $C$ is rational we have that $3\tau +\delta \leq p_a$. This gives $\delta+\sigma\geq 3(a+b)-ab$. The multiplicity of $C\cup C'$ is three in each point $P\in\mathscr{S}$, so $C\cdot C'\geq 2\delta + 2\sigma$. Therefore we find $$ \delta+\sigma\geq 3(a+b)-ab \geq \delta+\sigma\;. $$ So $C$ intersects $C'$ only in points $P\in\mathscr{S}$ and the blow up of these points separates both curves.
The case that $Q$ is a quadric cone with vertex outside $\mathscr{S}$ is handled in the same way. As $C$ is smooth outside $\mathscr{S}$ and there does not intersect $C'$ we conclude that $C$ does not pass through the vertex.
Finally we investigate the case that the vertex of $Q$ is a point $P\in \mathscr{S}$. Then $K=C\cup C'$ has multiplicity $6$ in $P$. Let $\overline Q$ be the blow up of $Q$ in the point $P$. Its Picard group is generated by $E$ and $f$, with $E^2=-1$, $E\cdot f=1$ and $f^2=0$; we have $K_{\overline Q}\sim -2E-4f$. The strict transform of $K$ is a curve of type $3E+12f$. Let $C$ have multiplicity $m$ in $P$, then its strict transform $\overline C$ is a curve of type $aE+(2a+m)f$, with $p_a(\overline C)=(a-1)(a+m-1)$. We have $\overline C'\sim (3-a)E+(12-2a-m)f$, so $\overline C\cdot \overline C'=12a+3m-2a(a+m)$. Let $\overline C$ have $\tau$ triple, $\delta$ double and $\sigma$ simple points in $\mathscr{S}\setminus\{P\}$. Then $3\tau +2\delta +\sigma= 4a+m+1$, $3\tau +\delta \leq p_a(\overline C)$ and $\overline C\cdot \overline C'\geq 2\delta + 2\sigma$. Therefore $$ \delta+\sigma\geq 6a+2m -a(a+m)\geq \delta+\sigma+\frac 12 m\;. $$ We conclude that $m=0$, so $C$ does not pass through $P$, and that $\tilde{C}$ is disjoint from $\tilde{C}'$. $\quad\square$
\par
\begin{corollary}\label{proposition:curves}
If $p_g\left(\smash{\tX}\right) =1$, then the degree of
every $(-1)$-curve is one of $\{2,4,5,6,7,8\}$.
Moreover there are at most
$6$ such disjoint curves.
If $p_g\left(\smash{\tX}\right) =2$ there are at most two $(-1)$-curves
of degree 2 or 4 and if $p_g\left(\smash{\tX}\right) \geq 3$, there is
at most one $(-1)$-curve of degree $2$.
\end{corollary}
{\noindent\it Proof:\enspace } In first case all $(-1)$-curves are contained in the unique canonical curve of degree $12$. Moreover $c_1\brtX^2=-3$ and $p_g\left(\smash{\tX}\right)\neq 0$, so $\tilde{X}$ has at least three $(-1)$-curves, which are disjoint by proposition \ref{proposition:disjoint}. In the second case the base locus of the adjoint system is a curve of degree $\leq 4$. In the last case the base locus of of the adjoint system is a curve of degree $\leq 3$. Now the proposition follows because there are no $(-1)$-curves of degree 1 or 3. $\quad\square$
\par
We can now determine $p_g\left(\smash{\tX}\right)$ for $\nu=9, 10$.
\begin{corollary}\label{corollary:nine}
If $\nu=9$, then $p_g\left(\smash{\tX}\right)=1$.
\end{corollary}
{\noindent\it Proof:\enspace } Let $\nu=9$, then $p_g\left(\smash{\tX}\right)\geq 1$. If $p_g\left(\smash{\tX}\right)\geq 2$ corollary \ref{proposition:curves} implies that $\tilde{X}$ has at most two $(-1)$-curves, which contradicts $c_1\brtX^2=-3$ and $p_g\left(\smash{\tX}\right)\neq 0$. $\quad\square$
\par
\begin{corollary}\label{corollary:ten}
If $\nu=10$, then $p_g\left(\smash{\tX}\right)=0$.
\end{corollary}
{\noindent\it Proof:\enspace } Let $\nu=10$, then $c_1\brtX^2=-6$. So if $p_g\left(\smash{\tX}\right)>0$, then $\tilde{X}$ contains at least six $(-1)$-curves. Then $p_g\left(\smash{\tX}\right)=1$ by corollary \ref{proposition:curves} . The only possibility for six $(-1)$-curves is six conics $C_1$, \dots, $C_6$ which make up $K_{\smtX}$. Blowing down the six conics gives a minimal surface $\overline{X}$ with $c_1\left(\smash{\oX}\right)^2=0$, $c_2\left(\smash{\oX}\right)=12$ and $K_{\oX}=\cO_{\oX}$. But there is no such surface in the Enriques-Kodaira classification. $\quad\square$
\par
In fact, an $X$ with 10 triple points and $p_g\left(\smash{\tX}\right)=1$ would have an equation of the form $h_1\cdots h_6 +q^3$ where the $h_i$ define planes. Six planes intersect in 20 triple points. It is possible to choose 10 of them under the condition that no three lie on a line, but those points never lie on a quadric. We found our first example of a sextic with 9 triple points by taking $h_1\cdots h_6 +q^3$ with $q$ defining a quadric through 9 of the 10 points, chosen as required.
\par
Now let $C\subset\tilde{X}$ be a rational quartic curve such that $\tilde{C}\subset\tilde{X}$ is a smooth rational $(-1)$-curve. $C$ is contained in a smooth quadric surface and is either of type $(2,2)$ or $(1,3)$.
\begin{itemize}
\item If $C$ is of type $(2,2)$, then $C$ has one double
point in a triple point of $X$. Moreover $C$ passes simply
through seven other triple points. $C$ is the base locus
of a pencil of quadrics whose general member is smooth.
\item If $C$ is of type $(1,3)$, then $C$ is smooth and passes
simply through nine triple points of $X$.
\end{itemize}
The case of a quartic $(-1)$-curve of type $(1,3)$ turns out to be impossible.
\begin{lemma}\label{lemma:quartic}
Every quartic $(-1)$-curve on $X$ is of type $(2,2)$.
\end{lemma}
{\noindent\it Proof:\enspace } Assume that $C_1$ is a rational quartic $(-1)$-curve on $X$ of type $(1,3)$. Then $C_1$ is smooth and passes simply through nine triple points of $X$, so $\nu\in\{9,10\}$. Moreover $C_1$ is contained in a unique smooth quadric $Q=\{q=0\}$. We have two cases.
\boldmath {\bf Case $\nu=9$:}\unboldmath\enspace Then $Q$ is the unique canonical surface (corollary \ref{corollary:nine}). Every $(-1)$-curve is contained in the degree twelve curve $K=Q\cdot X$. No five triple points lie on a plane, so there exist no $(-1)$-conics. But $c_1\brtX^2=-3$ and $p_g\left(\smash{\tX}\right)=1$, so there are at least three $(-1)$-curves. By corollary \ref{proposition:curves}, the only possibility is three $(-1)$-curves: the curve $C_1$ of type $(1,3)$ and two other quartic $(-1)$ curves $C_2$ and $C_3$ of types $(3,1)$ and $(2,2)$. Both $C_1$ and $C_2$ have multiplicity one in all points of $\mathscr{S}$, whereas $C_3$ misses one triple point. This contradicts $K=C_1+C_2+C_3$ and $\operatorname{mult}(K,P)=3$ for all $P\in\mathscr{S}$.
\boldmath {\bf Case $\nu=10$:}\unboldmath\enspace By corollary \ref{corollary:ten} we have $p_g\left(\smash{\tX}\right)=0$, so $Q$ passes exactly through nine triple points. If $X=\{f=0\}$, then a general element of the pencil defined by $\alpha f+\beta q^3=0$ is a sextic with $\nu=9$ ordinary triple points and $C_1$ as $(-1)$-curve. Hence we are done using the first case.$\quad\square$
\par
As a further consequence we get the useful
\begin{corollary}\label{corollary:conics}
If $C_1$, $C_2\subset X$ are two different $(-1)$-conics,
then $C_1$ and $C_2$ meet in two distinct
triple points of $X$.
\end{corollary}
{\noindent\it Proof:\enspace } Let $H_i$ be the plane containing $C_i$, $i=1,2$. Then $H_i\cdot X=3C_i$, so $C_1\cap C_2\neq\emptyset$. Moreover $H_i$ is a tangent plane to $X$ at every point of $C_i$. Thus every point of $C_1\cap C_2$ is a singular point, i.e.~a triple point. If there is just one such point, then $\nu\geq 9$ and $p_g\left(\smash{\tX}\right)=11-\nu$, which contradicts one of the corollaries \ref{corollary:nine} and \ref{corollary:ten}. $\quad\square$
\par
Assume that $X$ has $\nu=9$ triple points $P_1$, \dots, $P_9$. Let $Q$ be the unique (irreducible) canonical quadric surface and let $K=Q\cdot X$ be the adjoint curve. The resolution $\tilde{X}$ has at least three disjoint $(-1)$-curves, which all are components of $K$. There are two main possibilities: either $K$ is the union of all $(-1)$-curves or not. In the first case blowing them down gives a minimal surface $\overline{X}$ with $K_{\oX}=\cO_{\oX}$, so $c_1\left(\smash{\oX}\right)^2=0$ and there are exactly three $(-1)$-curves with degrees $c_1$, $c_2$ and $c_3$. It follows from corollary \ref{proposition:curves} that up to permutation
\begin{equation*}
(c_1,c_2,c_3)\in\{(2,2,8),(2,4,6),(2,5,5),(4,4,4)\}.
\end{equation*}
In the second case we end up with an effective canonical divisor after blowing down. In this case up to permutation the possible degrees are
$$
\displaylines{
(2,2,2),(2,2,2,2),(2,2,2,2,2), (2,2,4),(2,2,2,4),
(2,2,5),(2,2,2,5),\cr(2,2,6),(2,2,2,6),(2,2,7),
(2,4,4),(2,4,5).}
$$
First we will rule out some cases. For the remaining cases, we will give explicit examples, when making
a tour from zero to ten triple points, studying all possible cases. \begin{proposition}\label{proposition:makenot_up}
If the $(-1)$-curves do not make up $K$, then there are exactly
three with degrees $(2,2,2)$ or $(2,2,4)$.
\end{proposition}
{\noindent\it Proof:\enspace } We exclude all other possibilities case by case.
Suppose first that there are three $(-1)$-conics. As there are only nine triple points the only possibility is that the three planes containing the conics have only one point in common and that each intersection line contains two triple points, while the remaining three points each lie in only one plane. Assume now that there is a fourth $(-1)$-conic. Its five triple points have to lie on the intersection triangle with the first three planes with at least two triple points in the vertices. This implies that there are three triple points on a line, thus excluding the cases $(2,2,2,2)$ and $(2,2,2,2,2)$. Also $(2,2,2,4)$ is not possible: a quartic $(-1)$-curve $C$ passes through $8$ triple points, and there is at least one plane $H_i$ containing five of them, but $C\cdot H_i=4$.
Case $(2,2,5)$: As $\operatorname{mult}(C_3,\mathscr{S})=11=9+2$ the curve $C_3$ has either one triple point or two double points in $\mathscr{S}$. But every irreducible degree five curve on $Q$ has arithmetic genus 0 or 2. Every double point drops the genus by at least one, every triple point by at least three (as $C_3$ does not pass through the vertex of $Q$ if $Q$ is singular). The only possibility is that $C_3$ has two double points. As $C_3$ is not a plane curve, it meets the plane of a $(-1)$-conic simply in the five triple points. But the two $(-1)$-conics contain together eight triple points, so $C_3$ can have at most one double point, contradiction. This excludes also the case $(2,2,2,5)$.
The cases $(2,2,6)$, $(2,2,2,6)$ and $(2,2,7)$ are similar.
Case $(2,4,4)$: Both $C_2$ and $C_3$ are of type $(2,2)$ and have a singular point in $\mathscr{S}$ outside $H_1$. Counting intersection points of $C_2$ and $C_3$ gives a count $\geq 9$ (with multiplicity). This contradicts $C_2\cdot C_3=8$.
The case $(2,4,5)$ is similar. $\quad\square$
\par
We see that under the conditions of the proposition the degree of a $(-1)$-curve is always even. We shall show that this also holds if the degrees sum up to $12$ by excluding the case $(2,5,5)$. It is possible to construct a reducible curve on $Q$ consisting of a curve of type $(1,1)$, $(2,3)$ and $(3,2)$ with the required intersection behaviour. So we need a different type of argument.
\begin{proposition}\label{proposition:make_up}
If the $(-1)$-curves make up $K$, then
\begin{equation*}
(c_1,c_2,c_3)\in\{(2,2,8),(2,4,6),(4,4,4)\}.
\end{equation*}
\end{proposition}
{\noindent\it Proof:\enspace } Let $(c_1,c_2,c_3)=(2,5,5)$, heading for a contradiction. Let $P_1$, \dots, $P_5$ be the five triple points on the conic $C_1$. On $\tilde{X}$ we have $3C_1\sim_{lin}H-E_1-\ldots-E_5$, so
\begin{equation*}
3(C_2+C_3)\sim_{lin} 5H-2(E_1+\cdots+E_5)-3(E_6+\cdots+E_9).
\end{equation*}
Therefore there exists a degree five surface $Y=\{g=0\}$ with multiplicity two at the points $P_1$, \dots, $P_5$ and triple points $P_6$, \dots, $P_9$ such that $f=hg-q^3$. Here $Q=\{q=0\}$ is the canonical quadric and $H=\{h=0\}$ is the plane containing $C_1=\{q=h=0\}$. As $Y$ intersects $Q$ in two irreducible curves of odd degree, $Y$ is itself irreducible. A plane through three triple points intersects $Y$ in the triangle formed by the triple points and a residual conic through them. If one of the five double points lies in the plane, the conic degenerates and has multiplicity two at the double point. But this would imply that three triple points of the original sextic lie on a line. Consider the reciprocal transformation centred in the triple points of $Y$. The image of $Y$ will be an irreducible cubic surface $Y'$ with five points of multiplicity two. Therefore $Y'$ has nonisolated singularities: it has a double line. So $Y$ itself has a double curve of degree at most three, passing through the five points $P_1$, \dots, $P_5$. This means that the conic $C_1$ is a component of the double line, so the sextic surface $X$ is singular along $C_1$. $\quad\square$
\par
\subsection{Sextics with seven or less triple points} \label{section:less}
For all surfaces $X$ with up to four triple points $\tilde{X}$ is minimal by proposition \ref{proposition:multiplicity}. No three triple points lie on a line, thus $p_g\left(\smash{\tX}\right)=10-\nu$. There are no constraints on the position of the triple points except that no three are on a line, so we get an equation of $X$ just by solving linear equations in the coefficients. If four triple points $P_1$, $P_2$, $P_3$, $P_4$ lie in a plane $H\subset\mathbb{P}^3$, then $X\cdot H$ is a degree six curve with four triple points and hence splits into three conics: $X\cdot H=C_1+C_2+C_3$. Then $X$ has an equation of the form
\begin{equation*}
hg+q_1q_2q_3=0
\end{equation*}
with $H=\{h=0\}$ and $C_i=\{h=q_i=0\}$, $i=1,2,3$. Here $g$ is a degree five form which vanishes in $P_1$, \dots, $P_4$ to the second order. In any case, the base locus of the system of adjoint surfaces consists only of the triple points.
Imposing a fifth triple point $P_5$ opens the possibility of a $(-1)$-curve. This happens if and only if $P_1$, \dots, $P_5$ lie on a plane $H$. Then $X\cdot H=3C$ for a nondegenerate conic $C$ and $\tilde{C}\subset\tilde{X}$ is the $(-1)$-curve. Every such surface has an equation of the form
\begin{equation*}
hg+q^3=0
\end{equation*}
with $H=\{h=0\}$ and $C=\{h=q=0\}$. Here $g$ is a degree five form vanishing doubly in $P_1$, \dots, $P_5$. Then the base locus of the system of adjoint surfaces is exactly $C$. If the five triple points do not lie in a plane we find $\{P_1$, \dots, $P_5\}$ as base locus. In any case $p_g\left(\smash{\tX}\right)=5$.
Let $P_6$ be another triple point. Since six triple points cannot lie on a conic the geometric genus will drop by one: $p_g\left(\smash{\tX}\right)=4$. We end up with the same cases as for $\nu=5$ yielding as base locus $C\cup\{P_6\}$ resp. $\{P_1,\ldots,P_6\}$.
Examples of sextics with $\nu\leq 6$ triple points and base locus $\{P_1,\ldots,P_{\nu}\}$ of the adjoint system can be given as follows. Let $Q_i=\{q_i=0\}$ be generators of the linear system of quadrics through $\{P_1,\ldots,P_{\nu}\}$, $i=0$, \dots, $10-\nu$. Then the general element of the linear system spanned by the mixed third powers of the $q_i$ has only triple points in $P_1$, \dots, $P_\nu$. For $\nu\leq5$ every surface is of this form. For $\nu=6$ the linear system of mixed third powers has dimension $\binom 63-1=19$ while the system of all sextics with triple points has dimension $\binom 96-1-6\cdot10=23$.
Now we go for a seventh triple point $P_7$. Again the geometric genus drops: $p_g\left(\smash{\tX}\right)=3$. The base locus cannot be a degree three curve by proposition \ref{proposition:cubic}. If $P_1$, \dots, $P_5$ lie on a conic $C$, the base locus is $C\cup\{P_6,P_7\}$. If not, the net of quadrics defined by $P_1$, \dots, $P_7$ has a zero-dimensional base of the form $\{P_1,\ldots,P_7,P\}$ for a eighth point $P\in\mathbb{P}^3$, which may be infinitely near to one of the points $P_1$, \dots, $P_7$. Now the mixed third powers of the $q_i$ have an additional singularity in $P$. They form a system of dimension $9$, which is four less than the dimension of the system of all sextics with triple points. To find an equation of such a surface it suffices to give one possibly reducible sextic not passing through $P$; the surface obtained by adding a general combination of third powers has then only $7$ triple points. As such an extra sextic we can take the product $g_1g_2$ of a cubic $g_1$ with nodes in $P_1$, \dots, $P_4$ passing simply through $P_5$, $P_6$, $P_7$ and a cubic $g_2$ with nodes in $P_5$, $P_6$, $P_7$ passing simply through $P_1$, \dots, $P_4$.
In all cases considered so far we end up with $c_1\brtX^2>0$ and $p_g\left(\smash{\tX}\right)\geq 3$, so $\tilde{X}$ is a surface of general type.
\subsection{Eight triple points}
We distinguish the sextics with eight triple points by their geometric genus and their $(-1)$-curves.
We can always choose seven of the eight points so that no five lie on a conic. These seven triple points $P_1$, \dots, $P_7$ (no three on a line, no five on a conic) determine a net of quadrics spanned by $Q_i=\{q_i=0\}$, $i=1,2,3$.
{\bf The case $p_g\left(\smash{\tX}\right)=3$:} this means that the eighth triple point $P_8$ is the eighth base point of the net. For a general ternary cubic form $f_3$ the surface $X=\{f_3(q_1,q_2,q_3)=0\}$ is a sextic with only triple points in $P_1$, \dots, $P_8$. Here $p_g\left(\smash{\tX}\right)=3$, thus $q\left(\smash{\tX}\right)=1$ and $\tilde{X}$ is minimal because $K$ is effective and has no fixed components, so $\tilde{X}$ is minimal properly elliptic. This elliptic surface is fibred over an elliptic curve, namely the plane elliptic curve given by $f_3=0$. The fibration is induced by the net, i.e. given by $(q_1,q_2,q_3)$ and each fibre is the base locus of a pencil of quadrics, in a way the reader easily can work out. Note that the elliptic resolutions of the triple points are sections of this fibration, and hence all isomorphic to the base. (This can also be seen by noting that the linear parts of the $q_i$'s at the basepoints are linearly independent.)
{\bf The case $p_g\left(\smash{\tX}\right)=2$:} we assume that $P_8$ is not a base point of the net. Let the pencil of quadrics through $P_1$, \dots, $P_8$ be spanned by $Q_1$ and $Q_2$. Let $C$ be the base locus of the pencil. For the $(-1)$-curves we can have four different cases:
\begin{itemize}
\item one $(-1)$-curve of type $(2,2)$,
\item two $(-1)$-conics,
\item one $(-1)$-conic and
\item no $(-1)$ curves at all.
\end{itemize}
{\bf One $(-1)$-curve of type $(2,2)$:} in this case the equation of $X$ has a very special form.
\begin{lemma}\label{lemma:equation}
If $\tilde{X}$ contains a quartic $(-1)$-curve $C$ of type $(2,2)$, then
the equation of $X$ has the form
\begin{equation*}
q_0g+q^3=0
\end{equation*}
with $Q_0=\{q_0=0\}$ a quadric cone through eight triple points
with vertex in one of them and
$Q=\{q=0\}$ a smooth quadric through the eight triple points.
The pencil of quadrics with base locus $C$ is spanned by
$Q$ and $Q_0$.
Moreover $Y=\{g=0\}$ is a quartic surface passing through the vertex
of $Q_0$ with seven double points in the other seven triple points.
\end{lemma}
{\noindent\it Proof:\enspace } Let $P_1\in\mathscr{S}$ be the double point of $C$ and let $P_2$, \dots, $P_7$ be the other triple points on $C$. Let $M$ be the pencil of quadrics with base locus $C$. The general element $Q\in M$ is smooth and intersects $X$ in $C$ and a residual curve $C_Q$ of type $(4,4)$ passing simply through $P_1$ and doubly through $P_2$, \dots, $P_7$. Since $C$ and $C_Q$ do not have a common component we have $C\cap C_Q = \{P_1,\ldots,P_8\}$ in view of $C\cdot C_Q=(2,2)\cdot(4,4)=16$. Now fix a point $P\in C\setminus\mathscr{S}$. There exists a $Q_0\in M$ which has contact to $X$ at $P$. In particular $P\in C_{Q_0}$, implying that $C$ and $C_{Q_0}$ have a common component. Hence $X\cdot Q_0=2C+C'$ for a curve $C'$ of type $(2,2)$. Thus $\operatorname{mult}(X\cdot Q_0,P_1)\geq4$, so $Q_0$ is singular in $P_1$. Hence $\operatorname{mult}(C',P_1)\geq 2$ and $X\cdot Q_0=3C'$ by Bezout. In view of $C\subset Q_0$ the quadric $Q_0$ has to be a quadric cone with vertex $P_1$. Then $X$ has an equation as demanded. $\quad\square$
\par
The condition that a quadric $\sum \lambda_i q_i$ in the net of quadrics through 7 points (in general position) is singular is that there exists a point $P$ in which all derivatives of $\sum \lambda_i q_i$ vanish. Eliminating the $\lambda_i$ gives that the maximal minors of $$ \def\part#1#2{\frac{\partial#1}{\partial#2}} \def\partx#1{\part {q_{#1}}{x} & \part{q_{#1}}{y} &
\part {q_{#1}}{z} &\part {q_{#1}}{w}}
\left| \matrix \partx 1 \\ \partx 2 \\ \partx 3\endmatrix
\right | $$ have to vanish. This locus is known as the Steiner curve of the net. For $Q$ we take a smooth quadric through the eight points. There exists a six parameter family of quartics through $P_8$ with only nodes in $P_1$, \dots, $P_7$ (quadrics in the net give a five parameter family, but the general quartic is not of that form). Let $Y=\{g=0\}$ be a general such quartic, then $X=\{q_0g+q^3=0\}$ is a sextic with eight triple points and one $(-1)$-curve $C$ of type $(2,2)$. Altogether this construction has 13 moduli: we find seven independent sextics with triple points in the given points, and the configuration of points has seven moduli. The minimal model $\overline{X}$ of $X$ has $c_1\left(\smash{\oX}\right)^2=1$, so $\tilde{X}$ is of general type.
Such a surface has a characterisation as the minimal model of a double cover of $F_2$ (the resolution of a quadric cone), branched along a 5-section disjoint from the minimal section (the node) and that section. If $S$ denotes a section with $S^2=2$, and $F$ a fibre, the branch curve is simply $5S+(S-2F)$. As $K=-2B$ we have $K+B=S-F=(S-2F)+F$ with $S-2F$ as fixed component. Thus the free part of the canonical pencil gives a genus two fibration.
A generic quadric in the pencil intersects the sextic in a $(6,6)$ curve which splits up into the fixed component (the $(2,2)$ curve) and a residual $(4,4)$ curve with seven double points, passing simply through the node $P_8$ of the $(2,2)$ curve. The genus of the desingularisation is indeed $3\times 3-7=2$, giving us the genus two pencil.
In fact if $\{q+tq_0=0\}$ is a quadric in the pencil, the residual curve is the intersection $\{q+tq_0=tq^2+g=0\}$. The canonical system on such a genus two curve is given by adjunction as the pencil of $(2,2)$ curves passing through the seven nodes. Thus the involution on the surface will be given as follows. For each point $P$ choose $t$ such $q+tq_0$ vanish at $P$, then consider the residual intersection with the base locus of the pencil of quadrics through $P_1$, \dots, $P_7$, $P$ and the quartic $\{tq^2+g=0\}$.
{\bf One $(-1)$-conic:} such a sextic can be obtained as the reciprocal transform of the previous sextic in $P_1$, $P_2$, $P_3$ and $P_8$ (which necessarily do not lie in one plane). The quadric $Q$ transforms into a smooth quadric $Q'=\{q'=0\}$, $Q_0$ transforms into a plane $H_0'=\{h_0'=0\}$ and $Y$ transforms into a quintic $Y'=\{g'=0\}$ with three triple points and five double points. So the transform $X'$ of $X$ satisfies the equation $h_0'g'+{q'}^3=0$. The image of $C$ is a $(-1)$-conic lying in the plane $H_0'$. Conversely, every such sextic can be transformed into one with a $(-1)$-curve of type $(2,2)$: just take as fundamental points the three triple points not on $H_0'$ and a fourth triple point on $H_0'$. The base locus of the pencil of quadrics consists of the $(-1)$-conic and another conic not contained in $X'$. This family again has 13 moduli.
{\bf Two $(-1)$-conics:} the sextics $X$ with two $(-1)$-conics $C_1$ and $C_2$ are easily identified as those satisfying an equation of the form $h_1h_2g+q^3=0$. Here $H_i=\{h_i=0\}$ are planes containing $C_i$, $i=1,2$. By corollary \ref{corollary:conics}, the two conics intersect in two triple points, say $P_1$ and $P_2$. Then $Y=\{g=0\}$ is a quartic surface through $P_1$ and $P_2$ with only nodes in $P_3$, \dots, $P_8$ and $Q=\{q=0\}$ is a general element of the pencil. There are ten linearly independent sextics with the eight triple points: four from the pencil spanned by $h_1h_2$ and $q$, another four of the form $h_1^2h_2^2q_1$ where $q_1$ is one of the four quadrics through $P_3$, \dots, $P_8$ and finally $h_1h_2^2k_1$, $h_1^2h_2k_2$ for cubics $k_1$ and $k_2$ through all eight points such that $k_i$ has double points in the triple points not contained in $h_i$, $i=1,2$. The point configuration has five moduli, so we get a 14 parameter family. Again every such sextic is of general type.
The minimal models will have $p_g=2$ and $c_1^2=2$ and come equipped with genus three fibrations. Those will be defined by, in analogy with the case of a $(-1)$ curve of type $(2,2)$, as residual $(4,4)$ curves, with nodes at $P_3\dots P_8$, passing through $P_1$, $P_2$. We leave it to the reader to work out the details.
{\bf No $(-1)$-curves:} it is not so immediate to construct such surfaces. We follow the classical construction for asyzygetic eight-nodal quartics \cite{cayley,rohn}, which we now recall. Seven points $P_1$, \dots, $P_7$ in general position define a net of quadrics spanned by smooth quadrics $Q_i=\{q_i=0\}$, $i=1,2,3$. All quartics with nodes in $P_1$, \dots, $P_7$ are given by $g+f_2(q_1,q_2,q_3)=0$, where $f_2$ is a ternary quadratic form and $g$ is a fixed nodal quartic, which we can take as the product of a cubic with nodes in $P_1$, \dots, $P_4$ passing simply through $P_5$, $P_6$, $P_7$ and a plane through $P_5$, $P_6$, $P_7$. A sufficient condition for an eighth singular point is that the derivatives $dg$, $dq_1$, $dq_2$ and $dq_3$ are linearly dependent. The vanishing of the determinant
\begin{equation*}
\begin{vmatrix}
\frac{\partial g}{\partial x} &
\frac{\partial q_1}{\partial x} &
\frac{\partial q_2}{\partial x} &
\frac{\partial q_3}{\partial x} \\[3pt]
\frac{\partial g}{\partial y} &
\frac{\partial q_1}{\partial y} &
\frac{\partial q_2}{\partial y} &
\frac{\partial q_3}{\partial y} \\[3pt]
\frac{\partial g}{\partial z} &
\frac{\partial q_1}{\partial z} &
\frac{\partial q_2}{\partial z} &
\frac{\partial q_3}{\partial z} \\[3pt]
\frac{\partial g}{\partial w} &
\frac{\partial q_1}{\partial w} &
\frac{\partial q_2}{\partial w} &
\frac{\partial q_3}{\partial w}
\end{vmatrix}
\end{equation*}
defines a degree six surface $\Delta$, called the Cayley dianode surface. It is the closure of the locus of points $P_8$ such that there exists a quartic surface with only nodes in $P_1$, \dots, $P_8$. In general the dianode surface has triple points in $P_1$, \dots, $P_7$ as its only singularities, but it can reducible: if four points lie in a plane this plane becomes a component.
We now look at sextics with seven triple points in general position. We find 14 such sextic equations: the ten third powers of the three quadrics
$q_1$, $q_2$ and $q_3$, three equations of the form $\{gq_i=0\}$ for a quartic
$g$ which is {\em not} a conic of the three quadrics and finally a sextic $\delta$ not of this form, for which one can take the Cayley dianode surface. The vanishing of all second derivatives in an eighth point $P_8$ gives ten linear equations in the 14 coefficients. Note however that the columns of the coefficient matrix are not linearly independent. We observe that the all second derivatives of a third power have the function itself as factor: $$ (f^3)_{ij}=3f(2f_if_j+ff_{ij})\;. $$ For a product we have $$ (f^2g)_{ij}= (2f_if_j+ff_{ij})g+f(2f_ig_j+2f_jg_i+f_{ij}g+fg_{ij}) $$ and a similar expression for the derivatives of $fgh$. Suppose that $f\neq0$. Then we can divide all partials $(f^3)_{ij}$ by $f$ and after subtracting the same multiple of $2f_if_j+ff_{ij}$ from all partials $(f^2g)_{ij}$ we can again divide by $f$. {}From $(fg^2)_{ij}$ we can get $2g_ig_j+gg_{ij}$. So if one of the quadrics does not vanish (i.e., the point is not the 8th base point of the net), the 10 columns of the third powers give at most 6 independent ones and our matrix reduces to a $10\times 10$ matrix. One obtains a determinant of degree 28, but the rank of the matrix is eight on the dianode surface, whose equation is double factor of the determinant. We are left with a surface $\Delta'$ of degree 16. It has as double curves the 21 lines joining the triple points, the 7 cubics though 6 of the 7 points and the Steiner curve of the net. The dianode surface $\Delta$ intersects $\Delta'$ exactly in its singular locus.
Now we get two possibilities to construct a sextic, by taking $P_8$ in $\Delta$ or in $\Delta'$. If $P_8$ is a general point of the Cayley dianode surface, there exists a quartic surface $Y=\{g=0\}$ with only nodes in $P_1$, \dots, $P_8$. Then a general linear combination of $gq_1$, $gq_2$, $q_1^3$, $q_1^2q_2$, $q_1q_2^2$, $q_2^3$ defines a sextic $X$ with only triple points in the eight points and containing the base locus of the pencil spanned by $Q_1$ and $Q_2$. The surface will be a (minimal) elliptic surface, whose elliptic pencil is given by the pencil of quadrics through the points $P_1$, \dots, $P_8$: the residual intersection on a smooth quadric of the pencil is a curve of type $(4,4)$ with $8$ double points. The point configuration having eight moduli, we get a 13 parameter family.
Choosing $P_8\in\Delta'$ general we obtain a sextic with eight triple points not containing parts of the base locus of the pencil. The number of parameters is now $8+4=12$. The surface is again a (minimal) elliptic surface, with elliptic pencil given by the pencil of quadrics: a general intersection is a $(6,6)$-curve with eight triple points, hence the geometric genus is one.
\subsection{Nine triple points}
Assume that $X$ has $\nu=9$ triple points $P_1$, \dots, $P_9$. Let $Q$ be the unique (irreducible) canonical quadric surface and let $K=Q\cdot X$ be the adjoint curve. By propositions \ref{proposition:make_up} and \ref{proposition:makenot_up} the resolution $\tilde{X}$ has exactly three disjoint $(-1)$-curves $C_1$, $C_2$, $C_3$. If $C_1+C_2+C_3=K$, blowing down $C_1$, $C_2$ and $C_3$ gives a minimal surface $\overline{X}$ with $c_1\left(\smash{\oX}\right)^2=0$, $c_2\left(\smash{\oX}\right)=24$ and $K_{\oX}=\cO_{\oX}$. Then $\tilde{X}$ is a $K3$ surface blown up in three points. Otherwise we end up with an effective canonical divisor after blowing down $C_1$, $C_2$ and $C_3$. This implies $\kod{(\tX)}=1$ and thus $\tilde{X}$ is the blowup of a minimal properly elliptic surface in three points. As it will have the same basic invariants as a $K3$ surface, it will be obtained from an elliptic such by a series of logarithmic transforms, i.e. making some elliptic
fibres multiple.
\subsubsection*{The sextic $K3$ surface}
Using corollary \ref{corollary:conics}, the multiplicities of the $(-1)$-curves in the nine triple points are easily found to be (up to a permutation of triple points) as in the following table.
\begin{table}[H]
\centering
\begin{tabular}{|l|l||c|c|c|c|c|c|c|c|c|}\hline
type &
&$P_1$&$P_2$&$P_3$&$P_4$&$P_5$&$P_6$&$P_7$&$P_8$&$P_9$\\\hline\hline
\multirow{3}{1.3cm}{$(4,4,4)$}
& $C_1$ & 2 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\\cline{2-11}
& $C_2$ & 0 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\cline{2-11}
& $C_3$ & 1 & 0 & 2 & 1 & 1 & 1 & 1 & 1 & 1 \\\hline\hline
\multirow{3}{1.3cm}{$(2,4,6)$}
& $C_1$ & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\\cline{2-11}
& $C_2$ & 0 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\cline{2-11}
& $C_3$ & 2 & 1 & 2 & 2 & 2 & 1 & 1 & 1 & 1 \\\hline\hline
\multirow{3}{1.3cm}{$(2,2,8)$}
& $C_1$ & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\\cline{2-11}
& $C_2$ & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 \\\cline{2-11}
& $C_3$ & 2 & 2 & 2 & 2 & 3 & 2 & 2 & 1 & 1 \\\hline\hline
\end{tabular}
\caption{multiplicities of the $(-1)$-curves}
\label{table:multK3}
\end{table}
\begin{proposition}
If $(c_1,c_2,c_3)=(4,4,4)$, then $X$ satisfies
an equation of the form
\begin{equation*}
q_1q_2q_3+q^3=0\;,
\end{equation*}
where $Q=\{q=0\}$ is the unique canonical surface. Each $Q_i=\{q_i=0\}$
is a quadric cone through eight triple points with vertex
in one of them and $C_i=Q\cdot Q_i$, $i=1$, $2$, $3$.
\end{proposition}
{\noindent\it Proof:\enspace } The three quartic $(-1)$-curves are of type $(2,2)$ by lemma \ref{lemma:quartic}. Lemma \ref{lemma:equation} guarantees the existence of three quadric cones $Q_i=\{q_i=0\}$ such that $X\cdot Q_i=3C_i$, $i=1,2,3$. Thus $X\cdot(Q_1+Q_2+Q_3)=3(C_1+C_2+C_3)$, hence a equation of $X$ and $q_1q_2q_3$ differ by the cube of a quadratic polynomial vanishing in all triple points. Since $p_g\left(\smash{\tX}\right)=1$, such polynomial defines the unique canonical surface. $\quad\square$
\par
Let $Q_i=\{q_i=0\}$ be the quadric cones with vertices $P_i$ such that $Q_i$ passes through $P_{i+1\bmod 3}$ but not through $P_{i+2 \bmod 3}$, $i=1,2,3$. We choose $P_1$, $P_2$ and $P_3$ as $(1\cn0\cn0\cn0)$, $(0\cn1\cn0\cn0)$ and $(0\cn0\cn1\cn0)$. If we require that the quadrics also pass through $(0\cn0\cn0\cn1)$ and $(1\cn1\cn1\cn1)$ their equations can be written (inhomogeneously) as \begin{align*}
q_1 &= z^2+a_1y+b_1z-(a_1+b_1+1)yz,\\
q_2 &= x^2+a_2z+b_2x-(a_2+b_2+1)xz,\\
q_3 &= y^2+a_3x+b_3y-(a_3+b_3+1)yx,
\end{align*} where the $a_i$ and $b_i$ are constants.
The quadrics intersect in a zero-dimensional scheme of length $8$ containing at least six distinct points. To find the canonical surface $Q=\{q=0\}$ we have to pick six points $P_4$, \dots $P_{9}$. It is easier to specify the remaining scheme of length two. For a Zariski open set of the stratum it will consist of two points in general position, which we take as $(0\cn0\cn0\cn1)$ and $(1\cn1\cn1\cn1)$. Thus we require that $Q$ does not pass through these two points. To compute its equation $q=0$ we note that the $q_i$ lie in the ideal defining $(1\cn1\cn1\cn1)$; they can be written in vector form as
\begin{equation*}
\begin{pmatrix}
q_1 \\
q_2 \\
q_3
\end{pmatrix} =
\begin{pmatrix}
0 & (b_1+z)z & (a_1-z)y \\
(a_2-x)z & 0 & (b_2+x)x \\
(b_3+y)y & (a_3-y)x & 0
\end{pmatrix}
\begin{pmatrix}
1-x \\
1-y \\
1-z
\end{pmatrix}
\;.
\end{equation*}
In the triple points of $X$ the $q_i$ vanish, while $(x,y,z)\neq(1,1,1)$. So the determinant of the matrix vanishes. Dividing it by $xyz$ gives the inhomogeneous equation
\begin{equation*}
q=(a_1-z)(a_2-x)(a_3-y)+(b_1+z)(b_2+x)(b_3+y)
\end{equation*}
which is indeed the sought after quadric (note that the two terms $xyz$ cancel). The general element of the pencil $\alpha q_1q_2q_3+\beta q^3$ defines a sextic $X$ with nine triple points with multiplicities as in the first part of table \ref{table:multK3}.
A particular example is obtained by taking $b_i=0$, $a_i=-1$, so $(q_1,q_2,q_3)=(z^2-y,x^2-z,y^2-x)$ and $P_{3+i}=(\eta^{4i},\eta^{2i},\eta^i)$ with $\eta$ a primitive seventh root of unity. The number of parameters in the construction is 7 (the $a_i$, $b_i$ and $(\alpha\mathop{\rm :}\beta)$).
Now consider the reciprocal transformation with fundamental points $P_1$, $P_2$, $P_4$ and $P_5$. In general the image $X'$ of $X$ will have nine ordinary triple points as only singularities. It can be checked that this is indeed the case for the particular example, where we take $(\eta^{4},\eta^{2},\eta)$ and $(\overline\eta^{4},\overline\eta^{2},\overline\eta)$ as third and fourth fundamental point. Then the image $H_1'$ of $Q_1$ is a plane, the image $Q_2'$ of $Q_2$ is a quadric cone and the image $Y_3'$ of $Q_3$ is a cubic surface with four nodes. The image $Q'$ of $Q$ being again a quadric, the sextic $X'$ is given by an equation of the form
\begin{equation*}
h_1'q_2'g_3'+{q'}^3=0.
\end{equation*}
Note that $C_1$, $C_2$ and $C_3$ are mapped onto $(-1)$-curves $C_1'$, $C_2'$ and $C_3'$ of degree $2$, $4$ and $6$.
Surfaces with an equation of this type form a seven dimensional family. The nodes of the cubic, the vertex of the quadric and two points in the plane can be specified, while the last two are then to be found among the four other intersection points of the cubic, the cone and the plane. The dimension of the Zariski tangent space to the equisingular stratum in the specific example is $15+7$, so we obtain a full seven parameter family of sextics with nine triple points and $(c_1,c_2,c_3)=(2,4,6)$.
Now let $X$ have $(c_1,c_2,c_3)=(2,4,6)$ and an equation of the form $h_1q_2g+q^3=0$, where $h_1$ determines a plane $H_1$, $q_2$ a quadric cone $Q_2$ and $g_3$ a four-nodal cubic $Y_3$. Consider the reciprocal transform with fundamental points $P_2$, $P_5$, $P_6$ and $P_7$ as in table \ref{table:multK3}. Continuing with the specific example above, one can take the images of the points $(\eta,\eta^4,\eta^2)$ and $(\overline\eta,\overline\eta^4,\overline\eta^2)$ as $P_6$ and $P_7$. The reciprocal image has nine ordinary triple points, so again this property holds on an open dense set in the parameter space of all sextics with $(c_1,c_2,c_3)=(2,4,6)$. Here the image $H_1'$ of $H_1$ is again a plane, the image $H_2'$ of $Q_2$ is also a plane and the image $Y'_4$ of $Y_3$ is a quartic surface with one triple point and six double points. The image $Q'$ of $Q$ is again a quadric, so the image $X'$ of $X$ is given by an equation of the form
\begin{equation*}
h_1'h_2'g'_4+{q'}^3=0.
\end{equation*}
This time the curves $C_1$, $C_2$ and $C_3$ are mapped onto $(-1)$-curves $C_1'$, $C_2'$ and $C_3'$ of degrees $2$, $2$ and $8$.
Once more we check that this is a full seven parameter family of sextic surfaces with nine triple points and $(c_1,c_2,c_3)=(2,2,8)$. A direct construction of the family starts with seven points in general position, of which we choose one as the triple point for the quartic $Y_4$ and divide the remaining six into two groups of three, each determining a plane $H_i$. The intersection $Y_4\cap H_1 \cap H_2$ consists of four points. Two of them can be triple points for a sextic (remember that three are not allowed on a line) in the pencil $\alpha q^3 + \beta h_1h_2g_4$. An explicit example starts coordinate vertices and the points $(\lambda:1:1:1)$, $(1:\lambda:1:1)$ and $(1:1:\lambda:1)$. We get $$ \displaylines{ h_1= t,\qquad h_2=x+y+z-(\lambda+2)t \cr \qquad g_4=\lambda(\lambda+1)(\lambda+2)(xy+xz+yz)^2-(2\lambda+1)^2(\lambda+1)xyz(x+y+z)
\cr
-\lambda(2\lambda+1)(xy+xz+yz)(x+y+z)t+(2\lambda+1)^2(\lambda+2)xyzt\;.\qquad\cr} $$ The intersection line $H_1\cap H_2$ is now a double tangent of $Y_4\cap H_1$ so we take the two points of tangency as last two triple points. We find $$ q=(2+\lambda)(xy+xz+yz)-(2\lambda+1)(\lambda+1)(x+y+z)t \;. $$
There is a certain amount of choice in the fundamental points of the reciprocal transformations. They depend also on the position of the triple points. So we can conclude that we obtain correspondences between our families of sextics with triple points, whose exact nature we did not determine. We will say that our families are `related via reciprocal transformations'.
We summarise our findings.
\begin{theorem}\label{theorem:K3}
For every $(c_1,c_2,c_3)\in\{(2,2,8),(2,4,6),(4,4,4)\}$ there
exists a seven parameter family of sextic surfaces with
nine triple points and three $(-1)$-curves of degrees
$c_1$, $c_2$ and $c_3$. The three families are
related via reciprocal transformations.
For every such surface $X$ its minimal desingularisation
$\tilde{X}$ is a $K3$ surface blown up in three points.
Moreover $X$ satisfies an equation of the form
\begin{equation*}
\begin{array}{l@{\quad}l}
q_1q_2q_3+q^3=0 & \text{if $(c_1,c_2,c_3)=(4,4,4)$.}\\[1ex]
h_1q_2g_3+q^3=0 & \text{if $(c_1,c_2,c_3)=(2,4,6)$,}\\[1ex]
h_1h_2g_4+q^3=0 & \text{if $(c_1,c_2,c_3)=(2,2,8)$.}
\end{array}
\end{equation*}
Here $Q=\{q=0\}$ is the unique canonical surface.
The three exceptional curves are in each case obtained
as the locus where one of the three forms in the product
and $q$ vanish.
In the last case $g_4$ defines a
quartic surface with a triple point
six double points.
In the second case, $g_3$ defines a four nodal cubic.
\end{theorem}
It may now be amusing to digress on the geometry of the $K3$ surfaces obtained. Consider the case $(4,4,4)$. Let $E_i$ be the image in the minimal $K3$ surface of the exceptional curve in the resolution of $P_i$. Notice that $E_i^2=2$ if $i\leq3$ and $E_i^2=0$ otherwise. Furthermore for $i\neq j$ we have $E_i\cdot E_j=2$ if $i,j\leq 3$ while $E_i\cdot E_j=3$ otherwise. For future reference let us denote the case $E^2=0$ as the first type and $E^2=2$ as the second type, and by slight abuse of terminology, also the corresponding triple points.
It is now easy to write down $$ 2H=\sum_iE_i-8(C_1+C_2+C_3) $$ and from the above it is straightforward to check that $(2H)^2=24$. Notice also that $\sum_i E_i$ is an even divisor.
Conversely given such a configuration of curves $E_i$ in the Picard group we need to choose the points $c_i$ to be blown up carefully. Any divisor $E$ with $E^2=2$ on a $K3$ surface determines a net, and thus a Jacobian curve of the net, corresponding to the curve-locus of singular points of singular members. Each $E_i$, $i\leq3$ determines such a curve $J_i$. The point $c_1$ has to be chosen on $J_1$. The point $c_2$
has to lie on the intersection of $J_2$ with some element in $|E_1|$ singular at $c_1$ and we expect only a finite number of such choices. Furthermore $c_3$ has to lie on the intersection of $J_3$ with some
element of $|E_2|$ singular at $c_2$, and finally some element of $|E_3|$ singular at $c_3$ should pass through $c_1$. This indicates that $c_1$ should be chosen with care, and you thus expect only a finite number of configurations of points $c_i$. Further conditions are that the remaining $E_i$ also pass through the points $c_i$. We expect a 11-dimensional family of $K3$ surfaces with the appropriate sublattices, and therefore four independent conditions. Their exact nature remains mysterious.
To consider the two remaining cases, one writes down respectively $$ 2H=\sum_iE_i-4C_1-8C_2-12C_3 $$ and $$ 2H=\sum_iE_i-4C_1-4C_2-16C_3 $$ with different intersection matrices, and similar conditions on the points $c_i$; in the last case there will also be a third type of elliptic curve ($E_5^2=6$).
To see how those are related to the first case we note (once again) the fact that the intersection of a sextic with a plane passing through exactly three triple point gives an elliptic curve $F$ with $F^2=-3$ after the resolution. Given a tetrahedron of triple points, thus means replacing the exceptional vertices with the elliptic curves of the faces.
To make this explicit return to the first case $(4,4,4)$. Let us denote by $i$, $j$, $k$ different integers strictly greater than three, and $m$ an integer less or equal to three. We get four cases for $F$ namely $$ F=H+3(C_1+C_2+C_3)-E_i-E_j-E_k $$ or $$ F=H+3(C_1+C_2+C_3)-E_1-E_2-E_3 $$ with $F\cdot C_m=1$ and also $$ F=H+4C_1+2C_2+3C_3-E_1-E_j-E_k $$ or $$ F=H+4C_1+3C_2+2C_3)-E_1-E_2-E_k $$ with $F\cdot C_1=0$, $F\cdot C_2=2$, $F\cdot C_3=1$ and $F\cdot C_1=0$, $F\cdot C_2=1$, $F\cdot C_3=2$ respectively.
Thus in the first two cases we get elliptic curves $F$ of the first type, while in the last two cases, curves of the second type.
In the case $(4,4,4)$ we have three curves of the second type and six of the first. If we choose a tetrahedron with no triple points of the second type, or three, the number of elliptic curves of first or second type will not change. However if there are one or two triple points of the second type, after the transformation the number of each type will change from $6,3$ to $4,5$ landing us in the case $(2,4,6)$. We leave it to the reader to continue the analysis and show how we can get from situation $(2,4,6)$ back to $(4,4,4)$ or to $(2,2,8)$.
\subsubsection*{The sextic properly elliptic surface}
Now we turn to the remaining cases where $C_1+C_2+C_3$ does not make up $K$. Using corollary \ref{corollary:conics} again, the multiplicities of the $(-1)$-curves in the nine triple points are easily found to be (up to a permutation of triple points) as in the following table.
\begin{table}[H]
\centering
\begin{tabular}{|l|l||c|c|c|c|c|c|c|c|c|}\hline
type &
&$P_1$&$P_2$&$P_3$&$P_4$&$P_5$&$P_6$&$P_7$&$P_8$&$P_9$\\\hline\hline
\multirow{3}{1.3cm}{$(2,2,2)$}
& $C_1$ & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\\cline{2-11}
& $C_2$ & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 \\\cline{2-11}
& $C_3$ & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 \\\hline\hline
\multirow{3}{1.3cm}{$(2,2,4)$}
& $C_1$ & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\\cline{2-11}
& $C_2$ & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 \\\cline{2-11}
& $C_3$ & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 2 \\\hline\hline
\end{tabular}
\caption{multiplicities of the $(-1)$-curves}
\label{table:multelliptic}
\end{table} Let us construct a sextic $X$ with nine triple points and $(c_1,c_2,c_3)=(2,2,2)$. Then $X$ has an equation of the form $f=h_1h_2h_3g+q^3=0$. So take three planes in general position $H_i=\{h_i=0\}$, $i=1,2,3$ and let $L_1=H_1\cap H_2$, $L_2=H_1\cap H_3$ and $L_3=H_2\cap H_3$ be the lines of intersection. On each line $L_i$ choose two different points $P_{2i-1}$, $P_{2i}$. Finally choose points $P_{i+7}\in H_i$ not on the lines. The nine points determine a unique quadric $Q=\{q=0\}$. Let $H_4=\{h_4=0\}$ be the plane through $P_7$, $P_8$ and $P_9$. The reducible cubic $qh_4=0$ is an element of the pencil of cubics through all points with double points in $P_7$, $P_8$ and $P_9$. Let $Y=\{g=0\}$ be another such cubic. Then the general element of the net
\begin{equation*}
\alpha q^3 + \beta h_1h_2h_3qh_4 + \gamma h_1h_2h_3 g
\end{equation*} has only isolated triple points at the nine points.
Note that the canonical divisor will be a $(3,3)$ curve with three nodes, the intersection of $Y$ with $Q$. This is an elliptic curve and constitutes a reduced multiple fibre $F_0$. As $K=(m-1)F_0$ we conclude $m=2$. (In fact we have, as $X$ has the same invariants as a $K3$ surface, $K=\sum_i(m_i-1)F_{m_i}$, where all the multiple fibres $F_{m_i}$ are fixed components of the canonical divisor) Thus we have blown up a so called `fake $K3$' surface, obtained from an elliptic $K3$ surface, by making a fibre double. Thus in particular $P_2=2$ as $2K=F$. In fact the bicanonical divisors are given by quartics with nodes at the nine triple points. We can write down two independent such quartics, namely $q^2$ and $h_1h_2h_3h_4$. The pencil spanned by those cuts out the residual to the configuration of the three exceptional conics (with multiplicity two). As we have noted $q^2$ cuts out the double fibre $F_0$, while the other quartic will cut out the elliptic curve $F$ (intersection with the plane $h_4$) along with the three exceptional divisors. Thus $F$ is a general fibre, which has been blown up three times.
It could be interesting to investigate the position of the resolution curves $E_i$ (corresponding to the triple points $P_i$). Three of those $E_7$, $E_8$, $E_9$ span the plane $h_4$ and thus each of them intersects $F$ three times. They also intersect $F_0$ twice (corresponding to its nodes). Furthermore each of them intersects exactly one of the exceptional divisors, namely the one, whose plane the corresponding triple point happens to lie on. Thus in the minimal model these curves are elliptic four-sections, thus self-intersection $-2$. The remaining six intersect $F_0$ simply, hence are bi-sections, meaning self-intersection $-1$. Each of those intersects exactly two exceptional divisors and in the minimal model they will intersect $F$ twice rather than being disjoint from it, as in the resolution.
The reader could easily work out the intersection matrix of those nine curves and from that write down the hyperplane section. A similar analysis could be made for the second case (to be discussed below), and the relation between the two via the reciprocal transformation, elucidated by considering the possible tetrahedrons of triple points.
In order to search for a tenth triple point we need explicit equations. After a change of coordinates we may assume that the planes are the sides of the coordinate tetrahedron. The remaining coordinate transformations are given by diagonal matrices. We take $P_7=(0\cn1\mathop{\rm :}\lambda\cn0)$, $P_8=(\mu\cn0\cn1\cn0)$ and $P_9=(1\mathop{\rm :}\nu\cn0\cn0)$. A cubic through these points and also through $(0\cn0\cn0\cn1)$ is
\begin{align*}
g =\, & w^2(a_1x+a_2y+a_3z) \\
& +w( a_4x(\nu x - y - \mu\nu z)
+a_5y(\lambda y - \lambda\nu x - z)
+a_6z(\mu z - x - \mu\lambda z))\\
& +(\nu x- y - \mu\nu z)(\lambda y - \lambda\nu x - z)(\mu z -x -\mu\lambda z),
\end{align*}
while a quadric through $P_7$, $P_8$ and $P_9$ is
\begin{align*}
q =\, & b_4b_5b_6w^2
+ w(b_1b_4x+b_2b_5y+b_3b_6z) \\
& + b_4x(\nu x - y - \mu\nu z)
+ b_5y(\lambda y - \lambda\nu x - z)
+ b_6z(\mu z - x - \mu\lambda z),
\end{align*}
where the parameters are chosen with hindsight. The condition that $Y$ and $Q$ intersect the coordinate axes in the same points leads to easy equations between the coefficients. We get
\begin{align*}
g =\, & w^2(\lambda\nu b_5b_6x
+\lambda\mu b_4b_6y
+\mu\nu b_4b_5z) \\
&{} + w\big( \lambda b_1x(\nu x - y - \mu\nu z)
+ \mu b_2y(\lambda y - \lambda\nu x - z)
+ \nu b_3z(\mu z - x - \mu\lambda z)\big)\\
&{} +(\nu x - y - \mu\nu z)
(\lambda y - \lambda\nu x - z)
(\mu z - x -\ mu\lambda z)\;.
\end{align*}
The formulas contain nine parameters, three of which can be removed by coordinate transformations. The family depends therefore on eight moduli.
We can specialise to a symmetric equation by taking $\lambda=\mu=\nu=1$, $b_1=b_2=b_3=b$ and $b_4=b_5=b_6=\sqrt{a}$. With $a=b=1$ the surface $X=\{xyzg+q^3=0\}$ is a sextic with only nine triple points.
In this example we can compute the tangent space to the equisingular stratum to have dimension $15+8$. This shows that our construction fills up a whole component. Note that the multiplicities of the $(-1)$-curves are just as in table \ref{table:multelliptic}.
Consider the reciprocal transformation centred in $P_6$, $P_7$, $P_8$ and $P_9$. The image $H_1'$ of $H_1$ is a plane, the image $H_2'$ of $H_2$ is a plane, the image $Q_3'$ of $H_3$ is a quadric cone and the image $Q_4'$ of $K$ is a smooth quadric. So the image $X'$ of $X$ is given by an equation
\begin{equation*}
\alpha {q'}^3 + \beta h_1'h_2'q_1'q'+ \gamma h_1'h_2'q_1'q_2' =0\;.
\end{equation*}
The images $C_1'$, $C_2'$ and $C_3'$ of $C_1$, $C_2$ and $C_3$ are $(-1)$-curves of degrees $2$, $2$ and $4$. As before we see that the two cases $(c_1,c_2,c_3)=(2,2,2)$ and $(c_1,c_2,c_3)=(2,2,4)$ are related by reciprocal transformations. So there exists an eight parameter family of sextics with nine triple points and $(c_1,c_2,c_3)=(2,2,4)$.
Now the canonical divisor will be a smooth $(2,2)$ curve. It is again a fibre occurring with multiplicity two.
We summarise:
\begin{theorem}\label{theorem:elliptic}
For every $(c_1,c_2,c_3)\in\{(2,2,2),(2,2,4)\}$ there
exists an eight parameter family of sextic surfaces with
nine triple points and three $(-1)$-curves of degrees
$c_1$, $c_2$ and $c_3$. The two families are related
via reciprocal transformations.
For every such surface $X$ its
minimal desingularisation $\tilde{X}$ is a minimal properly
elliptic surface blown up in three points.
Moreover $X$ satisfies an equation of the form
\begin{equation*}
\begin{array}{l@{\quad}l}
h_1h_2h_3g+q^3=0 & \text{if $(c_1,c_2,c_3)=(2,2,2)$,}\\[1ex]
h_1h_2q_3g+q^3=0 & \text{if $(c_1,c_2,c_3)=(2,2,4)$.}
\end{array}
\end{equation*}
Here $Q=\{q=0\}$ is the unique canonical surface.
The three exceptional curves are in each case obtained
as the locus where the three forms in the product
and $q$ vanish.
In the first case $g$ defines a
cubic with three double points.
In the second case, $g$ defines a smooth quadric.
\end{theorem}
\subsection{Ten triple points}
Let $X$ be a sextic with $\nu=10$ triple points. As for the type of $\tilde{X}$ in the classifi\-cation, we have the
\begin{proposition}
If $\nu=10$, then $X$ is rational.
\end{proposition}
{\noindent\it Proof:\enspace } We will show that $\kod{(\tX)}=-\infty$, then the result follows from the Enriques-Kodaira classification.
Assume that $\kod{(\tX)}=0$. Then $\tilde{X}$ would be an Enriques surface $\overline{X}$ blown up in six points. Hence $P_2\left(\smash{\tX}\right)=1$, so there exists a unique quartic bicanonical surface $Y$ intersecting $X$ in a degree 24 curve $D$ made up by the six $(-1)$-curves $C_1$, \dots, $C_6$ of degrees $c_1$, \dots, $c_6$ (remember $2K_{\oX}=\cO_{\oX}$). On the one hand we get $\operatorname{mult}(D,\mathscr{S})=2(c_1+\ldots+c_6)+6=54$ from proposition \ref{proposition:multiplicity}. On the other hand $2K_{\smtX}\sim_{lin}4H-2E$, hence $\operatorname{mult}(D,\mathscr{S})=10\cdot2\cdot3=60$, contradiction.
Now assume that $\kod{(\tX)}\geq 1$, then $P_2\left(\smash{\tX}\right)\geq 2$. So we have at least two quartic bicanonical surfaces $Y_1$ and $Y_2$ intersecting in a degree 16 curve $D$. We must have $\operatorname{mult}(D,\mathscr{S})=10\cdot 2\cdot 2=40$. There exists a decomposition $D=D_1+D_2$ with $D_1\subset X$ and no component of $D_2\neq 0$ is contained in $X$. Let $d_i=\deg D_1$, $i=1,2$. Then $\operatorname{mult}(D,\mathscr{S})=\operatorname{mult}(D_1,\mathscr{S})+\operatorname{mult}(D_2,\mathscr{S})\leq 5d_1/2+2d_2<40$, contradiction. $\quad\square$
\par
Every sextic with ten triple points is a specialisation of a family of sextics with nine triple points: if $Q=\{q=0\}$ is the unique quadric through nine out of the ten triple points, the general element of the pencil $\alpha f+\beta q^3=0$ is a sextic with nine triple points, where $f$ is a defining equation for $X$. A sextic with $\nu=10$ is likely to be found in any of the five families with nine triple points described above, as a triple point gives seven conditions. However the equations on the coefficients become rather formidable, and we have only succeeded in one case by imposing extra symmetry. We start with the first family of properly elliptic surfaces with equations
\begin{equation*}
\alpha q^3 + \beta xyzqw + \gamma xyz g
\end{equation*}
where the planes $H_1=\{x=0\}$, \dots, $H_4=\{w=0\}$ are the faces of the coordinate tetrahedron and the cubic and quadric are as given above. We use the remaining freedom in coordinate transformations to place the putative tenth triple point in $(1\cn1\cn1\cn1)$. We compute in the affine chart $w=1$.
The condition for a triple point is then that the function, its derivatives and the second order derivatives vanish at $(1,1,1)$. This gives ten equations which are linear in $\alpha$, $\beta $ and $\gamma$, so we may eliminate them: the maximal minors of the coefficient matrix have to vanish. We have
\begin{align*}
\frac{\partial\, xyzg}{\partial x} &= yzg + xyzg_x, \qquad
\frac{\partial^2\, xyzg}{\partial x^2} =
2yzg_x + xyzg_{xx} \quad\text{and}\\
& \frac{\partial^2 \,xyzg}{\partial x\,\partial y}
= zg +xzg_x+yzg_y+xyz g_{xy}.
\end{align*}
Now we plug in $x=y=z=1$. From $g$ we get
\begin{align*}
&\lambda\nu b_5b_6+\lambda\mu b_4b_6+\mu\nu b_4b_5\\
&+\lambda b_1(\nu - 1 - \mu\nu)
+\mu b_2(\lambda - \lambda\nu - 1)
+\nu b_3(\mu - 1 - \mu\lambda)\\
&+(\nu - 1 - \mu\nu)
(\lambda - \lambda\nu - 1)
(\mu - 1 - \mu\lambda),
\end{align*}
an expression which we continue to denote by $g$. We get also expressions for all derivatives. Likewise we have
\begin{align*}
q =\,& b_4b_5b_6 + (b_1b_4 + b_2b_5 + b_3b_6) \\
& + b_4(\nu - 1 - \mu\nu )
+ b_5(\lambda - \lambda\nu - 1 )
+ b_6(\mu - 1 - \mu\lambda ) \;.
\end{align*}
Moreover
\begin{align*}
\frac{\partial q^3}{\partial x} & = 3q^2q_x,\qquad
\frac{\partial^2 q^3}{\partial x^2} = 3q^2q_{xx}+6qq_x^2 \quad\text{and}\\
& \frac{\partial^2 q^3}{\partial x\partial y} = 3q^2q_{xy}+6qq_xq_y.
\end{align*}
All these are divisible by $q$. After dividing by $q$ our matrix has the following form:
\begin{equation*}
\begin{pmatrix}
q^2& 3qq_x & \ldots & 3qq_{xx}+6q_x^2 &\ldots& 3qq_{xy}+6q_xq_y & \ldots\\
q & q +q_x & \ldots & 2q_x+q_{xx} &\ldots& q+q_x+q_y+q_{xy} & \ldots\\
g & g +g_x & \ldots & 2g_x+g_{xx} &\ldots& g+g_x+g_y+g_{xy} & \ldots
\end{pmatrix}
\end{equation*}
We simplify this matrix by subtracting $3q$ times the second row from the first row to remove all second derivatives from the first row. After that we apply only column operations. A computation reveals that $q_{xx}+2\nu q_{xy} +\nu^2 q_{yy}=0$ and also $g_{xx}+2\nu g_{xy} +\nu^2 g_{yy}=0$. Analogous equations hold for the other second partials. After multiplying the column containing $q_{xy}$ by $\nu$ and further column operations we get
\begin{equation*}
\begin{pmatrix}
-2q^2 & -q^2 & \ldots & 2q^2-6qq_x & \ldots & p_{xy} & \ldots \\
q & q_x & \ldots & q_{xx} & \ldots & 0 & \ldots \\
g & g_x & \ldots & g_{xx} & \ldots & 0 & \ldots
\end{pmatrix},
\end{equation*}
where
\begin{equation*}
p_{xy} = (\nu^2+\nu+1)q^2-3(\nu+1)q(\nu q_y+q_x)+3(\nu q_y+q_x)^2\;.
\end{equation*}
Now the entries in the columns with two zeroes have to vanish, for otherwise $\alpha=0$ and the equation for the sextic is divisible by $xyz$. We obtain the three equations
\begin{align*}
(\nu^2+\nu+1)q^2-3(\nu+1)q(\nu q_y+q_x)+3(\nu q_y+q_x)^2 &= 0\,, \\
(\mu^2+\mu+1)q^2-3(\mu+1)q(\mu q_x+q_z)+3(\mu q_x+q_z)^2 &= 0\,,\\
(\lambda^2+\lambda+1)q^2-3(\lambda+1)q(\lambda q_z+q_y)+3(\lambda q_z+q_y)^2 &= 0\,.
\end{align*}
This shows that the locus we are after consists of several components. The discriminant of the first equation, as a quadratic form in $q$ and $\nu q_y+q_x$, equals $-3(\nu-1)^2$, which implies that no solution is defined over $\mathbb{R}$. In principle $q_x$ is now expressible in terms of $\lambda$, $\mu$, $\nu$ and $q$, so the first row becomes divisible by $q^2$. We may divide by $q^2$ because there is no quadric through all ten triple points.
This simplification is not enough to solve the equations. To obtain manageable equations we impose symmetry. We take $\lambda=\mu=\nu$, $b_1=b_2=b_3=b$ and $b_4=b_5=b_6=\sqrt{a}$. This gives
\begin{align*}
g =\,& \lambda^2a(x+y+z) +
b\left(x(\lambda x-y-\lambda^2z)+
y(\lambda y-\lambda^2x-z)+
z(\lambda z-x-\lambda^2y)\right) \\
& +(\lambda x-y-\lambda^2 z)(\lambda y-\lambda^2x -z)(\lambda z -x-\lambda^2y)
\end{align*}
and after dividing by $\sqrt a$
\begin{align*}
q =\, & a+b(x+y+z)
+x(\lambda x- y - \lambda^2 z) + y (\lambda y -\lambda^2x - z) +z(\lambda z -x -\lambda^2y).
\end{align*}
The second derivatives evaluated in $(1,1,1)$ give only four different equations due to the symmetry in $x$, $y$ and $z$. Our three equations reduce to
\begin{equation*}
\frac14(\lambda-1)^2q^2+3(\lambda+1)^2(q_x-\frac12q)^2=0\,.
\end{equation*}
Substituting the values of $q$ and $q_x$ gives up to a constant
\begin{equation*}
3(\lambda-1)^2(b+a/3-\lambda^2+\lambda-1)^2+(\lambda+1)^2(b+a+\lambda^2-\lambda+1)^2 =0\,.
\end{equation*}
The first seven columns of our matrix above reduce to three independent ones. After multiplication of the last column by $(\lambda+1)^2$ we can use the equation above to eliminate $q_x$. Upon division by $\lambda$ we get
\begin{equation*}
\begin{pmatrix}
-2q^2 & -q^2 & -q^2 \\
q & q_x & q_{xy}- q_{xx} \\
g & g_x & g_{xy}- g_{xx}&
\end{pmatrix}\,.
\end{equation*}
As $q\neq 0$ we get as second equation
\begin{equation*}
q(g_x+g_{xx}-g_{xy})-(q_x+q_{xx}-q_{xy})g
-2q_x(g_{xx}-g_{xy})+2(q_{xx}-q_{xy})g_x
\end{equation*}
and by subtracting a suitable multiple of the first equation it becomes divisible by $(\lambda^2-1)^2$, giving as final equations
\begin{align*}
& a^2+3ab+3b^2-3a\lambda = 0\,,\\
& 3(\lambda-1)^2(b+a/3-\lambda^2+\lambda-1)^2+(\lambda+1)^2(b+a+\lambda^2-\lambda+1)^2 =0\,.
\end{align*}
These equations define a reducible curve, but no component is defined over $\mathbb{Q}$. To get a specific example note that in characteristic $31$ there is a solution $\lambda=2$, $a=9$ and $b=-11$. For these values of the parameters one finds explicit points. A {\it Macaulay\/} computation shows that there is in fact a unique sextic with only isolated triple points in the ten points. This shows that for a general solution over $\mathbb{C}$ of the equations above a unique surface with ten triple points exists.
Computing the dimension of the tangent space to the equisingular stratum for the specific example gives $15+3$. However if one looks at the number of equations and the number of variables it seems that in the general case the solution space has to be one-dimensional.
If one leaves out $P_7$, $P_8$ or $P_9$, one finds a pencil of sextics with nine triple points in the remaining ones, which contains as reducible curve two conics with five points in it and a degree eight curve with one triple point and six nodes. This is a surface with nine triple points and $(c_1,c_2,c_3)=(2,2,8)$! So the sextic with ten triple points lies in the closure of several families. We get solutions in the closure of other families by applying Cremona transformations.
One could also start out with the family $\alpha q^3+\beta q_1q_2q_3$. The surfaces with ten triple points in this family are reciprocally related with the surfaces in the other families with the same number of parameters. As the Cremona transformation depends on the position of the points it might be possible that one finds a real surface in this family. Unfortunately the corresponding equations are too difficult to solve.
Altogether we have proved the following
\begin{theorem}\label{theorem:ten}
For every triple
\begin{equation*}
(c_1,c_2,c_3)\in
\{(2,2,2),(2,2,4),(2,2,8),(2,4,6),(4,4,4)\},
\end{equation*}
the closure of
the seven parameter family of sextics with $\nu=9$ triple points
and $(-1)$-curves of degrees $c_1$, $c_2$ and $c_2$
contains at least a one parameter family of rational sextics with
ten triple points.
\end{theorem}
\begin{corollary}
$\mu_3(6)=10$.
\end{corollary}
\subsection{Summary}
\begin{theorem}\label{theorem:sextics}
The sextics with triple points fall into 18 classes according
to the following table.
\begin{table}[H]
\centering
\begin{math}
\begin{array}{|c||c|c|c|c|c|c|c|c|c|l|}\hline
\nu&c_1^2&c_2&\chi& p_g&q&b_2 &h^{1,1}&\#(-1)&\kappa&\overline{X}\\\hline\hline 0 & 24 &108 &11 & 10 & 0 & 106 & 86 & 0 & 2 & \text{general type} \\\hline 1 & 21 & 99 &10 & 9 & 0 & 97 & 79 & 0 & 2 & \text{general type} \\\hline 2 & 18 & 90 & 9 & 8 & 0 & 88 & 72 & 0 & 2 & \text{general type} \\\hline 3 & 15 & 81 & 8 & 7 & 0 & 79 & 65 & 0 & 2 & \text{general type} \\\hline 4 & 12 & 72 & 7 & 6 & 0 & 70 & 58 & 0 & 2 & \text{general type} \\\hline 5 & 9 & 63 & 6 & 5 & 0 & 61 & 51 & 1 & 2 & \text{general type} \\\hline 5 & 9 & 63 & 6 & 5 & 0 & 61 & 51 & 0 & 2 & \text{general type} \\\hline 6 & 6 & 54 & 5 & 4 & 0 & 52 & 44 & 1 & 2 & \text{general type} \\\hline 6 & 6 & 54 & 5 & 4 & 0 & 52 & 44 & 0 & 2 & \text{general type} \\\hline 7 & 3 & 45 & 4 & 3 & 0 & 43 & 37 & 1 & 2 & \text{general type} \\\hline 7 & 3 & 45 & 4 & 3 & 0 & 43 & 37 & 0 & 2 & \text{general type} \\\hline 8 & 0 & 36 & 3 & 2 & 0 & 34 & 30 & 1 & 2 & \text{general type} \\\hline 8 & 0 & 36 & 3 & 2 & 0 & 34 & 30 & 2 & 2 & \text{general type} \\\hline 8 & 0 & 36 & 3 & 2 & 0 & 34 & 30 & 0 & 1 & \text{elliptic} \\\hline 8 & 0 & 36 & 3 & 3 & 1 & 36 & 32 & 0 & 1 & \text{elliptic} \\\hline 9 & -3 & 27 & 2 & 1 & 0 & 25 & 23 & 3 & 1 & \text{elliptic} \\\hline 9 & -3 & 27 & 2 & 1 & 0 & 25 & 23 & 3 & 0 & K3 \\\hline 10& -6 & 18 & 1 & 0 & 0 & 16 & 16 & &-\infty&\text{rational} \\\hline
\end{array}
\end{math}
\end{table}
\noindent
All numbers denote invariants of the corresponding surface and
$\#(-1)$ denotes the number of $(-1)$-curves
distinguishing $\tilde{X}$ from its minimal model $\overline{X}$.
\end{theorem}
\section{Higher degree}
Surfaces of degree $d\geq7$ with many triple points are surfaces of general type by corollary \ref{corollary:minimal}. It is, as with surface with many ordinary double points, very difficult to find explicit examples of high degree with many ordinary triple points. A septic surface ($d=7$) can have at most 17 triple points by the spectrum bound.
We construct a one parameter family of septics with 16 triple points. The symmetric group $S_4$ acts on the polynomial ring $C[x,y,z,w]$ by permutation of the variables. The $\mathbb{C}$ vector space of all $S_4$-symmetric polynomials of degree seven has dimension eleven and is generated by the polynomials
\begin{equation*}
\sigma_1^7,\,
\sigma_1^5\sigma_2,\,
\sigma_1^4\sigma_3,\,
\sigma_1^3\sigma_2^2,\,
\sigma_1^3\sigma_4,\,
\sigma_1^2\sigma_2\sigma_3,\,
\sigma_1\sigma_2^3,\,
\sigma_1\sigma_2\sigma_4,\,
\sigma_1\sigma_3^2,\,
\sigma_2^2\sigma_3,\,
\sigma_3\sigma_4\,.
\end{equation*}
Here $\sigma_i$ denotes the $i$-th elementary symmetric polynomial, $i=1$, \dots, $4$. Now take as 16 triple points the $S_4$-orbit of length four generated by $P_1=(1\cn0\cn0\cn0)$ consisting of the vertices of the coordinate tetrahedron and an orbit of twelve points generated by a point $R_1=(\lambda\mathop{\rm :}\mu\mathop{\rm :}\nu\mathop{\rm :}\nu)$.
For a $S_4$-symmetric septic, the condition to have a triple point in $P_1$ implies that the coefficients of the first four polynomials vanish. Imposing a triple point in $R_1$ gives 10 equations in 7 coefficients, which by symmetry reduce to seven equations. This system of linear equations gives an $7\times7$ matrix whose determinant is up to a constant
\begin{align*}
\nu^5 &
(\lambda-\mu)^4
(\lambda-\nu)^5
(\mu-\nu)^5
(\lambda+\nu)
(\mu+\nu)
(\lambda+\mu+2\nu)^4\\
& \cdot (\lambda\mu-\nu^2)
(2\lambda\mu+\lambda\nu+\mu\nu)
(\lambda\mu+2\lambda\nu+2\mu\nu+\nu^2)^3. \end{align*}
It is easily checked that all solutions except $\lambda+\nu=0$ or equivalently $\mu+\nu=0$ correspond to either degenerate surfaces or to orbits with less than twelve points. For $\lambda+\nu=0$ the orbit of triple points is generated by $(-\nu\mathop{\rm :}\mu\mathop{\rm :}\nu\mathop{\rm :}\nu)$. The coefficients of the symmetric polynomials are now easily determined. For general values of $(\mu\mathop{\rm :}\nu)\in\mathbb{P}^1$ one has indeed a septic with 16 isolated ordinary triple points. Computing the dimension of the tangent space to the equisingular stratum gives one, so all surfaces in the family have $S_4$-symmetry. There is no irreducible surface with 17 triple points in this family.
\begin{theorem}
The general element of the
one parameter family of $S_4$-symmetric septics given by
\begin{align*}
(\mu-\nu)^3\nu(\sigma_1^2\sigma_2\sigma_3 &-
\sigma_1\sigma_3^2 - \sigma_1^3\sigma_4)
-(\mu+\nu)\nu^3\sigma_1\sigma_2^3
-(\mu+\nu)(\mu^3-\nu^3)\sigma_2^2\sigma_3\\
& +(\mu+\nu)(\mu-\nu)^2(\mu+2\nu)\sigma_1\sigma_2\sigma_4
+(\mu+\nu)(\mu-\nu)^3\sigma_3\sigma_4 =0
\end{align*}
has 16 ordinary triple points as its only singularities.
\end{theorem}
\begin{corollary}
$16\leq\mu_3(7)\leq 17$.
\end{corollary}
\begin{remark}
Whenever 16 points are invariant under the symmetric group $S_4$,
it is tempting to ask if they form a Kummer configuration.
This is not the case here.
\end{remark}
\noindent
\parindent=0pt
\small Addresses of the authors:\\ Stephan Endra\ss\ ({\tt [email protected]})\\ Micronas GmbH, Hans-Bunte-Stra\ss e 19, D 79108 Freiburg, Germany\\ Ulf Persson ({\tt [email protected]}), Jan Stevens ({\tt [email protected]})\\ Matematik, Chalmers tekniska h\"ogskola, SE 412 96 G\"oteborg, Sweden
\end{document} | arXiv |
Coincidence point
In mathematics, a coincidence point (or simply coincidence) of two functions is a point in their common domain having the same image.
This article is about the technical mathematical concept of coincidence. For numerical curiosities, see mathematical coincidence.
Formally, given two functions
$f,g\colon X\rightarrow Y$
we say that a point x in X is a coincidence point of f and g if f(x) = g(x).[1]
Coincidence theory (the study of coincidence points) is, in most settings, a generalization of fixed point theory, the study of points x with f(x) = x. Fixed point theory is the special case obtained from the above by letting X = Y and taking g to be the identity function.
Just as fixed point theory has its fixed-point theorems, there are theorems that guarantee the existence of coincidence points for pairs of functions. Notable among them, in the setting of manifolds, is the Lefschetz coincidence theorem, which is typically known only in its special case formulation for fixed points.[2]
Coincidence points, like fixed points, are today studied using many tools from mathematical analysis and topology. An equaliser is a generalization of the coincidence set.[3]
References
1. Granas, Andrzej; Dugundji, James (2003), Fixed point theory, Springer Monographs in Mathematics, New York: Springer-Verlag, p. xvi+690, doi:10.1007/978-0-387-21593-8, ISBN 0-387-00173-5, MR 1987179.
2. Górniewicz, Lech (1981), "On the Lefschetz coincidence theorem", Fixed point theory (Sherbrooke, Que., 1980), Lecture Notes in Math., vol. 886, Springer, Berlin-New York, pp. 116–139, doi:10.1007/BFb0092179, MR 0643002.
3. Staecker, P. Christopher (2011), "Nielsen equalizer theory", Topology and Its Applications, 158 (13): 1615–1625, arXiv:1008.2154, doi:10.1016/j.topol.2011.05.032, MR 2812471, S2CID 54999598.
| Wikipedia |
Bauer maximum principle
Bauer's maximum principle is the following theorem in mathematical optimization:
Any function that is convex and continuous, and defined on a set that is convex and compact, attains its maximum at some extreme point of that set.
It is attributed to the German mathematician Heinz Bauer.[1]
Bauer's maximum principle immediately implies the analogue minimum principle:
Any function that is concave and continuous, and defined on a set that is convex and compact, attains its minimum at some extreme point of that set.
Since a linear function is simultaneously convex and concave, it satisfies both principles, i.e., it attains both its maximum and its minimum at extreme points.
Bauer's maximization principle has applications in various fields, for example, differential equations[2] and economics.[3]
References
1. Bauer, Heinz (1958-11-01). "Minimalstellen von Funktionen und Extremalpunkte". Archiv der Mathematik (in German). 9 (4): 389–393. doi:10.1007/BF01898615. ISSN 1420-8938. S2CID 120811485.
2. Kružík, Martin (2000-11-01). "Bauer's maximum principle and hulls of sets". Calculus of Variations and Partial Differential Equations. 11 (3): 321–332. doi:10.1007/s005260000047. ISSN 1432-0835. S2CID 122781793.
3. Manelli, Alejandro M.; Vincent, Daniel R. (2007-11-01). "Multidimensional mechanism design: Revenue maximization and the multiple-good monopoly" (PDF). Journal of Economic Theory. 137 (1): 153–185. doi:10.1016/j.jet.2006.12.007. hdl:10419/74262. ISSN 0022-0531.
| Wikipedia |
Design of coherence-aware channel indication and prediction for rate adaptation
Yongjiu Du1,
Pengda Huang2,
Yan Shi ORCID: orcid.org/0000-0002-7034-877X3,
Dinesh Rajan3 &
Joseph Camp3
A number of rate adaptation protocols have been proposed using instantaneous channel quality to select the physical layer data rate. However, the indication of channel quality varies widely across platforms from simply a received signal strength level to a measurement of signal-to-noise ratio (SNR) across sub-carriers, with each channel quality indicator having differing levels of measurement error. Moreover, due to fast channel variations, even aggressive channel probing fails to offer an up-to-date notion of channel quality. In this paper, we propose a coherence-aware Channel Indication and Prediction algorithm for Rate Adaptation (CIPRA) and evaluate it analytically and experimentally, considering both the effects of measurement errors and the staleness of channel quality indicators. CIPRA uses the minimum mean square error (MMSE) method and first-order prediction. Our evaluation shows that CIPRA jointly considers the time interval over which the prediction will occur and the coherence time of the channel to determine the optimal window size for previous channel quality indicator measurements. Also, we demonstrate that CIPRA outperforms existing methods in terms of prediction fidelity and throughput via experimental results. By combining a strong channel indicator with the coherence-aware MMSE first-order channel prediction algorithm, CIPRA nearly doubles the throughput achieved in the field from the indication and prediction method currently used by off-the-shelf WiFi interfaces.
Channel fluctuations often exist in wireless communication systems and present great challenges in selecting the best data rate or modulation coding scheme (MCS) for communication. When there is a change in direction of the transmitter/receiver, antenna elevation and polarization, interference from nearby devices, or scatter distributions, the channel quality can vary, resulting in fluctuations in signal reception, even within the same environment. Depending on the magnitude of the variation, the received signal strength (RSS) can drastically change the link performance [1–4]. Rate adaptation protocols can be implemented to combat the fading channels and achieve high spectrum efficiency by dynamically changing the data rate according to the channel quality.
Rate adaptation protocols that depend upon packet success/failure information have been implemented in commercial equipment and widely discussed [5–10]. However, these loss-based protocols usually require tens of data frames to develop a reasonable estimate of the channel conditions. Other factors such as hidden terminals and interference can lead to inappropriate conclusions about the causes for the losses [11]. Thus, for fast-fading channels (e.g., in vehicular networks), these protocols cannot accurately track the changing channels, resulting in wrong selections of MCS. Moreover, packet-level estimation is too coarse to achieve an accurate estimation of the instantaneous channel conditions. With interfering sources, the transmitter can not distinguish the reason for packet loss and jitter (i.e., whether it is from a poor channel or interference). These factors contribute to frequent under-selection of the transmission rate by loss-based protocols [11].
To enable fast-fading channel tracking, various channel-indicator-based rate adaptation protocols have been proposed [12–15]. It is well known that for a certain MCS, there is a theoretical bit error rate (BER) versus signal-to-noise ratio (SNR) relationship [16]. SNR values can be reported to the transmitter at the physical layer (PHY)-frame level to enable selection of the optimal rate in fast-fading channels. Some of the SNR-based rate adaptation protocols leverage the received signal strength indicator (RSSI) to calculate the SNR. However, for different commercial wireless network devices, the RSSI varies significantly for the same received signal strength. Moreover, in the presence of interference and noise, the estimates of the signal power reported from RSSI measurements can be highly distorted [17]. In [13], the authors propose a SoftRate metric to indicate channel quality and select the optimal rate for transmission. However, this scheme requires a soft-in, soft-out decoder to calculate the SoftRate indicator, which is not available for most of the existing network transceivers and may also be resource-intensive for the design of future transceivers. AccuRate, a constellation-quality based indicator is provided in [15]. Nevertheless, the hardware cost and complexity of AccuRate is also high, often precluding the calculation of the channel indicator in the implementation. The Effective SNR metric [14] leverages the channel response in the frequency domain and the noise variance on each sub-carrier to predict the best rate to transmit a packet. The information required is already available in several wireless network interfaces.
In our previous work [12], we consider RSSI as the unique channel indicator under limited comparison matrices, which has been shown to be inaccurate and inconsistent across vendors [13]. In this work, in addition to the original channel indicator of RSSI, we study and analyze two more advanced channel indicators: SNR and Effective SNR. We implement channel prediction and rate adaptation based on these two advanced metrics to evaluate their performance in combination with traditional channel prediction methods and our proposed channel prediction method, showing significant throughput improvement. We additionally analyze the prediction error performance for our proposed algorithm at various time intervals, predicting the importance of our work on next-generation protocol design with higher frequency bands and tighter timing parameters. At last, we implement the Minstrel rate adaptation and compare our proposed rate adaptation method with it, showing that our proposed rate adaptation method outperforms Minstrel significantly.
Typically, in SNR-based rate adaptation, the receiver decides the rate of the next transmitted packet according to the measured SNR of the current packet, whether the measurement originates from the RTS/CTS or DATA/ACK exchange [11]. Several potential problems exist with this mechanism. First, if the channel changes quickly or there is a large time interval between two adjacent packets, the SNR reported by the last transmission may not accurately represent the instantaneous channel quality. Second, the SNR reported during the last transmission may not be accurate due to measurement errors. Even for slowly varying channels, there might be rate over-selection or under-selection due to channel quality estimation errors.
The rate selection problems caused by channel quality estimation errors have been studied [16, 18]. In order to increase the accuracy of rate selection, both works as leverage-filtering techniques to reduce channel quality estimation errors. However, the delay incurred in filtering can make rate selection even more stale to track with ongoing channel quality changes. To address the channel quality staleness problem, channel prediction has been extensively studied [19–23]. However, most work assumes perfectly accurate channel measurements to predict the future channel state, which may not be possible in hardware. Moreover, the estimation error can make the prediction highly erroneous if using the reported value from the last transmission.
In this paper, we propose a coherence-aware MMSE first-order prediction algorithm that takes into account both the measurement inaccuracy and measurement staleness. Prediction intervals and channel coherence time are jointly considered to select the optimal size of prediction window. We perform simulation studies as well as in-field experimentation on emulated and in-field channels, respectively. For our analysis, we implement an IEEE 802.11 PHY system on WARP (Wireless Open-Access Research Platform), an field programmable gate array (FPGA)-based platform [11]. We also implement and compare three different channel indicators, RSSI, SNR, and Effective SNR, and combine each with the proposed prediction algorithm to investigate the indicator's impact on the performance of channel prediction. To generate repeatable channel effects, we test our design on a channel emulator and compare the three indicators in diverse channel conditions. In addition to lab experiments, we also conduct field tests to show the in situ throughput improvements provided by our algorithm.
The main contributions of our work are as follows:
We propose a coherence-aware MMSE first-order channel quality prediction algorithm, which takes into account both the measurement errors and staleness of the channel quality. This scheme adapts the measurement processing parameters to the Doppler shift to achieve good performance under various mobility scenarios.
We implement and evaluate a family of the most commonly used channel quality indicators for rate adaptation, including RSSI, SNR, and Effective SNR.
We analyze different channel prediction approaches and compare them in terms of prediction errors as well as over-selection and under-selection probabilities for rate adaptation.
We present and implement a Doppler shift estimation method based on LCR (level-crossing rate) with a homogeneous window to remove the effect of channel quality measurement errors, achieving a good balance of complexity and accuracy.
We implement the existing channel prediction algorithm and the proposed algorithm on WARP and experimentally compare them in terms of system throughput through both repeatable channel emulator tests and in-field experiments. By combining Effective SNR with the coherence-aware MMSE first-order channel prediction algorithm, channel indication and prediction for rate adaptation (CIPRA) achieves up to a 98% throughput improvement in the field over the indication and prediction method currently used in off-the-shelf cards.
The rest of this paper is organized as follows. Section 2 introduces the online Doppler shift estimation method and discusses various channel quality indicators. Then, we describe the framework of our proposed CIPRA algorithm and compare its performance with conventional prediction algorithms in Section 3. In Section 4, we introduce the hardware setup for CIPRA evaluation and provide numerical results in Section 5. Finally, we conclude our work in Section 6.
Background and related work for channel indication and prediction
In this section, we first introduce the channel fading model used in this work. Then, we describe a Doppler shift estimation method for our platform implementation. Finally, in addition to the typical performance metrics developed to indicate the channel quality, such as RSSI and SNR, we study and analyze a more advanced channel quality indicator: Effective SNR.
Channel characteristics
Wireless channel quality is often affected by changing environments and interference. With the transmit signal power fixed, channel quality can be quantified by the received signal quality. We use a Rayleigh fading channel model in the following analysis and simulation. The normalized auto-correlation function, R(τ), of a Rayleigh fading channel with motion at a constant velocity is a zeroth-order Bessel function of the first kind [24, 25]:
$$ R(\tau)=J_{0}(2\pi f_{d} \tau) $$
Here, τ is the time delay, and fd is the maximum Doppler shift. The auto-correlation functions of a Rayleigh fading channel with a maximum Doppler shift of 10 Hz and 20 Hz are shown in Fig. 1. This auto-correlation reflects the statistical dependence between the channel gains at different times, which is leveraged in the prediction.
Auto-correlation function of a Rayleigh fading channel with a maximum Doppler shift of 10 Hz
Online doppler shift estimation
In this work, we introduce a Doppler shift estimation method that we implement and evaluate on our hardware platform. This method can be applied to our channel prediction algorithm discussed in Section 3. In general, Doppler shift estimation can leverage channel estimates, LCR, a maximum likelihood function, or correlation function [26]. LCR-based Doppler shift estimation achieves a good balance between complexity and accuracy. For Rayleigh fading channels, LCR is expressed as [27]:
$$ \text{LCR}=\sqrt{2 \pi}f_{d}\rho e^{-\rho^{2}} $$
Here, ρ is the threshold normalized to the root mean square (RMS) signal level [27]. For a fixed Doppler shift, the LCR achieves its maximum value, LCRmax, when \(\rho =\sqrt {0.5}\), and is given by
$$ \text{LCR}_{\text{max}}=\sqrt{\pi}e^{-0.5}f_{d} $$
In hardware, the received signal is also corrupted by additive noise. The LCR resulting from the noise usually leads to over-estimation of the channel level-crossing rate. In [28], an fast Fourier transform (FFT)-based Doppler-adaptive noise suppression method is proposed to remove the effect of additive noise on LCR-based Doppler shift estimation. However, the FFT/IFFT processing is computationally expensive, requiring approximately \(\frac {34}{9}N\log _{2} N\) real multiplications and additions for an N-point FFT/IFFT [29]. In this paper, we create a homogeneous-window method to avoid over-estimation caused by the additive noise. The main steps to this method are the following:
Pick a threshold value from a pre-defined threshold set and compare the RSSI samples with this threshold. If the RSSI value of sample i is greater than the selected threshold for each indicator, ci=1. Otherwise, ci=0, meaning that the RSSI is below the selected threshold.
Apply a sliding time window τ to the results in step 1. If ci==1 for all the samples in window τ, we denote the system state Si=1. If ci==0 for all the samples in window τ, we denote the system state Si=−1. Otherwise, Si=0. For multiple adjacent system state samples with the same value, only record one sample.
Calculate the derivative of the state vector recorded in step 2 and count the number of transitions of the derivative from negative to positive in one second, denoted by n.
Repeat step 1 to step 3 for all the values in the pre-defined threshold set, and finally, find the maximum value of n.
To decide the time window τ, we jointly consider the RSSI sample period and the Doppler shift range we want to estimate. In the IEEE 802.11 standard, the RSSI is reported every packet. For the maximum Doppler shift range of 10 to 100 Hz, τ of 3 ms achieves a normalized square error of 0.003 in our experiments using the channel emulator. Figure 2 shows the Doppler shift estimation by using the WARP board and the channel emulator (both are described in Section 5).
Doppler shift estimation on WARP
Channel quality indicators
RSSI: a poor channel indicator
Wireless channel quality is often affected by changing environments and interference. With the transmit signal power fixed, channel quality can be evaluated by the received signal quality. The most accessible channel quality indicator is RSSI [17]. RSSI is a relative value with off-the-shelf devices in which vendors usually use arbitrary scales from 0 to maximum-RSSI, where maximum-RSSI is vendor-specific. RSSI is often not associated with any particular power scale and not required to be of any particular accuracy or precision [30]. Hence, the received signal strength numbers reported by a network interface are inconsistent across vendors and should not be assumed to be representative of a particular channel state.
We now carry out experiments to evaluate the reliability of using RSSI as the channel indicator. We use one WARP as the transmitter to send a signal with 10 MHz bandwidth and - 62 dBm power and directly connect it to another WARP receiver with a coaxial cable. The reported RSSI values from our WARP receiver are shown in Fig. 3. We can see that, even with the same transmit power and channel state, the reported RSSI values can vary as much as 14 dB. Even if we use a filter to reduce the variance of the reported RSSI, other factors may also greatly affect the RSSI-based channel quality prediction accuracy. The general system architecture for the signal path is shown in Fig. 4. Any error or interference generated by the components in group 1 may vastly affect the received BER, but may not affect the RSSI value. Any error or interference generated by the components in group 2 may affect the RSSI value, but may not significantly affect the BER. Therefore, for different transmitter and receiver pairs, system components internally have different performances. The factors that might affect the channel quality estimation accuracy using RSSI include the following:
Phase noise. With the same RSSI, there might be different phase noises caused by the jitter of the clocks on both the transmitter and the receiver, leading to different values of BER. However, the phase noise may not affect the received signal power level.
Amplifier non-linearity. For orthogonal frequency-division (OFDM) systems, the non-linearity of both the transmit amplifier and the receive amplifier often cause inter-carrier interference (ICI) [31]. ICI may not affect the RSSI, but often affects the received BER performance.
RSSI signal noise. RSSI is measured in the transceiver and output in the form of an analog signal, which often suffers from noise and interference on the board.
RSSI analog to digital converter (ADC) performance. In a system, an ADC is typically used to convert the RSSI signal from the analog to digital domain. The noise on the board, the resolution of the ADC, and the reference voltage stability of the ADC may all affect the digitized RSSI value.
RSSI sample duration. In the IEEE 802.11 standard, RSSI is calculated during the preamble of a PHY frame. The limited duration of the preamble can not guarantee an accurate RSSI estimate.
Raw RSS values reported with the same transmit power and channel gain
Signal path from the transmitter to the receiver
Considering all the effects listed above, with the same received BER, the receivers often report different RSSI values for different transmitter and receiver pairs. We compare the RSSI values reported (shown in Fig. 5) from two different receivers but with the same signal source and emulated channel. Even with the same received signal, there is about 1.5-dB difference of reported RSSI between receivers on average. As a result, the RSSI inconsistency may severely handicap an optimal rate selection decision.
RSS values reported by different receivers (one shown above the other) with the same level of achieved BER
SNR: a more reliable channel indicator
Most of the soft decoders need the SNR or noise variance to calculate the decision probabilities of the demodulated signals. There are diverse SNR estimation methods for OFDM systems. In [32], the author proposes iterative SNR estimation based on pilot sub-carriers in an 802.11n system. In [33], an algorithm based on finding the difference between a noisy received sample in the frequency domain and the best hypothesis of the noiseless sample is proposed to estimate the SNR. In this work, we implement an SNR estimation method based on the Schmidl-Cox algorithm [34] because of its high accuracy and low complexity. In addition, we use the IEEE 802.11 PHY frame as the frame structure in this work, as shown in Fig. 6. One frame is composed of a preamble, a header symbol, and the number of OFDM symbols forming the data payload. One preamble consists of one short preamble and one long preamble [35]. In our design, we take advantage of the known training symbols (two identical OFDM symbols with a 1/2 symbol prefix) in the long preamble to calculate the SNR of the received packet. A detailed calculation can be found in the Appendix 1. We show the SNR distribution plotted in Fig. 7. The SNR has a mean value of 16 dB, with a standard deviation of 0.91 dB. Comparing the RSSI distribution which has a standard deviation of 1.5 dB, we can show that SNR is more accurate than the RSSI as the channel quality indicator.
IEEE 802.11 PHY frame structure
Distribution of estimated SNR
Effective SNR: a robust channel indicator
Although the SNR estimation is more tightly bounded and consistent with the channel state than the RSSI, a frequency-selective fading channel may disturb the mapping from SNR to BER. However, in a multi-path channel, the frequency-selective characteristics may result in a higher BER than a flat-fading channel with the same SNR, which corresponds to a lower Effective SNR than the actual SNR (demonstrated in Fig. 8). To solve this problem, Halperin [14] proposed the Effective SNR metric to improve the rate selection accuracy in multi-path fading channels. Instead of averaging the SNR on all the sub-carriers for SNR-based rate adaptation algorithms, systems implementing the Effective SNR not only average the BER on all the sub-carriers but also obtain an equivalent SNR with the same BER as narrow-band systems. This process can be represented by [14, Eqs. (1) and (2)]:
$$ \text{BER}_{e\ f\ f,\ k} =\frac{1}{52}\sum\limits_{s=1}^{S}{\text{BER}_{k}(\rho_{s})} $$
Different frequency-selective patterns with the same SNR value
$$ \rho_{e\ f\ f,\ k} = \text{BER}^{-1}_{k}(\text{BER}_{e\ f\ f,\ k}) $$
Here, ρs is the SNR on sub-carrier s. Assuming the average SNR is ρ and the channel gain on sub-carrier s is Hs, then ρs can be calculated as \(\rho _{s} =\rho \frac {|H_{s}|^{2}}{\bar {|H|^{2}}}\), where \(\bar {|H|^{2}}\) is the mean square of the channel gain across all the sub-carriers.
In an additive white Gaussian noise (AWGN) channel, the relationship between SNR and BER varies among different modulations [14, 36], as shown in Table 1. In the following discussion and experimental evaluation, we see that Effective SNR outperforms the other channel indicators due to its ability to capture both time-selective and frequency-selective fading effects.
Table 1 BER function with different modulations
Design of CIPRA
In this section, we analyze different channel prediction algorithms and propose an advanced algorithm to keep the transmitter in step with the fluctuating channel quality. In order to improve prediction accuracy, we take into account both the measurement error of the channel indicator and the staleness of the channel quality reported by the receiver.
Existing prediction methods
The wireless channel usually changes continuously and randomly with time, which makes the implementation of accurate closed-form characterization challenging. Nevertheless, a Rayleigh fading channel model is a good approximation and agrees well with empirical observations for mobile communications [25]. In order to select the optimal rate, the transmitter constantly needs the channel quality measurement from the receiver. Prior SNR-based protocols have frequently used the channel quality measured from the last packet transmission to the pertinent receiver [18]. Whether the last packet is a probe packet from the RTS/CTS exchange or simply the last data packet, these channel indicator measurements are stale in fast-fading channels. There are several mechanisms to make full use of the previous channel quality measurements for rate adaptation.
For this mechanism, the transmitter simply copies the channel measurements from the last packet transmission as the predicted value of the ongoing channel quality [18]. In particular, Follower can simply be denoted as \( \hat {\gamma }_{n,\text {follower}}=\gamma _{n-1}\), where \(\hat {\gamma }_{n,\text {follower}}\) is the estimate of the ongoing channel quality and γn−1 is the channel quality measurement reported during the last packet transmission. This estimate has minimal complexity but suffers from both measurement errors and staleness of the past channel indicator values.
There are three main kinds of moving average methods: simple moving average, linear weighted moving average (LWMA), and exponential weighted moving average (EWMA) [23]. Simple moving average is the unweighted mean of the previous points in a window size of w. The estimated channel quality \(\hat {\gamma }_{n,\text {MA}}\), is denoted as
$$ \hat{\gamma}_{n,\text{MA}}=\frac{\gamma_{n-1}+\gamma_{n-2}+ \cdots +\gamma_{n-w}}{w} $$
Simple moving average method reduces the effect of the measurement errors, while making the staleness effect more serious than the Follower method.
For LWMA, weight factors are assigned to the past measurements in a linear progression with a window size of w. The estimated value \(\hat {\gamma }_{n,\text {LWMA}}\), can be expressed as
$$ \hat{\gamma}_{n,\text{LWMA}}=\frac{w \gamma_{n-1}+(w-1) \gamma_{n-2}+ \cdots +\gamma_{n-w}}{(w+1)w/2} $$
LWMA puts greater weight on more recent measurements, which results in a balance between the prediction staleness and the measurement errors.
For EWMA, the weight of the measurements decreases by a factor of δ.
$$ \hat{\gamma}_{n,\text{EWMA}}=\delta \gamma_{n-1} +(1-\delta)\hat{\gamma}_{n-1} $$
EWMA reduces the number of previous measurements to one and has less computational complexity.
Each of the moving average methods reduces the effect of the measurement errors but introduces more severe staleness effect than the Follower method.
Linear prediction
By assuming that the channel quality indicator has a constant first-order derivative across three adjacent transmissions [23], we can predict the ongoing channel quality from the last two channel measurements.
$$ \hat{\gamma}_{n,\text{linear}}=\gamma_{n-1}+ \Delta \gamma (t_{n}-t_{n-1}), \quad \Delta \gamma = \frac{\gamma_{n-1}-\gamma_{n-2}}{t_{n-1}-t_{n-2}} $$
Here, γn−1 and γn−2 are the channel measurements at time tn−1 and tn−2, respectively. This method is more robust to the prediction staleness. However, the errors of the past measurements may make Δγ twice as noisy, leading to a prediction with an intolerable noise level in some cases.
Coherence-aware MMSE first-order prediction
Our previous discussion reveals that some of the methods are more robust to the measurement errors, while others are more robust to the prediction staleness. For a good prediction, both the measurement errors and staleness should be considered. In this section, we propose a coherence-aware MMSE first-order prediction.
In Section 2.1, we introduced the Rayleigh fading model and its auto-correlation. When conducting the prediction, we need the previous measurements within a time window T. From Fig. 1, we see differing dependence between measurements with the same maximum Doppler shift and differing time delay, or with the same time delay and differing maximum Doppler shift. Thus, when selecting the time window T, we should take the Doppler shift into account. We can denote T as \(T=\frac {\beta }{f_{d}}\), where fd is the Doppler shift that can be estimated by the method we proposed in Section 2. β is a constant factor. In our simulation and experiments, we select β=0.064, which empirically achieves the highest prediction fidelity.
Assuming that within the time window T, there are w channel measurements γn−1,γn−2,⋯,γn−w at time tn−1,tn−2,⋯,tn−w, respectively. From all the measurements within the window, we perform a linear regression first-order curve fit with the constraint of minimum square errors. To do so, we first assume the objective first order curve is f(t)=at+b, where a and b are parameters to be calculated. Then, we have
$$ \gamma\prime_{n-i}=f(t_{n-i})=a t_{n-i}+b \quad \text{\(i=1,2,\cdots, w\)} $$
The detailed process of computing a and b in (10) can be found in the Appendix 2.
We know that the fading channel does not strictly maintain a constant first-order derivative, especially for long intervals between packets. In a more extreme case, if the interval between the last packet and the ongoing packet exceeds a certain value, the prediction may be uncorrelated with the real channel quality. Considering this interval, we use a weighting factor δ to weight the pre-prediction and the channel quality with the maximum probability. Consequently, the estimated channel quality \(\hat {\gamma _{n}}\) is:
$$ \hat{\gamma_{n}}=\delta (t_{n}-t_{n-1}) \cdot \hat{\gamma_{n}} \prime + (1-\delta(t_{n}-t_{n-1}))\cdot \bar{\gamma} $$
$$ \delta (t) = \left\{ \begin{array}{l l} 1-t \cdot f_{d} & \quad \text{ if~} t<\frac{1}{f_{d}} \\ 0 & \quad \text{otherwise}\\ \end{array} \right. $$
In (12), fd is the maximum Doppler shift, and \(\bar {\gamma }\) is the channel quality that has the greatest statistical probability of occurrence. For computational simplicity, we approximate \(\bar {\gamma }\) as the mean value of the channel quality measurements during the last 10 s.
Note that within the time window T, there is the probability of w≤2. When w=2, our algorithm turns out to be the Linear prediction. Similarly, if w=1, our algorithm matches the Follower mechanism. For the case of w=0, we choose the maximum probability channel quality \(\bar {\gamma }\) as our prediction.
In Fig. 9, we show the result of the MMSE first-order channel quality prediction. There is a -15 dB measurement error compared to the channel quality. The reconstructed channel response approaches the theoretical curve well. The square errors of the prediction is about -38 dB compared to the theoretical one, which means a -23 dB accuracy gain.
Channel quality reconstruction using MMSE first-order prediction
In order to evaluate the prediction performance of our proposed algorithm under various time intervals between packets, we control the delay between the data packets decoded for channel estimation and the channel feedback packets received by the transmitter. We use a Rayleigh fading channel model to compare our proposed algorithm with three other mechanisms (Follower, EWMA, and Linear prediction), as shown in Fig. 10. We set a Doppler shift of 10 Hz and set the channel measurement error to -20 dB compared to the average channel quality. We can see that the prediction performances based on the time interval seem to follow the same pattern for all prediction algorithms: the prediction error increases as the time interval increases, as shown in Fig. 10a. Our evaluation shows that CIPRA presents the least prediction error of all prediction algorithms. Although similar patterns can be found at both over-selection probability, under-selection probability, and wrong selection probability, as shown in Fig. 10b, c, and d, it is interesting to note that the performance of CIPRA becomes identical to that of Linear prediction at a time interval above 3.6 ms. Considering that the MAC layer design in current and future IEEE 802.11 standards requires only smaller time interval between packets, we predict that the impact of CIPRA will only increase with the development of higher frequency band protocols such as those with millimeter wavelengths and smaller timing parameters. For example, a minimum interval between transmission packets, called Short Inter Frame Space (SIFS), has shortened from 28 μs in IEEE 802.11-1997 to 16 μs in IEEE 802.11n to 3 μs in WiGig/IEEE 802.11ad. Moreover, many timing parameters of the backoff process for the carrier sense and channel access are also reduced.
a–d Channel quality prediction performances comparison
The computational complexity of the proposed algorithm is higher than the other algorithms discussed above. However, it takes less than 1 μs on the PowerPC embedded on WARP (we use an 80-MHz clock frequency for the PowerPC), which is much less than the DIFS/SIFS time of the transmission. As a result, it does not affect the system throughput.
In this section, we describe the implementation of our CIPRA algorithm on the WARP board, a fully-customized, cross-layer software-defined radio (SDR) platform. Moreover, we use the Azimuth ACE-MX channel emulator to generate controllable channel effects, which allows repeatability of wireless channels over which to test diverse protocols.
The experiments are carried out using the WARP board, a useful wireless communication system supporting a fully-customized, cross-layer design [11]. We conduct our experimental evaluation based on a full OFDM physical layer design per the IEEE 802.11 PHY frame structure. The design operates in real-time, transmitting and receiving wide-band signals. We implement complete real-time signal processing, synchronization, and control systems in the fabric of the FPGA on WARP.Footnote 1
Channel emulator
In addition to the practical in-field wireless channels, we use the Azimuth ACE-MX channel emulator to generate repeatable and controllable channel effects, which isolates the impact of interference and approximates well complex over-the-air channels. Figure 11 illustrates our experimental setup. We use one WARP as the transmitter and another WARP as the receiver. We connect the transmitter and the receiver with RF cabling to the channel emulator. The transmitter sends data packets periodically to the receiver. The receiver measures the channel and feeds the channel indicator back to the transmitter with ACK packets following controllable time interval.
Channel emulator based evaluation system
In our evaluation with the channel emulator, we set the packet size to 1536 bytes. We use a two-tap Rayleigh fading channel with an average SNR of 15 dB. Both taps have a 0-dB relative attenuation, and the time delay between the taps is 0.5 μs. The resulting throughput is shown in Table 2 and indicates that Effective SNR performs best among all three channel indicators. The EWMA method performs better with less of a Doppler shift because there is less staleness when the Doppler shift is low. With an increasing Doppler shift, the linear method becomes comparatively better because of the increasing staleness of Follower and EWMA. SNR-based rate adaptation mechanisms on off-the-shelf devices use the RSSI metric and Follower prediction. Thus, with a Doppler shift of 10 Hz, CIPRA achieves a throughput improvement of 18% over the off-the-shelf configuration. If further combined with the advanced effective SNR, a total improvement of 33% could be achieved over the off-the-shelf method. Later, we show that in-field experimental results exceed these gains as the channels become more complex.
Table 2 Throughput with different combinations of channel indicator and prediction methods
Experimental evaluation and discussion
In this section, we implement the existing channel prediction algorithm and the proposed algorithm on WARP and experimentally compare them in terms of system throughput through both indoor pedestrian experiments and outdoor vehicular experiments.
Indoor pedestrian test
We conduct our indoor experiment on the eighth floor of the SMU Expressway Tower (floor plan shown in Fig. 12). Before evaluating the performance of our proposed algorithm with other mechanisms, we take actions to reduce the ambient interference power in our selected measurement locations by disabling or shielding interfering WiFi access points. Yet, the interference cannot be fully eliminated due to the existence of institutional WiFi access points that are beyond our control. We set up two transmitter/receiver pairs, which are operating independently, but simultaneously. Two transceivers use orthogonal channels: 2484 MHz (Ch. 14) and 2462 MHz (Ch. 11), respectively. For each experiment, we run CIPRA on a tx/rx pair, and one of the other three methods on the other tx/rx pair. We use RSSI with all four prediction methods to measure the throughput. We also provide additional experiments combining CIPRA and effective SNR to obtain the throughput. We co-locate the transmit antennas and co-locate the receive antennas to ensure the two links have very similar channels. For each comparison, we flip the links back and forth for each experimental trial to remove any unfair advantage between the two channels. The transmitters are located on the table in the lab, and we select four office rooms to put the mobile receivers, as shown in Fig. 12. We randomly move the receiver nodes in each office to create time-varying channels. We show the average throughput of the four methods on the four different locations in Fig. 13. CIPRA greatly outperforms other methods at all four locations with up to a 66% throughput improvement, with an improved prediction algorithm from the currently one used in practice. When further combined with the advanced Effective SNR, a maximum improvement of 98% could be achieved over the off-the-shelf method, nearly doubling the achieved throughput. In addition, compared with the experimental results using channel emulator, our evaluation demonstrates the performance improvement of CIPRA in real environments even with the impact of interference.
The eighth floor layout of SMU Expressway Tower
The indoor experiment result
Outdoor vehicular test
In addition to lab experiments, we also conduct in-field experiments to show the throughput improvement provided by our algorithm for in situ highly mobile scenarios. We perform outdoor experiments in the parking lot of the SMU Expressway Tower (shown in Fig. 14). The transmitter and receiver settings are the same as with the indoor experiments. We place the transmitters in the entrance of the tower and the receivers in a car with the antennas mounted on the roof. We drive the car along the path shown in Fig. 14, with an average speed of 32 km/h. We also switch the channels of the two tx/rx pairs to remove the unfairness of different channels for each comparison. We show the average throughput of the four methods in Table 3. Due to a higher Doppler shift in the vehicular environment, the Linear method has improved performance. CIPRA also outperforms the three other methods with up to 50% of the throughput improvement. When further combined with Effective SNR, a total improvement of 67% could be achieved over the off-the-shelf method.
The experimental environment outside of SMU Expressway Tower
Table 3 Throughput result for outdoor experiment
Throughput comparison between CIPRA and Minstrel
Minstrel is reported to be one of the best rate adaptation methods based on packet loss/success [37]. We implemented Minstrel according to the specifications in [37] and evaluated the throughput of Minstrel and CIPRA with the same experimental settings as in Section 4.2. We use SNR as the channel indicator for CIPRA and train the CIPRA rate adaptation threshold with a Doppler shift of 10 Hz, corresponding to a walking velocity for WiFi 2.4 GHz band. The Doppler shift can be approximately by
$$ f_{d} = \frac{vf_{c}}{c} $$
Here, v is velocity and c is the light speed. Then, we apply the same SNR threshold in all the different Doppler shift cases for CIPRA. We show the throughput of CIPRA and Minstrel in Fig. 15. We can see that, with a very low Doppler shift, CIPRA and Minstrel have similar throughput. However, as the Doppler shift increases, Minstrel degrades faster than CIPRA. This is explained by the long statistical time that composes rate decisions in Minstrel that prevent it from adapting as quickly to fast-fading channels.
In this paper, we proposed a coherence-aware MMSE first-order prediction algorithm (CIPRA), which considered both the measurement inaccuracy and staleness. Prediction intervals and channel coherence time were jointly considered to select the optimal prediction window. We also implemented a Doppler shift estimation method to assist our prediction algorithm. We compared CIPRA to the traditional channel quality prediction for rate adaptation protocols, performing experiments on an FPGA-based platform over emulated and in-field wireless channels. We showed that our proposed algorithm can provide better prediction fidelity and results in nearly double the throughput versus the current configuration in off-the-shelf devices in the field. Lastly, we estimated that the benefits brought by CIPRA will only increase with the development of next-generation protocols such as those with higher frequencies and higher packet rate.
his appendix presents the SNR computation for IEEE 802.11 OFDM systems. As shown in Fig. 6, the long preamble is composed of two identical OFDM symbols with a 1/2 symbol prefix. Denote the OFDM symbol in the long preamble as X(k),k=0,1,...,K−1, where k is the sample index in the time domain of one symbol. According to the frame structure specified in [35], the long preamble S(n) can be expressed as
$$ S(n)=X\left(\left(n+\frac{K}{2}\right)\text{mod} (K)\right), \quad n=0,1,...,\frac{5}{2}K-1 $$
where n is the sample index in the long preamble.
We assume this signal passes through a multi-path channel with a time spread of L, \(0\leq L\leq \frac {K}{4}\). Moreover, we make the assumption that the channel remains constant during one frame slot. Then, the received signal Y(n) with additive white Gaussian noise is
$$\begin{array}{*{20}l} Y(n)&=\sum\limits_{l=0}^{L-1} S(n-l)h(l) +Z(n) \\ &=\sum\limits_{l=0}^{L-1} X\left(\left(n-l+\frac{K}{2}\right)\text{mod} (K)\right)h(l)+Z(n) \end{array} $$
where \(n=0,1,...,\frac {5}{2}K+l-2\). It is then straightforward that
$$\begin{array}{*{20}l} Y(n+K)&=\sum\limits_{l=0}^{L-1} X\left(\left(n-l+\frac{3K}{2}\right)\text{mod} (K))h(l\right)\\ &\quad+Z(n+K) \\ &=\sum\limits_{l=0}^{L-1} X\left(\left(n-l+\frac{K}{2}\right)\text{mod} (K)\right)h(l)\\ &\quad+Z(n+K) \\ &=Y(n)+Z(n+K)-Z(n) \end{array} $$
We then compute the cross-correlation between the last symbol and the first symbol as
$$\begin{array}{*{20}l} P_{s}=&\frac{1}{K}\sum\limits_{n=\frac{K}{2}}^{\frac{3}{2}K-1} Y^{*}(n)Y(n+K) \\ =&\frac{1}{K}\sum\limits_{n=\frac{K}{2}}^{\frac{3}{2}K-1} \left(|Y(n)|^{2}+Y^{*}(n)(Z(n+K)-Z(n))\right) \end{array} $$
Because Y(n) and Z(n) are uncorrelated, we have
$$ \begin{aligned} E(P_{s})&=E\left[\frac{\sum\limits_{n=\frac{K}{2}}^{\frac{3}{2}K-1} \left(Y^{*}(n)(Z(n+K)-Z(n))\right.}{K}\right.\\ &\quad \left.\frac{\left.+|Y(n)|^{2}\right)}{} {\vphantom{\sum\limits_{n=\frac{K}{2}}}}\right]\\ &=\frac{1}{K}\sum\limits_{n=\frac{K}{2}}^{\frac{3}{2}K-1} E\left(|Y(n)|^{2}\right) \end{aligned} $$
We then compute the auto-correlation on the first symbol as
$$\begin{array}{*{20}l} P_{t}=&\frac{1}{K}\sum\limits_{n=\frac{K}{2}}^{\frac{3}{2}K-1} Y^{*}(n)Y(n) \\ =&\frac{1}{K}\sum\limits_{n=\frac{K}{2}}^{\frac{3}{2}K-1} (|Y(n)|^{2}+2Y^{*}(n)Z(n)+|Z(n)|^{2}) \end{array} $$
Since Y(n) and Z(n) are uncorrelated, we have
$$\begin{array}{*{20}l} E(P_{t})&=E\left[\frac{1}{K}\sum\limits_{n=\frac{K}{2}}^{\frac{3}{2}K-1} \left(|Y(n)|^{2}+2Y^{*}(n)Z(n)\right.\right. \\ &\quad +\Bigg.\left.|Z(n)|^{2}\right) \Bigg ] \\ &=\frac{1}{K}\sum\limits_{n=\frac{K}{2}}^{\frac{3}{2}K-1} E\left(|Y(n)|^{2}+|Z(n)|^{2}\right) \end{array} $$
Then, Ps is the estimated signal power, and Pt is the estimated total power. The noise variance will be Pt−Ps. Also, we can calculate the SNR as SNR=\(\frac {P_{s}}{P_{t}-P_{s}}\).
We provide the parameter computation for the first-order SNR prediction. The sum of the square errors between the samples on the curve γ′n−i and the actual measurements γn−i are
$$ E=\sum\limits_{i=1}^{w} {\left(\gamma\prime_{n-i}-\gamma_{n-i}\right)^{2}} $$
Our objective is to find the value of a and b when E achieves its minimum value. We can expand (21) as
$$\begin{array}{*{20}l} E&=\sum\limits_{i=1}^{w} \left(a^{2} t^{2}_{n-i} +2a(b-\gamma_{n-i})t_{n-i}+(b-\gamma_{n-i})^{2}\right) \\ &=a^{2} \sum\limits_{i=1}^{w} t^{2}_{n-i} + 2ab \sum\limits_{i=1}^{w} t_{n-i} -2a\sum\limits_{i=1}^{w} \gamma_{n-i} t_{n-i}\\ & \quad -2b\sum\limits_{i=1}^{w} \gamma_{n-i}+\sum\limits_{i=1}^{w} \gamma^{2}_{n-i}+\sum\limits_{i=1}^{w} b^{2}\\ \end{array} $$
Let us use the following notation for ease of expression: \(\alpha _{1}=\sum \nolimits _{i=1}^{w} t^{2}_{n-i}, \alpha _{2}=\sum \nolimits _{i=1}^{w} t_{n-i}, \alpha _{3}=\sum \nolimits _{i=1}^{w} \gamma _{n-i} t_{n-i}, \alpha _{4}=\sum \nolimits _{i=1}^{w} \gamma _{n-i}, \alpha _{5}=\gamma ^{2}_{n-i}, \alpha _{6}=w\). Then, we can simply express (22) as
$$\begin{array}{*{20}l} E&=\alpha_{1} a^{2} + 2\alpha_{2} ab-2\alpha_{3}a-2\alpha_{4} b+ \alpha_{5} +\alpha_{6} b^{2} \end{array} $$
To find the minimum value of E, we take its derivative in terms of a and b, respectively. Then, we force both the derivatives to 0 to obtain the following pair of equations:
$$ \left \{ \begin{aligned} \alpha_{1}a+\alpha_{2}b-\alpha_{3}&=0 \\ \alpha_{2}a+\alpha_{6}b-\alpha_{4}&=0 \\ \end{aligned} \right. $$
From (22), we know that E≥0 for all a and b, which means that there exists a minimum value of E. From (22), we can see that E is a convex function of a or b. As a result, the solution of a and b in the above equation pairs will enable E to achieve its minimum value. With a and b obtained, we can obtain our pre-estimate of \(\hat {\gamma _{n}} \prime \) as
$$ \hat{\gamma_{n}} \prime=f(t_{n})=at_{n}+b $$
While existing OFDM-based models have existed on the WARP repository (http://warp.rice.edu), they use System Generator to create the design, whereas our design is completely based upon Verilog HDL for system control and efficiency of compilation and real-time operation.
ACD:
AWGN:
Additive white Gaussian noise
Bit error rate
CIPRA:
Coherence-aware channel indication and prediction algorithm for rate adaptation
EWMA:
Exponential weighted moving average
FFT:
Fast Fourier transform
FPGA:
Field programmable gate array
Inter-carrier interference
IFFT:
Inverse fast Fourier transform
LCR:
Level-crossing rate
LWMA:
Linear weighted moving average
MCS:
Modulation coding scheme
MMSE:
Minimum mean square error
NLOS:
None line of sight
Orthogonal frequency-division
PHY:
Received signal strength
RSSI:
Received signal strength indicator
SDR:
SIFT:
Short Inter Frame Space
SNR:
WARP:
Wireless open-access research platform
Wireless fidelity
Y. Shi, R. Enami, J. Wensowitch, J. Camp, Uabeam: Uav-based beamforming system analysis with in-field air-to-ground channels, 2018 15th Annual IEEE International Conference on Sensing, Communication, and Networking (SECON), (2018).
X. Liu, M. Jia, Z. Na, W. Lu, F. Li, Multi-modal cooperative spectrum sensing based on dempster-shafer fusion in 5g-based cognitive radio, IEEE ACCESS, (2018).
X. Liu, X. Zhang, M. Jia, L. Fan, W. Lu, X. Zhai, 5g-based green broadband communication system design with simultaneous wireless information and power transfer, IEEE ACCESS, (2018).
X. Liu, M. Jia, X. Zhang, W. Lu, A novel multi-channel internet of things based on dynamic spectrum sharing in 5g communication, IEEE Internet of Things Journal, (2018).
J. C. Bicket, Bit-rate selection in wireless networks, Massachusetts Institute of Technology Library, (2005).
A. Ilori, Z. Tang, J. He, Y. Li, Throughput-based rate adaptation algorithm for IEEE 802.11 vehicle networks. Netw. IET. 4(2), 111–118 (2015).
C. -Y. Li, C. Peng, P. Cheng, S. Lu, X. Wang, F. Ren, T. Wang, An energy efficiency perspective on rate adaptation for 802.11n nic. Mob. Comput. IEEE Trans.PP(99), 1–1 (2015).
Z. Dong, P. Fan, E. Panayirci, X. Lei, Conditional power and rate adaptation for MQAM/OFDM systems under CFO with perfect and imperfect channel estimation errors. Veh. Technol. IEEE Trans.64(11), 5042–5055 (2015).
G. Wang, S. Zhang, K. Wu, Q. Zhang, L. M. Ni, TiM: Fine-grained rate adaptation in WLANs. Mob. Comput. IEEE Trans.15(3), 748–761 (2016).
C. -Y. Li, C. Peng, S. Lu, X. Wang, R. Chandra, Latency-aware rate adaptation in 802.11n home networks, IEEE INFOCOM, (2015).
J. Camp, E. Knightly, Modulation rate adaptation in urban and vehicular environments: Cross-layer implementation and experimental evaluation. Netw. IEEE/ACM Trans.18(6), 1949–1962 (2010).
Y. Du, P. Huang, D. Rajan, J. Camp, Cipra: Coherence-aware channel indication and prediction for rate adaptation, International Wireless Communications and Mobile Computing Conference (IWCMC), (2013).
M. Vutukuru, H. Balakrishnan, K. Jamieson, Cross-layer wireless bit rate adaptation, ACM SIGCOMM, (2009).
D. Halperin, W. Hu, A. Sheth, D. Wetherall, Predictable 802.11 packet delivery from wireless channel measurements, ACM SIGCOMM, (2010).
S. Sen, N. Santhapuri, R. R. Choudhury, S. Nelakuditi, Accurate: Constellation based rate estimation in wireless networks, SENIX NSDI, (2010).
G. Judd, X. Wang, P. Steenkiste, Efficient channel-aware rate adaptation in dynamic environments, the 6th international conference on Mobile systems, applications, and services, (2008).
Y. Shi, Improving the performance of RSS detection using wireless open-source platforms, Texas Symposium on Wireless and Microwave Circuits and Systems (TWMCS), (2015).
P. G. Xi Chen, D. Qiao, Ram: Rate adaptation in mobile environments. Mob. Comput. IEEE Trans.11(3), 464–477 (2012).
S. Abdallah, S. D. Blostein, Rate adaptation using long range channel prediction based on discrete prolate spheroidal sequences, Signal Processing Advances in Wireless Communic ations (SPAWC), (2014).
C. Min, N. Chang, J. Cha, J. Kang, MIMO-OFDM downlink channel prediction for IEEE 802.16e systems using Kalman filter, IEEE 2007 Wireless Communications and 848 Networking Conference (WCNC), (2007).
K. Farkas, K. Ruf, T. Hossmann, B. Plattner, Pattern matching based link quality prediction in wireless mobile ad hoc networks, ACM International Conference on Modelling, Analysis and Simulation of Wireless and Mobile Systems (MSWiM), (2006).
A. Wapf, M. R. Souryal, Measuring indoor mobile wireless link quality, IEEE ICC, (2009).
P. Casari, B. Tomasi, K. Pelekanakis, M. Chitre, M. Zorzi, Performance evaluation of SNR prediction schemes in acoustic communication systems using variable-rate modulation, IEEE Underwater Acoustic Measurement (UAM), (2011).
W. C. Jakes, Microwave mibile commmunications (Wiley, New York, 1975).
G. L. Stuber, Principles of mobile communications (MA:Kluwer, Norwell, 2000).
W. Zhou, W. H. Lam, A novel method of Doppler shift estimation for OFDM systems, IEEE Military Communications Conference (MILCOM), (2008).
T. S. Rappaport, Wireless communications: Principles and practice (PTR, Prentice Hall, 2001).
G. Park, D. Hong, C. Kang, Level crossing rate estimation with Doppler adaptive noise suppression technique in frequency domain, IEEE Vehicular Technology Conference VTC, (2003).
S. G. Johnson, M. Frigo, A modified split-radix FFT with fewer arithmetic operations. Signal. Process. IEEE Trans.55(1), 111–119 (2007).
Wild Packets White Paper (2018). http://www.wildpackets.com/elements/whitepapers/Converting_Signal_Strength.pdf.
C. Fernandes, J. Mota, Favier, Gerard, Cancellation of nonlinear inter-carrier interference in ofdm systems with nonliear power-amplifiers (Springer, Berlin/Heidelberg, 2010).
M. L. Huang, J. Lee, S. Park, An iterative SNR estimator for link adaptation in IEEE 802.11n System, the 23rd International Technical Conference on Circuits/Systems, Computers and Communications, (2008).
T. Yucek, H. Arslan, MMSE noise power and SNR estimation for OFDM systems, Sarnoff Symposium, (2006).
Z. Cvetkovic, V. Tarokh, S. Yoon, On frequency offset estimation for OFDM. Wirel. Commun. IEEE Trans.12(3), 1062–1072 (2013).
J. Goldberg, IEEE standard for W LAN- part 11: Wireless LAN medium access control (MAC) and physical layer (PHY) specifications, IEEE Standard Committee (2016).
A. Goldsmith, Wireless communications, Cambridge University Press, (2005).
J. Otaku, Minstrel Rate Adaptation Algorithm Documentation. http://madwifiproject.org/browser/madwifi/trunk/ath_rate/minstrel/minstrel.txt.
Broadcom Limited, Irvine, CA, USA
Yongjiu Du
Samsung Research America, Plano, TX, USA
Pengda Huang
Department of Electrical Engineering, Southern Methodist University, Dallas, TX, USA
Yan Shi
, Dinesh Rajan
& Joseph Camp
Search for Yongjiu Du in:
Search for Pengda Huang in:
Search for Yan Shi in:
Search for Dinesh Rajan in:
Search for Joseph Camp in:
YD and PH conceived the framework design and performed the data analysis. YS improved the work and took charge of all the work of paper submission. DR and JC reviewed and revised the manuscript. All authors have read and approved the final manuscript.
Correspondence to Yan Shi.
Du, Y., Huang, P., Shi, Y. et al. Design of coherence-aware channel indication and prediction for rate adaptation. J Wireless Com Network 2019, 201 (2019). https://doi.org/10.1186/s13638-019-1517-y
Rate adaptation
Channel indication
Channel coherence
MMSE | CommonCrawl |
Improving the odds of drug development success through human genomics: modelling study
Genetic drug target validation using Mendelian randomisation
Amand F. Schmidt, Chris Finan, … Aroon D. Hingorani
Advancing the use of genome-wide association studies for drug repurposing
William R. Reay & Murray J. Cairns
Using human genetics to improve safety assessment of therapeutics
Keren J. Carss, Aimee M. Deaton, … Jing Yuan
Phenotypes associated with genes encoding drug targets are predictive of clinical trial side effects
Phuong A. Nguyen, David A. Born, … Lucas D. Ward
Network and pathway expansion of genetic disease associations identifies successful drug targets
Aidan MacNamara, Nikolina Nakic, … Alex Gutteridge
Improving target assessment in biomedical research: the GOT-IT recommendations
Christoph H. Emmerich, Lorena Martinez Gamboa, … Michael J. Parnham
Predictive validity in drug discovery: what it is, why it matters and how to improve it
Jack W. Scannell, James Bosley, … J. Mark Treherne
Phenome-wide association studies across large population cohorts support drug target validation
Dorothée Diogo, Chao Tian, … Heiko Runz
Repurposing drugs to treat cardiovascular disease in the era of precision medicine
Mena Abdelsayed, Eric J. Kort, … Mark Mercola
Aroon D. Hingorani ORCID: orcid.org/0000-0001-8365-00811,2,
Valerie Kuan ORCID: orcid.org/0000-0001-7873-69721,2 na1,
Chris Finan1,2,
Felix A. Kruger3,
Anna Gaulton ORCID: orcid.org/0000-0003-2634-74004,
Sandesh Chopade1,2,
Reecha Sofat2,5,
Raymond J. MacAllister6,
John P. Overington ORCID: orcid.org/0000-0002-5859-10641,7,
Harry Hemingway ORCID: orcid.org/0000-0003-2279-06242,5,
Spiros Denaxas2,5,
David Prieto ORCID: orcid.org/0000-0001-5001-00615,9 na1 &
Juan Pablo Casas8
Target identification
Lack of efficacy in the intended disease indication is the major cause of clinical phase drug development failure. Explanations could include the poor external validity of pre-clinical (cell, tissue, and animal) models of human disease and the high false discovery rate (FDR) in preclinical science. FDR is related to the proportion of true relationships available for discovery (γ), and the type 1 (false-positive) and type 2 (false negative) error rates of the experiments designed to uncover them. We estimated the FDR in preclinical science, its effect on drug development success rates, and improvements expected from use of human genomics rather than preclinical studies as the primary source of evidence for drug target identification. Calculations were based on a sample space defined by all human diseases – the 'disease-ome' – represented as columns; and all protein coding genes – 'the protein-coding genome'– represented as rows, producing a matrix of unique gene- (or protein-) disease pairings. We parameterised the space based on 10,000 diseases, 20,000 protein-coding genes, 100 causal genes per disease and 4000 genes encoding druggable targets, examining the effect of varying the parameters and a range of underlying assumptions, on the inferences drawn. We estimated γ, defined mathematical relationships between preclinical FDR and drug development success rates, and estimated improvements in success rates based on human genomics (rather than orthodox preclinical studies). Around one in every 200 protein-disease pairings was estimated to be causal (γ = 0.005) giving an FDR in preclinical research of 92.6%, which likely makes a major contribution to the reported drug development failure rate of 96%. Observed success rate was only slightly greater than expected for a random pick from the sample space. Values for γ back-calculated from reported preclinical and clinical drug development success rates were also close to the a priori estimates. Substituting genome wide (or druggable genome wide) association studies for preclinical studies as the major information source for drug target identification was estimated to reverse the probability of late stage failure because of the more stringent type 1 error rate employed and the ability to interrogate every potential druggable target in the same experiment. Genetic studies conducted at much larger scale, with greater resolution of disease end-points, e.g. by connecting genomics and electronic health record data within healthcare systems has the potential to produce radical improvement in drug development success rate.
Almost all small molecule drugs and bio-therapeutics (such as monoclonal antibodies) act by perturbing the function of proteins. Drug development is therefore predicated on identifying those proteins or 'targets' that both play a causal role in a disease and are also 'druggable', i.e. amenable to pharmacological action by small molecule compounds, peptides or monoclonal antibody therapeutics. The ensuing challenges are to develop compounds specific for the target, with favourable pharmacokinetics and an acceptable toxicity profile, to prove target engagement, and to demonstrate clinical efficacy and safety in humans (Supplementary Note 1).
The extent of these challenges is revealed in an overall failure rate in drug development of over 96%, including a 90% failure rate during clinical development1,2,3,4,5,6. Failure rates are highest for drugs with a new mechanism of action against a previously 'undrugged' protein, and for diseases (e.g. Alzheimer's disease) where the pathogenesis is poorly understood.
Consequences of expensive drug development failures for Pharma have included site closures, job losses, and pruned R&D budgets. Failed R&D also inflates the price of the few successful drugs that trickle through development programmes, which are priced so as to recoup the incurred cost of historical failures and provide shareholders with a return on their investment7. This cost is borne initially by healthcare providers but then transferred to citizens through health insurance premiums or taxation.
High failure rates also discourage real innovation in favour of derivative compounds with identical mechanisms of action to existing drugs ('me too drugs'), minor formulation changes, or drug combinations, which all enjoy the same level of patent protection as drugs with a truly innovative mechanism of action, where the development risk is greater8. The result is that some diseases have few, if any, effective therapies, whilst others have a surplus of similar medicines jockeying for a market share. However, since healthcare providers are increasingly sophisticated in their assessment of the value of new medicines, derivative drugs with marginal benefits are now less likely to be taken up by healthcare systems than they once were9.
Governments, who are conflicted in their need to ensure cost-efficient healthcare on the one hand, but to support the pharmaceutical sector as a major employer and taxpayer on the other, has explored schemes to reduce barriers to market access for selected drugs10,11,12, but such schemes do not address the root of the drug development problem.
These issues suggest the need for a fresh approach that directly addresses the reasons for high rates of drug development failure13,14,15.
Superseding poor pharmacokinetics and toxicity, lack of efficacy in the intended indication has recently emerged as the major reason for late stage drug development failure, usually established in a randomised controlled clinical trial (RCT), the final step in the drug development pipeline16,17,18,19,20,21. A failure of this type is effectively an expensive demonstration that the target plays no role in the disease.
The reason for the high rate of late stage failure from lack of efficacy can be traced to two system flaws:
Preclinical experiments in isolated systems (cells, tissue preparations, isolated organs) together with animal disease models, which are used for the identification and validation of drug targets to progress into clinical phase testing, turn out to be poorly predictive of human efficacy
The pivotal clinical experiment, the RCT, is the final step in the drug development pipeline, which means that risk accumulates as a development programme progresses inflating the cost of any failure
The poor predictive ability of preclinical studies for human efficacy (an aspect of the so-called 'reproducibility crisis' in laboratory science) can be attributed in part to correctable flaws in experimental design including infrequent use of randomisation and blinding22,23,24,25.
However, errors of statistical inference leading to a high false discovery (FDR) rate may be equally important.
It can be shown (Supplementary Note 2 and Table 1) that
$$FDR=\frac{\alpha (1-\gamma )}{(1-\beta )\,\gamma +\alpha \,(1-\gamma )}$$
Table 1 The relationship between α, β andγ, the true discovery rate (TDR) and the false discovery rate (FDR).
\(\gamma =\mathrm{proportion}\,\mathrm{of}\,\mathrm{true}\,\mathrm{target} \mbox{-} \mathrm{disease}\,\mathrm{relationships}\)
\(\beta =\mathrm{false} \mbox{-} \mathrm{negative}\,\mathrm{rate}\)
\(1-\beta =\mathrm{power}\,(\mathrm{detection}\,\mathrm{rate}\,\mathrm{for}\,{\rm{a}}\,\mathrm{real}\,\mathrm{effect})\)
\(\alpha =\mathrm{false} \mbox{-} \mathrm{positive}\,\mathrm{rate}\)
FDR gives the probability of no causal relationship given success was declared, by applying Bayes rule to the above quantities.
False discoveries likely greatly outnumber true discoveries in preclinical research26 because:
The proportion of true relationships available for discovery (γ) is greatly outweighed by the proportion of false ones (1 − γ)
The usual experimental false positive rate (α) of 0.05 leads to many false relationships being declared as real27,28,29,30,31,32
Studies are often too small to reliably detect real relationships because the power(1 − β) is often lower than that pre-specified at the study design stage. Over optimistic estimates of effect sizes also means that when true relationships are detected, the effect sizes will be overestimated30
The result is that seemingly promising but flawed target-disease indication hypotheses are liable to progress from preclinical into clinical phase development only to stumble expensively at phase 2 or 3 for lack of efficacy.
The high FDR in standard preclinical research could be reduced by routinely setting more stringent values for (1 − β) and α32. However, there is a penalty to pay in the requirement for larger sample sizes (Supplementary Note 2). This is outwardly at odds with the 3R principles that encourage reduction in the number of animals sacrificed in medical research. However, ultimately, a smaller number of larger but definitive preclinical experiments may utilise fewer animals than numerous small, equivocal experiments undertaken in pursuit of an eventually futile hypothesis.
Nevertheless, other aspects of preclinical experimentation are unalterable: the proportion of true relationships available for discovery (γ) is fixed; experiments in isolated systems will never be fully representative of the situation in the whole animal; nor will animal models of human disease ever be completely reliable predictors of human success. A different solution is needed to address these limitations.
Relationships between variation in the genome and normal development and behaviour, physiology, metabolism, and disease susceptibility, (collectively, the phenotype), have been progressively uncovered in the last two decades. This has been enabled, in large part, by a single research design – the genome wide association study (GWAS). But the GWAS design is also beginning to reveal its potential as a new resource for drug development. GWAS have 'rediscovered' the known treatment indication or mechanism-based adverse for around 70 of the 670 known targets of licensed drugs33. This observation suggests that new drug targets for diseases with few effective therapies could also be identified using the same approach. Retrospective analyses have shown that the probability of a gene being associated with a human disease given that it encodes an approved drug target is greater than expected by chance34. Studies using variants in genes encoding individual targets have accurately predicted success or failure in RCTs35,36, helped separate mechanism-based from off-target actions of new drugs37,38, and identified new treatment indications and repurposing opportunities for established drugs39 (Supplementary Information). Genetic prediction of pharmacological action has been shown to encompass both small molecule drugs and bio therapeutics, on proteomics and metabolomics40, as well as physiological biomarkers and disease end-points. Collectively, these examples illustrate the potential of genetics and genomics to address the nub of the drug development problem: matching the right drug target with the right disease through GWAS (target identification); and delineating the diverse impacts of perturbing an individual target on a wide range of outcomes (target validation).
GWAS overcome many of the design flaws inherent in standard preclinical testing in isolated cells, tissues and animal models. They are an experiment in the correct organism (the human); have the lowest false discovery rate in any field of biomedicine (Supplementary Note 3); provide the systematic, concurrent interrogation of every potential drug target on the condition of interest (rather than a few targets selected from a larger pool); and exploit the unique attributes of genetic variation (fixed and allocated at random), which mimics the design of the pivotal experiment in drug development, the RCT41,42,43,44.
Studies that exploit the naturally randomised allocation of genetic variants that instrument an exposure of interest for causal inference have been termed Mendelian randomisation studies. Where the exposure of interest is the protein encoded by a specific gene and this is a drug target, the paradigm has been referred to as Mendelian randomisation for drug target validation (see Supplementary Information, Ref 1), since it was inspired by, and represents a special case of the Mendelian randomisation paradigm, which was applied initially to help determine the causal relevance of environmental exposures or disease related biomarkers45. A GWAS study can be considered to be a type of Mendelian randomisation analysis for drug target validation where variants in every gene encoding a drug target are interrogated for their association with a disease at the same time. This is made possible because naturally occurring variants in or around a gene (whether common or rare, coding or non-coding) are ubiquitous in the genome. Those that influence expression or activity of the encoded protein can, through their associations with biomarkers and disease end-points, anticipate the effect of pharmacological action on the same protein where this is druggable. Such an approach is disease agnostic, though it may be unsuited to aspects of cancer drug development, where somatic rather than germ line mutations perturb the targets of interest, or to the development of anti-infective drugs, in cases where the therapeutic drug target is in the pathogen rather than the human host.
In this paper, we develop a new conceptual framework and apply simple probabilistic reasoning to (a) explain why failure and inefficiency in orthodox preclinical drug development is the norm, and success the exception; and (b) estimate the probability of development success given the gene encoding the drug target is associated with the corresponding disease.
Since drug development depends on identifying proteins that play a causal role in a disease of interest, we introduce the concept of a sample space spanned by all human diseases – the 'disease-ome' – represented as columns; and all protein coding genes – 'the protein coding genome'– represented as rows. The result is a matrix of unique gene- (or equivalently protein-) disease pairings (Fig. 1).
Sample space (NG × NT) defined by 10,000 human diseases (columns) and 20,000 protein coding genes (rows). Expanded region comprising 1/10,000tℎ of the whole sample space is enlarged: (a) based on 10th causative genes per disease); (b) (based on 100 causative genes per disease); and c (based on 1000 causative genes per disease). Each cell represents a unique gene-disease pairing. Dark blue cells indicate causal gene-disease pairings, light blue cells druggable gene-disease pairings, with red cells indicating causal and druggable gene disease pairings.
We focus on common (multifactorial) human diseases of potential therapeutic interest that have both genetic and environmental contribution (Supplementary Note 4). We assume subsets of all the proteins encoded in the genome (Supplementary Note 5) play a causal role in any disease (Supplementary Note 6), and that only certain proteins are amenable to targeting by small molecule drugs or bio-therapeutics, leading to the concept of the 'druggable genome: the set of genes encoding actual or potential targets of drugs (Supplementary Note 7).
We therefore establish some definitions.
\(\{G\}\,{\rm{is}}\,{\rm{the}}\,{\rm{set}}\,{\rm{of}}\,{\rm{protein}}-{\rm{coding}}\,{\rm{genes}}\)
\(\{D\}\,{\rm{is}}\,{\rm{the}}\,{\rm{set}}\,{\rm{of}}\,{\rm{common}}\,{\rm{human}}\,{\rm{diseases}}\)
\(\{GD\}\,{\rm{is}}\,{\rm{the}}\,{\rm{set}}\,{\rm{of}}\,{\rm{all}}\,{\rm{possible}}\,{\rm{gene}}\,({\rm{or}}\,{\rm{protein}})-{\rm{disease}}\,{\rm{pairs}}\)
\(\{C\}\,{\rm{is}}\,{\rm{the}}\,{\rm{set}}\,{\rm{of}}\,{\rm{causal}}\,{\rm{genes}}\,{\rm{for}}\,{\rm{a}}\,{\rm{given}}\,{\rm{disease}}\)
\(\{CD\}\,{\rm{is}}\,{\rm{the}}\,{\rm{set}}\,{\rm{of}}\,{\rm{all}}\,{\rm{causal}}\,{\rm{gene}}-{\rm{disease}}\,{\rm{pairs}}\)
{T}isthesetofgenesencodingdruggabletargets: the druggable genome
Based on arguments rehearsed in Supplementary Notes 4–7 (see also Table S1 and Fig. 2), we set the following parameters:
Venn diagram illustrating the (a) the probabilities of selecting and (b) the number of causal, druggable gene-disease pair (\(CD\cap TD\)), a druggable gene disease pair (TD) and a causal, gene disease pair (CD) from 200 × 106 gene disease pairings, 100 causal genes per disease and 4000 druggable genes from the 20,000 in the genome. (Not to scale).
\({N}_{G}=\,{\rm{Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{protein}}-{\rm{coding}}\,{\rm{genes}}=20,\,000\)
\({N}_{D}\,={\rm{Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{complex}}\,{\rm{human}}\,{\rm{diseases}}=10,\,000\)
\({N}_{GD}={\rm{Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{gene}}-{\rm{disease}}\,{\rm{pairs}}=10,\,000\times 20,\,000=200\times {10}^{6}\)
\(C={\rm{the}}\,{\rm{number}}\,{\rm{of}}\,{\rm{causal}}\,{\rm{genes}}\,{\rm{in}}\,{\rm{a}}\,{\rm{given}}\,{\rm{disease}}\)
\(\bar{C}={\rm{the}}\,{\rm{average}}\,{\rm{number}}\,{\rm{of}}\,{\rm{causal}}\,{\rm{genes}}\,{\rm{per}}\,{\rm{disease}}=100\)
\({N}_{CD}={\rm{Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{causal}}\,{\rm{gene}}-{\rm{disease}}\,{\rm{pairs}}=100\times 10,\,000=1\times {10}^{6}\)
\({N}_{T}={\rm{Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{genes}}\,{\rm{encoding}}\,{\rm{druggable}}\,{\rm{targets}}=4000\)
We next formalise assumptions on which we base the subsequent calculations. Although some of the assumptions are oversimplifications, and exceptions can be identified from current drugs and diseases, they help to estimate certain 'base-case' probabilities. In Supplementary Note 8, we dissect these parameters and assumptions, and explore the impact of any modifications on our estimates.
Assumption 1: Each gene encodes a unique protein with a single function
Assumption 2: A given protein can influence the risk of more than one disease
Assumption 3: The probability of a protein influencing the pathogenesis of one disease is independent of the probability that it influences any other
Assumption 4: Drug treatments for human disease target proteins encoded in the germ line (We exclude drug targets encoded by the abnormal genome of cancer cells as well as antimicrobials, which typically target proteins encoded in the genomes of pathogens. For further discussion, see Supplementary Note 8).
Assumption 5: The probability that a protein affects disease pathogenesis and the probability the protein can be targeted by a drug is independent
Assumption 6: Inaccurate target selection is the exclusive reason for clinical phase drug development failure
Assumption 7: DNA sequence variants in and around a gene encoding a drug target that alter expression or activity of the encoded protein (cis-acting variants), are ubiquitous in the genome
Assumption 8: The association of cis-acting variants with biomarkers and disease end-points in a population genetic study accurately predict the effects of pharmacological modification of the encoded target in a clinical trial
Assumption 9: Genotyping arrays used in GWAS provide comprehensive, appropriately powered coverage of the genome, and associations discovered at any one gene are independent of those detected at any other gene
We use simple frequencies, binomial or hypergeometric distributions, and 2 × 2 tables to calculate a range of metrics relevant to drug development success, and to compare target identification based on standard preclinical models with target identification through GWAS.
Part A. Target identification through orthodox preclinical development
False discovery rate in preclinical science and drug development success rate
Ioannidis27 and others have provided empirical evidence from many research fields of extremely high rates of false discovery, leading to pervasive unreliability of the evidence base used to inform drug development46. In Bayesian terms, the prior probability of correctly pairing a causal gene (or protein) with a disease may be close to that of the background probability of a success in a random pick from the sample space.
Let us assume as a start point that this is the case. Then, using assumptions 1–3, the probability (Pc) that any gene- (or, equivalently, any protein)-disease pairing selected at random from the set of all possible gene-disease pairs {GD} also belongs to the set of causal gene-disease pairs {CD} is given by:
$${P}_{C}=\frac{{N}_{CD}}{{N}_{GD}}$$
$${P}_{C}=\frac{\bar{C}}{{N}_{G}}$$
Using either equation, and taking \(\bar{C}=100\); PC = 0.005
If \(\bar{C}=1000,{P}_{C}\,\)= 0.05
If \(\bar{C}=10,\,{P}_{C}\)= 0.0005
As follows from Eq. 3, Pc is independent of the number of diseases under consideration, as long as \(\bar{C}\,\) is constant.
PC can also be interpreted as the proportion of causal relationships amongst all possible gene-disease pairings, and can hence be represented as γC, the proportion of causal protein-disease relationships available for discovery (Supplementary Note 2).
$${P}_{C}={\gamma }_{C}$$
If preclinical experiments are initiated based on target-disease pairings drawn at random from the sample space, where \(\bar{C}=100\); γC = 0.005; α = 0.05; and (1 − β) = 0.8, then using Eq. 1,
$$FDR=\frac{\alpha (1-\gamma )}{(1-\beta )\,\gamma +\alpha \,(1-\gamma )}=92.6 \% $$
This FDR estimate is very close to that made previously by Ioannidis26 and also close the observed rate of drug development failure. We return to this point in a later section.
A priori probability of accurate drug target identification
Only a portion of the genome encodes proteins readily accessible to small molecule drugs, monoclonal antibodies or peptides that currently comprise the major chemical categories of medicines.
The probability(PT) of selecting a druggable gene (protein)-disease pairing at random is given by:
$${P}_{T}=\frac{{N}_{T}}{{N}_{G}}$$
$${P}_{T}=\frac{4,000}{20,000}=0.2$$
To estimate the probability PCT of selecting a disease-causing and druggable protein-disease pairing at random from the sample space, we take the probability that a protein affects disease pathogenesis and the probability the protein can be targeted by a drug to be independent (Assumption 5).
$${P}_{CT}={P}_{c}\times {P}_{T}$$
$$\begin{array}{l}{P}_{CT}=0.005\times 0.2\\ \,{P}_{CT}=0.001\end{array}$$
Corresponding probabilities and counts for scenarios in which \(\bar{C}=100,\,{\rm{and}}\,\bar{C}=1000\) are shown in Figs. S1 and S2 and Table S2. Note that these probabilities are independent of ND, the total number of diseases under consideration.
Following the arguments presented previously (Eq. 4), PCT can also be interpreted as γCT, the proportion of causal, druggable gene-disease pairs from the sample set of all gene-disease pairings.
From Eq. 1, with \(\bar{C}=100\), γCT = 0.001, α = 0.05; and (1 − β) = 0.8 the FDR for druggable and causal protein disease pairings is estimated as 98.4% (Table 1).
However, the probability of more direct interest is that of identifying a druggable, disease-causing gene having already specified the disease of therapeutic interest. Since we assume the probability of a protein influencing the pathogenesis of one disease is independent of the probability that it influences any other (Assumption 3) PC, PT and PCT are the same for each individual disease, as they are for the sample space overall.
For any given disease, with C causal genes, we can therefore write:
$$\begin{array}{rcl}{P}_{c} & = & \frac{C}{{N}_{G}}\\ {P}_{T} & = & \frac{{N}_{T}}{{N}_{G}}\\ {P}_{CT} & = & {P}_{c}\times {P}_{T}=(\frac{C}{{N}_{G}})(\frac{{N}_{T}}{{N}_{G}})\end{array}$$
These estimates can now be used to re-assort all genes in the genome from a therapeutic perspective for any given disease (Fig. 3).
Re-assorted 'therapeutic genome' of a hypothetical disease (d1). The 20,000 protein coding genes are organised into 100 causal and 19,900 non-causal genes. Causal genes are further subdivided into 20 that are also druggable and 80 that are not. Of the 20 causal, druggable genes, 3 are the targets of licensed drugs for the treatment of d1. Of the non-causal genes, 3980 are druggable but not causal for d1. The right hand panel indicates the expected number of true and false positive genes (including druggable genes) expected in a GWAS of d1 undertaken with a sample size that provides power, 1 − β = 0.8 and type 1 error rate of α = 5 × 10−8 at all loci.
For example, in a hypothetical disease (d1), where C = 100, the expected number of causal and druggable genes is given by:
$${P}_{CT}\times {N}_{G}=(\frac{100}{20,\,000})(\frac{4000}{20,\,000})\times 20,\,000=\,20$$
C − 20 = 80 causal genes would therefore be categorized as non-druggable. Of the NG − C = 19,900 non-causal genes, one fifth (\(\frac{{N}_{T}}{{N}_{G}}\times 19,900=3980\)) would be expected to be druggable but not causal in disease d1 (though of course some could be causal and of therapeutic interest in a different disease). The remaining 19,900 − 3980 = 15,920 genes would be classified as neither causal for d1, nor druggable.
Table S2 illustrates the influence of different estimates of C on PC(γC) and PCT(γCT).
Based on Eqs. 3–7, we can also write
$${\gamma }_{CT}=(\frac{C}{{N}_{G}})(\frac{{N}_{T}}{{N}_{G}}\,)$$
This equation suggests routes by which the a priori probability of accurate drug target identification might be increased. C is not amenable to manipulation, being largely determined by evolutionary forces; NG is also fixed; however, NT could be increased by developing technologies that allow a broader range of gene products to be targeted therapeutically. The development of therapeutic monoclonal antibodies has already increased NT by permitting targeting of proteins that were not previously amenable to a small molecule therapeutic strategy.
γCT could also be increased by constraining the sample space to the druggable genome. We could then write:
$${\gamma }_{CT}=(\frac{C}{{N}_{G}})(\frac{{N}_{T}}{{N}_{T}}\,)=(\frac{C}{{N}_{G}})$$
If C = 100, \({\gamma }_{CT}=\frac{100}{20,000}=0.005\)
Thus, the simple expedient of focusing target identification on the 4000 druggable genes, rather than all 20,000 protein-coding genes, increases γCT by a factor of five from 0.001 to 0.005: among the set of druggable genes, all causal genes are automatically both causal and druggable.
Alternatively, if it were possible, hypothetically, to reliably remove genes considered to have a low or no probability of playing a causal role in the disease of interest, i.e. focusing on the set {NC'}, where:
\(\{{N}_{C^{\prime} }\},=\)set of likely to be causal genes in the disease of interest
We could then write:
$${\gamma }_{CT}={P}_{CT}=(\frac{C}{{N}_{{C}^{\text{'}}}})(\frac{{N}_{T}}{{N}_{G}}\,)$$
If it were possible, hypothetically, to reliably remove genes considered to have a low or no probability of playing a causal role in the disease of interest, i.e. focusing on the set of causal genes, then:
$$\mathop{\mathrm{lim}}\limits_{{N}_{C^{\prime} }\to C}\,[(\frac{C}{{N}_{C^{\prime} }})(\frac{{N}_{T}}{{N}_{G}}\,)]\to (\frac{C}{C}\,)(\frac{{N}_{T}}{{N}_{G}}\,)=0.2$$
In the limit, among an exclusively causal set of genes, the probability of being causal and druggable is simply the probability of being druggable (Assumption 5). Eliminating non-causal while retaining causal genes is the crux of the target identification problem. We show later why GWAS (or whole genome or exome sequencing studies) address this issue as an inherent feature of their study design.
A posteriori estimates of true and false relationships explored in contemporary drug development
If the vast majority of research findings are false26, then the proportion of target-disease indication pairings studied in drug development should be close to that from a random pick from all possible target-indication pairs.
To estimate if this is the case, we use reported preclinical and success rates2,21 to make a posteriori estimates of the proportion of true target-disease relationships explored in preclinical and clinical phase development. We compare these a posteriori estimates to the a priori estimates based on a random pick of target-disease pairings in the sample space.
To facilitate the calculations, we reduce drug development to a two-stage process: a preclinical component (stage 1), whose function is to predict target-disease pairings destined for clinical phase success, and a clinical component (stage 2), whose function is to evaluate target-disease pairings brought forward from stage 1. Success in stage 2 is thus dependent on the predictive performance of stage 1. Since clinical phase drug development failure due to incorrect target specification accounts for around two in every three late-stage failures2,21, we utilize a further simplifying assumption (Assumption 6) that inaccurate target selection is the exclusive reason for clinical phase (stage 2) drug development failure.
Key variables in the following section are indexed by the lower-case suffix pc to denote preclinical and the lower-case suffix c to denote clinical stage development. Possible outcomes from pre-clinical and clinical phase development are summarized Table 2, where:
Table 2 The relationship α, β, and γ TP, TN, FP FN, and the declared success rate (s) in preclinical and clinical drug development (see text for details).
\(\gamma ={\rm{proportion}}\,{\rm{of}}\,{\rm{true}}\,\mathrm{target}-\mathrm{disease}\,{\rm{relationships}}\)
TP = true positive rate
FP = false positive rate
TN = true negative rate
FN = false negative rate
S = declared success rate
1 − S = declared failure rate
TDR = true discovery rate
If a clinical phase drug development programme follows every declared preclinical success, the proportion of true target disease relationships in clinical phase development is equivalent to the preclinical true discovery rate, so we can write:
$${\gamma }_{c}=TD{R}_{pc},\,({\rm{where}}\,TD{R}_{pc}=\frac{T{P}_{pc}}{{S}_{pc}})$$
It can be also be shown, by substitution and re-arrangement (Supplementary Note 9) that;
$$TD{R}_{c}=\frac{T{P}_{c}}{{S}_{c}}=\frac{TD{R}_{pc}\,(1-{\beta }_{c})}{TD{R}_{pc}(1-{\beta }_{c})+{\alpha }_{c}(1-TD{R}_{pc})}$$
By further substitution and re-arrangement (see Supplementary Note 9):
$$TD{R}_{C}=\frac{1}{1+(\frac{{\alpha }_{c}}{1-{\beta }_{c}})(\frac{{\alpha }_{pc}}{1-{\beta }_{pc}})(\frac{1-{\gamma }_{pc}}{{\gamma }_{pc}})}$$
Equation 10 illustrates that the clinical phase true discovery rate can be resolved mathematically into terms that encompass clinical phase power and experimental false positive rate \(({\rm{the\; term}}\frac{{\alpha }_{c}}{1-{\beta }_{c}})\), preclinical phase power and experimental false positive rate \(({\rm{the\; term}}\frac{{\alpha }_{pc}}{1-{\beta }_{pc}})\), and the true relationships available for discovery \(({\rm{the\; term}}\frac{1-{\gamma }_{pc}}{{\gamma }_{pc}})\). In this sense, Eq. 10 can be conceived as a mathematical summary of the probabilities and parameters determining drug development success. Equation 10 expresses TDRC as the odds of a randomly chosen drug being effective, the Bayes factor provided by a preclinical discovery, and the Bayes factor provided by a clinical discovery.
Using the calculations elaborated in Supplementary Note 9, and based on published 'success rates' for preclinical (Spc = 0.4)2 and clinical development (Sc = 0.1)2,22 and assuming values of α = 0.05 and 1 − β = 0.8, in both preclinical and clinical development, we estimateγc = 0.0667 and γpc = 0.03335; at αpc = 0.386 and FDRpc = 0.933.
Figure 4 illustrates values of γpc and αpc for a range of values for 1 − βpc from 0.2 to 0.8,using a fixed value of γc = 0.0667. For values of 1 − βpc in this range, values for γpc lie in the range 0.033 to 0.133, representing between a 6.5-fold to 26.5-fold enrichment in the proportion of true relationships actually studied in preclinical drug development over a random pick from a sample space demarcated by all diseases and the druggable genome (γpc = 0.005). Although these enrichment rates for established preclinical drug development might appear substantial, this degree of enrichment is insufficient to prevent a large proportion of false target-disease relationships being pursued during clinical phase development. This accounts for the low rates of clinical success. It also raises the possibility that a large proportion of declared clinical successes are actually themselves false discoveries, as illustrated by estimated values of TDRc (Table 2).
Back calculation of proportion of true target-disease relationships (γpc) studied in preclinical development, inferred from observed rates of clinical success (SC = 0.1) and preclinical success (Spc = 0.4). Estimates of γpc assume power in clinical phase development(1 − βc) = 0.8 and false positive rate in clinical development, αc = 0.05, so that the proportion of true target-disease relationships in clinical development, γc = 0.0667. The graph shows estimates of γpc (red line) for a range of values for power (1 − βpc) in preclinical development and corresponding estimates of the preclinical false positive rate, αpc (blue line). (See text for details).
Parallel development programmes for a single success
Pursuing multiple drug development programmes in parallel, each pursuing a different target, recognizing that the majority will fail, is a common, though inefficient strategy in contemporary drug development. For example, 1120 unique pipeline drug programmes for Alzheimer's disease were initiated across the industry in the period 1995–201447.
Around 4 in 100(0.04) preclinical drug development programmes yield licensed drugs. However, this estimate is based on the success rates of compounds rather than targets. The success in early development of a first-in-class molecule for a given disease indication is often followed by a flurry of development programmes, distributed across several companies, based on the same target and disease indication. The consequence is that multiple drugs may emerge, all in the same class. Using the ChEMBL database, we estimate a median of 2 (mean of 4) licensed drugs per efficacy target (Fig. 5). Therefore, the overall developmental success rate for targets could be around half that of compounds i.e. 2 in 100(0.02).
Distribution of number of licensed drug compounds per target.
With an overall developmental success rate for targets of 0.02, how many parallel programmes (N) should be pursued in order to have a 90% chance of at least one success?
Assuming all programmes are independent, the probability of all N programmes failing is:
$${(1-{P}_{s})}^{N}$$
where Ps = with in programme success rate
A 90% probability of at least 1 success equates to a 10% probability of no success in any programme (i.e. a 10% probability of all programmes failing). Therefore:
$${(1-{P}_{s})}^{N}=0.1$$
If Ps = 0.02
$$N=\frac{\log \,0.1}{\log \,(1-0.02)}=114$$
Thus, 114 parallel, independent programmes, should be pursued on average, to have a 90% probability of at least one developmental success; 34 programmes to have an 50% (evens) chance of at least one success. Values of N for a range of hypothetical values of Psare shown in Table S3.
Impact of a target selection step in orthodox preclinical drug development
Logistics and cost preclude orthodox (non-genomic) pre-clinical studies based on cells, tissues and animal models from evaluating the potential causal role of every protein in every disease. This imposes a selection step in drug development in which a subset of targets must first be prioritized for inclusion in preclinical drug development programmes. By contrast, as we elaborate later, a GWAS is capable of interrogating every target in parallel, without a selection step.
This selection step in standard preclinical drug development introduces two constraints. First, it results in slow progress in the investigation of target-disease indication hypotheses. To illustrate, the sample space spanned by the druggable genome and human diseases contains NT × ND = 40 × 106 unique druggable gene (or protein target)-disease pairs, of which 0.005 × (40 × 106) = 200,000 would be expected to be causal \(({\rm{if}}\,\bar{C}=100)\). A recent survey estimated only 15, 101 unique human target-indication pairings have been studied in drug development programmes over the last two decades, representing just 0.04% of this theoretical sample space48.
The second constraint is illustrated by a further probability consideration. The probability that 0, 1, 2, … A causal targets occurs in a sample of size N (where each member of the sample corresponds to an independent development programme based on a different drug target –disease indication pairing), drawn without replacement from the pool of 4000 druggable genes (proteins), of which C are causal for the disease of interest, is given by the hypergeometric distribution where:
$$P(A)=\frac{(\begin{array}{c}C\\ A\end{array})(\begin{array}{c}4,\,000-C\,\\ N-A\end{array})}{(\begin{array}{c}4,\,000\\ N\end{array})}$$
The expected number of causal, druggable targets E(A) in the sample of development programmes is given by:
$$E(A)=N(\frac{C}{4,\,000}),\,{\rm{with}}\,{\rm{SD}}=\sqrt{\frac{N\,C\,(4,\,000-{\rm{C}})(4,\,000-N)}{4,\,{000}^{2}(4,\,000-1)}}$$
Expected values for A based on a range of values of N and C are shown in Table S3. Four preclinical development outcomes are therefore possible: (a) one or more true positives is correctly identified with no false positives; (b) a mixture of one or more true and false positives emerge; (c) there are no positive findings; or, (d) in a worst-case scenario, one or more false positive results emerge with no true positives.
Unless N is very large (e.g. 200 independent preclinical programmes proceeding in parallel, each evaluating a different target), there is a very low probability of a causal, druggable target being included in the set of programmes selected for preclinical studies, based on a random pick. Let us assume one nominally positive target is pursued for clinical development under the three scenarios that generate positive findings from preclinical studies (regardless of whether they are true or false positives), and that correct target selection is the only barrier to eventual drug development success (Assumption 9). Under the first scenario, clinical development will always be successful, under the second it will sometimes be successful and under the fourth never successful. The overall probabilities of eventual development success are given by equations in Supplementary Note 10 and the results are shown in Tables S4 and S5 and Fig. 6. With 20 causal, druggable targets to find, increasing the number of parallel preclinical programmes from 20 to 50 to 200 has a modest impact on drug development success if these are picked from the full set of 4000 druggable proteins. The expected number of true positives will only be greater than the number of false positives if the set of targets in the sampling frame is relatively low (<400 targets) and all causal, druggable targets are retained in the sample. This emphasises the need for very strong priors before embarking on a drug development programme.
Probability of orthodox drug development success according to the number of candidate targets in the initial sampling frame (left panel) and the number of parallel preclinical development programmes pursued (right panel). The calculations assume there are 4000druggable genes and 20 causal, druggable targets per disease.
Probability of repurposing success
It would appear attractive to identify new disease indications for drugs that failed to show efficacy for the original indication, but which have proved safe in man; or to expand indications for a drug already effective in one disease to another condition (Table S6). However, repurposing or indication expansion relies on the assumption that different diseases share at least some common drug targets. How likely is this?
The probability of repurposing success can be considered from three perspectives:
How many diseases are likely to be influenced by the perturbation of a single therapeutic target?
How many diseases need to be considered for at least one pair of diseases to share a common therapeutic target, under the assumption of independence?
How many diseases need to be studied to find at least one that will be affected by pharmacological perturbation of a particular target of interest?
Diseases influenced by perturbation of a single protein: We showed previously in equation 2 (\({\rm{assuming}}\,\bar{C}=100\), ND = 10,000, and NG = 20,000):
$${P}_{C}=\frac{{N}_{CD}}{{N}_{GD}}=\frac{C}{{N}_{G}}=0.005$$
With PC = 0.005 the expected number diseases (ED) affected by any given gene (with standard deviation SD) is given by:
$${E}_{D}={P}_{C}\times {N}_{D}=0.005\times 10,000=50$$
$${S}_{D}=\sqrt{(1-{P}_{C})\times {P}_{C}\times {N}_{D}}=\sqrt{0.995\times 0.005\times 10,\,000}=7$$
ED declines the fewer diseases (ND) under consideration, or if \(\bar{C} < 100\) (see Table S2). Since the estimate of ED should be precisely the same for a gene encoding a druggable as a non-druggable target, under Assumption 5, it can be inferred that even the most specific of medicines is likely to influence a range of conditions; leading either to mechanism-based adverse effects, efficacy in more than one condition, or some combination of the two. In fact, under the assumptions above, we are 95% confident that perturbation of a therapeutic target will affect between 36 and 64 diseases and only 1 in 1000 targets would affect 28 or fewer conditions.
Shared therapeutic targets: Consider two diseases. If we assume \(\bar{C}=100\), the first disease in the pair could have any 100 of the 20,000 genes in the genome in its causal set. The probability of the second disease sharing a number x1 of the 100 genes already involved in the first disease is given by the hypergeometric distribution:
$$P({x}_{1})=\frac{(\begin{array}{c}100\\ {x}_{1}\end{array})(\begin{array}{c}20000-100\\ 100-{x}_{1}\end{array})}{(\begin{array}{c}20000\\ 100\end{array})}$$
So, the probability that the two diseases do not share any causal gene is:
$$P({x}_{1}=0)=\frac{(\begin{array}{c}100\\ 0\end{array})(\begin{array}{c}20000-100\\ 100-0\end{array})}{(\begin{array}{c}20000\\ 100\end{array})}=0.605$$
If we study a third disease, the probability of that disease sharing x2 of the 200 genes involved in the previous two diseases would be:
So, the probability of the third disease not sharing a single gene with the other two (x2 = 0) is:
So the total probability of the three diseases not sharing any of the genes is:
$$P({x}_{1}=0)\times P({x}_{2}=0)=0.605\times 0.365=0.221$$
With four diseases, the probability of none of them sharing a gene is <5%, and for eight diseases it is less than 1 in a million: it is almost certain that at least two diseases from this pool of eight, will share at least one common susceptibility gene.
Number of diseases that need to be studied to identify at least one that is affected by perturbation of a given target: The answer to the third question follows the same reasoning as that used previously to estimate the number of drug development programmes that need to be pursued in parallel to have at least a 90% or greater chance of at least one development success. With PC = 0.005(i.e. focusing on the druggable genome), 460 diseases would need to be studied to have ≥90% chance of identifying at least one condition that is causally affected by perturbation of a particular target of interest. When \(\bar{C}=1000\), the number of diseases that need to be studied is 45.
Despite these considerations, the ultimate challenge for repurposing remains the same as that for de novo drug development: knowing precisely which targets are important in which diseases and therefore which targets are shared among a set of diseases of interest. We show in the next section how a human genomic approach to drug development is well placed to address this critical issue.
Part B. Target identification through GWAS
Design features of GWAS that address the major contributions to drug development failure are: (1) investigation of humans, not animal models; (2) a much more stringent∝ value (typically 5 × 10−8) than is routine in orthodox preclinical studies49; (3) concurrent interrogation of every drug target in parallel obviating the need for a selection step; and, (4), the naturally randomised allocation of genetic variants that mimics the design of a randomised controlled trial.
To attempt to quantify potential efficiency gains from using GWAS rather than standard preclinical models for drug target identification, we review the number of licensed drug targets already 'rediscovered' by GWAS; estimate the expected 'yield' of drug targets from a well powered GWAS in a disease of interest; and the predictive accuracy of GWAS for drug target identification, compared to the conventional preclinical study-based approach.
Rediscovery of licensed drug target-disease indications by a GWAS
Examples of the apparently sporadic 'rediscovery' by GWAS of drug targets already exploited for the treatment of the corresponding disease, as well as rediscoveries of the known mechanism-based adverse effects of several drug classes are included in Table 3 and a linked paper33. Are such rediscoveries serendipitous or predictable?
Table 3 (following pages). Illustrative examples of mapping SNPs curated in the GWAS catalogue to genomic linkage dis-equilibrium (LD) intervals containing targets of licensed and clinically used drugs (adapted with modification from.Finan C, Gaulton A, et al. Sci. Translational Med. 2017 Mar 29; 9(383). pii: eaag1166. doi: 10.1126/scitranslmed.aag1166).
Among diseases with at least one licensed drug treatment, the total number of targets exploited by such drugs will vary. For example, nine drug classes (corresponding to nine different drug targets) contain compounds currently licensed for the treatment of type 2 diabetes but only two therapeutic classes contain compounds licensed for treatment of dementia. We can safely assume, from the efficacy of these drugs, that their targets (along with others, yet to be identified) play a causal role in the course of those diseases.
Consider the hypothetical disease (d1), for which g1, g2 … gn independent genes encode targets of drugs that have already been licensed on the basis of proven efficacy in the condition. Let us assume that a GWAS in disease d1 utilises a genotyping array with adequate coverage of all nlicensed drug target genes, that the probability of missing such a target is the false negative rate(β) and therefore there is a probability ((1 − β1), (1 − β2) … (1 − βn)) of detecting the genetic association at each of these loci. Thus (1 − βi) is the power (or the detection rate) for a real effect of gene giin disease d1.
We consider testing for a genetic association at the locus encoding each drug target in each hypothetical GWAS of d1 to be an independent trial (Assumption 7), where success equates to detection of an association at the locus and failure to overlooking the association. If there are 3 licensed drug targets in disease d1 available for rediscovery, and the power to detect true associations is the same at all 3 target loci i.e. (1 − β1) = (1 − β2) = (1 − β3) = (1 − β). A GWAS in d1 might detect 0, 1, 2 orall 3 of the known drug targets, and the probability that each of these situations occurs is given by the binomial distribution:
$$P\,(x)=(\begin{array}{c}{n}_{1}\\ x\end{array}){(1-\beta )}^{x}{\beta }^{{n}_{1}-x}$$
\(P\,(x)={\rm{the}}\,{\rm{probability}}\,{\rm{of}}\,{\rm{detecting}}\,x\,{\rm{licensed}}\,{\rm{drug}}\,{\rm{targets}}\)
\({n}_{1}={\rm{the}}\,{\rm{number}}\,{\rm{of}}\,{\rm{licensed}}\,{\rm{drug}}\,{\rm{targets}}\,{\rm{in}}\,{\rm{disease}}\,{d}_{1}\)
\({n}_{1}-x={\rm{the}}\,{\rm{number}}\,{\rm{of}}\,{\rm{undetected}}\,{\rm{licensed}}\,{\rm{drug}}\,{\rm{targets}}\,{\rm{in}}\,{\rm{disease}}\,{d}_{1}\)
\(\beta ={\rm{Type}}\,{\rm{II}}\,({\rm{false}}\,{\rm{negative}})\,{\rm{error}}\,{\rm{rate}}\,{\rm{at}}\,{\rm{each}}\,{\rm{genetic}}\,{\rm{locus}}\)
If β = 0.2, the probability (P) that a GWAS in disease d1:
Detects none of the three licensed drug target genes, P(x = 0) = β3 = 0.008
Detects only one of the three licensed drug target genes but misses the remaining two, P(x = 1) = 3β2(1 − β) = 0.096
Detects only two of the three licensed drug target genes but misses the other, P(x = 2) = 3β(1 − β)2 = 0.384
Detects all three licensed drug target genes, P(x = 3) = (1 − β)3 = 0.512
Detects at least one of the three licensed drug target genes, P(x > 0) = 1 − β3 = 1 − 0.008 = 0.992
In general, if power at all loci in a GWAS of a disease dis (1 − β) and there are nd licensed drug targets to rediscover, the expected number of drug targets rediscovered (Ed) and its standard deviation (Sd) will be given by:
$${E}_{d}={n}_{d}\,(1-\beta )$$
$${S}_{d}=\sqrt{{n}_{d}\,\beta \,(1-\beta )}$$
In the worked example, we would therefore expect 2.4(SD = 0.7) of the 3 possible licensed drug targets to be rediscovered, on average.
Suppose we do one GWAS for each of K different diseases (d1, d2 … dK) where, for each disease, the number of licensed targets available for rediscovery is (n1, n2, … nK). If we assume that the power to detect an association at gene i encoding the target of licensed drug is the same for all drug targets in all GWAS j, regardless of disease (i.e. (1 − βi,j) = (1 − β) for all i and j), then the expected number of true drug target-indication rediscoveries (ET) across the K GWAS would be the sum of the expected rediscoveries in each GWAS. Therefore:
$${E}_{T}={E}_{1}+{E}_{2}+\ldots +{E}_{K}$$
$${E}_{T}=(1-\beta ){n}_{1}+(1-\beta ){n}_{2}+\ldots +(1-\beta ){n}_{K}$$
$${E}_{T}=(1-\beta )({n}_{1}+{n}_{2}+\ldots +{n}_{K})$$
$${E}_{T}=(1-\beta ){N}_{K}$$
NK = (n1 + n2 + … + nK) = the total number of licensed drug targets for K diseases
Dividing and multiplying the above equation by K, we obtain:
$${E}_{T}=K(1-\beta ){N}_{K}/K$$
$${E}_{T}=K(1-\beta )\bar{n}$$
Where;
\(\bar{n}\) = NK/K = the average number of targets of licensed drugs per disease
The standard deviation (SDT) is given by:
$$S{D}_{T}=\sqrt{\beta (1-\beta )\,\bar{n}\,K}$$
Suppose a GWAS was done for each of 200 different diseases, each with power (1 − β) = 0.8 to detect each true licensed target, and \(\bar{n}\) = 3(i.e. an average of 3 targets per disease and NK = \(\bar{n}\)K = 600 potentially re-discoverable target-disease combinations in total).
The total number of licensed drug target rediscoveries from the combined dataset would be expected to be:
$${E}_{T}=(1-\beta ){N}_{K}=480$$
$$S{D}_{T}=\sqrt{0.2\times 0.8\times 600}=9.8$$
Values of ET for a range of plausible values of β and \(\bar{n}\), given K = 200 are provided in Table S7.
It seems reasonable to ask if the number of licensed drug target rediscoveries already made by GWAS is close to that expected from these arguments. However, the answer is not straightforward. It requires enumerating the number of GWAS that have already been done for conditions that correspond to either a treatment indication or a mechanism based adverse effect for at least one licensed drug target, and counting the total number of licensed drug targets represented across all these conditions (since some diseases may be connected with multiple licensed drug targets). Different disease terminologies used to catalogue GWAS, drug indications and adverse effects hamper these efforts. There is also a requirement to make strong assumptions about the average power of eligible GWAS to detect a true association at a gene encoding a licensed drug target.
However, the question can also be inverted: given the observed number of rediscoveries, what was the average power of GWAS to rediscover loci encoding licensed drug targets for the same indication or through a known mechanism-based adverse effect? We previously reported that GWAS to 2015 had encompassed 315 unique MeSH disease terms and led to the 'rediscovery' of 74 of the 670 or so known licensed drug targets, either through treatment indication, or mechanism-based adverse effect association33.
To estimate average power, we use:
$${E}_{T}=K(1-\,\beta )\,\bar{n}$$
$$(1-\,\beta )=\frac{{E}_{T}}{\bar{n}\,K}$$
$$(1-\beta )=\frac{74}{\bar{n}\times 315}$$
$$(1-\beta )=\frac{74}{315}\times \frac{1}{\bar{n}\,}$$
$$(1-\beta )=\frac{0.23}{\bar{n}\,}$$
If \(\,\bar{n}=1,(1-\beta )=0.23\)
If \(\bar{n} < 1,(1-\beta ) > 0.23\) (as would be the case if some GWAS concerned diseases with no licensed drug target available for rediscovery)\(\mathrm{If}\,\bar{n} > 1,(1-\beta ) < 0.23\)
Despite the modest estimated average power, the discovery by GWAS of around 74 of the 670 or so known licensed targets, suggests the approach shows promise as a means of identifying target-disease indication pairings more systematically in the future, particularly if power were to be enhanced. We return to this point in a later section.
Estimated yield of druggable targets from a GWAS
In the previous section, we discussed the rediscovery of known licensed drug targets by GWAS. In this section, we discuss the potential for GWAS to specify new drug targets for common diseases prospectively.
For example, take the hypothetical disease (d1), where C = 100, and the expected number of causal and druggable genes is 20. Assuming a GWAS in d1interrogates each of the causal protein-coding genes with power (1 − β) = 0.8, the expected number of causal, druggable targets (ECT,d1) identified by such a GWAS is given by:
$${E}_{CT,d1}={n}_{CT,d1}(1-\beta )$$
(where nCT,d1is the true number of causal, druggable targets in d1)
$${E}_{CT,1}=20\times 0.8=16$$
$$S{D}_{CT,1}=\sqrt{{n}_{CT,d1}\,\beta \,(1-\beta )}=1.8$$
The probability of a GWAS detecting x = 0, 1, 2, 3, 4, … all 20 of the available causal, druggable targets is again given by the binomial distribution:
$$P\,(x)=(\begin{array}{c}{n}_{CT,d1}\\ x\end{array}){(1-\beta )}^{x}{(\beta )}^{{n}_{CT,d1}-x}$$
P(x) is the probability of detecting x causal, druggable targets
nCT,d1 is the number of causal, druggable targets in disease d1 (20 in this example)
nCT,d1 − x is the number of causal, druggable targets not detected in the GWAS
(1 − β) is the power of the GWAS to detect a true association at a genetic locus (set at 0.8 in this analysis and assumed to be homogeneous for all loci)
In summary, with C = 100, PC = 0.005, PT = 0.2, i.e. PCT = 0.001,a GWAS with power 1 − β = 0.8 at all loci would be expected to discover 16 (SD1.8) of the 20 available, causal, druggable targets, on average. Moreover, it would be extremely unlikely that a GWAS with (1 − β = 0.8) at all loci, would discover fewer than 10druggable targets.
The exceedingly stringent type 1 error rate (α) incorporated in GWAS (e.g. 5 × 10−8) also makes the probability of even one false target discovery being present among the declared associations very low indeed (Fig. 3). These calculations suggest that adequately powered GWAS (designed with appropriate consideration of the distribution of genetic effect sizes, sample size and comprehensive coverage of sequence variation in protein coding genes) should provide a highly accurate and reliable way of specifying drug targets for human diseases, addressing the high FDR problem that underpins inefficiency in drug development.
Comparison of orthodox preclinical drug development vs. human genomics as a predictive test for drug development success
Consider orthodox non-genomic preclinical (stage 1) drug development programmes with base case parameters defined by the sample space, NG × ND where:
\({N}_{G}={\rm{Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{protein}}-{\rm{coding}}\,{\rm{genes}}=20,\,000\)
\({N}_{D}={\rm{Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{complex}}\,{\rm{human}}\,{\rm{diseases}}=10,\,000\)
\(\bar{C}={\rm{Average}}\,{\rm{number}}\,{\rm{of}}\,{\rm{causal}}\,{\rm{genes}}\,{\rm{per}}\,{\rm{disease}}=100\)
\({N}_{T}={\rm{Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{genes}}\,{\rm{encoding}}\,{\rm{druggable}}\,{\rm{targets}}=4,\,000\)
From Eq. 7, we can infer that the proportion of causal and druggable target-disease indication pairs available for rediscovery is;
$${\gamma }_{pc}=(\frac{\bar{C}}{{N}_{G}})(\frac{{N}_{T}}{{N}_{G}}\,)=(\frac{100}{20,\,000})(\frac{4,\,000}{20,\,000}\,)=0.001$$
Setting αpc and βpc to 0.05 and 0.2 respectively, see previous note, and assuming it were somehow possible to evaluate every protein in every disease in such studies, then TDRpc = 0.016 and FDRpc = 0.984.TDRpc increases to 0.14 and the FDRpc falls to 0.86 if \(\bar{C}=1000\) \(({\gamma }_{pc}=\frac{1}{100})\), but the corresponding values are 0.002 and 0.998 if \(\bar{C}=10\) \(({\gamma }_{pc}=\frac{1}{10,000})\) (Table 4).
Table 4 A priori estimates of preclinical (pc), clinical (c) and overall (o) drug development success contrasting orthodox (non-genomic) with genomic approaches.
In striking contrast, with the same sample space but a genomic approach to target identification, where (1 − β) = 0.8, α = 5 × 10−8 and all 20,000 targets encoded by the genome are, by definition, interrogated simultaneously, TDRpc = 0.999, and FDRpc = 0.001. This is a reversal of TDRpc and FDRpcvalues when compared to the orthodox (non-genomic) preclinical approach. The performance of genomic studies for target identification, based on these values of α and 1 − β, is little affected by 100-fold differences in \(\overline{C\,}\) andγpc (Table 4).
As we showed previously, if sampling were restricted to the a sample space demarcated by the druggable genome, NT × ND, where;
\({N}_{D}\,={\rm{Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{complex}}\,{\rm{human}}\,{\rm{diseases}}=10,000\)
\({N}_{T}={\rm{Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{genes}}\,{\rm{encoding}}\,{\rm{druggable}}\,\mathrm{targets}\,=4000\)
\({N}_{TD}={\rm{Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{possible}}\,{\rm{druggable}}\,{\rm{gene}}-{\rm{disease}}\,{\rm{pairs}}=4,000\times 10,000=40\times {10}^{6}\)
$${\gamma }_{pc}=(\frac{\bar{C}}{{N}_{G}})(\frac{{N}_{T}}{{N}_{T}}\,)=(\frac{100}{20,000}\,)(\frac{4,000}{4000}\,)=0.005$$
Focusing orthodox (non-genomic) preclinical studies on this restricted sample space (with conventional values for α and (1 − β) marginally increases the TDRpc(from 0.016 to 0.08) and reduces FDRpc but also only marginally (from 0.998 to 0.920). Applying the genomic approach in the same sample space, where (1 − β) = 0.8, and α = 5 × 10−8, and all 4,000 druggable targets encoded by the genome are interrogated simultaneously, the already high TDRpc increases to 0.9999, and the already low FDRpc would fall further to 0.0001 (Table 4).
Based on Assumption 7 (DNA sequence variants in and around a gene encoding a drug target that alter expression or activity of the encoded protein (cis-acting variants), are ubiquitous in the genome) the approach of applying the usual type 1 error rate (α) used in a GWAS (5 × 10−8) but to association tests undertaken on only the 2% or so of the genome occupied by protein coding genes (or perhaps 0.5% of the genome occupied by genes encoding druggable targets) should reduce the multiple testing burden by about 50-fold compared to a standard GWAS, where association tests are undertaken genome wide. Moreover, the use of gene rather than SNP based association testing (e.g. using Predixscan50, VEGAS51 and FastBAT52) would also help mitigate the multiple testing burden.
It might be argued that TDRpc and Spc in conventional (non-genomic) preclinical pipelines could also be enhanced by simply setting a more stringent false positive rate in experiments involving cells, tissues and animal models. This is correct, but the change would have practical consequences. Very substantial increases in sample size would be required to maintain power. However, attending to the type 1 error rate issue alone fails to address the problem of the questionable validity of many animal models of human disease. It is also predicated on being able to evaluate every protein in every disease, a task we know to be beyond the capability of orthodox (non-genomic) preclinical studies based on cells, tissues and animal models.
Turning now to clinical (stage 2) development, αc and 1 − βc are typically set to 0.05 and 0.8 respectively, so it is also possible to examine the influence of variation in γpc, αpc and βpc on preclinical (Spc), clinical (Sc) and overall success (So = Spc × Sc), using Eqs. 9 and 10. The results are summarised in Table 4.
For orthodox (non-genomic) preclinical development, with sampling from the whole genome (where \(\bar{C}=100,\,1-{\beta }_{pc}=0.8\), \({\alpha }_{pc}=0.05,\,{\gamma }_{pc}=\frac{1}{1000\,})\), Spc = 0.05(TDRpc = 0.016; FDRpc = 0.984) and Sc = 0.06(TDRc = 0.2; FDRc = 0.8) giving an overall declared drug development success rate So = Spc × Sc = 0.003 (Table 4).
With the same parameters \((\bar{C}=100,\,{\gamma }_{pc}=\frac{1}{1000\,})\), but with the genomic approach replacing orthodox non-genomic preclinical programmes, Spc = 0.0008(TDRpc = 0.99994; FDRpc = 0.00006), Sc = 0.79995(TDRc = 0.999996; FDRc = 0.000004), and So = 0.00064. It may at first seem surprising that Spc (and So) is actually lower for genomic than orthodox (non-genomic) stage 1 development, because of a higher stage 1 'failure' rate. However, a stage 1 'failure' in a GWAS simply refers to a null association with the disease of interest of a specific gene (from all 20,000 evaluated in a single study), which is very different from the expensive failure of a lengthy orthodox preclinical development programme focusing on a single target at a time. The high 'failure rate' (i.e. high rate of null associations) in GWAS reflects the much more stringent αpc in this type of study design, which results in a much lower FDRpc and much higher TDRpc. Since TDRpc = γc, the GWAS design ensures fewer false relationships are carried forward into clinical development when compared to the non-genomic approach. Consequently, TDRc is much increased with the genomic (compared to non-genomic) preclinical target identification.
Summary of findings
In summary, the calculations indicate that a genomic approach to preclinical target validation has the potential to reverse the probability of drug development success when compared to the established (non-genomic) approach.
Drug development success has previously been constrained by:
The apparently widespread contamination of the scientific literature by false discoveries, which undermines the validity of the hypotheses used to prioritise the selection of drug targets for different diseases;
The poor predictive accuracy of orthodox preclinical studies, arising due to shortfalls in design and animal-human differences in pathophysiology;
The limitation of such preclinical studies in only being able to study a handful of targets at a time, imposing a need for selecting only a subset of all possible targets
The system flaw in drug development that sees the definitive target validation step (the RCT) deferred to the end of the drug development pipeline.
With reasonable assumptions about the number of protein coding genes, druggable proteins and human diseases, and using probabilistic reasoning, we estimated that the observed success rate in drug development \(( \sim \frac{4}{100}\) for compounds; \( \sim \frac{2}{100}{\rm{for\; targets}})\) only marginally exceeds the probability \((\frac{1}{200})\) of correctly selecting a causal, druggable protein-disease pair through a random pick from a sample space defined by the 4,000 genes that are predicted to encode druggable targets and 10,000 diseases, assuming an average of 100 causal genes per disease. With a target success rate of \(\frac{2}{100}\), based on the orthodox (non-genomic) approach to target selection and validation, over 100 independent drug development programmes for each disease need to proceed in parallel to have a 90% probability of even one success.
Based on reported clinical and preclinical success rates, and making reasonable assumptions about values of clinical phase type 1 and type 2 error rates (αc and βc),we also found evidence that the proportion of true target disease relationships studied in preclinical development is small, that these form only the minor proportion of nominally positive findings that are brought forward in to clinical phase studies. This likely contributes to the high preclinical false discovery rate and low clinical phase success rate.
Even applying the assumption that the probability of a protein influencing the pathogenesis of one disease is independent of the probability of it influencing any other, we show that it is highly likely that even small groups of diseases taken at random share at least one common target. This implies numerous opportunities should exist for therapeutic repurposing, but also that even highly specific modification of any target still runs a high risk of mechanism-based adverse effects. The balance between the two remains to be discovered. However, knowledge of the effect of target-specific perturbation on multiple disease outcomes currently remains incomplete because the orthodox approach to target identification and validation is neither systematic nor comprehensive.
In contrast to established non-genomic, approaches to preclinical drug development, GWAS deliver a methodical and reliable means of specifying the correct drug targets for a disease, provided that the genotyping arrays that are deployed have sufficient coverage of the druggable genome, and that the studies are adequately powered. GWAS differ from established non-genomic preclinical experiments for target identification in that the evidence source is the human not an animal model; the false positive (type 1) error rate is low (typically set at 5 × 10−8); every potential drug target is interrogated in parallel (not just a selected subset); and the study design shares features of an RCT, the pivotal step in drug development. For these reasons, we suggest that genetic studies will soon be universally regarded as an indispensable, though not exclusive element of drug development for common diseases. By improving the efficiency and reliability of target identification, GWAS and similar genetic study designs offer the potential to overturn the currently poor odds of success currently beleaguering drug development.
Implications for drug development
Despite the opportunities highlighted by this paper, GWAS are yet to be optimally designed or sufficiently widely deployed to maximise their potential for drug development. Most genotyping arrays used in early GWAS provided incomplete coverage of variation in genes encoding druggable targets. To address this, we recently assembled variant content for the Illumina DrugDev genotyping array, designed to for low-cost, high-volume genotyping of samples to support genetic association studies for drug target selection and validation ('druggable GWAS')33.
The range of diseases studied has also been limited. The 400 or so unique diseases and biomarkers tackled by GWAS so far represents only a fraction of the thousands of disease terms listed by classification systems or ontologies, or that are observed in electronic health record datasets (Supplementary Note 4). Sample sizes in most GWAS may also have been too small to detect all contributing genes and all relevant drug targets.
GWAS up to now have also typically been undertaken one disease at a time using investigator-led, research-funded case collections. Yet, when the findings are collated, the same genetic loci or even variants are seen to contribute to more than one disorder, a phenomenon referred to as 'pleiotropy'53. Pleiotropy can arise through a number of mechanisms, but where explained by the involvement of the same protein in the pathogenesis of different diseases, it unveils opportunities to repurpose therapies ineffective in one condition for another, to expand indications for already effective therapies, and to identify potential mechanism-based adverse effects of target perturbation. Undertaking GWAS one disease at a time, while efficient for accumulating large numbers of cases with a particular condition, is inefficient for the investigation of pleiotropy as a means of target validation and developing repurposing hypotheses.
To realise the full potential of genomics for drug target identification and validation, comprehensive capture of variation in the genome (by sequencing or genotyping) needs to be connected to the diversity of human phenotype at even larger scale than now, with attention to multiple biological layers and disease end-points. There are several routes to achieving this.
Amalgamating large cohort studies and consortia across the globe
GWAS in population based research cohort studies allows interrogation of multiple phenotypes in the same dataset. Such studies are well placed to evaluate genetic associations with mRNA and protein expression, with metabolite level and measures of organs and systems function. Even when obtained in different datasets, information of this type can be connected using a variety of statistical methods, because natural genetic variation (unaffected by disease and allocated at random) provides a fixed anchor point, exploiting the central dogma of the molecular biology that posits a unidirectional flow of information from DNA to RNA to protein54 and, via downstream mechanisms, to disease. In recognition of this, the Global Genomic Medicine Collaborative (G2MC) is gathering information on large cohorts worldwide55.
Embedding genomics in whole healthcare systems
However, cases of common diseases accrue slowly in cohort studies, such that power to detect the effects of common variants on such conditions may be limited. This is partly addressed by meta-analysis of summary level data from the many existing cohorts and consortia, and through the ongoing assimilation of data from very large national biobanks56. Nevertheless, additional effort will also be required to increase the scale, breadth and depth of disease outcomes captured. An efficient approach would be to embed genomic analysis within the healthcare setting so that information on natural genetic variation could be linked to the wealth of laboratory, imaging, and diagnostic data captured routinely during each clinical episode to provide insight both on disease aetiology and to unveil new drug targets57.
Some population cohort and healthcare genomics initiatives of this type are beginning, some in conjunction with Pharma (Table 5), but if their use is to be expanded, funders, healthcare providers, patients and populations will need to be convinced of the benefits of this new model for drug development. Legitimate concerns about data security and the secondary use of data also need to be addressed, an issue to which we return later. If successful, a new model of drug development might supervene because population and healthcare data typically resides outside the domain of the pharmaceutical industry within the academic and healthcare sectors, which, in many countries, are wholly or substantially state-run. In turn, this would dictate that a new funding and delivery structure might need to be established, at least for the component of drug development that relates to target identification and validation.
Table 5 Selected examples of Academia, Pharma, and Pharma-Academia initiatives concerning genomics and drug development.
There would be additional benefits from such an effort. We have focused here mainly on GWAS for matching targets to a disease (target identification). However, in related work (see Appendix 1) we (and others) have shown that the principle can also be used to anticipate the spectrum of effects of pharmacological action on a specific target on biomarkers, disease surrogates and clinically relevant disease end-points (sometimes called phenome wide association analyses; PheWAS) for target validation (Fig. 7). PheWAS (or Mendelian randomisation for drug target validation) has been used to accurately predict phase 3 trial outcomes, distinguish on- from off- target effects of drugs, correctly identify detailed biomarker profiles of therapeutic response, and to identify repurposing opportunities for licensed therapies. This underscores the view that such studies are not just useful for target identification but can also for inform drug development programmes from start to finish by indicating biomarkers of therapeutic response to measure in phase 1/2 clinical studies, and the relevant spectrum of clinical outcomes that should be ascertained in clinical trials. The incorporation of outcomes in clinical trials that are anticipated to be affected by pharmacological action on a particular target (target-specific outcomes of both efficacy and safety) would represent a departure from the current norm where end-points in a particular therapeutic area tend to be uniform regardless of the target being evaluated. Genetic information could also be useful for compound optimisation since the profile of biomarker effects of a SNP in a gene encoding a drug target should be those of a clean drug with no off-target actions. Where compounds are developed that have actions that are distinct from those observed in a genetic study, these may be off-target effects, and suggest that a more specific compound may need to be developed before the programme progresses. By the same principle, PheWAS would inform which clinical efficacy and safety end-points should be specified as outcomes in RCTs of compounds against a specified target. The spectrum of outcomes could differ from target to target, even for two targets being evaluated for the same primary disease indication. RCTs would need to be powered for both safety and efficacy outcomes, so that the balance between the benefits and any risk of target modification can be quantified before licensing. It should reduce the problem of mechanism-based side effects only emerging post marketing. This would also ensure that RCTs do not fail for failure to select the correct end-points, or because of the contamination of composite end-points (and thereby dilution of any treatment effect) by inclusion of outcomes that are unaffected by target modification.
Study designs relevant to drug target identification and validation based on human genomics: (a) conventional genome-wide association analysis in which variation in 20,000 genes is tested against a single disease; (b) phenome wide association analysis of a gene encoding a drug target in which variation in a single druggable gene is evaluated against many (all) diseases; (c) druggable genome and phenome wide association analysis; and (d) whole genome and phenome wide association analysis.
There are a number of inherent assumptions and limitations to the approach we describe. We provide an extensive discussion of these issues in Supplementary Note 8. In brief, we justify our estimates of the number of human disease entities, protein coding genes, genes encoding druggable targets and the likely number of causal genes critical to the pathogenesis of common diseases. We have assumed that each gene encodes a single protein with a unique function; that a protein can influence the risk of more than one disease; that the probability that a gene influences one disease is independent of the probability that it influences another; that the probability of a protein being causal for a disease and druggable is independent; that variants in a gene encoding a drug target that affect expression or function are ubiquitous in the genome and can accurately predict the effect of pharmacological action on the same protein;, and that these variants are adequately captured by commonly used genotyping arrays. We discuss the validity of all these assumptions and the impact that the failure of these assumptions would have on the inferences that we draw in Supplementary Note 8.
Finally, most common disease genetic association studies that might inform drug development that have been performed to date have been undertaken in population-based longitudinal cohorts or case-control control datasets, where cases typically represent the first occurrence of a condition (e.g. a coronary heart disease event). However, first-in-class agents for many other common conditions, are tested or used initially patients with established disease, for prevention of disease progression or recurrence58. Mendelian randomization studies for target identification and validation in longitudinal clinical cohorts with established disease are few, currently limited by the available datasets, and also perhaps by potential biases arising from survivorship of, or indexing by, an initial event, that may limit inferences that can be drawn59. Nevertheless, the rediscovery by GWAS of over 70 drug targets suggests that genes influencing disease onset can, in many (but perhaps not all) cases, provide useful insight on targetable pathways for prevention of progression or recurrence of common conditions.
The fundamental problem in contemporary drug development has been the unreliability of target identification leading to low development success rates, inefficiency and escalating cost to healthcare users. Genomics now provides a tool to address the problem directly by accurate identification of proteins that both play a controlling role in a disease and which are amenable to targeting by drugs. Maximising the opportunities arising from this paradigm requires the wider use of genomics in the healthcare setting and with this, the active participation of healthcare users in drug development. The democratisation of drug development through human genomics could have the consequence of reducing wasted investment, increasing value for investors and, eventually, reducing drug price inflation for healthcare providers. It might also provide the sorely needed stimulus for true drug development innovation, to the benefit of patients, health systems, business and society.
Data sharing is not applicable to this article as no datasets were generated or analysed during the current stud.
Paul, S. M. et al. How to improve RD productivity: The pharmaceutical industry's grand challenge. Nature Reviews Drug Discovery, https://doi.org/10.1038/nrd3078 (2010).
Hay, M., Thomas, D. W., Craighead, J. L., Economides, C. & Rosenthal, J. Clinical development success rates for investigational drugs. Nat. Biotechnol., https://doi.org/10.1038/nbt.2786 (2014).
Munos, B. Lessons from 60 years of pharmaceutical innovation. Nature Reviews Drug Discovery, https://doi.org/10.1038/nrd2961 (2009).
Pammolli, F., Magazzini, L. & Riccaboni, M. The productivity crisis in pharmaceutical R&D. Nat. Rev. Drug Discov., https://doi.org/10.1038/nrd3405 (2011).
Scannell, J. W., Blanckley, A., Boldon, H. & Warrington, B. Diagnosing the decline in pharmaceutical R&D efficiency. Nature Reviews Drug Discovery, https://doi.org/10.1038/nrd3681 (2012).
Kola, I. & Landis, J. Can the pharmaceutical industry reduce attrition rates? Nat. Rev. Drug Discov., https://doi.org/10.1038/nrd1470 (2004).
The price of failure - Pharmaceuticals. Available at: https://www.economist.com/business/2014/11/27/the-price-of-failure. (Accessed: 3rd July 2019).
Hitchings, A. W., Baker, E. H. & Khong, T. K. Making medicines evergreen. BMJ (Online), https://doi.org/10.1136/bmj.e7941 (2012).
NICE|The National Institute for Health and Care Excellence.
PRIME: priority medicines|European Medicines Agency. Available at: https://www.ema.europa.eu/en/human-regulatory/research-development/prime-priority-medicines (Accessed: 3rd July 2019).
Apply for the early access to medicines scheme (EAMS) - GOV.UK. Available at: https://www.gov.uk/guidance/apply-for-the-early-access-to-medicines-scheme-eams (Accessed: 3rd July 2019).
Darrow, J. J., Avorn, J. & Kesselheim, A. S. The FDA Breakthrough-Drug Designation — Four Years of Experience. N. Engl. J. Med., https://doi.org/10.1056/nejmhpr1713338 (2018).
Moors, E. H. M., Cohen, A. F. & Schellekens, H. Towards a sustainable system of drug development. Drug Discovery Today, https://doi.org/10.1016/j.drudis.2014.03.004 (2014).
Kola, I. The state of innovation in drug development. Clinical Pharmacology and Therapeutics, https://doi.org/10.1038/sj.clpt.6100479 (2008).
Berndt, E. R., Nass, D., Kleinrock, M. & Aitken, M. Decline in economic returns from new drugs raises questions about sustaining innovations. Health Aff., https://doi.org/10.1377/hlthaff.2014.1029 (2015).
Arrowsmith, J. Trial watch: Phase II failures: 2008–2010. Nature Reviews Drug Discovery, https://doi.org/10.1038/nrd3439 (2011).
Arrowsmith, J. Trial watch: Phase III and submission failures: 2007–2010. Nature Reviews Drug Discovery, https://doi.org/10.1038/nrd3375 (2011).
Arrowsmith, J. & Miller, P. Trial Watch: Phase II and Phase III attrition rates 2011–2012. Nature Reviews Drug Discovery, https://doi.org/10.1038/nrd4090 (2013).
Naci, H. & Ioannidis, J. P. A. How Good Is "Evidence" from Clinical Studies of Drug Effects and Why Might Such Evidence Fail in the Prediction of the Clinical Utility of Drugs? Annu. Rev. Pharmacol. Toxicol., https://doi.org/10.1146/annurev-pharmtox-010814-124614 (2014).
Cook, D. et al. Lessons learned from the fate of AstraZeneca's drug pipeline: A five-dimensional framework. Nature Reviews Drug Discovery, https://doi.org/10.1038/nrd4309 (2014).
Hwang, T. J. et al. Failure of investigational drugs in late-stage clinical development and publication of trial results. JAMA Intern. Med., https://doi.org/10.1001/jamainternmed.2016.6008 (2016).
Lindner, M. D. Clinical attrition due to biased preclinical assessments of potential efficacy. Pharmacology and Therapeutics, https://doi.org/10.1016/j.pharmthera.2007.05.002 (2007).
Macleod, M. R. et al. Risk of Bias in Reports of In Vivo Research: A Focus for Improvement. PLoS Biol., https://doi.org/10.1371/journal.pbio.1002273 (2015).
Perel, P. et al. Comparison of treatment effects between animal experiments and clinical trials: Systematic review. Br. Med. J., https://doi.org/10.1136/bmj.39048.407928.BE (2007).
Henderson, V. C., Kimmelman, J., Fergusson, D., Grimshaw, J. M. & Hackam, D. G. Threats to Validity in the Design and Conduct of Preclinical Efficacy Studies: A Systematic Review of Guidelines for In Vivo Animal Experiments. PLoS Med., https://doi.org/10.1371/journal.pmed.1001489 (2013).
Ioannidis, J. P. A. Why most published research findings are false. PLoS Medicine, https://doi.org/10.1371/journal.pmed.0020124 (2005).
Halsey, L. G., Curran-Everett, D., Vowler, S. L. & Drummond, G. B. The fickle P value generates irreproducible results. Nat. Methods, https://doi.org/10.1038/nmeth.3288 (2015).
Goodman, S. N. Toward evidence-based medical statistics. 1: The P value fallacy. Annals of Internal Medicine, https://doi.org/10.7326/0003-4819-130-12-199906150-00008 (1999).
Sterne, J. A. C., Smith, G. D. & Cox, D. R. Sifting the evidence—what's wrong with significance tests? BMJ, https://doi.org/10.1136/bmj.322.7280.226 (2001).
Colquhoun, D. An investigation of the false discovery rate and the misinterpretation of p-values. R. Soc. Open Sci., https://doi.org/10.1098/rsos.140216 (2014).
Article ADS MathSciNet PubMed PubMed Central Google Scholar
Colquhoun, D. The reproducibility of research and the misinterpretation of p-values. R. Soc. open Sci. 4, 171085 (2017).
Article MathSciNet PubMed PubMed Central Google Scholar
Ioannidis, J. P. A. The proposal to lower P value thresholds to .005. JAMA - Journal of the American Medical Association, https://doi.org/10.1001/jama.2018.1536 (2018).
Finan, C. et al. The druggable genome and support for target identification and validation in drug development. Sci. Transl. Med. 9 (2017).
Nelson, M. R. et al. The support of human genetic evidence for approved drug indications. Nat. Genet., https://doi.org/10.1038/ng.3314 (2015).
Casas, J. P. et al. PLA2G7 Genotype, lipoprotein-associated phospholipase A2 activity, and coronary heart disease risk in 10 494 cases and 15 624 controls of european ancestry. Circulation 121 (2010).
Holmes, M. V. et al. Secretory phospholipase A<inf>2</inf>-IIA and cardiovascular disease: A mendelian randomization study. J. Am. Coll. Cardiol. 62 (2013).
Sofat, R. et al. Separating the mechanism-based and off-target actions of cholesteryl ester transfer protein inhibitors with CETP gene polymorphisms. Circulation 121 (2010).
Swerdlow, D. I. et al. HMG-coenzyme A reductase inhibition, type 2 diabetes, and bodyweight: Evidence from genetic analysis and randomised trials. Lancet 385 (2015).
Swerdlow, D. I. et al. The interleukin-6 receptor as a target for prevention of coronary heart disease: A mendelian randomisation analysis. Lancet 379 (2012).
Würtz, P. et al. Metabolomic Profiling of Statin Use and Genetic Inhibition of HMG-CoA Reductase. J. Am. Coll. Cardiol. 67 (2016).
Hingorani, A. & Humphries, S. Nature's randomised trials. Lancet 366 (2005).
Thanassoulis, G. & O'Donnell, C. J. Mendelian randomization: Nature's randomized trial in the post-genome era. JAMA - Journal of the American Medical Association, https://doi.org/10.1001/jama.2009.812 (2009).
Swerdlow, D. I. et al. Selecting instruments for Mendelian randomization in the wake of genome-wide association studies. Int. J. Epidemiol. 45 (2016).
Plenge, R. M. Disciplined approach to drug discovery and early development. Sci. Transl. Med., https://doi.org/10.1126/scitranslmed.aaf2608 (2016).
Ebrahim, S. & Davey, S G. Mendelian randomization: Can genetic epidemiology help redress the failures of observational epidemiology? Hum. Genet., https://doi.org/10.1007/s00439-007-0448-6 (2008).
Prinz, F., Schlange, T. & Asadullah, K. Believe it or not: how much can we rely on published data on potential drug targets? Nat. Rev. Drug Discov., https://doi.org/10.1038/nrd3439-c1 (2011).
Calcoen, D., Elias, L. & Yu, X. What does it take to produce a breakthrough drug? Nat. Rev. Drug Discov., https://doi.org/10.1038/nrd4570 (2015).
Shih, H. P., Zhang, X. & Aronov, A. M. Drug discovery effectiveness from the standpoint of therapeutic mechanisms and indications. Nat. Rev. Drug Discov., https://doi.org/10.1038/nrd.2017.194 (2018).
Dudbridge, F. & Gusnanto, A. Estimation of significance thresholds for genomewide association scans. Genet. Epidemiol., https://doi.org/10.1002/gepi.20297 (2008).
Gamazon, E. R. et al. A gene-based association method for mapping traits using reference transcriptome data. Nat. Genet., https://doi.org/10.1038/ng.3367 (2015).
Liu, J. Z. et al. A versatile gene-based test for genome-wide association studies. Am. J. Hum. Genet., https://doi.org/10.1016/j.ajhg.2010.06.009 (2010).
Bakshi, A. et al. Fast set-based association analysis using summary data from GWAS identifies novel gene loci for human complex traits. Sci. Rep., https://doi.org/10.1038/srep32894 (2016).
Solovieff, N., Cotsapas, C., Lee, P. H., Purcell, S. M. & Smoller, J. W. Pleiotropy in complex traits: Challenges and strategies. Nature Reviews Genetics, https://doi.org/10.1038/nrg3461 (2013).
Crick, F. Central dogma of molecular biology. Nature, https://doi.org/10.1038/227561a0 (1970).
G.P., P. Global initiatives in genomics: The global genomic medicine collaborative (G2MC) and the genomic medicine alliance (GMA). Public Health Genomics, https://doi.org/10.1159/000493202 LK - http://bb2sz3ek3z.search.serialssolutions.com?sid=EMBASE&issn=16628063&id=doi:10.1159%2F000493202&atitle=Global+initiatives+in+genomics%3A+The+global+genomic+medicine+collaborative+%28G2MC%29+and+the+genomic+medicine+alliance+%28GMA%29&stitle=Public+Health+Genomics&title=Public+Health+Genomics&volume=21&issue=&spage=8&epage=&aulast=Patrinos&aufirst=G.P.&auinit=G.P.&aufull=Patrinos+G.P.&coden=&isbn=&pages=8-&date=2018&auinit1=G&auinitm=P (2018).
UK Biobank. Available at: https://www.ukbiobank.ac.uk/ (Accessed: 4th January 2019).
Denny, J. C., Bastarache, L. & Roden, D. M. Phenome-Wide Association Studies as a Tool to Advance Precision Medicine. Annu. Rev. Genomics Hum. Genet., https://doi.org/10.1146/annurev-genom-090314-024956 (2016).
Paternoster, L., Tilling, K. & Davey Smith, G. Genetic epidemiology and Mendelian randomization for informing disease therapeutics: Conceptual and methodological challenges. PLoS Genetics, https://doi.org/10.1371/journal.pgen.1006944 (2017).
Hu, Y. J. et al. Impact of Selection Bias on Estimation of Subsequent Event Risk. Circ. Cardiovasc. Genet., https://doi.org/10.1161/CIRCGENETICS.116.001616 (2017).
A.D.H. and H.H. are NIHR Senior Investigators and supported by UCL Hospitals NIHR Biomedical Research Centre, the UCL BHF Research Accelerator, Rosetrees Trust. J.P.O. is an employee of Medicines Discovery Catapult, a UK non-profit aimed at supporting the discovery of novel medicines. SD holds an Alan Turing Fellowship Work at the Farr Institute of Health Informatics Research was funded by The Medical Research Council (K006584/1), in partnership with Arthritis Research UK, the British Heart Foundation, Cancer Research UK, the Economic and Social Research Council, the Engineering and Physical Sciences Research Council, the National Institute of Health Research, the National Institute for Social Care and Health Research (Welsh Assembly Government), the Chief Scientist Office (Scottish Government Health Directorates) and the Wellcome Trust. Work at the European Bioinformatics Institute is funded by Member States of the European Molecular Biology Laboratory.
These authors contributed equally: Valerie Kuan and David Prieto.
Institute of Cardiovascular Science, University College London, London, UK
Aroon D. Hingorani, Valerie Kuan, Chris Finan, Sandesh Chopade & John P. Overington
Health Data Research UK and UCL BHF Research Accelerator, London, UK
Aroon D. Hingorani, Valerie Kuan, Chris Finan, Sandesh Chopade, Reecha Sofat, Harry Hemingway & Spiros Denaxas
Benevolent AI, London, UK
Felix A. Kruger
European Molecular Biology Laboratory, European Bioinformatics Institute (EMBL-EBI), Wellcome Genome Campus, Cambridge, UK
Anna Gaulton
Institute of Health Informatics, University College London, London, UK
Reecha Sofat, Harry Hemingway, Spiros Denaxas & David Prieto
Dorset County Hospital NHS Foundation Trust, Dorchester, UK
Raymond J. MacAllister
Medicines Discovery Catapult, Mereside, Alderley Park, Alderley Edge, Cheshire, UK
John P. Overington
Massachusetts Veterans Epidemiology Research and Information Center (MAVERIC), Veterans Administration, Boston, MA, USA
Juan Pablo Casas
Applied Statistics in Medical Research Group, Catholic University of Murcia (UCAM), Murcia, Spain
David Prieto
Aroon D. Hingorani
Valerie Kuan
Chris Finan
Sandesh Chopade
Reecha Sofat
Harry Hemingway
Spiros Denaxas
A.D.H., J.P.C., R.S., A.G., R.J.M., J.P.O., S.D. and H.H. shaped the concepts explored in this paper; A.D.H., V.K., F.K. and D.P. did the calculations. S.C. and C.F. contributed to the research underpinning the assumptions underlying the calculations. A.D.H. wrote the first draft of the manuscript and all authors contributed to a critical revision and redrafting.
Correspondence to Aroon D. Hingorani.
Benevolent AI provided financial support in the form of salaries for two authors – Dr. Felix Kruger and Professor John Overington during part of the period covered by this work. Benevolent AI did not play a role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Supplementary Dataset
Hingorani, A.D., Kuan, V., Finan, C. et al. Improving the odds of drug development success through human genomics: modelling study. Sci Rep 9, 18911 (2019). https://doi.org/10.1038/s41598-019-54849-w
DOI: https://doi.org/10.1038/s41598-019-54849-w
An overview of Fintech applications to solve the puzzle of health care funding: state-of-the-art in medical crowdfunding
Laura Grassi
Simone Fantaccini
Financial Innovation (2022)
Enhancer promoter interactome and Mendelian randomization identify network of druggable vascular genes in coronary artery disease
Arnaud Chignon
Samuel Mathieu
Patrick Mathieu
Human Genomics (2022)
Jack W. Scannell
James Bosley
J. Mark Treherne
Nature Reviews Drug Discovery (2022)
Multi-channel GCN ensembled machine learning model for molecular aqueous solubility prediction on a clean dataset
Chenglong Deng
Li Liang
Haichun Liu
Molecular Diversity (2022)
COVID-19: impact on Public Health and hypothesis-driven investigations on genetic susceptibility and severity
Susana David
Guillermo Dorado
Helena Rebelo-de-Andrade
Immunogenetics (2022) | CommonCrawl |
S & P
Program (Compact)
Student PC
MAY 21-23, 2018 AT THE HYATT REGENCY, SAN FRANCISCO, CA
39th IEEE Symposium on
Sponsored by the IEEE Computer Society Technical Committee on Security and Privacy in cooperation with the International Association for Cryptologic Research
Registration and Reception
Session #1: Machine Learning
AI2: Safety and Robustness Certification of Neural Networks with Abstract Interpretation
Timon Gehr (ETH Zürich), Matthew Mirman (ETH Zürich), Dana Drachsler Cohen (ETH Zürich), Petar Tsankov (ETH Zürich), Swarat Chaudhuri (Rice University), Martin Vechev (ETH Zürich)
We present AI2, the first sound and scalable analyzer for deep neural networks. Based on overapproximation, AI2 can automatically prove safety properties (e.g., robustness) of realistic neural networks (e.g., convolutional neural networks).
The key insight behind AI2 is to phrase reasoning about safety and robustness of neural networks in terms of classic abstract interpretation, enabling us to leverage decades of advances in that area. Concretely, we introduce abstract transformers that capture the behavior of fully connected and convolutional neural network layers with rectified linear unit activations (ReLU), as well as max pooling layers. This allows us to handle real-world neural networks, which are often built out of those types of layers.
We present a complete implementation of AI2 together with an extensive evaluation on 20 neural networks. Our results demonstrate that: (i) AI2 is precise enough to prove useful specifications (e.g., robustness), (ii) AI2 can be used to certify the effectiveness of state-of-the-art defenses for neural networks, (iii) AI2 is significantly faster than existing analyzers based on symbolic analysis, which often take hours to verify simple fully connected networks, and (iv) AI2 can handle deep convolutional networks, which are beyond the reach of existing methods.
Manipulating Machine Learning: Poisoning Attacks and Countermeasures for Regression Learning
Matthew Jagielski (Northeastern University), Alina Oprea (Northeastern University), Battista Biggio (University of Cagliari, Italy; Pluribus One, Italy), Chang Liu (UC Berkeley), Cristina Nita-Rotaru (Northeastern University), Bo Li (UC Berkeley)
As machine learning becomes widely used for automated decisions, attackers have strong incentives to manipulate the results and models generated by machine learning algorithms. In this paper, we perform the first systematic study of poisoning attacks and their countermeasures for linear regression models. In poisoning attacks, attackers deliberately influence the training data to manipulate the results of a predictive model. We propose a theoretically-grounded optimization framework specifically designed for linear regression and demonstrate its effectiveness on a range of datasets and models. We also introduce a fast statistical attack that requires limited knowledge of the training process. Finally, we design a new principled defense method that is highly resilient against all poisoning attacks. We provide formal guarantees about its convergence and an upper bound on the effect of poisoning attacks when the defense is deployed. We evaluate extensively our attacks and defenses on three realistic datasets from health care, loan assessment, and real estate domains.
Stealing Hyperparameters in Machine Learning
Binghui Wang (ECE Department, Iowa State University),Neil Zhenqiang Gong (ECE Department, Iowa State University)
Hyperparameters are critical in machine learning, as different hyperparameters often result in models with significantly different performance. Hyperparameters may be deemed confidential because of their commercial value and the confidentiality of the proprietary algorithms that the learner uses to learn them. In this work, we propose attacks on stealing the hyperparameters that are learned by a learner. We call our attacks hyperparameter stealing attacks. Our attacks are applicable to a variety of popular machine learning algorithms such as ridge regression, logistic regression, support vector machine, and neural network. We evaluate the effectiveness of our attacks both theoretically and empirically. For instance, we evaluate our attacks on Amazon Machine Learning. Our results demonstrate that our attacks can accurately steal hyperparameters. We also study countermeasures. Our results highlight the need for new defenses against our hyperparameter stealing attacks for certain machine learning algorithms.
A Machine Learning Approach To Prevent Malicious Calls Over Telephony Networks
Huichen Li (Shanghai Jiao Tong University),Xiaojun Xu (Shanghai Jiao Tong University),Chang Liu (University of California, Berkeley),Teng Ren (TouchPal Inc.),Kun Wu (TouchPal Inc.),Xuezhi Cao (Shanghai Jiao Tong University),Weinan Zhang (Shanghai Jiao Tong University),Yong Yu (Shanghai Jiao Tong University),Dawn Song (University of California, Berkeley)
Malicious calls, i.e., telephony spams and scams, have been a long-standing challenging issue that causes billions of dollars of annual financial loss worldwide. This work presents the first machine learning-based solution without relying on any particular assumptions on the underlying telephony network infrastructures.
The main challenge of this decade-long problem is that it is unclear how to construct effective features without the access to the telephony networks' infrastructures. We solve this problem by combining several innovations. We first develop a TouchPal user interface on top of a mobile App to allow users tagging malicious calls. This allows us to maintain a large-scale call log database. We then conduct a measurement study over three months of call logs, including 9 billion records. We design 29 features based on the results, so that machine learning algorithms can be used to predict malicious calls. We extensively evaluate different state-of-the-art machine learning approaches using the proposed features, and the results show that the best approach can reduce up to 90% unblocked malicious calls while maintaining a precision over 99.99% on the benign call traffic. The results also show the models are efficient to implement without incurring a significant latency overhead. We also conduct ablation analysis, which reveals that using 10 out of the 29 features can reach a performance comparable to using all features.
Surveylance: Automatically Detecting Online Survey Scams
Amin Kharraz (University of Illinois Urbana-Champaign),William Robertson (Northeastern University),Engin Kirda (Northeastern University)
Online surveys are a popular mechanism for performing market research in exchange for monetary compensation. Unfortunately, fraudulent survey websites are similarly rising in popularity among cyber-criminals as a means for executing social engineering attacks. In addition to the sizable population of users that participate in online surveys as a secondary revenue stream, unsuspecting users who search the web for free content or access codes to commercial software can also be exposed to survey scams. This occurs through redirection to websites that ask the user to complete a survey in order to receive the promised content or a reward.
In this paper, we present SURVEYLANCE , the first system that automatically identifies survey scams using machine learning techniques. Our evaluation demonstrates that SURVEYLANCE works well in practice by identifying 8,623 unique websites involved in online survey attacks. We show that SURVEYLANCE is suitable for assisting human analysts in survey scam detection at scale. Our work also provides the first systematic analysis of the survey scam ecosystem by investigating the capabilities of these services, mapping all the parties involved in the ecosystem, and quantifying the consequences to users that are exposed to these services. Our analysis reveals that a large number of survey scams are easily reachable through the Alexa top 30K websites, and expose users to a wide range of security issues including identity fraud, deceptive advertisements, potentially unwanted programs (PUPs), malicious extensions, and malware .
Session Chair: Tudor Dumitras
Break (30 Minutes)
Session #2: Privacy
Privacy Risks with Facebook's PII-based Targeting: Auditing a Data Broker's Advertising Interface
Giridhari Venkatadri (Northeastern University),Athanasios Andreou (EURECOM),Yabing Liu (Northeastern University),Alan Mislove (Northeastern University),Krishna P. Gummadi (MPI-SWS),Patrick Loiseau (Univ. Grenoble Alpes, CNRS, Inria, Grenoble INP, LIG and MPI-SWS),Oana Goga (Univ. Grenoble Alpes, CNRS, Inria, Grenoble INP, LIG)
Sites like Facebook and Google now serve as de facto data brokers, aggregating data on users for the purpose of implementing powerful advertising platforms. Historically, these services allowed advertisers to select which users see their ads via targeting attributes. Recently, most advertising platforms have begun allowing advertisers to target users directly by uploading the personal information of the users who they wish to advertise to (e.g., their names, email addresses, phone numbers, etc.); these services are often known as custom audiences. Custom audiences effectively represent powerful linking mechanisms, allowing advertisers to leverage any PII (e.g., from customer data, public records, etc.) to target users.
In this paper, we focus on Facebook's custom audience implementation and demonstrate attacks that allow an adversary to exploit the interface to infer users' PII as well as to infer their activity. Specifically, we show how the adversary can infer users' full phone numbers knowing just their email address, determine whether a particular user visited a website, and de-anonymize all the visitors to a website by inferring their phone numbers en masse. These attacks can be conducted without any interaction with the victim(s), cannot be detected by the victim(s), and do not require the adversary to spend money or actually place an ad. We propose a simple and effective fix to the attacks based on reworking the way Facebook de-duplicates uploaded information. Facebook's security team acknowledged the vulnerability and has put into place a fix that is a variant of the fix we propose. Overall, our results indicate that advertising platforms need to carefully consider the privacy implications of their interfaces.
Anonymity Trilemma: Strong Anonymity, Low Bandwidth Overhead, Low Latency --- Choose Two
Debajyoti Das (Purdue University),Sebastian Meiser (University College London),Esfandiar Mohammadi (ETH Zurich),Aniket Kate (Purdue University)
This work investigates the fundamental constraints of anonymous communication (AC) protocols. We analyze the relationship between bandwidth overhead, latency overhead, and sender anonymity or recipient anonymity against the global passive (network-level) adversary. We confirm the trilemma that an AC protocol can only achieve two out of the following three properties: strong anonymity (i.e., anonymity up to a negligible chance), low bandwidth overhead, and low latency overhead.
We further study anonymity against a stronger global passive adversary that can additionally passively compromise some of the AC protocol nodes. For a given number of compromised nodes, we derive necessary constraints between bandwidth and latency overhead whose violation make it impossible for an AC protocol to achieve strong anonymity. We analyze prominent AC protocols from the literature and depict to which extent those satisfy our necessary constraints. Our fundamental necessary constraints offer a guideline not only for improving existing AC systems but also for designing novel AC protocols with non-traditional bandwidth and latency overhead choices.
Locally Differentially Private Frequent Itemset Mining
Tianhao Wang (Purdue University),Ninghui Li (Purdue University),Somesh Jha (University of Wisconsin-Madison)
The notion of Local Differential Privacy (LDP) enables users to respond to sensitive questions while preserving their privacy. The basic LDP frequent oracle (FO) protocol enables an aggregator to estimate the frequency of any value. But when each user has a set of values, one needs an additional padding and sampling step to find the frequent values and estimate their frequencies. In this paper, we formally define such padding and sample based frequency oracles (PSFO). We further identify the privacy amplification property in PSFO. As a result, we propose SVIM, a protocol for finding frequent items in the set-valued LDP setting. Experiments show that under the same privacy guarantee and computational cost, SVIM significantly improves over existing methods. With SVIM to find frequent items, we propose SVSM to effectively find frequent itemsets, which to our knowledge has not been done before in the LDP setting.
EyeTell: Video-Assisted Touchscreen Keystroke Inference from Eye Movements
Yimin Chen (Arizona State University),Tao Li (Arizona State University),Rui Zhang (University of Delaware),Yanchao Zhang (Arizona State University),Terri Hedgpeth (Arizona State University)
Keystroke inference attacks pose an increasing threat to ubiquitous mobile devices. This paper presents EyeTell, a novel video-assisted attack that can infer a victim's keystrokes on his touchscreen device from a video capturing his eye movements. EyeTell explores the observation that human eyes naturally focus on and follow the keys they type, so a typing sequence on a soft keyboard results in a unique gaze trace of continuous eye movements. In contrast to prior work, EyeTell requires neither the attacker to visually observe the victim's inputting process nor the victim device to be placed on a static holder. Comprehensive experiments on iOS and Android devices confirm the high efficacy of EyeTell for inferring PINs, lock patterns, and English words under various environmental conditions.
Understanding Linux Malware
Emanuele Cozzi (Eurecom),Mariano Graziano (Cisco Systems, Inc.),Yanick Fratantonio (Eurecom),Davide Balzarotti (Eurecom)
For the past two decades, the security community has been fighting malicious programs for Windows-based operating systems. However, the recent surge in adoption of embedded devices and the IoT revolution are rapidly changing the malware landscape. Embedded devices are profoundly different than traditional personal computers. In fact, while personal computers run predominantly on x86-flavored architectures, embedded systems rely on a variety of different architectures. In turn, this aspect causes a large number of these systems to run some variants of the Linux operating system, pushing malicious actors to give birth to "Linux malware."
To the best of our knowledge, there is currently no comprehensive study attempting to characterize, analyze, and understand Linux malware. The majority of resources on the topic are available as sparse reports often published as blog posts, while the few systematic studies focused on the analysis of specific families of malware (e.g., the Mirai botnet) mainly by looking at their network-level behavior, thus leaving the main challenges of analyzing Linux malware unaddressed.
This work constitutes the first step towards filling this gap. After a systematic exploration of the challenges involved in the process, we present the design and implementation details of the first malware analysis pipeline specifically tailored for Linux malware. We then present the results of the first large-scale measurement study conducted on 10,548 malware samples (collected over a time frame of one year) documenting detailed statistics and insights that can help directing future work in the area.
Session Chair: Carmela Troncoso
Session #3: Side Channels
Racing in Hyperspace: Closing Hyper-Threading Side Channels on SGX with Contrived Data Races
Guoxing Chen (The Ohio State University),Wenhao Wang (Indiana University Bloomington & SKLOIS, Institute of Information Engineering, Chinese Academy of Sciences),Tianyu Chen (Indiana University Bloomington),Sanchuan Chen (The Ohio State University),Yinqian Zhang (The Ohio State University),XiaoFeng Wang (Indiana University Bloomington),Ten-Hwang Lai (The Ohio State University),Dongdai Lin (SKLOIS, Institute of Information Engineering, Chinese Academy of Sciences)
In this paper, we present HYPERRACE, an LLVM-based tool for instrumenting SGX enclave programs to eradicate all side-channel threats due to Hyper-Threading. HYPERRACE creates a shadow thread for each enclave thread and asks the underlying untrusted operating system to schedule both threads on the same physical core whenever enclave code is invoked, so that Hyper-Threading side channels are closed completely. Without placing additional trust in the operating system's CPU scheduler, HYPERRACE conducts a physical-core co-location test: it first constructs a communication channel between the threads using a shared variable inside the enclave and then measures the communication speed to verify that the communication indeed takes place in the shared L1 data cache-a strong indicator of physical-core co-location. The key novelty of the work is the measurement of communication speed without a trustworthy clock; instead, relative time measurements are taken via contrived data races on the shared variable. It is worth noting that the emphasis of HYPERRACE's defense against Hyper-Threading side channels is because they are open research problems. In fact, HYPERRACE also detects the occurrence of exception- or interrupt-based side channels, the solutions of which have been studied by several prior works.
Grand Pwning Unit: Accelerating Microarchitectural Attacks with the GPU
Pietro Frigo (Vrije Universiteit Amsterdam), Kaveh Razavi (Vrije Universiteit Amsterdam), Cristiano Giuffrida (Vrije Universiteit Amsterdam), Herbert Bos (Vrije Universiteit Amsterdam)
Dark silicon is pushing processor vendors to add more specialized units such as accelerators to commodity processor chips. Unfortunately this is done without enough care to security. In this paper we look at the security implications of integrated Graphical Processor Units (GPUs) found in almost all mobile processors. We demonstrate that GPUs, already widely employed to accelerate a variety of benign applications such as image rendering, can also be used to "accelerate" microarchitectural attacks (i.e., making them more effective) on commodity platforms. In particular, we show that an attacker can build all the necessary primitives for performing effective GPU-based microarchitectural attacks and that these primitives are all exposed to the web through standardized browser ex- tensions, allowing side-channel and Rowhammer attacks from JavaScript. These attacks bypass state-of-the-art mitigations and advance existing CPU-based attacks: we show the first end-to- end microarchitectural compromise of a browser running on a mobile phone in under two minutes by orchestrating our GPU primitives. While powerful, these GPU primitives are not easy to implement due to undocumented hardware features. We describe novel reverse engineering techniques for peeking into the previously unknown cache architecture and replacement policy of the Adreno 330, an integrated GPU found in many common mobile platforms. This information is necessary when building shader programs implementing our GPU primitives. We conclude by discussing mitigations against GPU-enabled attackers.
SoK: Keylogging Side Channels
John Monaco (U.S. Army Research Laboratory)
The first keylogging side channel attack was discovered over 50 years ago when Bell Laboratory researchers noticed an electromagnetic spike emanating from a Bell 131-B2 teletype terminal. This spike, emitted upon each key press, enabled up to 75% of plaintext communications to be recovered in field conditions. Since then, keylogging attacks have come to leverage side channels emanating from the user's finger and hand movements, countless keyboard electromagnetic and acoustic emanations, microarchitectural attacks on the host computer, and encrypted network traffic. These attacks can each be characterized by the type of information the side channel leaks: a spatial side channel reveals physical key locations or the similarity between key pairs, and a temporal side channel leverages key press and release timings. We define and evaluate the performance of idealized spatial and temporal keylogging side channels and find that, under the assumption of typing English words, nontrivial information gains can be achieved even in the presence of substantial measurement error. For temporal side channels, we find that the information gained by different temporal features strongly correlates to typing speed and style. Finally, to help drive future research, we review the current state-of-the-art keylogging side channel attacks and discuss some of the mitigation techniques that can be applied.
FPGA-Based Remote Power Side-Channel Attacks
Mark Zhao (Cornell University),G. Edward Suh (Cornell University)
The rapid adoption of heterogeneous computing has driven the integration of Field Programmable Gate Arrays (FPGAs) into cloud datacenters and flexible System-on-Chips (SoCs). This paper shows that the integrated FPGA introduces a new security vulnerability by enabling software-based power side-channel attacks without physical proximity to a target system. We first demonstrate that an on-chip power monitor can be built on a modern FPGA using ring oscillators (ROs), and characterize its ability to observe the power consumption of other modules on the FPGA or the SoC. Then, we show that the RO- based FPGA power monitor can be used for a successful power analysis attack on an RSA cryptomodule on the same FPGA. Additionally, we show that the FPGA-based power monitor can observe the power consumption of a CPU on the same SoC, and demonstrate that the FPGA-to-CPU power side-channel attack can break timing-channel protection for a RSA program running on a CPU. This work introduces and demonstrates remote power side-channel attacks using an FPGA, showing that the common assumption that power side-channel attacks require specialized equipment and physical access to the victim hardware is not true for systems with an integrated FPGA.
Another Flip in the Wall of Rowhammer Defenses
Daniel Gruss (Graz University of Technology, Graz, Austria),Moritz Lipp (Graz University of Technology, Graz, Austria),Michael Schwarz (Graz University of Technology, Graz, Austria),Daniel Genkin (University of Pennsylvania and University of Maryland, USA),Jonas Juffinger (Graz University of Technology, Graz, Austria),Sioli O'Connell (University of Adelaide, Adelaide, Australia),Wolfgang Schoechl (Graz University of Technology, Graz, Austria),Yuval Yarom (University of Adelaide and Data61, Adelaide, Australia)
The Rowhammer bug allows unauthorized modification of bits in DRAM cells from unprivileged software, enabling powerful privilege-escalation attacks. Sophisticated Rowhammer countermeasures have been presented, aiming at mitigating the Rowhammer bug or its exploitation. However, the state of the art provides insufficient insight on the completeness of these defenses. In this paper, we present novel Rowhammer attack and exploitation primitives, showing that even a combination of all defenses is ineffective. Our new attack technique, one-location hammering, breaks previous assumptions on requirements for triggering the Rowhammer bug, i.e., we do not hammer multiple DRAM rows but only keep one DRAM row constantly open. Our new exploitation technique, opcode flipping, bypasses recent isolation mechanisms by flipping bits in a predictable and targeted way in userspace binaries. We replace conspicuous and memory-exhausting spraying and grooming techniques with a novel reliable technique called memory waylaying. Memory waylaying exploits system-level optimizations and a side channel to coax the operating system into placing target pages at attacker-chosen physical locations. Finally, we abuse Intel SGX to hide the attack entirely from the user and the operating system, making any inspection or detection of the attack infeasible. Our Rowhammer enclave can be used for coordinated denial-of-service attacks in the cloud and for privilege escalation on personal computers. We demonstrate that our attacks evade all previously proposed countermeasures for commodity systems.
Session Chair: Kevin Fu
Session #4: Computing on Hidden Data
EnclaveDB: A Secure Database using SGX
Christian Priebe (Imperial College London),Kapil Vaswani (Microsoft Research),Manuel Costa (Microsoft Research)
We propose EnclaveDB, a database engine that guarantees confidentiality, integrity, and freshness for data and queries. EnclaveDB guarantees these properties even when the database administrator is malicious, when an attacker has compromised the operating system or the hypervisor, and when the database runs in an untrusted host in the cloud. EnclaveDB achieves this by placing sensitive data (tables, indexes and other metadata) in enclaves protected by trusted hardware (such as Intel SGX). EnclaveDB has a small trusted computing base, which includes an in-memory storage and query engine, a transaction manager and pre-compiled stored procedures. A key component of EnclaveDB is an efficient protocol for checking integrity and freshness of the database log. The protocol supports concurrent, asynchronous appends and truncation, and requires minimal synchronization between threads. Our experiments using standard database benchmarks and a performance model that simulates large enclaves show that EnclaveDB achieves strong security with low overhead (up to 40% for TPC-C) compared to an industry strength in-memory database engine.
Oblix: An Efficient Oblivious Search Index
Pratyush Mishra (UC Berkeley),Rishabh Poddar (UC Berkeley),Jerry Chen (UC Berkeley),Alessandro Chiesa (UC Berkeley),Raluca Ada Popa (UC Berkeley)
Search indices are fundamental building blocks of many systems, and there is great interest in running them on encrypted data. Unfortunately, many known schemes that enable search queries on encrypted data achieve efficiency at the expense of security, as they reveal access patterns to the encrypted data.
In this paper we present Oblix, a search index for encrypted data that is oblivious (provably hides access patterns), is dynamic (supports inserts and deletes), and has good efficiency.
Oblix relies on a combination of novel oblivious-access techniques and recent hardware enclave platforms (e.g., Intel SGX). In particular, a key technical contribution is the design and implementation of doubly-oblivious data structures, in which the client's accesses to its internal memory are oblivious, in addition to accesses to its external memory at the server. These algorithms are motivated by hardware enclaves like SGX, which leak access patterns to both internal and external memory.
We demonstrate the usefulness of Oblix in several applications: private contact discovery for Signal, private retrieval of public keys for Key Transparency, and searchable encryption that hides access patterns and result sizes.
Improved Reconstruction Attacks on Encrypted Data Using Range Query Leakage
Marie-Sarah Lacharite (Royal Holloway, University of London),Brice Minaud (Royal Holloway, University of London),Kenneth G. Paterson (Royal Holloway, University of London)
We analyse the security of database encryption schemes supporting range queries against persistent adversaries. The bulk of our work applies to a generic setting, where the adversary's view is limited to the set of records matched by each query (known as access pattern leakage). We also consider a more specific setting where rank information is also leaked, which is inherent inherent to multiple recent encryption schemes supporting range queries. We provide three attacks.
First, we consider full reconstruction, which aims to recover the value of every record, fully negating encryption. We show that for dense datasets, full reconstruction is possible within an expected number of queries N log N + O(N), where N is the number of distinct plaintext values.
This directly improves on a quadratic bound in the same setting by Kellaris et al. (CCS 2016).
Second, we present an approximate reconstruction attack recovering all plaintext values in a dense dataset within a constant ratio of error, requiring the access pattern leakage of only O(N) queries.
Third, we devise an attack in the common setting where the adversary has access to an auxiliary distribution for the target dataset. This third attack proves highly effective on age data from real-world medical data sets. In our experiments, observing only 25 queries was sufficient to reconstruct a majority of records to within 5 years.
In combination, our attacks show that current approaches to enabling range queries offer little security when the threat model goes beyond snapshot attacks to include a persistent server-side adversary.
Bulletproofs: Short Proofs for Confidential Transactions and More
Benedikt Bünz (Stanford University),Jonathan Bootle (University College London),Dan Boneh (Stanford University),Andrew Poelstra (Blockstream),Pieter Wuille (Blockstream),Greg Maxwell
We propose Bulletproofs, a new non-interactive zero-knowledge proof protocol with very short proofs and without a trusted setup; the proof size is only logarithmic in the witness size. Bulletproofs are especially well suited for efficient range proofs on committed values: they enable proving that a committed value is in a range using only 2 log_2(n)+9 group and field elements, where n is the bit length of the range. Proof generation and verification times are linear in n.
Bulletproofs greatly improve on the linear (in n) sized range proofs in existing proposals for confidential transactions in Bitcoin and other cryptocurrencies. Moreover, Bulletproofs supports aggregation of range proofs, so that a party can prove that m commitments lie in a given range by providing only an additive O(log(m)) group elements over the length of a single proof. To aggregate proofs from multiple parties, we enable the parties to generate a single proof without revealing their inputs to each other via a simple multi-party computation (MPC) protocol for constructing Bulletproofs. This MPC protocol uses either a constant number of rounds and linear communication, or a logarithmic number of rounds and logarithmic communication. We show that verification time, while asymptotically linear, is very efficient in practice. The marginal cost of batch verifying 32 aggregated range proofs is less than the cost of verifying 32 ECDSA signatures. Bulletproofs build on the techniques of Bootle et al. (EUROCRYPT 2016). Beyond range proofs, Bulletproofs provide short zero-knowledge proofs for general arithmetic circuits while only relying on the discrete logarithm assumption and without requiring a trusted setup. We discuss many applications that would benefit from Bulletproofs, primarily in the area of cryptocurrencies. The efficiency of Bulletproofs is particularly well suited for the distributed and trustless nature of blockchains. The full version of this article is available on ePrint.
FuturesMEX: Secure, Distributed Futures Market Exchange
Fabio Massacci (University of Trento, IT),Chan Nam Ngo (University of Trento, IT),Jing Nie (University of International Business and Economics Beijing, CN),Daniele Venturi (University of Rome "La Sapienza", IT),Julian Williams (University of Durham, UK)
In a Futures-Exchange, such as the Chicago Mercantile Exchange, traders buy and sell contractual promises (futures) to acquire or deliver, at some future pre-specified date, assets ranging from wheat to crude oil and from bacon to cash in a desired currency. The interactions between economic and security properties and the exchange's essentially non-monotonic security behavior; a valid trader's valid action can invalidate other traders' previously valid positions, are a challenge for security research.
We show the security properties that guarantee an Exchange's economic viability (availability of trading information, liquidity, confidentiality of positions, absence of price discrimination, risk-management) and an attack when traders' anonymity is broken.
We describe all key operations for a secure, fully distributed Futures-Exchange, hereafter referred to as simply the "Exchange". Our distributed, asynchronous protocol simulates the centralized functionality under the assumptions of anonymity of the physical layer and availability of a distributed ledger. We consider security with abort (in absence of honest majority) and extend it to penalties. Our proof of concept implementation and its optimization (based on zk-SNARKs and SPDZ) demonstrate that the computation of actual trading days (along Thomson-Reuters Tick History DB) is feasible for low-frequency markets; however, more research is needed for high-frequency ones.
Implementing Conjunction Obfuscation under Entropic Ring LWE
David Bruce Cousins (Raytheon BBN Technologies),Giovanni Di Crescenzo (Applied Communication Sciences / Vencore Labs),Kamil Doruk Gür (NJIT Cybersecurity Research Center, New Jersey Institute of Technology),Kevin King (Massachusetts Institute of Technology),Yuriy Polyakov (NJIT Cybersecurity Research Center, New Jersey Institute of Technology),Kurt Rohloff (NJIT Cybersecurity Research Center, New Jersey Institute of Technology),Gerard W. Ryan (NJIT Cybersecurity Research Center, New Jersey Institute of Technology),Erkay Savaş (NJIT Cybersecurity Research Center, New Jersey Institute of Technology)
We address the practicality challenges of secure program obfuscation \revised{by implementing, optimizing, and experimentally assessing an approach to securely obfuscate conjunction programs proposed in [1]. Conjunction programs evaluate functions $f\left(x_1,\ldots,x_L\right) = \bigwedge_{i \in I} y_i$, where $y_i$ is either $x_i$ or $\lnot x_i$ and $I \subseteq \left[L\right]$, and can be used as classifiers. Our obfuscation approach satisfies distributional Virtual Black Box (VBB) security based on reasonable hardness assumptions, namely an entropic variant of the Ring Learning with Errors (Ring-LWE) assumption. Prior implementations of secure program obfuscation techniques support either trivial programs like point functions, or support the obfuscation of more general but less efficient branching programs to satisfy Indistinguishability Obfuscation (IO), a weaker security model. Further, the more general implemented techniques, rather than relying on standard assumptions, base their security on conjectures that have been shown to be theoretically vulnerable.
Our work is the first implementation of non-trivial program obfuscation based on polynomial rings. Our contributions include multiple design and implementation advances resulting in reduced program size, obfuscation runtime, and evaluation runtime by many orders of magnitude. We implement our design in software and experimentally assess performance in a commercially available multi-core computing environment. Our implementation achieves runtimes of 6.7 hours to securely obfuscate a 64-bit conjunction program and 2.5 seconds to evaluate this program over an arbitrary input. We are also able to obfuscate a 32-bit conjunction program with \revised{53 bits} of security in 7 minutes and evaluate the obfuscated program in 43 milliseconds on a commodity desktop computer, which implies that 32-bit conjunction obfuscation is already practical. Our graph-induced (directed) encoding implementation runs up to 25 levels, which is higher than previously reported in the literature for this encoding. Our design and implementation advances are applicable to obfuscating more general compute-and-compare programs and can also be used for many cryptographic schemes based on lattice trapdoors.
Session Chair: Yinqian Zhang
Poster Reception
Session #5: Understanding Users
Hackers vs. Testers: A Comparison of Software Vulnerability Discovery Processes
Daniel Votipka (University of Maryland),Rock Stevens (University of Maryland),Elissa Redmiles (University of Maryland),Jeremy Hu (University of Maryland),Michelle Mazurek (University of Maryland)
Identifying security vulnerabilities in software is a critical task that requires significant human effort. Currently, vulnerability discovery is often the responsibility of software testers before release and white-hat hackers (often within bug bounty programs) afterward. This arrangement can be ad-hoc and far from ideal; for example, if testers could identify more vulnerabilities, software would be more secure at release time. Thus far, however, the processes used by each group - and how they compare to and interact with each other - have not been well studied. This paper takes a first step toward better understanding, and eventually improving, this ecosystem: we report on a semi-structured interview study (n=25) with both testers and hackers, focusing on how each group finds vulnerabilities, how they develop their skills, and the challenges they face. The results suggest that hackers and testers follow similar processes, but get different results due largely to differing experiences and therefore different underlying knowledge of security concepts. Based on these results, we provide recommendations to support improved security training for testers, better communication between hackers and developers, and smarter bug bounty policies to motivate hacker participation.
Towards Security and Privacy for Multi-User Augmented Reality: Foundations with End Users
Kiron Lebeck (University of Washington),Kimberly Ruth (University of Washington),Tadayoshi Kohno (University of Washington),Franziska Roesner (University of Washington)
Immersive augmented reality (AR) technologies are becoming a reality. Prior works have identified security and privacy risks raised by these technologies, primarily considering individual users or AR devices. However, we make two key observations: (1) users will not always use AR in isolation, but also in ecosystems of other users, and (2) since immersive AR devices have only recently become available, the risks of AR have been largely hypothetical to date.
To provide a foundation for understanding and addressing the security and privacy challenges of emerging AR technologies, grounded in the experiences of real users, we conduct a qualitative lab study with an immersive AR headset, the Microsoft HoloLens. We conduct our study in pairs - 22 participants across 11 pairs - wherein participants engage in paired and individual (but physically co-located) HoloLens activities. Through semi-structured interviews, we explore participants' security, privacy, and other concerns, raising key findings. For example, we find that despite the HoloLens's limitations, participants were easily immersed, treating virtual objects as real (e.g., stepping around them for fear of tripping). We also uncover numerous security, privacy, and safety concerns unique to AR (e.g., deceptive virtual objects misleading users about the real world), and a need for access control among users to manage shared physical spaces and virtual content embedded in those spaces. Our findings give us the opportunity to identify broader lessons and key challenges to inform the design of emerging single- and multi-user AR technologies.
Computer Security and Privacy for Refugees in the United States
Lucy Simko (University of Washington),Ada Lerner (Wellesley College),Samia Ibtasam (University of Washington),Franziska Roesner (University of Washington),Tadayoshi Kohno (University of Washington)
In this work, we consider the computer security and privacy practices and needs of recently resettled refugees in the United States. We ask: How do refugees use and rely on technology as they settle in the US? What computer security and privacy practices do they have, and what barriers do they face that may put them at risk? And how are their computer security mental models and practices shaped by the advice they receive? We study these questions through in-depth qualitative interviews with case managers and teachers who work with refugees at a local NGO, as well as through focus groups with refugees themselves. We find that refugees must rely heavily on technology (e.g., email) as they attempt to establish their lives and find jobs; that they also rely heavily on their case managers and teachers for help with those technologies; and that these pressures can push security practices into the background or make common security "best practices'' infeasible. At the same time, we identify fundamental challenges to computer security and privacy for refugees, including barriers due to limited technical expertise, language skills, and cultural knowledge--for example, we find that scams as a threat are a new concept for many of the refugees we studied, and that many common security practices (e.g., password creation techniques and security questions) rely on US cultural knowledge. From these and other findings, we distill recommendations for the computer security community to better serve the computer security and privacy needs and constraints of refugees, a potentially vulnerable population that has not been previously studied in this context.
On Enforcing the Digital Immunity of a Large Humanitarian Organization
Stevens Le Blond (École Polytechnique Fédérale de Lausanne),Alejandro Cuevas (École Polytechnique Fédérale de Lausanne),Juan Ramón Troncoso-Pastoriza (École Polytechnique Fédérale de Lausanne),Philipp Jovanovic (École Polytechnique Fédérale de Lausanne),Bryan Ford (École Polytechnique Fédérale de Lausanne),Jean-Pierre Hubaux (École Polytechnique Fédérale de Lausanne)
Humanitarian action, the process of aiding individuals in situations of crises, poses unique information-security challenges due to natural or manmade disasters, the adverse environments in which it takes place, and the scale and multi-disciplinary nature of the problems. Despite these challenges, humanitarian organizations are transitioning towards a strong reliance on the digitization of collected data and digital tools, which improves their effectiveness but also exposes them to computer security threats. In this paper, we conduct a qualitative analysis of the computer-security challenges of the International Committee of the Red Cross (ICRC), a large humanitarian organization with over sixteen thousand employees, an international legal personality, which involves privileges and immunities, and over 150 years of experience with armed conflicts and other situations of violence worldwide. To investigate the computer security needs and practices of the ICRC from an operational, technical, legal, and managerial standpoint by considering individual, organizational, and governmental levels, we interviewed 27 field workers, IT staff, lawyers, and managers. Our results provide a first look at the unique security and privacy challenges that humanitarian organizations face when collecting, processing, transferring, and sharing data to enable humanitarian action for a multitude of sensitive activities. These results highlight, among other challenges, the trade offs between operational security and requirements stemming from all stakeholders, the legal barriers for data sharing among jurisdictions; especially, the need to complement privileges and immunities with robust technological safeguards in order to avoid any leakages that might hinder access and potentially compromise the neutrality, impartiality, and independence of humanitarian action.
The Spyware Used in Intimate Partner Violence
Rahul Chatterjee (Cornell Tech), Periwinkle Doerfler (NYU), Hadas Orgad (Technion), Sam Havron (Cornell Univ), Jackeline Palmer (Hunter College), Diana Freed (Cornell Tech), Karen Levy (Cornell Tech), Nicola Dell (Cornell Tech), Damon McCoy (NYU), Thomas Ristenpart (Cornell Tech)
Survivors of intimate partner violence increasingly report that abusers install spyware on devices to track their location, monitor communications, and cause emotional and physical harm. To date there has been only cursory investigation into the spyware used in such intimate partner surveillance (IPS). We provide the first in-depth study of the IPS spyware ecosystem. We design, implement, and evaluate a measurement pipeline that combines web and app store crawling with machine learning to find and label apps that are potentially dangerous in IPS contexts. Ultimately we identify several hundred such IPS-relevant apps. While we find dozens of overt spyware tools, the majority are "dual-use" apps - they have a legitimate purpose (e.g., child safety or anti-theft), but are easily and effectively repurposed for spying on a partner. We document that a wealth of online resources are available to educate abusers about exploiting apps for IPS. We also show how some dual-use app developers are encouraging their use in IPS via advertisements, blogs, and customer support services. We analyze existing anti-virus and anti-spyware tools, which universally fail to identify dual-use apps as a threat.
Session Chair: Sascha Fahl
Session #6: Programming Languages
Compiler-assisted Code Randomization
Hyungjoon Koo (Stony Brook University),Yaohui Chen (Northeastern University),Long Lu (Northeastern University),Vasileios P. Kemerlis (Brown University),Michalis Polychronakis (Stony Brook University)
Despite decades of research on software diversification, only address space layout randomization has seen widespread adoption. Code randomization, an effective defense against return-oriented programming exploits, has remained an academic exercise mainly due to i) the lack of a transparent and streamlined deployment model that does not disrupt existing software distribution norms, and ii) the inherent incompatibility of program variants with error reporting, whitelisting, patching, and other operations that rely on code uniformity. In this work we present compiler-assisted code randomization (CCR), a hybrid approach that relies on compiler-rewriter cooperation to enable fast and robust fine-grained code randomization on end-user systems, while maintaining compatibility with existing software distribution models. The main concept behind CCR is to augment binaries with a minimal set of transformation- assisting metadata, which i) facilitate rapid fine-grained code transformation at installation or load time, and ii) form the basis for reversing any applied code transformation when needed, to maintain compatibility with existing mechanisms that rely on referencing the original code. We have implemented a prototype of this approach by extending the LLVM compiler toolchain, and developing a simple binary rewriter that leverages the embedded metadata to generate randomized variants using basic block reordering. The results of our experimental evaluation demonstrate the feasibility and practicality of CCR, as on average it incurs a modest file size increase of 11.46% and a negligible runtime overhead of 0.28%, while it is compatible with link-time optimization and control flow integrity.
Protecting the Stack with Metadata Policies and Tagged Hardware
Nick Roessler (University of Pennsylvania), Andre DeHon (University of Pennsylvania)
The program call stack is a major source of exploitable security vulnerabilities in low-level, unsafe languages like C. In conventional runtime implementations, the underlying stack data is exposed and unprotected, allowing programming errors to turn into security violations. In this work, we design novel metadata-tag based, stack-protection security policies for a general-purpose tagged architecture. Our policies specifically exploit the natural locality of dynamic program call graphs to achieve cacheability of the metadata rules that they require. Our simple Return Address Protection policy has a performance overhead of 1.2% but just protects return addresses. The two richer policies we present, Static Authorities and Depth Isolation, provide object-level protection for all stack objects. When enforcing memory safety, our Static Authorities policy has a performance overhead of 5.7% and our Depth Isolation policy has a performance overhead of 4.5%. When enforcing data-flow integrity (DFI), in which we only detect a violation when a corrupted value is read, our Static Authorities policy has a performance overhead of 3.6% and our Depth Isolation policy has a performance overhead of 2.4%. To characterize our policies, we provide a stack threat taxonomy and show which threats are prevented by both prior work protection mechanisms and our policies.
Impossibility of Precise and Sound Termination-Sensitive Security Enforcements
Minh Ngo (INRIA, France),Frank Piessens (imec-DistriNet, KU Leuven, Belgium),Tamara Rezk (INRIA, France)
An information flow policy is termination-sensitive if it imposes that the termination behavior of programs is not influenced by confidential input. Termination-sensitivity can be statically or dynamically enforced. On one hand, existing static enforcement mechanisms for termination-sensitive policies are typically quite conservative and impose strong constraints on programs like absence of while loops whose guard depends on confidential information. On the other hand, dynamic mechanisms can enforce termination-sensitive policies in a less conservative way. Secure Multi-Execution (SME), one of such mechanisms, was even claimed to be sound and precise in the sense that the enforcement mechanism will not modify the observable behavior of programs that comply with the termination-sensitive policy. However, termination-sensitivity is a subtle policy, that has been formalized in different ways. A key aspect is whether the policy talks about actual termination, or observable termination.
This paper proves that termination-sensitive policies that talk about actual termination are not enforceable in a sound and precise way. For static enforcements, the result follows directly from a reduction of the decidability of the problem to the halting problem. However, for dynamic mechanisms the insight is more involved and requires a diagonalization argument.
In particular, our result contradicts the claim made about SME. We correct these claims by showing that SME enforces a subtly different policy that we call indirect termination-sensitive noninterference and that talks about observable termination instead of actual termination. We construct a variant of SME that is sound and precise for indirect termination-sensitive noninterference. Finally, we also show that static methods can be adapted to enforce indirect termination-sensitive information flow policies (but obviously not precisely) by constructing a sound type system for an indirect termination-sensitive policy.
Static Evaluation of Noninterference using Approximate Model Counting
Ziqiao Zhou (University of North Carolina at Chapel Hill), Zhiyun Qian (University of California, Riverside), Michael K. Reiter (University of North Carolina at Chapel Hill), Yinqian Zhang (The Ohio State University)
Noninterference is a definition of security for secret values provided to a procedure, which informally is met when attacker-observable outputs are insensitive to the value of the secret inputs or, in other words, the secret inputs do not "interfere" with those outputs. This paper describes a static analysis method to measure interference in software. In this approach, interference is assessed using the extent to which different secret inputs are consistent with different attacker-controlled inputs and attacker-observable outputs, which can be measured using a technique called model counting. Leveraging this insight, we develop a flexible interference assessment technique for which the assessment accuracy quantifiably grows with the computational effort invested in the analysis. This paper demonstrates the effectiveness of this technique through application to several case studies, including leakage of: search-engine queries through auto-complete response sizes; secrets subjected to compression together with attacker-controlled inputs; and TCP sequence numbers from shared counters.
DEEPSEC: Deciding Equivalence Properties in Security Protocols -- Theory and Practice
Vincent Cheval (Inria Nancy & Loria),Steve Kremer (Inria Nancy & Loria),Itsaka Rakotonirina (Inria Nancy & Loria)
Automated verification has become an essential part in the security evaluation of cryptographic protocols. Recently, there has been a considerable effort to lift the theory and tool support that existed for reachability properties to the more complex case of equivalence properties. In this paper we contribute both to the theory and practice of this ver- ification problem. We establish new complexity results for static equivalence, trace equivalence and labelled bisimilarity and provide a decision procedure for these equivalences in the case of a bounded number of sessions. Our procedure is the first to decide trace equivalence and labelled bisimilarity exactly for a large variety of cryptographic primitives-those that can be represented by a subterm convergent destructor rewrite system. We implemented the procedure in a new tool, DEEPSEC. We showed through extensive experiments that it is significantly more efficient than other similar tools, while at the same time raises the scope of the protocols that can be analysed.
Session Chair: Deian Stefan
Session #7: Networked Systems
Distance-Bounding Protocols: Verification without Time and Location
Sjouke Mauw (CSC/SnT, University of Luxembourg),Zach Smith (CSC, University of Luxembourg),Jorge Toro-Pozo (CSC, University of Luxembourg),Rolando Trujillo-Rasua (SnT, University of Luxembourg)
Distance-bounding protocols are cryptographic protocols that securely establish an upper bound on the physical distance between the participants. Existing symbolic verification frameworks for distance-bounding protocols consider timestamps and the location of agents. In this work we introduce a causality-based characterization of secure distance-bounding that discards the notions of time and location. This allows us to verify the correctness of distance-bounding protocols with standard protocol verification tools. That is to say, we provide the first fully automated verification framework for distance-bounding protocols. By using our framework, we confirmed known vulnerabilities in a number of protocols and discovered unreported attacks against two recently published protocols.
Sonar: Detecting SS7 Redirection Attacks With Audio-Based Distance Bounding
Christian Peeters (University of Florida),Hadi Abdullah (University of Florida),Nolen Scaife (University of Florida),Jasmine Bowers (University of Florida),Patrick Traynor (University of Florida),Bradley Reaves (North Carolina State University),Kevin Butler (University of Florida)
The global telephone network is relied upon by billions every day. Central to its operation is the Signaling System 7 (SS7) protocol, which is used for setting up calls, managing mobility, and facilitating many other network services. This protocol was originally built on the assumption that only a small number of trusted parties would be able to directly communicate with its core infrastructure. As a result, SS7 --- as a feature --- allows all parties with core access to redirect and intercept calls for any subscriber anywhere in the world. Unfortunately, increased interconnectivity with the SS7 network has led to a growing number of illicit call redirection attacks. We address such attacks with Sonar, a system that detects the presence of SS7 redirection attacks by securely measuring call audio round-trip times between telephony devices. This approach works because redirection attacks force calls to travel longer physical distances than usual, thereby creating longer end-to-end delay. We design and implement a distance bounding-inspired protocol that allows us to securely characterize the round-trip time between the two endpoints. We then use custom hardware deployed in 10 locations across the United States and a redirection testbed to characterize how distance affects round trip time in phone networks. We develop a model using this testbed and show Sonar is able to detect 70.9% of redirected calls between call endpoints of varying attacker proximity (300--7100 miles) with low false positive rates (0.3%). Finally, we ethically perform actual SS7 redirection attacks on our own devices with the help of an industry partner to demonstrate that Sonar detects 100% of such redirections in a real network (with no false positives). As such, we demonstrate that telephone users can reliably detect SS7 redirection attacks and protect the integrity of their calls.
OmniLedger: A Secure, Scale-Out, Decentralized Ledger via Sharding
Eleftherios Kokoris-Kogias (École Polytechnique Fédérale de Lausanne),Philipp Jovanovic (École Polytechnique Fédérale de Lausanne),Linus Gasser (École Polytechnique Fédérale de Lausanne),Nicolas Gailly (École Polytechnique Fédérale de Lausanne),Ewa Syta (Trinity College),Bryan Ford (École Polytechnique Fédérale de Lausanne)
Designing a secure permissionless distributed ledger (blockchain) that performs on par with centralized payment processors, such as Visa, is a challenging task. Most existing distributed ledgers are unable to scale-out, i.e., to grow their total processing capacity with the number of validators; and those that do, compromise security or decentralization. We present OmniLedger, a novel scale-out distributed ledger that preserves long- term security under permissionless operation. It ensures security and correctness by using a bias-resistant public-randomness protocol for choosing large, statistically representative shards that process transactions, and by introducing an efficient cross- shard commit protocol that atomically handles transactions affecting multiple shards. OmniLedger also optimizes performance via parallel intra-shard transaction processing, ledger pruning via collectively-signed state blocks, and low-latency "trust-but- verify" validation for low-value transactions. An evaluation of our experimental prototype shows that OmniLedger's throughput scales linearly in the number of active validators, supporting Visa-level workloads and beyond, while confirming typical transactions in under two seconds.
Routing Around Congestion: Defeating DDoS Attacks and Adverse Network Conditions via Reactive BGP Routing
Jared M Smith (University of Tennessee, Knoxville),Max Schuchard (University of Tennessee, Knoxville)
In this paper, we present Nyx, the first system to both effectively mitigate modern Distributed Denial of Service (DDoS) attacks regardless of the amount of traffic under adversarial control and function without outside cooperation or an Internet redesign. Nyx approaches the problem of DDoS mitigation as a routing problem rather than a filtering problem. This conceptual shift allows Nyx to avoid many of the common shortcomings of existing academic and commercial DDoS mitigation systems. By leveraging how Autonomous Systems (ASes) handle route advertisement in the existing Border Gateway Protocol (BGP), Nyx allows the deploying AS to achieve isolation of traffic from a critical upstream AS off of attacked links and onto alternative, uncongested, paths. This isolation removes the need for filtering or de-prioritizing attack traffic. Nyx controls outbound paths through normal BGP path selection, while return paths from critical ASes are controlled through the use of specific techniques we developed using existing traffic engineering principles and require no outside coordination. Using our own realistic Internet-scale simulator, we find that in more than 98% of cases our system can successfully route critical traffic around network segments under transit-link DDoS attacks; a new form of DDoS attack where the attack traffic never reaches the victim AS, thus invaliding defensive filtering, throttling, or prioritization strategies. More significantly, in over 95% of those cases, the alternate path provides complete congestion relief from transit-link DDoS. Nyx additionally provides complete congestion relief in over 75% of cases when the deployer is being directly attacked.
Tracking Ransomware End-to-end
Danny Yuxing Huang (Princeton University),Maxwell Matthaios Aliapoulios (New York University),Vector Guo Li (University of California, San Diego),Luca Invernizzi (Google),Elie Bursztein (Google),Kylie McRoberts (Google),Jonathan Levin (Chainalysis),Kirill Levchenko (University of California, San Diego),Alex C. Snoeren (University of California, San Diego),Damon McCoy (New York University)
Ransomware is a type of malware that encrypts the files of infected hosts and demands payment, often in a crypto-currency like Bitcoin. In this paper, we create a measurement framework that we use to perform a large-scale, two-year, end-to-end measurement of ransomware payments, victims, and operators. By combining an array of data sources, including ransomware binaries, seed ransom payments, victim telemetry from infections, and a large database of bitcoin addresses annotated with their owners, we sketch the outlines of this burgeoning ecosystem and associated third-party infrastructure. In particular, we are able to trace the financial transactions, from the acquisition of bitcoins by victims, through the payment of ransoms, to the cash out of bitcoins by the ransomware operators. We find that many ransomware operators cashed out using BTC-e, a now-defunct Bitcoin exchange. In total we are able to track over $16 million USD in likely ransom payments made by 19,750 potential victims during a two-year period. While our study focuses on ransomware, our methods are potentially applicable to other cybercriminal operations that have similarly adopted Bitcoin as their payment channel.
Session Chair: Cristina Nita-Rotaru
Session #8: Program Analysis
The Rise of the Citizen Developer: Assessing the Security Impact of Online App Generators
Marten Oltrogge (CISPA, Saarland University),Erik Derr (CISPA, Saarland University),Christian Stransky (CISPA, Saarland University),Yasemin Acar (Leibniz University Hannover),Sascha Fahl (Leibniz University Hannover),Christian Rossow (CISPA, Saarland University),Giancarlo Pellegrino (CISPA, Saarland University, Stanford University),Sven Bugiel (CISPA, Saarland University),Michael Backes (CISPA, Saarland University)
Mobile apps are increasingly created using online application generators (OAGs) that automate app development, distribution, and maintenance. These tools significantly lower the level of technical skill that is required for app development, which makes them particularly appealing to citizen developers, i.e., developers with little or no software engineering background. However, as the pervasiveness of these tools increases, so does their overall influence on the mobile ecosystem's security, as security lapses by such generators affect thousands of generated apps. The security of such generated apps, as well as their impact on the security of the overall app ecosystem, has not yet been investigated.
We present the first comprehensive classification of commonly used OAGs for Android and show how to fingerprint uniquely generated apps to link them back to their generator. We thereby quantify the market penetration of these OAGs based on a corpus of 2,291,898 free Android apps from Google Play and discover that at least 11.1% of these apps were created using OAGs. Using a combination of dynamic, static, and manual analysis, we find that the services' app generation model is based on boilerplate code that is prone to reconfiguration attacks in 7/13 analyzed OAGs. Moreover, we show that this boilerplate code includes well-known security issues such as code injection vulnerabilities and insecure WebViews. Given the tight coupling of generated apps with their services' backends, we further identify security issues in their infrastructure. Due to the blackbox development approach, citizen developers are unaware of these hidden problems that ultimately put the end-users sensitive data and privacy at risk and violate the user's trust assumption. A particular worrisome result of our study is that OAGs indeed have a significant amplification factor for those vulnerabilities, notably harming the health of the overall mobile app ecosystem.
Learning from Mutants: Using Code Mutation to Learn and Monitor Invariants of a Cyber-Physical System
Yuqi Chen (Singapore University of Technology and Design),Christopher M. Poskitt (Singapore University of Technology and Design),Jun Sun (Singapore University of Technology and Design)
Cyber-physical systems (CPS) consist of sensors, actuators, and controllers all communicating over a network; if any subset becomes compromised, an attacker could cause significant damage. With access to data logs and a model of the CPS, the physical effects of an attack could potentially be detected before any damage is done. Manually building a model that is accurate enough in practice, however, is extremely difficult. In this paper, we propose a novel approach for constructing models of CPS automatically, by applying supervised machine learning to data traces obtained after systematically seeding their software components with faults ("mutants"). We demonstrate the efficacy of this approach on the simulator of a real-world water purification plant, presenting a framework that automatically generates mutants, collects data traces, and learns an SVM-based model. Using cross-validation and statistical model checking, we show that the learnt model characterises an invariant physical property of the system. Furthermore, we demonstrate the usefulness of the invariant by subjecting the system to 55 network and code-modification attacks, and showing that it can detect 85% of them from the data logs generated at runtime.
Precise and Scalable Detection of Double-Fetch Bugs in OS Kernels
Meng Xu (Georgia Institute of Technology),Chenxiong Qian (Georgia Institute of Technology),Kangjie Lu (University of Minnesota),Michael Backes (CISPA Helmholtz Center i.G.),Taesoo Kim (Georgia Institute of Technology)
During system call execution, it is common for operating system kernels to read userspace memory multiple times (multi-reads). A critical bug may exist if the fetched userspace memory is subject to change across these reads, i.e., a race condition, which is known as a double-fetch bug. Prior works have attempted to detect these bugs both statically and dynamically. However, due to their improper assumptions and imprecise definitions regarding double-fetch bugs, their multi-read detection is inherently limited and suffers from significant false positives and false negatives. For example, their approach is unable to support device emulation, inter-procedural analysis, loop handling, etc. More importantly, they completely leave the task of finding real double-fetch bugs from the haystack of multi-reads to manual verification, which is expensive if possible at all.
In this paper, we first present a formal and precise definition of double-fetch bugs and then implement a static analysis system -Deadline - to automatically detect double-fetch bugs in OS kernels. Deadline uses static program analysis techniques to systematically find multi-reads throughout the kernel and employs specialized symbolic checking to vet each multi-read for double-fetch bugs. We apply Deadline to Linux and FreeBSD kernels and find 23 new bugs in Linux and one new bug in FreeBSD. We further propose four generic strategies to patch and prevent double-fetch bugs based on our study and the discussion with kernel maintainers.
CollAFL: Path Sensitive Fuzzing
Shuitao Gan (State Key Laboratory of Mathematical Engineering and Advanced Computing),Chao Zhang (Tsinghua University),Xiaojun Qin (State Key Laboratory of Mathematical Engineering and Advanced Computing),Xuwen Tu (State Key Laboratory of Mathematical Engineering and Advanced Computing),Kang Li (Cyber Immunity Lab),Zhongyu Pei (Tsinghua University),Zuoning Chen (National Research Center of Parallel Computer Engineering and Technology)
Coverage-guided fuzzing is a widely used and ef- fective solution to find software vulnerabilities. Tracking code coverage and utilizing it to guide fuzzing are crucial to coverage- guided fuzzers. However, tracking full and accurate path coverage is infeasible in practice due to the high instrumentation overhead. Popular fuzzers (e.g., AFL) often use coarse coverage information, e.g., edge hit counts stored in a compact bitmap, to achieve highly efficient greybox testing. Such inaccuracy and incompleteness in coverage introduce serious limitations to fuzzers. First, it causes path collisions, which prevent fuzzers from discovering potential paths that lead to new crashes. More importantly, it prevents fuzzers from making wise decisions on fuzzing strategies.
In this paper, we propose a coverage sensitive fuzzing solution CollAFL. It mitigates path collisions by providing more accurate coverage information, while still preserving low instrumentation overhead. It also utilizes the coverage information to apply three new fuzzing strategies, promoting the speed of discovering new paths and vulnerabilities. We emented a prototype of CollAFL based on the popular fuzzer AFL and evaluated it on 24 popular applications. The results showed that path collisions are common, i.e., up to 75% of edges could collide with others in some applications, and CollAFL could reduce the edge collision ratio to nearly zero. Moreover, armed with the three fuzzing strategies, CollAFL outperforms AFL in terms of both code coverage and vulnerability discovery. On average, CollAFL covered 20% more program paths, found 320% more unique crashes and 260% more bugs than AFL in 200 hours. In total, CollAFL found 157 new security bugs with 95 new CVEs assigned.
T-Fuzz: fuzzing by program transformation
Hui Peng (Purdue University), Yan Shoshitaishvili (Arizona State University), Mathias Payer (Purdue University)
Abstract-Fuzzing is a simple yet effective approach to discover software bugs utilizing randomly generated inputs. However, it is limited by coverage and cannot find bugs hidden in deep execution paths of the program because the randomly generated inputs fail complex sanity checks, e.g., checks on magic values, checksums, or hashes. To improve coverage, existing approaches rely on imprecise heuristics or complex input mutation techniques (e.g., symbolic execution or taint analysis) to bypass sanity checks. Our novel method tackles coverage from a different angle: by removing sanity checks in the target program. T-Fuzz leverages a coverage-guided fuzzer to generate inputs. Whenever the fuzzer can no longer trigger new code paths, a light-weight, dynamic tracing based technique detects the input checks that the fuzzer-generated inputs fail. These checks are then removed from the target program. Fuzzing then continues on the transformed program, allowing the code protected by the removed checks to be triggered and potential bugs discovered. Fuzzing transformed programs to find bugs poses two challenges: (1) removal of checks leads to over-approximation and false positives, and (2) even for true bugs, the crashing input on the transformed program may not trigger the bug in the original program. As an auxiliary post-processing step, T-Fuzz leverages a symbolic execution-based approach to filter out false positives and reproduce true bugs in the original program. By transforming the program as well as mutating the input, T-Fuzz covers more code and finds more true bugs than any existing technique. We have evaluated T-Fuzz on the DARPA Cyber Grand Challenge dataset, LAVA-M dataset and 4 real-world programs (pngfix, tiffinfo, magick and pdftohtml). For the CGC dataset, T-Fuzz finds bugs in 166 binaries, Driller in 121, and AFL in 105. In addition, found 3 new bugs in previously-fuzzed programs and libraries.
Angora: Efficient Fuzzing by Principled Search
Peng Chen (ShanghaiTech University),Hao Chen (University of California, Davis)
Abstract-Fuzzing is a popular technique for finding software bugs. However, the performance of the state-of-the-art fuzzers leaves a lot to be desired. Fuzzers based on symbolic execution produce quality inputs but run slow, while fuzzers based on random mutation run fast but have difficulty producing quality inputs. We propose Angora, a new mutation-based fuzzer that outperforms the state-of-the-art fuzzers by a wide margin. The main goal of Angora is to increase branch coverage by solving path constraints without symbolic execution. To solve path constraints efficiently, we introduce several key techniques: scalable byte-level taint tracking, context-sensitive branch count, search based on gradient descent, and input length exploration. On the LAVA-M data set, Angora found almost all the injected bugs, found more bugs than any other fuzzer that we compared with, and found eight times as many bugs as the second-best fuzzer in the program who. Angora also found 103 bugs that the LAVA authors injected but could not trigger. We also tested Angora on eight popular, mature open source programs. Angora found 6, 52, 29, 40 and 48 new bugs in file, jhead, nm, objdump and size, respectively. We measured the coverage of Angora and evaluated how its key techniques contribute to its impressive performance.
Session Chair: Thorsten Holz
Digital Forensics, Digital Futures - SADFE 2018
Michael Losavio (University of Louisville), Glenn Dardick (Embry-Riddle Aeronautic University), Abe Baggili (University of New Haven)
Droplet: Decentralized Authorization for IoT Data Streams
Hossein Shafagh (ETH Zurich)
Efficiently Authenticated Data Storage with Blockchain
Yuzhe (Richard) Tang, Zihao Xing, Ju Chen (Syracuse University)m Cheng Xu, Jianliang Xu (HKBU)
Encouraging Diversity in Security and Privacy Research and a report on GREPSEC: A Workshop for Women in Computer Security Research
Terry Benzel (University of Southern California - ISI), Hilarie Orman (Purple Streak)
IEEE SecDev 2018
Rob Cunningham, Dinara Doyle, Daphne Yao
Impact Analysis of Vulnerabilities on Business Processes in a Cloud Environment
Anoop Singhal (NIST), Peng Liu (Penn State University)
Kangacrypt 2018
Yuval Yarom (University of Adelaide and Data61)
Let "The Hulk" Protect Your Personal Information
Nicholas Micallef, Gaurav Misra (University of New South Wales)
Processing Publicly Disclosed Personal Data According to the GDPR - A Nole in the Privacy Regulation Framework
Gianluigi Maria Riva (University College Dublin)
Towards Image Privacy against Automated Classifiers
Arezoo Rajabi, Rakesh B. Bobba (Oregon State University)
seL4-US Center of Excellence Grand Opening
Jason Li (Intelligent Automation Inc)
S&P TC Business Meeting
Session #9: Web
FP-STALKER: Tracking Browser Fingerprint Evolutions Along Time
Antoine Vastel (University of Lille / INRIA),Pierre Laperdrix (INSA / INRIA),Walter Rudametkin (University of Lille / INRIA),Romain Rouvoy (University of Lille / INRIA)
Browser fingerprinting has emerged as a technique to track users without their consent. Unlike cookies, fingerprinting is a stateless technique that does not store any information on devices, but instead exploits unique combinations of attributes handed over freely by browsers. The uniqueness of fingerprints allows them to be used for identification. However, browser fingerprints change over time and the effectiveness of tracking users over longer durations has not been properly addressed.
In this paper, we show that browser fingerprints tend to change frequently-from every few hours to days-due to, for example, software updates or configuration changes. Yet, despite these frequent changes, we show that browser fingerprints can still be linked, thus enabling long-term tracking.
FP-STALKER is an approach to link browser fingerprint evolutions. It compares fingerprints to determine if they originate from the same browser. We created two variants of FP-STALKER, a rule-based variant that is faster, and a hybrid variant that exploits machine learning to boost accuracy. To evaluate FP-STALKER , we conduct an empirical study using 98,598 fingerprints we collected from 1, 905 distinct browser instances. We compare our algorithm with the state of the art and show that, on average, we can track browsers for 54.48 days, and 26 % of browsers can be tracked for more than 100 days.
Study and Mitigation of Origin Stripping Vulnerabilities in Hybrid-postMessage Enabled Mobile Applications
Guangliang Yang (Texas A&M; University),Jeff Huang (Texas A&M; University),Guofei Gu (Texas A&M; University),Abner Mendoza (Texas A&M; University)
postMessage is popular in HTML5 based web apps to allow the communication between different origins. With the increasing popularity of the embedded browser (i.e., WebView) in mobile apps (i.e., hybrid apps), postMessage has found utility in these apps. However, different from web apps, hybrid apps have a unique requirement that their native code (e.g., Java for Android) also needs to exchange messages with web code loaded in WebView. To bridge the gap, developers typically extend postMessage by treating the native context as a new frame, and allowing the communication between the new frame and the web frames. We term such extended postMessage "hybrid postMessage" in this paper. We find that hybrid postMessage introduces new critical security flaws: all origin information of a message is not respected or even lost during the message delivery in hybrid postMessage. If adversaries inject malicious code into WebView, the malicious code may leverage the flaws to passively monitor messages that may contain sensitive information, or actively send messages to arbitrary message receivers and access their internal functionalities and data. We term the novel security issue caused by hybrid postMessage "Origin Stripping Vulnerability" (OSV).
In this paper, our contributions are fourfold. First, we conduct the first systematic study on OSV. Second, we propose a lightweight detection tool against OSV, called OSV-Hunter. Third, we evaluate OSV-Hunter using a set of popular apps. We found that 74 apps implemented hybrid postMessage, and all these apps suffered from OSV, which might be exploited by adversaries to perform remote real-time microphone monitoring, data race, internal data manipulation, denial of service (DoS) attacks and so on. Several popular development frameworks, libraries (such as the Facebook React Native framework, and the Google cloud print library) and apps (such as Adobe Reader and WPS office) are impacted. Lastly, to mitigate OSV from the root, we design and implement three new postMessage APIs, called OSV-Free. Our evaluation shows that OSV-Free is secure and fast, and it is generic and resilient to the notorious Android fragmentation problem. We also demonstrate that OSV-Free is easy to use, by applying OSV-Free to harden the complex "Facebook React Native" framework. OSV-Free is open source, and its source code and more implementation and evaluation details are available online.
Mobile Application Web API Reconnaissance: Web-to-Mobile Inconsistencies & Vulnerabilities
Abner Mendoza (Texas A&M; University),Guofei Gu (Texas A&M; University)
Modern mobile apps use cloud-hosted HTTP-based API services and heavily rely on the Internet infrastructure for data communication and storage. To improve performance and leverage the power of the mobile device, input validation and other business logic required for interfacing with web API services are typically implemented on the mobile client. However, when a web service implementation fails to thoroughly replicate input validation, it gives rise to inconsistencies that could lead to attacks that can compromise user security and privacy. Developing automatic methods of auditing web APIs for security remains challenging.
In this paper, we present a novel approach for automatically analyzing mobile app-to-web API communication to detect inconsistencies in input validation logic between apps and their respective web API services. We present our system, \sysname, which implements a static analysis-based web API reconnaissance approach to uncover inconsistencies on real world API services that can lead to attacks with severe consequences for potentially millions of users throughout the world. Our system utilizes program analysis techniques to automatically extract HTTP communication templates from Android apps that encode the input validation constraints imposed by the apps on outgoing web requests to web API services. WARDroid is also enhanced with blackbox testing of server validation logic to identify inconsistencies that can lead to attacks.
We evaluated our system on a set of 10,000 popular free apps from the Google Play Store. We detected problematic logic in APIs used in over 4,000 apps, including 1,743 apps that use unencrypted HTTP communication. We further tested 1,000 apps to validate web API hijacking vulnerabilities that can lead to potential compromise of user privacy and security and found that millions of users are potentially affected from our sample set of tested apps.
Enumerating Active IPv6 Hosts for Large-scale Security Scans via DNSSEC-signed Reverse Zones
Kevin Borgolte (University of California, Santa Barbara),Shuang Hao (University of Texas at Dallas),Tobias Fiebig (Delft University of Technology),Giovanni Vigna (University of California, Santa Barbara)
Security research has made extensive use of exhaustive Internet-wide scans over the recent years, as they can provide significant insights into the overall state of security of the Internet, and ZMap made scanning the entire IPv4 address space practical. However, the IPv4 address space is exhausted, and a switch to IPv6, the only accepted long-term solution, is inevitable. In turn, to better understand the security of devices connected to the Internet, including in particular Internet of Things devices, it is imperative to include IPv6 addresses in security evaluations and scans. Unfortunately, it is practically infeasible to iterate through the entire IPv6 address space, as it is 2^96 times larger than the IPv4 address space. Therefore, enumeration of active hosts prior to scanning is necessary. Without it, we will be unable to investigate the overall security of Internet-connected devices in the future.
In this paper, we introduce a novel technique to enumerate an active part of the IPv6 address space by walking DNSSEC-signed IPv6 reverse zones. Subsequently, by scanning the enumerated addresses, we uncover significant security problems: the exposure of sensitive data, and incorrectly controlled access to hosts, such as access to routing infrastructure via administrative interfaces, all of which were accessible via IPv6. Furthermore, from our analysis of the differences between accessing dual-stack hosts via IPv6 and IPv4, we hypothesize that the root cause is that machines automatically and by default take on globally routable IPv6 addresses. This is a practice that the affected system administrators appear unaware of, as the respective services are almost always properly protected from unauthorized access via IPv4.
Our findings indicate (i) that enumerating active IPv6 hosts is practical without a preferential network position contrary to common belief, (ii) that the security of active IPv6 hosts is currently still lagging behind the security state of IPv4 hosts, and (iii) that unintended IPv6 connectivity is a major security issue for unaware system administrators.
Tracking Certificate Misissuance in the Wild
Deepak Kumar (University of Illinois, Urbana-Champaign),Zhengping Wang (University of Illinois, Urbana-Champaign),Matthew Hyder (University of Illinois, Urbana-Champaign),Joseph Dickinson (University of Illinois, Urbana-Champaign),Gabrielle Beck (University of Michigan),David Adrian (University of Michigan),Joshua Mason (University of Illinois, Urbana-Champaign),Zakir Durumeric (University of Michigan),J. Alex Halderman (University of Michigan),Michael Bailey (University of Illinois, Urbana-Champaign)
Certificate Authorities (CAs) regularly make mechanical errors when issuing certificates. To quantify these errors, we introduce ZLint, a certificate linter that codifies the policies set forth by the CA/Browser Forum Baseline Requirements and RFC 5280 that can be tested in isolation. We run ZLint on browser-trusted certificates in Censys and systematically analyze how well CAs construct certificates. We find that the number errors has drastically reduced since 2012. In 2017, only 0.02% of certificates have errors. However, this is largely due to a handful of large authorities that consistently issue correct certificates. There remains a long tail of small authorities that regularly issue non-conformant certificates. We further find that issuing certificates with errors is correlated with other types of mismanagement and for large authorities, browser action. Drawing on our analysis, we conclude with a discussion on how the community can best use lint data to identify authorities with worrisome organizational practices and ensure long-term health of the Web PKI.
A Formal Treatment of Accountable Proxying over TLS
Karthikeyan Bhargavan (INRIA de Paris, France),Ioana Boureanu (Univ. of Surrey, SCCS, UK),Antoine Delignat-Lavaud (Microsoft Research, UK),Pierre-Alain Fouque (Univ. of Rennes 1, IRISA, France),Cristina Onete (Univ. of Limoges, XLIM, CNRS, France)
Much of Internet traffic nowadays passes through active proxies, whose role is to inspect, filter, cache, or trans- form data exchanged between two endpoints. To perform their tasks, such proxies modify channel-securing protocols, like TLS, resulting in serious vulnerabilities. Such problems are exacerbated by the fact that middleboxes are often invisible to one or both endpoints, leading to a lack of accountability. A recent protocol, called mcTLS, pioneered accountability for proxies, which are authorized by the endpoints and given limited read/write permissions to application traffic.
Unfortunately, we show that mcTLS is insecure: the protocol modifies the TLS protocol, exposing it to a new class of middlebox-confusion attacks. Such attacks went unnoticed mainly because mcTLS lacked a formal analysis and security proofs. Hence, our second contribution is to formalize the goal of accountable proxying over secure channels. Third, we propose a provably-secure alternative to soon-to-be-standardized mcTLS: a generic and modular protocol-design that care- fully composes generic secure channel-establishment protocols, which we prove secure. Finally, we present a proof-of-concept implementation of our design, instantiated with unmodified TLS 1.3, and evaluate its overheads.
Session Chair: Nikita Borisov
Session #10: Authentication
Secure Device Bootstrapping without Secrets Resistant to Signal Manipulation Attacks
Nirnimesh Ghose (University of Arizona), Loukas Lazos (University of Arizona), Ming Li (University of Arizona)
In this paper, we address the fundamental problem of securely bootstrapping a group of wireless devices to a hub, when none of the devices share prior associations (secrets) with the hub or between them. This scenario aligns with the secure deployment of body area networks, IoT, medical devices, industrial automation sensors, autonomous vehicles, and others. We develop VERSE, a physical-layer group message integrity verification primitive that effectively detects advanced wireless signal manipulations that can be used to launch man-in-the-middle (MitM) attacks over wireless. Without using shared secrets to establish authenticated channels, such attacks are notoriously difficult to thwart and can undermine the authentication and key establishment processes. VERSE exploits the existence of multiple devices to verify the integrity of the messages exchanged within the group. We then use VERSE to build a bootstrapping protocol, which securely introduces new devices to the network. Compared to the state-of-the-art, VERSE achieves in-band message integrity verification during secure pairing using only the RF modality without relying on out-of-band channels or extensive human involvement. It guarantees security even when the adversary is capable of fully controlling the wireless channel by annihilating and injecting wireless signals. We study the limits of such advanced wireless attacks and prove that the introduction of multiple legitimate devices can be leveraged to increase the security of the pairing process. We validate our claims via theoretical analysis and extensive experimentations on the USRP platform. We further discuss various implementation aspects such as the effect of time synchronization between devices and the effects of multipath and interference. Note that the elimination of shared secrets, default passwords, and public key infrastructures effectively addresses the related key management challenges when these are considered at scale.
Do You Feel What I Hear? Enabling Autonomous IoT Device Pairing using Different Sensor Types
Jun Han (Carnegie Mellon University),Albert Jin Chung (Carnegie Mellon University),Manal Kumar Sinha (Carnegie Mellon University),Madhumitha Harishankar (Carnegie Mellon University),Shijia Pan (Carnegie Mellon University),Hae Young Noh (Carnegie Mellon University),Pei Zhang (Carnegie Mellon University),Patrick Tague (Carnegie Mellon University)
Context-based pairing solutions increase the usability of IoT device pairing by eliminating any human involvement in the pairing process. This is possible by utilizing on-board sensors (with same sensing modalities) to capture a common physical context (e.g., ambient sound via each device's microphone). However, in a smart home scenario, it is impractical to assume that all devices will share a common sensing modality. For example, a motion detector is only equipped with an infrared sensor while Amazon Echo only has microphones. In this paper, we develop a new context-based pairing mechanism called Perceptio that uses time as the common factor across differing sensor types. By focusing on the event timing, rather than the specific event sensor data, Perceptio creates event fingerprints that can be matched across a variety of IoT devices. We propose Perceptio based on the idea that devices co-located within a physically secure boundary (e.g., single family house) can observe more events in common over time, as opposed to devices outside. Devices make use of the observed contextual information to provide entropy for Perceptio's pairing protocol. We design and implement Perceptio, and evaluate its effectiveness as an autonomous secure pairing solution. Our implementation demonstrates the ability to sufficiently distinguish between legitimate devices (placed within the boundary) and attacker devices (placed outside) by imposing a threshold on fingerprint similarity. Perceptio demonstrates an average fingerprint similarity of 94.9% between legitimate devices while even a hypothetical impossibly well-performing attacker yields only 68.9% between itself and a valid device.
On the Economics of Offline Password Cracking
Jeremiah Blocki (Purdue University),Benjamin Harsha (Purdue University),Samson Zhou (Purdue University)
We develop an economic model of an offline password cracker which allows us to make quantitative predictions about the fraction of accounts that a rational password attacker would crack in the event of an authentication server breach. We apply our economic model to analyze recent massive password breaches at Yahoo!, Dropbox, LastPass and AshleyMadison. All four organizations were using key-stretching to protect user passwords. In fact, LastPass' use of PBKDF2-SHA256 with $10^5$ hash iterations exceeds 2017 NIST minimum recommendation by an order of magnitude. Nevertheless, our analysis paints a bleak picture: the adopted key-stretching levels provide insufficient protection for user passwords. In particular, we present strong evidence that most user passwords follow a Zipf's law distribution, and characterize the behavior of a rational attacker when user passwords are selected from a Zipf's law distribution. We show that there is a finite threshold which depends on the Zipf's law parameters that characterizes the behavior of a rational attacker --- if the value of a cracked password (normalized by the cost of computing the password hash function) exceeds this threshold then the adversary's optimal strategy is always to continue attacking until each user password has been cracked. In all cases (Yahoo!, Dropbox, LastPass and AshleyMadison) we find that the value of a cracked password almost certainly exceeds this threshold meaning that a rational attacker would crack all passwords that are selected from the Zipf's law distribution (i.e., most user passwords). This prediction holds even if we incorporate an aggressive model of diminishing returns for the attacker (e.g., the total value of $500$ million cracked passwords is less than $100$ times the total value of $5$ million passwords). On a positive note our analysis demonstrates that memory hard functions (MHFs) such as SCRYPT or Argon2i can significantly reduce the damage of an offline attack. In particular, we find that because MHFs substantially increase guessing costs a rational attacker will give up well before he cracks most user passwords and this prediction holds even if the attacker does not encounter diminishing returns for additional cracked passwords. Based on our analysis we advocate that password hashing standards should be updated to require the use of memory hard functions for password hashing and disallow the use of non-memory hard functions such as BCRYPT or PBKDF2.
A Tale of Two Studies: The Best and Worst of YubiKey Usability
Joshua Reynolds (University of Illinois at Urbana-Champaign), Trevor Smith (Brigham Young University), Ken Reese (Brigham Young University), Luke Dickinson (Brigham Young University), Scott Ruoti (MIT Lincoln Laboratory), Kent Seamons (Brigham Young University)
Two-factor authentication (2FA) significantly improves the security of password-based authentication. Recently, there has been increased interest in Universal 2nd Factor (U2F) security keys-small hardware devices that require users to press a button on the security key to authenticate. To examine the usability of security keys in non-enterprise usage, we conducted two user studies of the YubiKey, a popular line of U2F security keys. The first study tasked 31 participants with configuring a Windows, Google, and Facebook account to authenticate using a YubiKey. This study revealed problems with setup instructions and workflow including users locking themselves out of their operating system or thinking they had successfully enabled 2FA when they had not. In contrast, the second study had 25 participants use a YubiKey in their daily lives over a period of four weeks, revealing that participants generally enjoyed the experience. Conducting both a laboratory and longitudinal study yielded insights into the usability of security keys that would not have been evident from either study in isolation. Based on our analysis, we recommend standardizing the setup process, enabling verification of success, allowing shared accounts, integrating with operating systems, and preventing lockouts.
When Your Fitness Tracker Betrays You: Quantifying the Predictability of Biometric Features Across Contexts
Simon Eberz (University of Oxford),Giulio Lovisotto (University of Oxford),Andrea Patanè (University of Oxford),Marta Kwiatkowska (University of Oxford),Vincent Lenders (armasuisse),Ivan Martinovic (University of Oxford)
Attacks on behavioral biometrics have become increasingly popular. Most research has been focused on presenting a previously obtained feature vector to the biometric sensor, often by the attacker training themselves to change their behavior to match that of the victim. However, obtaining the victim's biometric information may not be easy, especially when the user's template on the authentication device is adequately secured. As such, if the authentication device is inaccessible, the attacker may have to obtain data elsewhere. In this paper, we present an analytic framework that enables us to measure how easily features can be predicted based on data gathered in a different context (e.g., different sensor, performed task or environment). This framework is used to assess how resilient individual features or entire biometrics are against such cross-context attacks. In order to be able to compare existing biometrics with regard to this property, we perform a user study to gather biometric data from 30 participants and ?ve biometrics (ECG, eye movements, mouse movements, touchscreen dynamics and gait) in a variety of contexts. We make this dataset publicly available online. Our results show that many attack scenarios are viable in practice as features are easily predicted from a variety of contexts. All biometrics include features that are particularly predictable (e.g., amplitude features for ECG or curvature for mouse movements). Overall, we observe that cross-context attacks on eye movements, mouse movements and touchscreen inputs are comparatively easy while ECG and gait exhibit much more chaotic cross-context changes.
Session Chair: Gang Tan
Session #11: Cryptography
vRAM: Faster Verifiable RAM With Program-Independent Preprocessing
Yupeng Zhang (University of Maryland),Daniel Genkin (University of Maryland and University of Pennsylvania),Jonathan Katz (University of Maryland),Dimitrios Papadopoulos (Hong Kong University of Science and Technology),Charalampos Papamanthou (University of Maryland)
We study the problem of verifiable computation (VC) for RAM programs, where a computationally weak verifier outsources the execution of a program to a powerful (but untrusted) prover. Existing efficient implementations of VC protocols require an expensive preprocessing phase that binds the parties to a single circuit. (While there are schemes that avoid preprocessing entirely, their performance remains significantly worse than constructions with preprocessing.) Thus, a prover and verifier are forced to choose between two approaches: (1) Allow verification of arbitrary RAM programs, at the expense of efficiency, by preprocessing a universal circuit which can handle all possible instructions during each CPU cycle; or (2) Sacrifice expressiveness by preprocessing an efficient circuit which is tailored to the verification of a single specific RAM program.
We present vRAM, a VC system for RAM programs that avoids both the above drawbacks by having a preprocessing phase that is entirely circuit-independent (other than an upper bound on the circuit size). During the proving phase, once the program to be verified and its inputs are chosen, the circuit-independence of our construction allows the parties to use a smaller circuit tailored to verifying the specific program on the chosen inputs, i.e., without needing to encode all possible instructions in each cycle. Moreover, our construction is the first with asymptotically optimal prover overhead; i.e., the work of the prover is a constant multiplicative factor of the time to execute the program.
Our experimental evaluation demonstrates that vRAM reduces the prover's memory consumption by 55-110x and its running time by 9-30x compared to existing schemes with universal preprocessing. This allows us to scale to RAM computations with more than 2 million CPU cycles, a 65x improvement compared to the state of the art. Finally, vRAM has performance comparable to (and sometimes better than) the best existing scheme with program-specific preprocessing despite the fact that the latter can deploy program-specific optimizations (and has to pay a separate preprocessing cost for every new program).
Doubly-efficient zkSNARKs without trusted setup
Riad S. Wahby (Stanford), Ioanna Tzialla (New York University), Abhi Shelat (Northeastern), Justin Thaler (Georgetown), Michael Walfish (New York University)
We present a zero-knowledge argument for NP with low communication complexity, low concrete cost for both the prover and the verifier, and no trusted setup, based on standard cryptographic assumptions. Communication is proportional to d log G (for d the depth and G the width of the verifying circuit) plus the square root of the witness size. When applied to batched or data-parallel statements, the prover's runtime is linear and the verifier's is sub-linear in the verifying circuit size, both with good constants. In addition, witness-related communication can be reduced, at the cost of increased verifier runtime, by leveraging a new commitment scheme for multilinear polynomials, which may be of independent interest. These properties represent a new point in the tradeoffs among setup, complexity assumptions, proof size, and computational cost. We apply the Fiat-Shamir heuristic to this argument to produce a zero-knowledge succinct non-interactive argument of knowledge (zkSNARK) in the random oracle model, based on the discrete log assumption, which we call Hyrax. We implement Hyrax and evaluate it against five state-of-the-art baseline systems. Our evaluation shows that, even for modest problem sizes, Hyrax gives smaller proofs than all but the most computationally costly baseline, and that its prover and verifier are each faster than three of the five baselines.
xJsnark: A Framework for Efficient Verifiable Computation
Ahmed Kosba (University of Maryland),Charalampos Papamanthou (University of Maryland),Elaine Shi (Cornell University)
Many cloud and cryptocurrency applications rely on verifying the integrity of outsourced computations, in which a verifier can efficiently verify the correctness of a computation made by an untrusted prover. State-of-the-art protocols for verifiable computation require that the computation task be expressed as arithmetic circuits, and the number of multiplication gates in the circuit is the primary metric that determines performance. At the present, a programmer could rely on two approaches for expressing the computation task, either by composing the circuits directly through low-level development tools; or by expressing the computation in a high-level program and rely on compilers to perform the program-to-circuit transformation. The former approach is difficult to use but on the other hand allows an expert programmer to perform custom optimizations that minimize the resulting circuit. In comparison, the latter approach is much more friendly to non-specialist users, but existing compilers often emit suboptimal circuits.
We present xJsnark, a programming framework for verifiable computation that aims to achieve the best of both worlds: offering programmability to non-specialist users, and meanwhile automating the task of circuit size minimization through a combination of techniques. Specifically, we present new circuit-friendly algorithms for frequent operations that achieve constant to asymptotic savings over existing ones; various globally aware optimizations for short- and long- integer arithmetic; as well as circuit minimization techniques that allow us to reduce redundant computation over multiple expressions. We illustrate the savings in different applications, and show the framework's applicability in developing large application circuits, such as ZeroCash, while minimizing the circuit size as in low-level implementations.
PIR with Compressed Queries and Amortized Query Processing
Sebastian Angel (UT Austin and NYU), Hao Chen (Microsoft Research), Kim Laine (Microsoft Research), Srinath Setty (Microsoft Research)
Private information retrieval (PIR) is a key building block in many privacy-preserving systems. Unfortunately, existing constructions remain very expensive. This paper introduces two techniques that make the computational variant of PIR (CPIR) more efficient in practice. The first technique targets a recent class of CPU-efficient CPIR protocols where the query sent by the client contains a number of ciphertexts proportional to the size of the database. We show how to compresses this query, achieving size reductions of up to 274X. The second technique is a new data encoding called probabilistic batch codes (PBCs). We use PBCs to build a multi query PIR scheme that allows the server to amortize its computational cost when processing a batch of requests from the same client. This technique achieves up to 40× speedup over processing queries one at a time, and is significantly more efficient than related encodings. We apply our techniques to the Pung private communication system, which relies on a custom multi-query CPIR protocol for its privacy guarantees. By porting our techniques to Pung, we find that we can simultaneously reduce network costs by 36× and increase throughput by 3X.
Secure Two-party Threshold ECDSA from ECDSA Assumptions
Jack Doerner (Northeastern University),Yashvanth Kondi (Northeastern University),Eysa Lee (Northeastern University),abhi shelat (Northeastern University)
The Elliptic Curve Digital Signature Algorithm (ECDSA) is one of the most widely used schemes in deployed cryptography. Through its applications in code and binary authentication, web security, and cryptocurrency, it is likely one of the few cryptographic algorithms encountered on a daily basis by the average person. However, its design is such that executing multi-party or threshold signatures in a secure manner is challenging: unlike other, less widespread signature schemes, secure multi-party ECDSA requires custom protocols, which has heretofore implied reliance upon additional cryptographic assumptions such as the Paillier encryption scheme.
We propose new protocols for multi-party ECDSA key-generation and signing with a threshold of two, which we prove secure against malicious adversaries in the random oracle model using only the Computational Diffie-Hellman Assumption and the assumptions already implied by ECDSA itself. Our scheme requires only two messages, and via implementation we find that it outperforms the best prior results in practice by a factor of 55 for key generation and 16 for signing, coming to within a factor of 12 of local signatures. Concretely, two parties can jointly sign a message in just over two milliseconds.
Session Chair: David Evans
Session #12: Devices
Speechless: Analyzing the Threat to Speech Privacy from Smartphone Motion Sensors
S Abhishek Anand (University of Alabama at Birmingham),Nitesh Saxena (University of Alabama at Birmingham)
According to recent research, motion sensors available on current smartphone platforms may be sensitive to speech signals. From a security and privacy perspective, this raises a serious concern regarding sensitive speech reconstruction, and speaker or gender identification by a malicious application having unrestricted access to motion sensor readings, without using the microphone.
In this paper, we revisit this important line of research and closely inspect the effect of speech on smartphone motion sensors, in particular, gyroscope and accelerometer. First, we revisit the previously studied scenario (Michalevsky et al.; USENIX Security 2014), where the smartphone shares a common surface with a loudspeaker (with subwoofer) generating speech signals. We observe some effect on the motion sensor signals, which may indeed allow speaker and gender recognition to an extent. However, we also argue that the recorded effect on the sensor readings is possibly from conductive vibrations through the shared surface instead of direct acoustic vibrations due to speech as perceived in previous work. Second, we further extend the previous work by analyzing the effect of speech produced by (1) other less powerful speakers like the in-built laptop and smartphone speakers, and (2) live humans. Our experiments show that in-built laptop speakers were only able to affect the accelerometer when the laptop and the motion sensor shared a surface. Smartphone speakers were not found to be powerful enough to invoke a response in the motion sensors through aerial vibrations. We also report that in the presence of live human speech, we did not notice any effect on the motion sensor readings.
Our results have two-fold implications. First, human-rendered speech seems potentially incapacitated to trigger smartphone motion sensors within the limited sampling rates imposed by the smartphone operating systems. Second, it seems that even machine-rendered speech may not be powerful enough to affect smartphone motion sensors through the aerial medium, although it may induce vibrations through a conductive surface that these sensors, especially accelerometer, could pick up if a relatively powerful speaker is used. Overall, our results suggest that smartphone motion sensors may pose a threat to speech privacy only in some limited scenarios.
Crowd-GPS-Sec: Leveraging Crowdsourcing to Detect and Localize GPS Spoofing Attacks
Kai Jansen (Ruhr-University Bochum),Matthias Schäfer (University of Kaiserslautern),Daniel Moser (ETH Zurich),Vincent Lenders (armasuisse),Christina Pöpper (New York University Abu Dhabi),Jens Schmitt (University of Kaiserslautern)
The aviation industry's increasing reliance on GPS to facilitate navigation and air traffic monitoring opens new attack vectors with the purpose of hijacking UAVs or interfering with air safety. We propose Crowd-GPS-Sec to detect and localize GPS spoofing attacks on moving airborne targets such as UAVs or commercial airliners. Unlike previous attempts to secure GPS, Crowd-GPS-Sec neither requires any updates of the GPS infrastructure nor of the airborne GPS receivers, which are both unlikely to happen in the near future. In contrast, Crowd-GPS-Sec leverages crowdsourcing to monitor the air traffic from GPS-derived position advertisements that aircraft periodically broadcast for air traffic control purposes. Spoofing attacks are detected and localized by an independent infrastructure on the ground which continuously analyzes the contents and the times of arrival of these advertisements. We evaluate our system with real-world data from a crowdsourced air traffic monitoring sensor network and by simulations. We show that Crowd-GPS-Sec is able to globally detect GPS spoofing attacks in less than two seconds and to localize the attacker up to an accuracy of 150 meters after 15 minutes of monitoring time.
SoK: "Plug & Pray" Today - Understanding USB Insecurity in Versions 1 through C
Jing Tian (University of Florida),Nolen Scaife (University of Florida),Deepak Kumar (University of Illinois at Urbana-Champaign),Michael Bailey (University of Illinois at Urbana-Champaign),Adam Bates (University of Illinois at Urbana-Champaign),Kevin Butler (University of Florida)
USB-based attacks have increased in complexity in recent years. Modern attacks now incorporate a wide range of attack vectors, from social engineering to signal injection. To address these challenges, the security community has responded with a growing set of fragmented defenses. In this work, we survey and categorize USB attacks and defenses, unifying observations from both peer-reviewed research and industry. Our systematization extracts offensive and defensive primitives that operate across layers of communication within the USB ecosystem. Based on our taxonomy, we discover that USB attacks often abuse the trust-by-default nature of the ecosystem, and transcend different layers within a software stack; none of the existing defenses provide a complete solution, and solutions expanding multiple layers are most effective. We then develop the first formal verification of the recently released USB Type- C Authentication specification, and uncover fundamental flaws in the specification's design. Based on the findings from our systematization, we observe that while the spec has successfully pinpointed an urgent need to solve the USB security problem, its flaws render these goals unattainable. We conclude by outlining future research directions to ensure a safer computing experience with USB.
Blue Note: How Intentional Acoustic Interference Damages Availability and Integrity in Hard Disk Drives and Operating Systems
Connor Bolton (University of Michigan),Sara Rampazzi (University of Michigan),Chaohao Li (Zhejiang University),Andrew Kwong (University of Michigan),Wenyuan Xu (Zhejiang University),Kevin Fu (University of Michigan)
Intentional acoustic interference causes unusual errors in the mechanics of magnetic hard disk drives in desktop and laptop computers, leading to damage to integrity and availability in both hardware and software such as file system corruption and operating system reboots. An adversary without any special purpose equipment can co-opt built-in speakers or nearby emitters to cause persistent errors. Our work traces the deeper causality of these risks from the physics of materials to the I/O request stack in operating systems for audible and ultrasonic sound. Our experiments show that audible sound causes the head stack assembly to vibrate outside of operational bounds; ultrasonic sound causes false positives in the shock sensor, which is designed to prevent a head crash.
The problem poses a challenge for legacy magnetic disks that remain stubbornly common in safety critical applications such as medical devices and other highly utilized systems difficult to sunset. Thus, we created and modeled a new feedback controller that could be deployed as a firmware update to attenuate the intentional acoustic interference. Our sensor fusion method prevents unnecessary head parking by detecting ultrasonic triggering of the shock sensor.
The Cards Aren't Alright: Detecting Counterfeit Gift Cards Using Encoding Jitter
Nolen Scaife (University of Florida),Christian Peeters (University of Florida),Camilo Velez (University of Florida),Hanqing Zhao (University of Florida),Patrick Traynor (University of Florida),David Arnold (University of Florida)
Gift cards are an increasingly popular payment platform. Much like credit cards, gift cards rely on a magnetic stripe to encode account information. Unlike credit cards, however, the EMV standard is entirely infeasible for gift cards due to compatibility and cost. As such, much of the fraud that has plagued credit cards has started to move towards gift cards, resulting in billions of dollars of loss annually. In this paper, we present a system for detecting counterfeit magnetic stripe gift cards that does not require the original card to be measured at the time of manufacture. Our system relies on a phenomenon known as jitter, which is present on all ISO/IEC-standard magnetic stripe cards. Variances in bit length are induced by the card encoding hardware and are difficult and expensive to reduce. We verify this hypothesis with a high-resolution magneto-optical microscope, then build our detector using inexpensive, commodity card readers. We then partnered with Walmart to evaluate their gift cards and distinguished legitimate gift cards from our clones with up to 99.3% accuracy. Our results show that measurement and detection of jitter increases the difficulty for adversaries to produce undetectable counterfeits, thereby creating significant opportunity to reduce gift card fraud.
Session Chair: Matthew Hicks
NITRD Panel: ML/AI and Cybersecurity
Machine Learning has become an indispensable technology that allows us to extract insights from vast quantities of data in many industries and applications. Advances in areas such as perception, language recognition, medical diagnosis, or self-driving have been spectacular. In security, ML has been instrumental in identifying threats, attacks, and abnormal activities. However, ML algorithms have not been designed to operate in the presence of adversaries. Furthermore, securing and defending ML systems is very hard because we lack theoretical tools for developing principled ML defenses. Can ML/AI give defenders the upper ground or are we consigned to another security whack-a-mole? Panelists will discuss Federal Government's research in ML/AI and cybersecurity and issues to drive further R&D.
Dr. Kenneth Calvert, Division Director, NSF/CISE
Dr. Ahmad Ridley, Senior Researcher, NSA
Ms. Sharothi Pikar, Associate Director for Cyber Strategies, OUSD(R&E), DoD
Moderator: Dr. Tomas Vagoun, National Coordination Office for NITRD
Symposium/Workshops Bridging Reception
Speed Mentoring
Workshops Breakfast
Workshops Opening Remarks
Workshops Session #1
Workshops Break (30 Minutes)
Workshops Lunch
Workshops Closing Remarks | CommonCrawl |
Robust fuzzy programming
Robust fuzzy programming (ROFP) is a powerful mathematical optimization approach to deal with optimization problems under uncertainty. This approach is firstly introduced at 2012 by Pishvaee, Razmi & Torabi[1] in the Journal of Fuzzy Sets and Systems. ROFP enables the decision makers to be benefited from the capabilities of both fuzzy mathematical programming and robust optimization approaches. At 2016 Pishvaee and Fazli[2] put a significant step forward by extending the ROFP approach to handle flexibility of constraints and goals. ROFP is able to achieve a robust solution for an optimization problem under uncertainty.
Definition of robust solution
Robust solution is defined as a solution which has "both feasibility robustness and optimality robustness; Feasibility robustness means that the solution should remain feasible for (almost) all possible values of uncertain parameters and flexibility degrees of constraints and optimality robustness means that the value of objective function for the solution should remain close to optimal value or have minimum (undesirable) deviation from the optimal value for (almost) all possible values of uncertain parameters and flexibility degrees on target value of goals".[2]
Classification of ROFP methods
As fuzzy mathematical programming is categorized into Possibilistic programming and Flexible programming, ROFP also can be classified into:[2]
1. Robust possibilistic programming (RPP)
2. Robust flexible programming (RFP)
3. Mixed possibilistic-flexible robust programming (MPFRP)
The first category is used to deal with imprecise input parameters in optimization problems while the second one is employed to cope with flexible constraints and goals. Also, the last category is capable to handle both uncertain parameters and flexibility in goals and constraints.
From another point of view, it can be said that different ROFP models developed in the literature can be classified in three categories according to degree of conservatism against uncertainty. These categories include:[1]
1. Hard worst case ROFP
2. Soft worst case ROFP
3. Realistic ROFP
Hard worst case ROFP has the most conservative nature among ROFP methods since it provides maximum safety or immunity against uncertainty. Ignoring the chance of infeasibility, this method immunizes the solution for being infeasible for all possible values of uncertain parameters. Regarding the optimality robustness, this method minimizes the worst possible value of objective function (min-max logic). On the other hand, Soft worst case ROFP method behaves similar to hard worst case method regarding optimality robustness, however does not satisfy the constraints in their extreme worst case. Lastly, realistic method establishes a reasonable trade-off between the robustness, the cost of robustness and other objectives such as improving the average system performance (cost-benefit logic).
Applications
ROFP is successfully implemented in different practical application areas such as the following ones.
• Supply chain management such as the work by Pishvaee et al.[1] which addresses the design of a social responsible supply chain network under epistemic uncertainty.
• Healthcare management such as the works by Zahiri et al.[3] and Mousazadeh et al.[4] which consider the planning of an organ transplantation network and a pharmaceutical supply chain, respectively.
• Energy planning such as Bairamzadeh et al.[5] which uses a multi-objective possibilistic programming model to deal with the design of a bio-ethanol production-distribution network. Also in another research, Zhou et al.[6] developed a robust possibilistic programming model to deal with the planning problem of municipal electric power system.
• Sustainability such as Xu and Huang[7] which employ ROFP to cope with an air quality management problem.
References
1. Pishvaee, M. S.; Razmi, J.; Torabi, S. A. (2012-11-01). "Robust possibilistic programming for socially responsible supply chain network design: A new approach". Fuzzy Sets and Systems. Theme : Operational Research. 206: 1–20. doi:10.1016/j.fss.2012.04.010.
2. Pishvaee, Mir Saman; Fazli Khalaf, Mohamadreza (2016-01-01). "Novel robust fuzzy mathematical programming methods". Applied Mathematical Modelling. 40 (1): 407–418. doi:10.1016/j.apm.2015.04.054.
3. Zahiri, Behzad; Tavakkoli-Moghaddam, Reza; Pishvaee, Mir Saman (2014-08-01). "A robust possibilistic programming approach to multi-period location–allocation of organ transplant centers under uncertainty". Computers & Industrial Engineering. 74: 139–148. doi:10.1016/j.cie.2014.05.008.
4. Mousazadeh, M.; Torabi, S. A.; Zahiri, B. (2015-11-02). "A robust possibilistic programming approach for pharmaceutical supply chain network design". Computers & Chemical Engineering. 82: 115–128. doi:10.1016/j.compchemeng.2015.06.008.
5. Bairamzadeh, Samira; Pishvaee, Mir Saman; Saidi-Mehrabad, Mohammad (2015-12-22). "Multiobjective Robust Possibilistic Programming Approach to Sustainable Bioethanol Supply Chain Design under Multiple Uncertainties". Industrial & Engineering Chemistry Research. 55 (1): 237–256. doi:10.1021/acs.iecr.5b02875.
6. Zhou, Y.; Li, Y.P.; Huang, G.H. (2015-12-15). "A robust possibilistic mixed-integer programming method for planning municipal electric power systems". International Journal of Electrical Power & Energy Systems. 73: 757–772. doi:10.1016/j.ijepes.2015.06.009.
7. Xu, Ye; Huang, Guohe (2015-10-15). "Development of an Improved Fuzzy Robust Chance-Constrained Programming Model for Air Quality Management". Environmental Modeling & Assessment. 20 (5): 535–548. doi:10.1007/s10666-014-9441-3.
| Wikipedia |
\begin{document}
\titlerunning{Coderivative-Based Semi-Newton Method in Nonsmooth Difference Programming} \title{Coderivative-Based Semi-Newton Method\\ in Nonsmooth Difference Programming \thanks{Research of the first author was partially supported by the Ministry of Science, Innovation and Universities of Spain and the European Regional Development Fund (ERDF) of the European Commission, Grant PGC2018-097960-B-C22, and by the Generalitat Valenciana, grant AICO/2021/165. Research of the second author was partially supported by the USA National Science Foundation under grants DMS-1808978 and DMS-2204519, by the Australian Research Council under Discovery Project DP-190100555, and by the Project 111 of China under grant D21024. Research of the third author was partially supported by grants: Fondecyt Regular 1190110 and Fondecyt Regular 1200283.}} \subtitle{} \author{Francisco J. Arag\'{o}n-Artacho \and \mbox{Boris S. Mordukhovich} \and \mbox{Pedro P\'erez-Aros}}
\institute{Francisco J. Arag\'{o}n-Artacho \at Department of Mathematics, University of Alicante, Alicante, Spain\\ \email{[email protected]} \and Boris S. Mordukhovich \at Department of Mathematics, Wayne State University, Detroit, Michigan 48202, USA\\ \email{[email protected]} \and Pedro P\'erez-Aros \at Instituto de Ciencias de la Ingenier\'ia, Universidad de O'Higgins, Rancagua, Chile\\ \email{[email protected]}}
\date{\today}
\maketitle
\begin{abstract} This paper addresses the study of a new class of nonsmooth optimization problems, where the objective is represented as a difference of two generally nonconvex functions. We propose and develop a novel Newton-type algorithm to solving such problems, which is based on the coderivative generated second-order subdifferential (generalized Hessian) and employs advanced tools of variational analysis. Well-posedness properties of the proposed algorithm are derived under fairly general requirements, while constructive convergence rates are established by using additional assumptions including the Kurdyka--{\L}ojasiewicz condition. We provide applications of the main algorithm to solving a general class of nonsmooth nonconvex problems of structured optimization that encompasses, in particular, optimization problems with explicit constraints. Finally, applications and numerical experiments are given for solving practical problems that arise in biochemical models, constrained quadratic programming, etc., where advantages of our algorithms are demonstrated in comparison with some known techniques and results. \end{abstract}\vspace*{-0.05in}
\keywords{Nonsmooth difference programming \and generalized Newton methods \and global convergence \and convergence rates \and variational analysis \and generalized differentiation }\vspace*{-0.05in}
\subclass{49J53, 90C15, 9J52}\vspace*{-0.2in}
\section{Introduction}\label{intro}\vspace*{-0.05in}
The primary mathematical model considered in this paper is described by \begin{equation}\label{EQ01} \min_{x\in \mathbb{R}^n} \varphi(x):=g(x)-h(x), \end{equation} where $g:\mathbb{R}^n \to \mathbb{R}$ is of {\em class $\mathcal{C}^{1,1}$} (i.e., the collection of $\mathcal{C}^1$-smooth functions with locally Lipschitzian derivatives), and where $h: \mathbb{R}^n \to \mathbb{R}$ is a locally Lipschitzian and {\em prox-regular} function; see below. Although \eqref{EQ01} is a problem of unconstrained optimization, it will be shown below that a large class of constrained optimization problems can be reduced to this form. In what follows, we label the optimization class in \eqref{EQ01} as problems of {\em difference programming}.
The difference form \eqref{EQ01} reminds us of problems of {\em DC $($difference of convex$)$ programming}, which have been intensively studied in optimization with a variety of practical applications; see, e.g., \cite{Aragon2020,Artacho2019,AragonArtacho2018,Oliveira_2020,hiriart,Toh,Tao1997,Tao1998,Tao1986} and the references therein. However, we are not familiar with a systematic study of the class of difference programming problems considered in this paper.
Our main goal here is to develop an efficient numerical algorithm to solve the class of difference programs \eqref{EQ01} with subsequent applications to nonsmooth and nonconvex problems of particular structures, problems with geometric constraints, etc. Furthermore, the efficiency of the proposed algorithm and its modifications is demonstrated by solving some practical models for which we conduct numerical experiments and compare the obtained results with previously known developments and computations by using other algorithms. The proposed algorithm is of a {\em regularized damped Newton type} with a {\em novel choice of directions} in the iterative scheme providing a {\em global convergence} of iterates to a stationary point of the cost function. At the {\em first order}, the novelty of our algorithm, in comparison with, e.g., the most popular {\em DCA algorithm} by Tao et al. \cite{Tao1997,Tao1998,Tao1986} and its {\em boosted} developments by Arag\'on-Artacho et al.\cite{Aragon2020,Artacho2019,AragonArtacho2018,MR4078808} in DC programming, is that instead of a convex subgradient of $h$ in \eqref{EQ01}, we now use a {\em limiting subgradient} of $-h$. No second-order information on $h$ is used in what follows. Concerning the other function $g$ in \eqref{EQ01}, which is nonsmooth of the {\em second-order}, our algorithm replaces the classical Hessian matrix by the {\em generalized Hessian/second-order subdifferential} of $g$ in the sense of Mordukhovich \cite{m92}. The latter construction, which is defined as the coderivative of the limiting subdifferential has been well recognized in variational analysis and optimization due its comprehensive calculus and explicit evaluations for broad classes of extended-real-valued functions arising in applications. We refer the reader to, e.g., \cite{chhm,dsy,Helmut,hmn,hos,hr,2020arXiv200910551D,MR3823783,MR2191744,mr,os,yy} and the bibliographies therein for more details. Note also that the aforementioned generalized Hessian has already been used in differently designed algorithms of the Newton type to solve optimization-related problems of different nonsmooth structures in comparison with \eqref{EQ01}; see \cite{Helmut,2020arXiv200910551D,jogo,2021arXiv210902093D,BorisEbrahim}. Having in mind the discussions above, we label the main algorithm developed in this paper as the {\em regularized coderivative-based damped semi-Newton method} (abbr.\ RCSN).
The rest of the paper is organized as follows. Section~\ref{sec:2} recalls constructions and statements from variational analysis and generalized differentiation, which are broadly used in the formulations and proofs of the major results. Besides well-known facts, we present here some new notions and further elaborations.
In Section~\ref{sec:3}, we design our {\em main RCSN algorithm}, discuss each of its steps, and establish various results on its performance depending on imposed assumptions whose role and importance are illustrated by examples. Furthermore, Section~\ref{sec:4} employs the {\em Kurdyka-{\L}ojasiewicz {\rm(KL)} property} of the cost function to establish quantitative convergence rates of the RCSN algorithm depending on the exponent in the KL inequality.
Section~\ref{sec:5} addresses the class of (nonconvex) problems of {\em structured optimization} with the cost functions given in the form $f(x)+\psi(x)$, where $f\colon\mathbb{R}^n\to\mathbb{R}$ is a twice continuously differentiable function with a Lipschitzian Hessian (i.e., of class ${\cal C}^{2,1}$), while $\psi\colon\mathbb{R}^n\to\Bar{\R}:=(-\infty,\infty]$ is an extended-real-valued prox-bounded function. By using the {\em forward-backward envelope} \cite{MR3845278} and the associated {\em Asplund function} \cite{asplund}, we reduce this class of structured optimization problems to the difference form \eqref{EQ01} and then employ the machinery of RCSN to solving problems of this type. As a particular case of RCSN, we design and justify here a new {\em projected-like Newton algorithm} to solve optimization problems with geometric constraints given by general closed sets.
Section~6 is devoted to implementations of the designed algorithms and {\em numerical experiments} in two different problems arising in practical modeling. Although these problems can be treated after some transformations by DCA-like algorithms, we demonstrate in this section numerical advantages of the newly designed algorithms over the known developments in both smooth and nonsmooth settings. The concluding Section~\ref{sec:7} summarizes the major achievements of the paper and discusses some directions of our future research.\vspace*{-0.4in}
\section{Tools of Variational Analysis and Generalized Differentiation}\label{sec:2}
Throughout the entire paper, we deal with finite-dimensional Euclidean spaces and use the standard notation and terminology of variational analysis and generalized differentiation; see, e.g., \cite{MR3823783,MR1491362}, where the reader can find the majority of the results presented in this section. Recall that $\mathbb{B}_r(x)$ stands for the closed ball centered at $x\in\mathbb{R}^n$ with radius $r> 0$ and that $\mathbb{N}:=\{1,2,\ldots\}$.
Given a set-valued mapping $F: \mathbb{R}^n \tto \mathbb{R}^m$, its {\em graph} is the set $ \operatorname{gph} F:=\big\{ (v,w) \in {\R^n}\times \R^{m}\;|\; w\in F(x)\big\}$, while the (Painlev\'e--Kuratowski) {\em outer limit} of $F$ at $x\in\mathbb{R}^n$ is defined by \begin{equation}\label{pk}
\mathop{{\rm Lim}\,{\rm sup}}_{u\to x}F(u):=\big\{y\in\mathbb{R}^m\;\big|\;\exists\,u_k\to x,\,y_k\to y,\;y_k\in F(u_k)\;\mbox{as}\;k\in\mathbb{N}\big\}. \end{equation}
For a nonempty set $C\subseteq\mathbb{R}^n$, the (Fr\'echet) {\em regular normal cone} and (Mordukhovich) {\em basic/limiting normal cone} at $x\in C$ are defined, respectively, by \begin{equation}\label{rnc} \begin{aligned}
\widehat{N}(x;C)=\widehat N_C(x):&=\Big\{ x^*\in \mathbb{R}^n\;\Big|\;\limsup\limits_{ u \overset{C}{\to} x } \Big\langle x^\ast,\frac{u - x}{\| u - x \| }\Big\rangle\leq 0\Big\},\\ N(x;C)=N_C(x):&=\mathop{{\rm Lim}\,{\rm sup}}\limits_{u \overset{C}{\to} x}\widehat{N}(u;C), \end{aligned} \end{equation} where ``$u\overset{C}{\to} x$'' means that $u \to x$ with $u \in C$. We use the convention $\widehat N(x;C)=N(x;C):=\emptyset$ if $x\notin C$. The {\em indicator function} $\delta_C(x)$ of $C$ is equal to 0 if $x\in C$ and to $\infty $ otherwise.
For a lower semicontinuous (l.s.c.) function $f:{\R^n}\to\overline{\mathbb{R}}$, its {\em domain} and {\em epigraph} are given by $\mbox{\rm dom}\, f := \{ x\in \mathbb{R}^n \mid f(x) < \infty \}$ and $\mbox{\rm epi}\, f:=\{ (x,\alpha) \in {\R^n}\times \mathbb{R} \;|\;f(x)\leq \alpha\},$ respectively. The {\em regular} and {\em basic subdifferentials} of $f$ at $x\in\mbox{\rm dom}\, f$ are defined by \begin{equation}\label{sub} \begin{aligned} \widehat{\partial} f(x)&:=\big\{ x^\ast\in \mathbb{R}^n\mid(x^\ast ,-1) \in \widehat{N}\big((x,f(x));\mbox{\rm epi}\, f\big)\big\},\\ \partial f(x) &:= \big\{ x^\ast\in \mathbb{R}^n\mid(x^\ast ,-1) \in N\big((x,f(x));\mbox{\rm epi}\, f\big)\big\}, \end{aligned} \end{equation} via the corresponding normal cones \eqref{rnc} to the epigraph. The function $f$ is said to be {\em lower/subdifferentially regular} at $\bar{x}\in\mbox{\rm dom}\, f$ if $\partial f(\bar{x}) =\widehat{\partial}f(\bar{x})$.
Given further a set-valued mapping/multifunction $F: {\R^n} \tto \R^{m}$, the {\em regular} and {\em basic coderivatives} of $F$ at $(x,y)\in\operatorname{gph} F$ are defined for all $y^*\in\R^{m}$ via the corresponding normal cones \eqref{rnc} to the graph of $F$, i.e., \begin{equation}\label{cod} \begin{aligned}
\widehat{D}^\ast F(x,y) (y^\ast)&:=\big\{ x^\ast \in {\R^n}\;\big|\;(x^\ast,-y^\ast) \in \widehat{N}\big( (x,y); \operatorname{gph} F\big)\big\},\\
{D}^\ast F(x,y) (y^\ast)&:=\big\{x^\ast \in {\R^n}\;\big|\;(x^\ast,-y^\ast) \in N\big( (x,y); \operatorname{gph} F\big)\big\}, \end{aligned} \end{equation} where $y$ is omitted if $F$ is single-valued at $x$. When $F$ is single-valued and locally Lipschitzian around $x$, the basic coderivative has the following representation via the basic subdifferential of the scalarization \begin{align}\label{coder:sub} {D}^\ast F(x) (y^\ast) = \partial \langle y^\ast, F\rangle (x), \text{ where } \langle y^\ast, F\rangle (x):= \langle y^\ast, F(x)\rangle. \end{align}
Recall that a set-valued mapping $F: \mathbb{R}^n \tto \mathbb{R}^m $ is {\em strongly metrically subregular} at $(\bar{x},\bar{y})\in \operatorname{gph} F$ if there exist $\kappa,\epsilon>0$ such that \begin{align}\label{def:stron_subreg}
\| x -\bar{x}\| \leq \kappa \| y - \bar{y}\|\; \text{ for all }\;(x,y) \in \mathbb{B}_\epsilon(\bar{x},\bar{y}) \cap\operatorname{gph} F. \end{align} It is well-known that this property of $F$ is equivalent to the {\em calmness} property of the inverse mapping $F^{-1}$ at $(\bar{y},\bar{x})$. In what follows, we use the calmness property of single-valued mappings $h\colon\mathbb{R}^n\to\mathbb{R}^m$ at $\bar{x}$ meaning that there exist positive numbers $\kappa$ and $\varepsilon>0$ such that \begin{equation}\label{calm}
\|h(x)-h(\bar{x})\|\leq\kappa\|x-\bar{x}\|\;\text{ for all }\;x\in\mathbb{B}_\epsilon(\bar{x}). \end{equation} The infimum of all $\kappa>0$ in \eqref{calm} is called the \emph{exact calmness bound} of $h$ at $\bar{x}$ and is denoted it by $\mbox{\rm clm}\, h(\bar{x})$. On the other hand, a multifunction $F\colon\mathbb{R}^n\tto\mathbb{R}^m$ is {\em strongly metrically regular} around $(\bar{x},\bar{y}) \in \operatorname{gph} F$ if its inverse $F^{-1}$ admits a single-valued and Lipschitz continuous localization around this point.
Along with the (first-order) basic subdifferential in \eqref{sub}, we consider the {\em second-order subdifferential/generalized Hessian} of $f: {\R^n} \to \overline{\mathbb{R}}$ at $x\in \mbox{\rm dom}\, f$ relative to $x^\ast\in\partial f(x)$ defined by \begin{equation}\label{2nd} \partial^2f(x,x^\ast) (v^\ast) = \left(D^\ast\partial f\right) (x,x^\ast)(v^\ast),\quad v^\ast\in {\R^n} \end{equation} and denoted by $\partial^2f(x) (v^\ast)$ when $\partial f(x)$ is a singleton. If $f$ is twice continuously differentiable ($\mathcal{C}^2$-smooth) around $x$, then $\partial^2 f(x)(v^\ast)=\{\nabla^2 f(x)v^\ast\}$.
Next we introduce an extension of the notion of positive-definiteness for multifunctions, where the the corresponding constant may not be positive.\vspace*{-0.1in}
\begin{definition}\label{def:lower-def} Let $F: \mathbb{R}^n \tto \mathbb{R}^n$ and $\xi\in\mathbb{R}$. Then $F$ is \emph{$\xi$-lower-definite} if \begin{align}\label{Stron-semide}
\langle y, x\rangle \geq \xi\|x\|^2\;\text{ for all }\;(x,y)\in\operatorname{gph} F. \end{align} \end{definition}
\begin{remark}\label{rem:definite} We can easily check the following:
(i) For any symmetric matrix $Q$ with the smallest eigenvalue $\lambda_{\min}(Q)$, the function $f(x)=Qx$ is $\lambda_{\min}(Q)$-lower-definite.
(ii) If a function $f: \mathbb{R}^n \to \overline{\mathbb{R}}$ is {\em strongly convex} with modulus $\rho>0$, (i.e., $ f -\frac{\rho }{2}\| \cdot \|^2$ is convex), it follows from \cite[Corollary~5.9]{MR3823783} that $\partial^2 f(x,x^\ast)$ is $\rho$-lower-definite for all $(x,x^\ast)\in\operatorname{gph}\partial f$.
(iii) If $F_1,F_2: \mathbb{R}^n \tto \mathbb{R}^n$ are $\xi_1$ and $\xi_2$-lower-definite, then the sum $F_1+F_2$ is $(\xi_1+\xi_2)$-lower-definite. \end{remark}\vspace*{-0.05in}
Recall next that a function $f: \mathbb{R}^n \to \overline{\mathbb{R}}$ is \emph{prox-regular} at $\bar{x}\in\mathbb{R}^n$ \emph{for} $\bar{v} \in \partial f(\bar{x})$ if it is l.s.c.\ around $\bar{x}$ and there exist $\epsilon>0$ and $r\geq 0$ such that \begin{align}\label{proregularity}
f(x') \geq f(x) + \langle v, x'-x\rangle - \frac{r}{2}\| x'- x\|^2 \end{align} whenever $x,x'\in\mathbb{B}_\epsilon(\bar{x})$ with $ f(x) \leq f(\bar{x}) + \epsilon$ and $v\in \partial f(x) \cap \mathbb{B}_\epsilon(\bar{v})$. If this holds for all $\bar{v} \in \partial f(\bar{x})$, $f$ is said to be \emph{prox-regular at} $\bar x$.\vspace*{-0.05in}
\begin{remark} The class of prox-regular functions has been well-recognized in modern variational analysis. It is worth mentioning that if $f$ is a locally Lipschitzian function around $\bar{x}$, then the following properties of $f$ are equivalent: (i) prox-regularity at $\bar{x}$, (ii) lower-$\mathcal{C}^2$ at $\bar{x}$, and (iii) primal-lower-nice at $\bar{x}$; see, e.g., \cite[Corollary 3.12]{MR2101873} for more details. \end{remark}\vspace*{-0.05in}
Given a function $f: \mathbb{R}^n \to \overline{\mathbb{R}}$ and $\bar{x}\in\mbox{\rm dom}\, f$, the {\em upper directional derivative} of $f$ at $\bar{x}$ with respect to $d\in\mathbb{R}^n$ is defined by \begin{equation}\label{UDD} f'(\bar{x};d):=\limsup\limits_{t\to 0^+} \frac{f(\bar{x}+td) - f(\bar{x})}{t}. \end{equation} The following proposition establishes various properties of prox-regular functions used below. We denote the convex hull of a set by ``co".
\begin{proposition}\label{Lemma:Dire01} Let $f: \mathbb{R}^n \to \overline{\mathbb{R}}$ be locally Lipschitzian around $\bar{x}$ and prox-regular at this point. Then $f$ is lower regular at $\bar{x}$, $\mbox{\rm co}\,\partial (-f)(\bar{x}) = -\partial f(\bar{x})$, and for any $d\in\mathbb{R}^n$ we have the representations \begin{align} \label{Dire01}
(-f)'(\bar x;d)=\inf\big\{\langle w, d\rangle\;\big|\;w\in \partial (-f)(\bar x)\big\}= \inf\big\{\langle w, d\rangle\;\big|\;w\in-\partial f(\bar{x})\big\}. \end{align} \end{proposition} \begin{proof} First we fix an arbitrary subgradient $\bar{v} \in \partial f(\bar{x})$
and deduce from \eqref{proregularity} applied to $x=\bar{x}$ and $v=\bar{v}$ that \begin{equation*}
f(x') \geq f(\bar{x}) +\langle \bar{v}, x'-\bar{x}\rangle -\frac{r}{2} \| x' - \bar{x}\|^2\;\text{ for all }\;x'\in\mathbb{B}_\epsilon(\bar{x}). \end{equation*} Passing to the limit as $x' \to \bar{x}$ tells us that \begin{align*}
\liminf_{x'\to \bar{x}} \frac{ f(x') - f(\bar{x}) - \langle \bar{v}, x'-\bar{x}\rangle }{ \| x' - \bar{x}\| } \geq 0, \end{align*} which means that $\bar{v}\in\widehat\partial f(\bar{x})$ and thus shows that $f$ is lower regular at $\bar{x}$. By the Lipschitz continuity of $f$ around $\bar{x}$ and the convexity of the set $\widehat{\partial} f(\bar{x})$, we have that $\widehat{\partial} f(\bar{x})=\partial f(\bar{x}) =\mbox{\rm co}\, \partial f(\bar{x}) = \overline{\partial } f(\bar{x})$, where $\overline{\partial }$ denotes the (Clarke) {\em generalized gradient}. It follows from $ \overline{\partial } (-f)(\bar{x}) =-\overline{\partial} f(\bar{x})$ that $ \overline{\partial } (-f)(\bar{x}) =-\partial f(\bar{x})$, which implies therefore that $\partial (-f)(\bar{x}) \subseteq -\partial f(\bar{x})$.
Pick $v\in \partial (-f)(\bar{x})$, $d\in \mathbb{R}^n$ and find by the prox-regularity of $f$ at $\bar{x}$ for $-v\in\partial f(\bar{x})$ that there exists $r>0$ such that \begin{align*}
\langle v,d \rangle+\frac{rt}{2}\| d\|^2 \geq \frac{-f(\bar x+td) + f(\bar x)}{t} \end{align*} if $t>0$ is small enough. This yields $(-f)'(\bar x;d) \leq \langle v, d \rangle $ for all ${v}\in \partial (-f)(\bar x) $ and thus verifies the inequality ``$\leq $'' in the first representation of \eqref{Dire01}.
To prove the opposite inequality therein, take $t_k \to 0^+$ such that \begin{align*} \lim_{k\to \infty} \frac{ -f(\bar{x} + t_k d ) + f(\bar{x}) }{ t_k}= (-f)'(\bar x;d). \end{align*} Employing the mean value theorem from \cite[Corollary~4.12]{MR3823783}) gives us \begin{align*} f(\bar{x} + t_k d ) - f(\bar{x}) = t_k \langle v_k,d\rangle \text{ for some } v_k \in \partial f(\bar{x} + \lambda_k t_k d) \text{ with } \lambda_k \in ( 0,1). \end{align*} It follows from the Lipschitz continuity of $f$ that $\{v_k\}$ is bounded, and so we can assume that $v_k \to \bar{v}\in \partial f(\bar{x})$. Therefore, \begin{equation*} \begin{array}{ll}
(-f)'(\bar x;d)=\langle -\bar{v}, d\rangle \geq \inf\big\{\langle w, d\rangle\;\big|\; w \in -\partial f(\bar{x})\big\} \\
=\inf\big\{\langle w, d\rangle\;\big|\;w \in \mbox{\rm co}\, \partial (-f)(\bar{x}) \big\}=\inf\big\{\langle w, d\rangle\;\big|\;w \in \partial (-f)(\bar{x}) \big\}, \end{array} \end{equation*} which verifies \eqref{Dire01} and completes the proof of the proposition. \end{proof}\vspace*{-0.05in}
Next we define the notion of stationary points for problem \eqref{EQ01} the finding of which is the goal of our algorithms.
\begin{definition}\label{def:stationary} Let $\varphi=g-h$ be the cost function in \eqref{EQ01}, where $g$ is of class $\mathcal{C}^{1,1}$ around some point $\bar{x}$, and where $h$ is locally Lipschitzian around $\bar{x}$ and prox-regular at this point. Then $\bar{x}$ is a \emph{stationary point} of \eqref{EQ01} if $0\in\partial\varphi(\bar{x})$.
\end{definition}\vspace*{-0.2in}
\begin{remark}
The stationarity notion $0 \in \partial\varphi(\bar{x})$, expressed via
the limiting subdiffential, is known as the {\em M$($ordukhovich$)$-stationarity}. Since no other stationary points are considered in this paper, we skip ``M" in what follows. Observe from the subdifferential sum rule in our setting that $\bar{x}$ is a stationary point in \eqref{EQ01} if and only if $0\in\nabla g(\bar{x}) +\partial(-h)(\bar{x})$. Thus every stationary point $\bar{x}$ is a {\em critical point} in the sense that $0\in\nabla g(\bar{x}) - \partial h(\bar{x})$. By Proposition~\ref{Lemma:Dire01}, the latter can be equivalently described in terms of the generalized gradient and also via the {\em symmetric subdifferential} \cite{MR3823783} of $\varphi$ at $\bar{x}$ defined by \begin{equation}\label{sym} \partial^0\varphi(\bar{x}):=\partial\varphi(\bar{x})\cup\big(-\partial(-\varphi)(\bar{x})\big) \end{equation} which possesses the plus-minus symmetry $\partial^0(-\varphi(\bar{x}))=-\partial^0(\varphi(\bar{x}))$. When both $g$ and $h$ are convex, the classical DC algorithm~\cite{Tao1986,Tao1997} and its BDCA variant \cite{MR4078808} can be applied for solving problem~\eqref{EQ01}. Although these algorithms only converge to critical points, they can be easily combined as in \cite{Aragon2020} with a basic derivative-free optimization scheme to converge to {d-stationary points}, which satisfy $\partial h(\bar{x})=\{\nabla g(\bar{x})\}$ (or, equivalently, $\varphi'(\bar{x};d)=0$ for all $d\in\mathbb{R}^n$; see~\cite[Proposition~1]{Aragon2020}). In the DC setting, every local minimizer of problem~\eqref{EQ01} is a d-stationary point \cite[Theorem~3]{Toland1979}, a property which is stronger than the notion of stationarity in Definition~\ref{def:stationary}. \end{remark}\vspace*{-0.1in}
To proceed, recall that a mapping $f: U\to \mathbb{R}^m$ defined on an open set $U\subseteq \mathbb{R}^n $ is \emph{semismooth} at $\bar{x}$ if it is locally Lipschitzian around $\bar{x}$, directionally differentiable at this point, and the limit \begin{align*} \lim\limits_{A \in \tiny{\mbox{\rm co}\,} \overline{\nabla} f(\bar{x} + t u'), \atop u' \to u, t\to 0^+} A u' \end{align*}
exists for all $u \in \mathbb{R}^n$, where $\overline{\nabla} f(x) :=\{ A\;|\; \exists x_k \overset{D}{\to } x \text{ and } \nabla f(x_k) \to A \}$, and where $D$ is the set on which $f$ is differentiable; see \cite{MR1955649,MR3289054} for more details. We say that a function $g: \mathbb{R}^n \to \overline{\mathbb{R}}$ is {\em semismoothly differentiable} at $\bar{x}$ if $g$ is $\mathcal{C}^{1}$-smooth around $\bar{x}$ and its gradient mapping $\nabla g$ is semismooth at this point.
Recall further that a function $\psi: \mathbb{R}^n \to \overline{\mathbb{R}}$ is \emph{prox-bounded} if there exists $\lambda>0$ such that $\MoreauYosida{\psi}{\lambda}(x)>-\infty$ for some $x\in \mathbb{R}^n$, where $\MoreauYosida{\psi}{\lambda} :\mathbb{R}^n \to \overline{\mathbb{R}}$ is the \emph{Moreau envelope} of $\psi$ with parameter $\lambda>0$ defined by \begin{equation}\label{moreau}
\MoreauYosida{\psi}{\lambda} (x):=\inf_{ z \in \mathbb{R}^n }\Big\{ \psi(z) + \frac{1}{2\lambda} \| x-z\|^2\Big\}. \end{equation}
The number $\lambda_\psi:= \sup \{\lambda >0\;|\;\MoreauYosida{\psi}{\lambda}(x)>-\infty \text{ for some } x\in \mathbb{R}^n \}$ is called the \emph{threshold} of prox-boundedness of $\psi$. The corresponding \emph{proximal mapping} is the multifunction $\Prox{\psi}{\lambda} : \mathbb{R}^n \tto \mathbb{R}^n$ given by \begin{equation}\label{prox}
\Prox{\psi}{\lambda} (x):= \mathop{\rm argmin}_{z \in \mathbb{R}^n }\Big\{ \psi(z) + \frac{1}{2\lambda} \| x-z\|^2\Big\}. \end{equation}
Next we observe that that the Moreau envelope can be represented as a {\em DC function}. For any function $\varphi:\mathbb{R}^n\to\overline{\mathbb{R}}$, consider its \emph{Fenchel conjugate} \begin{equation*} \phi^*(x):=\sup_{z\in\mathbb{R}^n}\big\{\langle x,z\rangle-\phi(z)\big\}, \end{equation*} and for any $\psi\colon\mathbb{R}^n\to\Bar{\R}$ and $\lambda>0$, define the {\em Asplund function} \begin{equation}\label{asp}
\Asp{\lambda}{\psi}(x):=\sup\limits_{z\in \mathbb{R}^n}\Big\{ \frac{ 1}{\lambda} \langle z,x\rangle - \psi(z) - \frac{1}{ 2\lambda } \| z\|^2\Big\}=\Big(\psi + \frac{1}{ 2\lambda } \| \cdot\|^2\Big)^\ast(x), \end{equation} which is inspired by Asplund's study of metric projections in \cite{asplund}. The following proposition presents the precise formulation of the aforementioned statement and reveals some remarkable properties of the Asplund function \eqref{asp}.
\begin{proposition}\label{Lemma5.1} Let $\psi$ be a prox-bounded function with threshold $\lambda_\psi$. Then for every $\lambda \in (0,\lambda_\psi)$, we have the representation \begin{equation}\label{more-asp}
\MoreauYosida{\psi}{\lambda}(x)=\frac{1}{2\lambda} \| x\|^2 - \Asp{\lambda}{\psi}(x),\quad x\in\mathbb{R}^n, \end{equation} where the Asplund function is convex and Lipschitz continuous on $\mathbb{R}^n$. Furthermore, for any $x\in\mathbb{R}^n$ the following subdifferential evaluations hold: \begin{align}{ } \partial(-\Asp{\lambda}{\psi}) (x) &\subseteq- \frac{1}{\lambda} \Prox{\psi}{\lambda} (x),\label{eq_sub_eq01}\\ \partial \Asp{\lambda}{\psi}(x) &= \frac{1}{\lambda} \mbox{\rm co}\,\left( \Prox{\psi}{\lambda} (x)\right).\label{eq_sub_eq02} \end{align} Moreover, if $v\in \Prox{\psi}{\lambda} (x)$ is such that $v\notin \mbox{\rm co}\,\left( \Prox{\psi}{\lambda} (x)\backslash \{v\}\right)$, then the vector $-\frac{1}{\lambda}v$ belongs to $\partial(-\Asp{\lambda}{\psi}) (x)$. If in addition $f$ is of class $\mathcal{C}^{2,1}$ on $\mathbb{R}^n$, then the function $x\mapsto \Asp{\lambda}{\psi}( x - \lambda \nabla f(x))$ is prox-regular at any point $x\in\mathbb{R}^n$. \end{proposition}\vspace*{-0.03in} \begin{proof} Representation \eqref{more-asp} easily follows from definitions of the Moreau envelope and Asplund function. Due to the second equality in \eqref{asp}, the Asplund function is convex on $\mathbb{R}^n$. It is also Lipschitz continuous due its finite-valuedness on $\mathbb{R}^n$, which is induced by this property of the Moreau envelope. The subdifferential evaluations in \eqref{eq_sub_eq01} and \cite[Example~10.32]{MR1491362} and the subdifferential sum rule in \cite[Proposition~1.30]{MR3823783}) tell us that $\partial(-\Asp{\lambda}{\psi} )(x) = -\lambda^{-1}x + \partial ( \MoreauYosida{\psi}{\lambda}) (x)$ and $\partial \Asp{\lambda}{\psi} (x) = \lambda^{-1} x + \partial (-\MoreauYosida{\psi}{\lambda}) (x)$ for any $x\in\mathbb{R}^n$.
Take further $v\in \Prox{\psi}{\lambda} (x) $ with $-\frac{1}{\lambda}v \not\in\partial(-\Asp{\lambda}{\psi} ) (x)$ and show that $v \in {\rm co}\left( \Prox{\psi}{\lambda} (x)\backslash \{v\}\right)$. Indeed, it follows from \eqref{eq_sub_eq01} that \begin{align*} \partial(-\Asp{\lambda}{\psi} ) (x) &\subseteq- \frac{1}{\lambda} \Prox{\psi}{\lambda} (x) \backslash\{v\}. \end{align*} The Lipschitz continuity and convexity of $\Asp{\lambda}{\psi}$ implies that \begin{align}\label{eqLemma26} {\rm co}\,\partial(-\Asp{\lambda}{\psi} ) (x) = -\partial\Asp{\lambda}{\psi} (x) \end{align} by \cite[Theorem~3.57]{MR2191744}, which allows us to deduce from \eqref{eq_sub_eq02} and \eqref{eqLemma26} that \begin{equation*} {\rm co}\big(\Prox{\psi}{\lambda} (x)\big) = {\rm co}\big( \Prox{\psi}{\lambda} (x) \backslash \{v\}\big). \end{equation*} This verifies the inclusion $v \in \mbox{\rm co}\,\left( \Prox{\psi}{\lambda} (x)\backslash \{v\}\right)$ as claimed.
Observe finally that the function $x\mapsto \Asp{\lambda}{\psi}_{\lambda }( x - \lambda \nabla f(x))$ is the composition of the convex function $ \Asp{\lambda}{\psi}$ and the $\mathcal{C}^{1,1}$ mapping $x\mapsto x - \lambda \nabla f(x)$, which ensures by \cite[Proposition~2.3]{MR2069350} its prox-regularity at any point $x\in \mathbb{R}^n$. \end{proof}\vspace*{-0.05in}
The following remark discusses a useful representation of the basic subdifferential of the function $-\Asp{\lambda}{\psi}$ and other functions of this type.
\begin{remark}\label{asp-rem}
It is worth mentioning that the subdifferential $\partial (-\Asp{\lambda}{\psi})(x)$ can be expressed via the set $D:=\{x\in\mathbb{R}^n\;|\;\Asp{\lambda}{\psi}\;\mbox{ is differentiable at }\;x\}$ as follows: \begin{equation}\label{asp-rem1}
\partial (-\Asp{\lambda}{\psi})(x) =\big\{ v\in\mathbb{R}^n\;\big|\;\text{ there exists } x_k \overset{D}{\to} x \text{ and } \nabla \Asp{\lambda}{\psi} (x_k) \to -v\big\}. \end{equation} We refer to \cite[Theorem~10.31]{MR1491362} for more details. Note that we do not need to take the convex hull on the right-hand side of \eqref{asp-rem1} as in the case of the generalized gradient of locally Lipschitzian functions. \end{remark}
Finally, recall the definitions of the convergence rates used in the paper.\vspace*{-0.05in}
\begin{definition}\label{def:rates} Let $\{x_k\}$ be a sequence in $\mathbb{R}^n$ converging to $\bar{x}$ as $k \rightarrow \infty$. The convergence rate is said to be:
{\bf(i)} \emph{R-linear} if there exist $\mu \in(0,1), c>0$, and $k_0 \in \mathbb{N}$ such that $$
\left\|x_k-\bar{x}\right\| \leq c \mu^k\;\text { for all }\;k \geq k_0. $$
{\bf(ii)} \emph{Q-linear} if there exists $\mu\in(0,1)$ such that $$
\limsup_{k\to\infty}\frac{\left\|x_{k+1}-\bar{x}\right\|}{\left\|x_k-\bar{x}\right\|}=\mu. $$
{\bf(iii)} \emph{Q-superlinear} if it is Q-linear for all $\mu\in (0,1)$, i.e., if $$
\lim_{k\to\infty}\frac{\left\|x_{k+1}-\bar{x}\right\|}{\left\|x_k-\bar{x}\right\|}=0. $$
{\bf(iv)} \emph{Q-quadratic} if we have $$
\limsup_{k\to\infty}\frac{\left\|x_{k+1}-\bar{x}\right\|}{\left\|x_k-\bar{x}\right\|^2}<\infty. $$ \end{definition}\vspace*{-0.2in}
\section{Regularized Coderivative-Based Damped Semi-Newton Method in Nonsmooth Difference Programming}\label{sec:3}\vspace*{-0.05in}
The goal of this section is to justify the well-posedness and good performance of the novel algorithm RCSN under appropriate and fairly general assumptions. In the following remark, we discuss the difference between the choice of subgradients and hence of directions in RCSN and DC algorithms.\vspace*{0.05in}
Our main RCSN algorithm to find stationary points of nonsmooth problems \eqref{EQ01} of difference programming is labeled below as Algorithm~\ref{alg:1}.
\begin{algorithm}[h!] \begin{algorithmic}[1] \Require{$x_0 \in \mathbb{R}^n$, $\beta \in (0,1)$, $\zeta>0$, $t_{\min}>0 $, $\rho_{\max}>0$ and $\sigma\in(0,1)$.} \For{$k=0,1,\ldots$} \State Take $w_k\in \partial \varphi (x_k)$. If $ w_k=0$, STOP and return $x_k$. \State Choose $\rho_k\in[0,\rho_{\max}]$ and $d_k \in \mathbb{R}^n\backslash \{ 0\}$ such that \begin{align}
-w_k\in \partial^2 g(x_k)(d_k)+\rho_kd_k\quad\text{and}\quad \langle w_k,d_k\rangle\leq -\zeta\|d_k\|^2. \label{EQALG01} \end{align} \State Choose any $\overline{\tau}_k\geq t_{\min}$. Set $\overline{\tau}_k:=\tau_k$. \While{$\varphi(x_k + \tau_k d_k) > \varphi(x_k) +\sigma \tau_k \langle w_k , d_k\rangle $} \State $\tau_k = \beta \tau_k$. \EndWhile \State Set $x_{k+1}:=x_k + \tau_kd_k$. \label{step5} \EndFor \end{algorithmic} \caption{Regularized coderivative-based damped semi-Newton algorithm for nonsmooth difference programming}\label{alg:1} \end{algorithm}\vspace*{-0.05in}
\begin{remark}\label{rem:subgr} Observe that Step~2 of Algorithm~\ref{alg:1} selects $w_k\in\partial\varphi(x_k)=\nabla g(x_k)+\partial(-h)(x_k)$, which is equivalent to choosing $v_k:=w_k-\nabla g(x_k)$ in the basic subdifferential of $-h$ at $x_k$. Under our assumptions, the set $\partial(-h)(x_k)$ can be {\em considerably smaller} than $\partial h(x_k)$; see the proof of Proposition~\ref{Lemma:Dire01} and also Remark~\ref{asp-rem} above. Therefore, Step~2 differs from those in DC algorithms, which choose subgradients in $\partial h(x_k)$. The purpose of our development is to find a {\em stationary point} instead of a (classical) critical point for problem~\eqref{EQ01}. In some applications, Algorithm~\ref{alg:1} would not be implementable if the user only has access to subgradients contained in $\partial h(x_k)$ instead of $\partial(-h)(x_k)$. In such cases, a natural alternative to Algorithm~\ref{alg:1} would be a scheme replacing $w_k\in\partial\varphi(x_k)$ in Step~2 by $w_k:=\nabla g(x_k)+v_k$ with $v_k\in\partial h(x_k)$. Under the setting of our convergence results, the modified algorithm would find a critical point for problem~\eqref{EQ01}, which is not guaranteed to be stationary. \end{remark}\vspace*{-0.1in}
The above discussions are illustrated by the following example.\vspace*{-0.1in}
\begin{example}
Consider problem~\eqref{EQ01} with $g(x):=\frac{1}{2}x^2$ and $h(x):= |x|$. If an algorithm similar to Algorithm~\ref{alg:1} was run by using $x_0 =0$ as the initial point but choosing $w_0=\nabla g(x_0)+v_0$ with $v_0= 0 \in\partial h(0)$ (instead of $w_0\in\partial\varphi(x_0)$), it would stop at the first iteration and return $x=0$, which is a critical point, but not a stationary one. On the other hand, for any $w_0\in\partial \varphi(0)=\{-1,1\}$ we get $w_0\neq 0$, and so Algorithm~\ref{alg:1} will continue iterating until it converges to one of the two stationary points $-1/2$ and $1/2$, which is guaranteed by our main convergence result; see Theorem~\ref{The01} below. \end{example}\vspace*{-0.1in}
The next lemma shows that Algorithm~\ref{alg:1} is well-defined by proving the existence of a direction $d_k$ satisfying \eqref{EQALG01} in Step~3 for sufficiently large regularization parameters $\rho_k$.\vspace*{-0.1in}
\begin{lemma}\label{lemma1} Let $\varphi: \mathbb{R}^n \to \mathbb{R}$ be the objective function in problem~\eqref{EQ01} with $g\in\mathcal{C}^{1,1}$ and $h$ being locally Lipschitz around $\bar{x}$ and prox-regular at this point. Further, assume that $\partial^2 g(\bar{x})$ is $\xi$-lower-definite for some $\xi\in\mathbb{R}$ and consider a nonzero subgradient $w\in \partial \varphi(\bar{x})$. Then for any $\zeta>0$ and any $\rho\geq\zeta-\xi$, there exists a nonzero direction $d\in \mathbb{R}^n$ satisfying the inclusion \begin{align}\label{Eq002} -w \in\partial^2 g(\bar{x})(d)+\rho d. \end{align}\vspace*{-0.05in} Moreover, any nonzero direction from \eqref{Eq002} obeys the conditions:\\[1ex]
{\bf(i)}\label{lemma1b} $\varphi'(\bar{x}; d) \leq \langle w, d\rangle \leq -\zeta\| d\|^2$.\\[1ex] {\bf(ii)} \label{lemma1c} Whenever $\sigma \in (0,1)$, there exists $\eta >0$ such that \begin{align*}
\varphi(\bar{x} + \tau d ) < \varphi(\bar{x}) + \sigma \tau \langle w, d\rangle \leq \varphi(\bar{x})- \sigma \zeta \tau \|d\|^2\;\mbox{ when }\;\tau \in (0,\eta). \end{align*} \end{lemma} \begin{proof}
Consider the function $\psi(x):=g(x)+\langle w-\nabla g(\bar{x}),x\rangle+\frac{\rho}{2}\|x\|^2$ for which we clearly have that $\partial^2 \psi(\bar{x}) =\partial^2 g(\bar{x})+\rho I$, where $I$ denotes the identity mapping. This shows by Remark~\ref{rem:definite} that $\partial^2 \psi(\bar{x})$ is $(\xi+\rho)$-lower-definite, and thus it is $\zeta$-lower-definite as well. Since $\nabla \psi(\bar{x})=w\neq 0$ and $\zeta>0$, it follows from \cite[Proposition~3.1]{2021arXiv210902093D} (which requires $\psi$ to be $\mathcal{C}^{1,1}$ on $\mathbb{R}^n$, but actually only $\mathcal{C}^{1,1}$ around $\bar{x}$ is needed) that there exists a nonzero direction $d$ such that $-\nabla \psi(\bar{x})\in \partial^2 \psi (\bar{x})(d)$. This readily verifies \eqref{Eq002}, which yields in turn the second inequality in (i) due to Definition~\ref{def:lower-def}. On the other hand, we have by Proposition~\ref{Lemma:Dire01} the following:
\begin{align}\label{lemma1:INQ001}
\begin{aligned}
\varphi'(\bar{x}; d)=\lim\limits_{t \to 0^+}\frac{ g(\bar{x} +td) -g(\bar{x}) }{ t }+\limsup\limits_{t \to 0^+}\frac{ -h(\bar{x} +td) +h(\bar{x}) }{t } \\
= \langle \nabla g(\bar{x}), d \rangle + \inf\big\{\langle w, d\rangle\;\big|\;w\in -\partial h(\bar x)\big\}\leq \langle \nabla g(\bar{x}) + v, d\rangle \leq- \zeta \|d\|^2,
\end{aligned}
\end{align} where in the last estimate is a consequence of the second inequality in (i).
Finally, assertion (ii) follows directly from \eqref{lemma1:INQ001} and the definition of directional derivatives \eqref{UDD}. \end{proof}\vspace*{-0.2in}
\begin{remark} Under the $\xi$-lower-definiteness of $\partial^2 g(x_k)$, Lemma~\ref{lemma1} guarantees the existence of a direction $d_k$ satisfying both conditions in~\eqref{EQALG01} for all $\rho_k\geq \zeta-\xi$. {\em When $\xi$ is unknown}, it is still possible to implement Step~3 of the algorithm as follows. Choose first any initial value of $\rho\geq 0$, then compute a direction satisfying the inclusion in \eqref{EQALG01} and continue with Step~4 if the descent condition in \eqref{EQALG01} holds. Otherwise, increase the value of $\rho$ and repeat the process until the descent condition is satisfied. \end{remark}\vspace*{-0.05in}
The next example demonstrates that the {\em prox-regularity} of $h$ is {\em not a superfluous assumption} in Lemma~\ref{lemma1}. Namely, without it the direction $d$ used in Step~3 of Algorithm~\ref{alg:1} can even be an {\em ascent direction}.\vspace*{-0.03in}
\begin{example}\label{ex:failure} Consider the {\em least squares problem} given by $$
\min_{x\in\mathbb{R}^2} \frac{1}{2}(Ax-b)^2+\| x\|_1 - \|x\|_2,\quad x\in\mathbb{R}^2, $$
with $A:=[1,0]$ and $b:=1$. Denote $g(x):=\frac{1}{2}\|Ax-b\|^2$ and $h(x):=\| x\|_2 - \|x\|_1$. If we pick $\bar{x}:=(1,0)^T$, the function $h$ is not prox-regular at $\bar{x}$ because it is not lower regular at $\bar{x}$; see Proposition~\ref{Lemma:Dire01}. Indeed, $\widehat\partial h(\bar{x})=\emptyset$, while $$
\partial h(\bar{x})=\frac{\bar{x}}{\|\bar{x}\|}+\partial(-\|\cdot\|_1)(\bar{x})=\left\{\begin{pmatrix} 0\\ -1 \end{pmatrix},\begin{pmatrix} 0\\ 1 \end{pmatrix}\right\}. $$ Therefore, although $\nabla^2 g(\bar{x})=A^TA$ is $\lambda_{\min}(A^TA)$-lower-definite, the assumptions of Lemma~\ref{lemma1} are not satisfied. Due to the representation $$
\partial (-h)(\bar{x})=-\frac{\bar{x}}{\|\bar{x}\|}+\partial\|\cdot\|_1(\bar{x})=\left\{\begin{pmatrix} 0\\ v \end{pmatrix}\;\Bigg|\;v\in [-1,1]\right\}, $$ the choice of $v:=(0, 1)^T\in\partial (-h)(\bar{x})$ yields $w:=\nabla g(\bar{x})+v = (0, 1)^T\in\partial\varphi(\bar{x})$. For any $\rho>0$, inclusion \eqref{Eq002} gives us $d = (0,- 1/\rho)^T$. This is an ascent direction for the objective function $\varphi(x)=g(x)-h(x)$ at $\bar{x}$ due to $$ \varphi(\bar{x}+\tau d)=1+\frac{\tau}{\rho}-\sqrt{1+(\tau/\rho)^2}>\varphi(\bar{x})=0\;\mbox{ for all }\;\tau>0, $$ which illustrates that the prox-regularity is an essential assumption in Lemma~\ref{lemma1}. \end{example}\vspace*{-0.07in}
Algorithm~\ref{alg:1} either stops at a stationary point, or produces an infinite sequence of iterates. The convergence properties of the iterative sequence of our algorithm are obtained below in the main theorem of this section. Prior to the theorem, we derive yet another lemma, which establishes the following {\em descent property} for the difference of a $\mathcal{C}^{1,1}$ function and a prox-regular one.\vspace*{-0.05in}
\begin{lemma}\label{Lemma:01}
Let $\varphi(x) = g(x) -h(x)$, where $g$ is of class $\mathcal{C}^{1,1}$ around $\bar{x}$, and where $h$ is continuous around $\bar{x}$ and prox-regular at this point. Then for every $\bar{v} \in \partial h(\bar{x})$, there exist positive numbers $\epsilon$ and $r$ such that \begin{align*}
\varphi(y) \leq \varphi(x) + \langle \nabla g(x) - v , y-x \rangle +r \| y-x\|^2 \end{align*} whenever $x, y \in \mathbb{B}_{\epsilon}(\bar{x})$ and $v\in \partial h(x) \cap \mathbb{B}_{\epsilon}(\bar{v}) $. \end{lemma}\vspace*{-0.05in} \begin{proof} Pick any $\bar{v} \in \partial h(\bar{x})$ and deduce from the imposed prox-regularity and continuity of $h$ that there exist $\epsilon_1>0$ and $r_1>0$ such that \begin{align}\label{Lemma:01:Eq01}
-h(y) \leq -h(x) + \langle - v , y - x \rangle + r_1 \| y-x\|^2\;\mbox{ for all }\;x,y \in \mathbb{B}_{\epsilon_1}(\bar{x})
\end{align} and all $v\in \partial h(x)\cap \mathbb{B}_{\epsilon_1}(\bar{v}) $. It follows from the $\mathcal{C}^{1,1}$ property of $g$ by \cite[Lemma~A.11]{MR3289054} that there exist positive numbers $r_2$ and $\epsilon_2$ such that \begin{align}\label{Lemma:01:Eq02}
g(y) \leq g(x) + \langle \nabla g(x) , y-x \rangle + r_2 \| y- x\|^2\;\mbox{ for all }\;\mathbb{B}_{\varepsilon_2}.
\end{align}
Summing up the inequalities in \eqref{Lemma:01:Eq01} and \eqref{Lemma:01:Eq02} and defining $r:= r_1 +r_2$ and $\epsilon := \min\{ \epsilon_1,\epsilon_2\}$, we get that
\begin{align*}
g(y)- h(y) \leq g(x) - h(x) + \langle \nabla g(x)-v , y-x \rangle + r\| y-x\|^2
\end{align*} for all $x,y \in \mathbb{B}_\epsilon(\bar{x})$ and all $v\in \partial h(x) \cap \mathbb{B}_\epsilon(\bar{v})$. This completes the proof.
\end{proof}\vspace*{-0.1in}
Now we are ready to establish the aforementioned theorem about the performance of Algorithm~\ref{alg:1}.\vspace*{-0.05in}
\begin{theorem}\label{The01}
Let $\varphi: \mathbb{R}^n \to \overline{\mathbb{R}}$ be the objective function of problem~\eqref{EQ01} given by $\varphi = g-h$ with $\inf \varphi >-\infty$. Pick an initial point $x_0 \in \mathbb{R}^n$ and suppose that the sublevel set $\Omega:=\{x \in\mathbb{R}^n\;|\;\varphi(x) \leq \varphi(x_0)\}$ is closed. Assume also that:\\[0.5ex] {\bf(a)} \label{Theo01ass:a} The function $g$ is $\mathcal{C}^{1,1}$ around every $x\in\Omega$ and the second-order subdifferential $\partial^2 g(x)$ is $\xi$-lower-definite for all $x\in \Omega$ with some $\xi\in\mathbb{R}$.\\[1ex] {\bf(b)} The function $h$ is locally Lipschitzian and prox-regular on $\Omega$.\\[0.5ex] Then Algorithm~{\rm\ref{alg:1}} either stops at a stationary point, or produces sequences $\{x_k\} \subseteq \Omega$, $\{\varphi(x_k)\}$, $\{w_k\}$, $\{d_k\}$, and $\{\tau_k\}$ such that:\\[0.5ex] {\bf(i)} \label{The01a} The sequence $\{\varphi(x_k)\}$ monotonically decreases and converges.\\[0.5ex] {\bf(ii)}\label{The01b} If $\{x_{k_j}\}$ as $j\in \mathbb{N}$ is any bounded subsequence of $\{x_k\}$, then $\displaystyle\inf_{j\in \mathbb{N}}\tau_{k_j}>0$, \begin{align*}
\sum\limits_{j \in \mathbb{N}}\| d_{k_j}\|^2 < \infty,\; \sum\limits_{j\in \mathbb{N} } \| x_{k_j +1} - x_{k_j}\|^2< \infty,\;\text{ and }\;\sum\limits_{j\in \mathbb{N} } \|w_{k_j} \|^2<\infty. \end{align*} In particular, the boundedness of the entire sequence $\{x_k\}$ ensures that the set of accumulation points of $\{x_k\}$ is a nonempty, closed, and connected.\\[0.5ex] {\bf(iii)}\label{The01c} If $x_{k_j} \to \bar{x}$ as $j\to\infty$, then $\bar{x}$ is a stationary point of problem \eqref{EQ01} with the property $\varphi(\bar{x}) =\displaystyle\inf_{k\in \mathbb{N}} \varphi (x_k)$.\\[0.5ex] {\bf(iv)}\label{The01d} If $\{x_k\}$ has an isolated accumulation point $\bar{x}$, then the entire sequence $\{x_k\}$ converges to $\bar{x}$ as $k\to\infty$, where $\bar x$ is a stationary point of \eqref{EQ01}. \end{theorem}\vspace*{-0.05in} \begin{proof} If Algorithm~\ref{alg:1} stops after a finite number of iterations, then it clearly returns a stationary point. Otherwise, it produces an infinite sequence $\{x_k\}$. By Step~5 of Algorithm~\ref{alg:1} and Lemma~\ref{lemma1}, we have that $\inf \varphi\le\varphi(x_{k+1}) < \varphi(x_k) $ for all $k \in \mathbb{N}$, which proves assertion (i) and also shows that $\{x_k\} \subseteq \Omega$.
To proceed, suppose that $\{x_k\}$ has a bounded subsequence $\{x_{k_j}\}$ (otherwise there is nothing to prove) and split the rest of the proof into the {\em five claims}.\vspace*{0.03in}
\noindent\textbf{Claim~1:} \emph{The sequence $\{\tau_{k_j}\}$, associated with $\{x_{k_j}\}$ as $j\in\mathbb{N}$ and produced by Algorithm~{\rm\ref{alg:1}}, is bounded from below.}\\ Indeed, otherwise consider a subsequence $\{\tau_{\nu_i}\}$ of $\{\tau_{k_j}\}$ such that $\tau_{ \nu_i} \to 0^+$ as $i\to \infty$. Since $\{x_{k_j}\}$ is bounded, we can assume that $\{x_{\nu_i}\} $ converges to some point $\bar{x}$. By Lemma~\ref{lemma1}, we have that \begin{align}\label{EQ002}
-\langle w_{\nu_i}, d_{\nu_i} \rangle \geq \zeta \|d_{\nu_i}\|^2\;\mbox{ for all }\;i\in\mathbb{N}, \end{align} which yields by the Cauchy--Schwarz inequality the estimate \begin{align}\label{EQ002bis}
\|w_{\nu_i}\|\geq \zeta \| d_{\nu_i}\|,\quad i\in\mathbb{N}. \end{align} Since $\varphi$ is locally Lipschitzian and $w_{\nu_i} \in \partial\varphi (x_{{\nu_i}})$, we suppose without loss of generality that $w_{\nu_i}$ converges to some $\bar{w} \in \partial\varphi(\bar{x}) \subseteq \nabla g(\bar{x})- \partial h(\bar{x})$ as $i\to\infty$. It follows from \eqref{EQ002bis} that $\{d_{\nu_i}\}$ is bounded, and therefore $d_{\nu_i} \to\bar{d}$ along a subsequence. Since $\tau_{ \nu_i } \to 0^+$, we can assume that $\tau_{ \nu_i }<t_{\min}$ for all $i\in\mathbb{N}$, and hence Step~5 of Algorithm~\ref{alg:1} ensures the inequality \begin{align}\label{EQ003} \varphi(x_{\nu_i} + \beta^{-1}\tau_{\nu_i} d_{\nu_i}) > \varphi(x_{\nu_i}) +\sigma \beta^{-1}\tau_{\nu_i} \langle w_{\nu_i},d_{\nu_i}\rangle,\quad i\in\mathbb{N}. \end{align} Lemma~\ref{Lemma:01} gives us a constant $r>0$ such that \begin{align}\label{EQ004}
\varphi(x_{\nu_i} + \beta^{-1}\tau_{\nu_i} d_{\nu_i})\le\varphi(x_{\nu_i}) +\beta^{-1}\tau_{\nu_i} \langle w_{\nu_i} , d_{\nu_i}\rangle+r\beta^{-2}\tau_{\nu_i}^2\|d_{\nu_i}\|^2 \end{align} for all $i$ sufficiently large. Combining \eqref{EQ003},~\eqref{EQ004}, and \eqref{EQ002} tells us that \begin{equation*} \begin{array}{ll} \sigma \beta^{-1}\tau_{\nu_i} \langle w_{\nu_i} , d_{\nu_i}\rangle< \varphi(x_{\nu_i} + \beta^{-1}\tau_{\nu_i} d_{\nu_i}) - \varphi(x_{\nu_i})\\
\leq \beta^{-1}\tau_{\nu_i} \langle w_{\nu_i} , d_{\nu_i}\rangle+r\beta^{-2}\tau_{\nu_i}^2\|d_{\nu_i}\|^2\leq \beta^{-1}\tau_{\nu_i}\left(1-\displaystyle\frac{r}{\zeta\beta}\tau_{\nu_i}\right)\langle w_{\nu_i}, d_{\nu_i}\rangle \end{array} \end{equation*} for large $i$. Since $\langle w_{\nu_i} ,d_{\nu_i}\rangle<0$ by \eqref{EQ002}, we get that $\sigma >1 - \frac{r}{ \zeta\beta } \tau_{ \nu_i }$ for such $i$, which contradicts the choice of $\sigma \in (0,1)$ and thus verifies this claim.\vspace*{0.03in}
\noindent\textbf{Claim~2:} \emph{We have the series convergence $\sum_{j \in \mathbb{N}} \| d_{k_j}\|^2 < \infty $, $\sum_{j\in \mathbb{N} } \| x_{k_j +1} - x_{k_j}\|^2 < \infty$, and $\sum_{j\in \mathbb{N} } \| w_{k_j}\|^2 < \infty$.}\\ To justify this, deduce from Step~5 of Algorithm~\ref{alg:1} and Lemma~\ref{lemma1} that \begin{align*}
\sum\limits_{k\in \mathbb{N}} \zeta\tau_k \| d_k\|^2 \leq \frac{1}{\sigma } \Big( \varphi(x_0) - \inf_{k\in \mathbb{N}} \varphi(x_k)\Big). \end{align*}
It follows from Claim~1 that $\zeta\tau_{k_j} >\gamma >0$ for all $j\in \mathbb{N}$, which yields $\sum_{j\in \mathbb{N}}\| d_{k_j}\|^2 <\infty$. On the other hand, we have that $\| x_{k_j +1} - x_{k_j}\|= \tau_{k_j}\| d_{k_j}\|$, and again Claim~1 ensures that $\sum_{j\in \mathbb{N} } \| x_{k_j +1} - x_{k_j}\|^2 < \infty$. To proceed further, let $l_2 := \sup\{ \| d_{k_j}\|\;|\;\in \mathbb{N} \}$ and use the Lipschitz continuity of $\nabla g$ on the compact set ${\rm cl}\{x_{k_j}\;|\;{j\in \mathbb{N}}\} \subseteq \Omega$. Employing the subdifferential condition from \cite[Theorem~4.15]{MR3823783} together with the coderivative scalarization in \eqref{coder:sub}, we get by the standard compactness argument the existence of $l_3>0$ such that \begin{align*}
w \in \partial \langle d, \nabla g\rangle (x_{k_j})= \partial^2 g(x_{k_j})(d) \Longrightarrow \| w\| \leq l_3 \end{align*} for all $j\in\mathbb{N}$ and all $d\in \mathbb{B}_{l_2}(0)$. Therefore, it follows from the inclusion $-w_{k_j} \in \partial^2 g(x_{k_j})(d_{k_j}) +\rho_{k_j}d_{k_j}$ that we have \begin{align}\label{LipGrad}
\|w_{k_j} +\rho_{k_j}d_{k_j}\| \leq l_3 \| d_{k_j}\|\;\text{ for all large }\;j \in \mathbb{N}. \end{align}
Using finally the triangle inequality and the estimate $\rho_k\leq \rho_{\max}$ leads us to the series convergence $\sum_{j\in \mathbb{N} } \| w_{k_j} \|^2 < \infty$ as stated in Claim~2.\vspace*{0.03in}
\noindent\textbf{Claim~3:} \emph{If the sequence $\{x_k\}$ is bounded, then the set of its accumulation points is nonempty, closed and connected.}\\
Applying Claim~2 to the sequence $\{x_k\}$, we have the \emph{Ostrowski condition} $\lim_{k \to \infty }\| x_{k +1} - x_{k}\| = 0$. Then, the conclusion follows from \cite[Theorem~28.1]{Ostrowski1966}.
\noindent\textbf{Claim~4:} \emph{If $x_{k_j} \to \bar{x}$ as $j\to\infty$, then $\bar{x}$ is a stationary point of \eqref{EQ01} being such that $\varphi(\bar{x}) = \inf_{k\in \mathbb{N}} \varphi (x_k)$.} \\ By Claim 2, we have that the sequence $w_{k_j} \in \partial \varphi(x_{k_j})$ with $w_{k_j} \to 0$ as $j\to\infty$. The closedness of the basic subgradient set ensures that $0 \in \partial \varphi (\bar{x})$. The second assertion of the claim follows from the continuity of $\varphi$ at $\bar{x} \in \Omega$.
\noindent\textbf{Claim~5:} \emph{If $\{x_k\}$ has an isolated accumulation point~$\bar{x}$, then the entire sequence of $x_k$ converges to $\bar{x}$ as $k\to\infty$, and $\bar{x}$ is a stationary point of \eqref{EQ01}.} Indeed, consider any subsequence $x_{k_j} \to \bar{x}$. By Claim~4, $\bar{x}$ is a stationary point of \eqref{EQ01}, and it follows from Claim~2 that
$\lim_{j\to \infty} \| x_{k_j +1}-x_{k_j}\|=0$. Then we deduce from by \cite[Proposition~8.3.10]{MR1955649} that $x_k \to \bar{x}$ as $k\to\infty$, which completes the proof of theorem. \end{proof}\vspace*{-0.2in}
\begin{remark}\label{rem:theorem} Regarding Theorem~\ref{The01}, observe the following:
(i) If $h=0$, $g$ is of class $\mathcal{C}^{1,1}$, and $\xi>0$, then the results of Theorem~\ref{The01} can be found in \cite{2021arXiv210902093D}.
(ii) If $\xi\geq 0$, we can choose the regularization parameter $\rho_k:=c\|w_k\|$ and (a varying) $\zeta:=c\|w_k\|$ in~\eqref{EQALG01} for some $c>0$ to verify that assertions (i) and (iii) of Theorem~\ref{The01} still hold. Indeed, if $\{x_{k_j}\}$ converges to some $\bar{x}$, then $\{w_{k_j}\}$ is bounded by the Lipschitz continuity of $\varphi$. Hence the sequence $\{w_{k_j}\}$ converges to $0$. Otherwise, there exists $M>0$ and a subsequence of $\{w_{k_j}\}$ whose norms are bounded from below by $M$. Using the same argumentation as in the proof of Theorem~\ref{The01} with $\zeta=c M$, we arrive at the contradiction with $0$ being an accumulation point of of $\{w_{k_j}\}$. \end{remark}\vspace*{-0.05in}
When the objective function $\varphi$ is coercive and its stationary points are isolated, Algorithm~\ref{alg:1} converges to a stationary point because Theorem~\ref{The01}(iii) ensures that the set of accumulation points is connected. This property enables us to prove the convergence in some settings when even there exist nonisolated accumulation points; see the two examples below.\vspace*{-0.05in}
\begin{example} Consider the function $\varphi: \mathbb{R} \to \mathbb{R}$ given by \begin{align*} \varphi(x) & :=\int_0^x t^4 \sin\left(\frac{\pi}{t}\right) dt. \end{align*}
This function is clearly $\mathcal{C}^2$-smooth and coercive. For any starting point $x_0$, the level set $\Omega =\{ x\;|\;\varphi(x) \leq \varphi(x_0) \}$ is bounded, and hence there exists a number $\xi\in \mathbb{R}$ such that the functions $g(x) :=\varphi(x)$ and $h(x):=0$ satisfy the assumptions of Theorem~\ref{The01}. Observe furthermore that $\varphi$ is a DC function because it is $\mathcal{C}^2$-smooth; see, e.g., \cite{Oliveira_2020,hiriart}. However, it is not possible to write its DC decomposition with $g(x) = \varphi(x) + ax^2$ and $h(x)=ax^2$ for $a>0$, since there exists no scalar $a>0$ such that the function $g(x) = \varphi(x) + ax^2$ is convex on the entire real line.
It is easy to see that the stationary points of $\varphi$ are described by $S:=\left\{ \frac{1}{n}\;\big|\;n \in\mathbb{Z}\backslash\{ 0\} \right\}\cup\{ 0\}$. Moreover, if Algorithm~\ref{alg:1} generates an iterative sequence $\{x_k\}$ starting from $x_0$, then the accumulation points form by Theorem~\ref{The01}(ii) a nonempty, closed, and connected set $A \subseteq S$ . If $A=\{ 0\}$, the sequence $\{x_k\}$ converges to $\bar{x}=0$. If $A$ contains any point of the form $\bar{x}=\frac{1}{n}$, then it is an isolated point, and Theorem~\ref{The01}(iv) tells us that the entire sequence $\{x_k\}$ converges to that point, and consequently we have $A=\{\bar{x}\}$. \end{example}
\begin{example}\label{example3.10} Consider the function $\varphi: \mathbb{R}^n \to \mathbb{R}$ given by \begin{align*} \varphi(x):=\sum_{i=1}^n \varphi_i(x_i),\; \text{ where }\;\varphi_i(x_i):= g_i(x_i) - h_i(x_i) \\
\text{ with }\;g_i(x_i):= \frac{1}{2}x_i^2\; \text{ and }\;h_i(x_i):= |x_i| +\big| 1-|x_i|\,\big|. \end{align*} We can easily check that the function $\varphi$ is coercive and satisfies the assumptions of Theorem~\ref{The01} with $g(x):=\sum_{i=1}^n g_i(x_i)$, $h(x):= \sum_{i=1}^n h_i(x_i)$, and $\xi=1$. For this function, the points in the set $\{-2, -1,0,1,2\}^n$ are critical but not stationary. Moreover, the points in the set $\{-2,0,2 \}^n$ give the global minima to the objective function $\varphi$. Therefore, Algorithm~\ref{alg:1} leads us to global minimizers of $\varphi$ starting from any initial point.
\begin{figure}
\caption{ Plot of the function $\varphi_i $ in Example \ref{example3.10}}
\label{fig:screenshot001}
\end{figure} \end{example}\vspace*{-0.1in}
The following theorem establishes convergence rates of the iterative sequences in Algorithm~\ref{alg:1} under some additional assumptions.\vspace*{-0.05in}
\begin{theorem}\label{corSMR} Suppose in addition to the assumptions of Theorem~{\rm\ref{The01}}, that $\{x_k\}$ has an accumulation point $\bar{x}$ such that the subgradient mapping $\partial \varphi$ is strongly metrically subregular at $(\bar{x},0)$. Then the entire sequence $\{x_k\}$ converges to $\bar{x}$ with the Q-linear convergence rate for $\{\varphi(x_k)\}$ and the R-linear convergence rate for $\{x_k\}$ and $\{w_k\}$. If furthermore, $\xi>0$, $0<\zeta\leq\xi$, $\rho_k\to 0$, $\sigma \in(0,\frac{1}{2})$, $t_{\min}=1$, $g$ is semismoothly differentiable at $\bar{x}$, $h$ is of class $\mathcal{C}^{1,1}$ around $\bar{x}$, and $\mbox{\rm clm}\, \nabla h(\bar{x})=0$, then the rate of convergence of all the sequences above is at least Q-superlinear. \end{theorem}\vspace*{-0.05in} \begin{proof} We split the proof of the theorem into the following two claims.\vspace*{0.03in}
\noindent\textbf{Claim~1:} \emph{The rate of convergence of $\{\varphi(x_k)\}$ is at least Q-linear, while both sequences $\{x_k\}$ and $\{w_k\}$ converge at least R-linearly.}\\ Observe first that it follows from the imposed strong metric subregularity of $\partial\varphi$ that $\bar{x}$ is an isolated accumulation point, and so $x_k\to\bar{x}$ as $k\to\infty$ by Theorem~\ref{The01}(iii). Further, we get from \eqref{def:stron_subreg} that there exists $\kappa>0$ such that \begin{align}\label{eq:1}
\| x_k - \bar{x}\| \leq \kappa \| w_k\|\;\text{ for large }\;k \in \mathbb{N}, \end{align}
since $w_k\to 0$ as $k\to\infty$ by Theorem \ref{The01}(ii). Using \eqref{LipGrad} and the triangle inequality gives us $\ell > 0$ such that $ \| w_k\| \leq \ell \| d_k\| $ for sufficiently large $k\in\mathbb{N}$. Lemma~\ref{Lemma:01} yields then the cost function increment estimate \begin{align}\label{eq:2}
\varphi(x_k)-\varphi(\bar{x}) \le r\| x_k -\bar x\|^2\;\text{ for all large }\;k \in\mathbb{N}. \end{align}
By Step~5 of Algorithm~\ref{alg:1} and Lemma~\ref{lemma1}, we get that $\varphi(x_k)-\varphi(x_{k+1}) \geq \sigma \zeta\tau_k \| d_k \|^2 $ for large $k \in \mathbb{N}$. Remembering that $\inf_{k\in \mathbb{N}} \tau_k >0$, we deduce from Theorem \ref{The01}(ii) the existence of $\eta >0$ such that \begin{align}\label{eq:3}
\varphi(x_k)-\varphi(\bar{x}) - (\varphi(x_{k+1})- \varphi(\bar{x}) ) \geq \eta \| w_k\|^2 \end{align}
whenever $k$ large enough. Therefore, applying \cite[Lemma~7.2]{2021arXiv210902093D} to the sequences $\alpha_k := \varphi(x_k) - \varphi(\bar{x})$, $\beta_k:= \| w_k\|$, and $\gamma_k := \| x_k - \bar{x}\|$ with the positive constants $c_1:= \eta$, $c_2:= \kappa^{-1}$, and $c_3:= r$, we verify the claimed result.\vspace*{0.03in}
\noindent\textbf{Claim ~2: } \emph{Assuming that $\sigma \in (0,\frac{1}{2})$, $t_{\min}=1$, $g$ is semismoothly differentiable at $\bar{x}$, $h$ is of class $\mathcal{C}^{1,1}$ around $\bar{x}$, and $\mbox{\rm clm}\, \nabla h(\bar{x})=0$, we have that the rate of convergence for all the above sequences is at least Q-superlinear.}\\ Suppose without loss of generality that $h$ is differentiable at any $x_k\to\bar{x}$. It follows from the coderivative scalarization \eqref{coder:sub} and the basic subdifferential sum rule in \cite[Theorem~2.19]{MR3823783} valid under the imposed assumptions that \begin{align}\label{sublineal} \partial^2 g(x_k)(d_k) \subseteq \partial^2 g(x_k) (x_k + d_k - \bar{x}) + \partial^2 g(x_k) (-x_k + \bar{x}). \end{align} This yields the existence of $z_k \in \partial^2 g(x_k) (-x_k + \bar{x})+\rho_k(-x_k+\bar{x})$ such that \begin{align}\label{inclusion01} -\nabla g(x_k) +\nabla h(x_k) - z_k \in \partial^2 g(x_k) (x_k + d_k - \bar{x})+\rho_k(x_k + d_k - \bar{x}). \end{align} Moreover, the $(\xi+\rho_k)$-lower-definiteness of $\partial^2 g(x_k )+\rho_kI$ and the Cauchy--Schwarz inequality imply that
\begin{align*}
\| x_k + d_k -\bar{x}\|\leq \frac{1}{\xi+\rho_k} \| \nabla g(x_k) - \nabla h(x_k)+ z_k\|. \end{align*} Combining now the semismoothness of $\nabla g$ at $\bar{x}$ with the conditions $\nabla g(\bar{x})=\nabla h(\bar{x})$ and $\mbox{\rm clm}\, \nabla h(\bar{x})=0$ brings us to the estimates \begin{align*} \begin{array}{ll}
\| \nabla g(x_k) - \nabla h(x_k)+ z_k \|\leq \| \nabla g(x_k) -\nabla g(\bar{x})+ z_k+\rho_k(x_k-\bar{x})\|\\
+\rho_k\|x_k-\bar{x}\| + \|\nabla h(\bar{x})-\nabla h(x_k)\|= o(\| x_k - \bar{x}\|). \end{array} \end{align*}
Then we have $\| x_k + d_k -\bar{x}\|=o(\|x_k-\bar{x}\|)$ and deduce therefore from \cite[Proposition~8.3.18]{MR1955649} and Lemma~\ref{lemma1}(i) that \begin{align}\label{eqSLC01} \varphi(x_k + d_k) \leq \varphi(x_k) + \sigma \langle \nabla \varphi(x_k), d_k\rangle. \end{align} It follows from \eqref{eqSLC01} that $x_{k+1}=x_k + d_k$ if $k$ for large $k$. Applying \cite[Proposition~8.3.14]{MR1955649} yields the $Q$-superlinear convergence of $\{x_k\}$ to $\bar{x}$ as $k\to\infty$.
Finally, conditions \eqref{eq:1}--\eqref{eq:3} and the Lipschitz continuity of $\nabla \varphi$ around~$\bar{x}$ ensure the existence of $L>0$ such that \begin{align*}
\frac{\eta}{\kappa^2}\| x_k - \bar{x} \|^2 & \le\varphi(x_k) -\varphi(\bar{x}) \leq r \| x_k - \bar{x} \|^2, \\
\quad\frac{1}{\kappa}\| x_k - \bar{x} \| & \leq \| \nabla \varphi(x_k) \|\leq L\| x_k - \bar{x} \| \end{align*} for sufficiently large $k$, and therefore we get the estimates \begin{equation}\label{estimationsCOR} \begin{array}{ll}
\displaystyle\frac{\varphi(x_{k+1}) -\varphi(\bar{x}) }{\varphi(x_k) -\varphi(\bar{x})} \leq \kappa r\displaystyle\frac{ \| x_{k+1} - \bar{x} \|^2 }{ \| x_{k} -\bar{x} \|^2}, \\
\quad\;\displaystyle\frac{\| \nabla \varphi(x_{k+1}) \| }{ \| \nabla \varphi(x_{k})\|}\le\kappa L\displaystyle\frac{ \| x_{k+1} - \bar{x} \| }{ \| x_{k} -\bar{x}\|}, \end{array} \end{equation} which thus conclude the proof of the theorem. \end{proof}\vspace*{-0.25in}
\begin{remark}\label{rem:subregul} The property of {\em strong metric subregularity} of {\em subgradient mappings}, which is a central assumption of Theorem~\ref{corSMR}, has been well investigated in variational analysis, characterized via second-order growth and coderivative type conditions, and applied to optimization-related problems; see, e.g., \cite{ag,dmn,MR3823783} and the references therein. \end{remark}\vspace*{-0.05in}
The next theorem establishes the $Q$-superlinear and $Q$-quadratic convergence of the sequences generated by Algorithm~\ref{alg:1} provided that: $\xi>0$ (i.e., $\partial^2 g(x)$ is $\xi$-strongly positive-definite), $\rho_k=0$ for all $k\in \mathbb{N}$ (no regularization is used), $g$ is semismoothly differentiable at the cluster point $\bar{x}$, and the function $h$ can be expressed as the pointwise maximum of finitely many affine functions at $\bar{x}$, i.e., when there exist $(x^\ast_i, \alpha_i)_{i=1}^p \subseteq \mathbb{R}^n \times \mathbb{R}$ and $\epsilon >0$ such that \begin{align}\label{max_affine} h(x)=\max_{i=1,\ldots, p} \left\{ \langle x^\ast_i,x\rangle +\alpha_i \right\}\; \text{ for all }\;x\in \mathbb{B}_\epsilon(\bar{x}). \end{align}
\begin{theorem}\label{Cor:max_affine} In addition to the assumptions of Theorem~{\rm\ref{The01}}, suppose that $\xi>0$, $0<\zeta\leq\xi$, $\sigma \in (0,\frac{1}{2})$, $t_{\min}=1$, and $\rho_k=0$ for all $k\in\mathbb{N}$. Suppose also that the sequence $\{x_k\}$ generated by Algorithm~{\rm\ref{alg:1}} has an accumulation point $ \bar{x}$ at which $g$ is semismoothly differentiable and $h$ can be represented in form \eqref{max_affine}. Then we have the convergence $x_k\to\bar{x}$, $\varphi(x_k)\to\varphi(\bar{x})$, $w_k\to 0$, and $\nabla g(x_k)\to\nabla g(\bar{x})$ as $k\to\infty$ with at least $Q$-superlinear rate. If in addition $g$ is of class $\mathcal{C}^{2,1}$ around $\bar{x}$, then the rate of convergence is at least quadratic. \end{theorem}\vspace*{-0.05in} \begin{proof} Observe that by \eqref{max_affine} and \cite[Proposition~1.113]{MR2191744} we have the inclusion \begin{align}\label{FORMSUBD}
\partial (-h)(x)\subseteq \bigcup\big\{ -x^\ast_i\;\big|\;h(x)=\langle x^\ast_i,x\rangle +\alpha_i\big\} \end{align} for all $x$ near $\bar{x}$. The rest of the proof is split into the five claims below.\vspace*{0.03in}
\noindent\textbf{Claim~1:} \emph{The sequence $\{x_k\}$ converges to $\bar{x}$ as $k\to\infty$.}\\ Observe that $\bar{x}$ is an isolated accumulation point. Indeed, suppose on the contrary that there is a sequence $\{y_\nu\}$ of accumulation points of $\{x_k\}$ such that $y_\nu \to \bar{x}$ as $\nu\to\infty$ with $y_\nu \neq \bar{x}$ for all $\nu\in\mathbb{N}$. Since each $y_\nu$ is accumulation point of $\{x_k\}$, they are stationary points of $\varphi$. The ${\cal C}^1$-smoothness of $g$ ensures that $\nabla g(y_\nu) \to \nabla g(\bar{x})$ as $\nu\to\infty$, and so \eqref{FORMSUBD} yields $\nabla g(y_\nu)=x_{i_\nu}^\ast$ for large $\nu\in\mathbb{N}$. Since there are finitely many of $x_i^\ast$ in \eqref{max_affine}, we get that $\nabla g(y_\nu) = \nabla g(\bar{x})$ when $\nu$ is sufficiently large. Further, it follows from \cite[Theorem~5.16]{MR3823783} that the gradient mapping $\nabla g$ is strongly locally maximal monotone around $\bar{x}$, i.e., there exist positive numbers $\epsilon$ and $r$ such that \begin{align*}
\langle \nabla g(x)-\nabla g(y), x-y\geq r\| x -y\|^2\;\text{ for all } \;x,y\in\mathbb{B}_\epsilon(\bar{x}). \end{align*} Putting $x:=\bar{x}$ and $y:=y_\nu$ in the above inequality tells us that $\bar{x}= y_\nu$ for large $\nu \in \mathbb{N}$, which is a contradiction. Applying finally Theorem~\ref{The01}(iv), we complete the proof of this claim.\vspace*{0.03in}
\noindent\textbf{Claim 2:} \emph{The sequence $\{x_k\}$ converges to $\bar{x}$ as $k\to\infty$ at least $Q$-superlinearly.}\\ As $x_k\to\bar{x}$, we have by Theorem~\ref{The01}(ii) that $w_k-\nabla g(x_k)\to-\nabla g(\bar{x})$, and so it follows from \eqref{FORMSUBD} that there exists $i\in\{1,\ldots,p\}$ such that $h(\bar x ) = \langle x^\ast_i , \bar x\rangle +\alpha_i$, $h(x_k) = \langle x^\ast_i , x_k\rangle -\alpha_i$ and $w_k-\nabla g(x_k)=-\nabla g(\bar{x})=-x_i^\ast$ for all $k$ sufficiently large. Define the auxiliary function $\widehat{\varphi}:\mathbb{R}^n \to \overline{\mathbb{R}}$ by \begin{align}\label{aux:func} \widehat{\varphi} (x):= g(x) -\langle x^\ast_i , x\rangle -\alpha_i \end{align} and observe that $\widehat\varphi$ is $\mathcal{C}^{1,1}$ around $\bar{x}$ and semismoothly differentiable at this point. We have the equalities \begin{align}\label{eq_auxvarphi} \varphi (x_k) =\widehat{\varphi} (x_k),\;\varphi (\bar{x}) =\widehat{\varphi} (\bar{x}),\; \nabla \widehat{\varphi} (x_k)=w_k,\;\text{ and }\; \nabla \widehat{\varphi} (\bar{x})=0 \end{align} for large $k$. It follows from $\partial^2\widehat{\varphi} (x) = \partial^2 g(x)$ that the mapping $\partial^2\widehat{\varphi}(\bar{x})+\rho_kI$ is $(\xi+\rho_k)$-lower-definite. Using \eqref{sublineal} and \eqref{inclusion01} with the replacement of $g$ by $\widehat\varphi$ and taking \eqref{eq_auxvarphi} into account ensures the existence of $z_k \in \partial^2 \widehat{\varphi}(x_k) (-x_k + \bar{x}) +\rho_k(-x_k + \bar{x})$ satisfying the estimate
\begin{align*}
\| x_k + d_k -\bar{x}\| \leq \frac{1}{\xi+\rho_k} \| \nabla \widehat{\varphi}(x_k)- \nabla\widehat{\varphi}(\bar{x})+ z_k\|.
\end{align*}
The triangle inequality and the semismoothness of $\nabla\widehat \varphi$ at $\bar{x}$ yield \begin{align*}
\|\nabla \widehat{\varphi}(x_k)- \nabla \widehat{\varphi}(\bar{x})+ z_k\|&\leq \| \nabla \widehat{\varphi}(x_k)- \nabla \widehat{\varphi}(\bar{x})+ z_k+\rho_k(x_k-\bar{x})\|+\rho_k\|x_k-\bar{x}\|\\
&=o(\|x_k-\bar{x}\|), \end{align*}
which tells us that $\| x_k + d_k -\bar{x}\|=o(\| x_k - \bar{x}\|)$. Then it follows from\cite[Proposition~8.3.18]{MR1955649} and Lemma~\ref{lemma1}(i) above that \begin{align}\label{eqSLC} \widehat{\varphi}(x_k + d_k) \leq \widehat{\varphi}(x_k) + \sigma \langle \nabla\widehat{\varphi}(x_k),d_k\rangle \end{align} whenever $k$ is sufficiently large. Applying finally \cite[Proposition~8.3.14]{MR1955649} verifies the claimed $Q$-superlinear convergence of $\{x_k\}$ to $\bar{x}$.\vspace*{0.03in}
\noindent\textbf{Claim~3:} \emph{The gradient mapping of $\widehat \varphi$ from \eqref{aux:func} is strongly metrically regular around $(\bar{x},0)$ and hence strongly metrically subregular at this point.}\\ Using the $\xi$-lower-definiteness of $\partial^2 \widehat\varphi (\bar{x})$ and the pointbased coderivative characterization of strong local maximal monotonicity given in \cite[Theorem~5.16]{MR3823783}, we verify this property for $\nabla \widehat \varphi$ around $\bar{x}$. Then \cite[Corollary~5.15]{MR3823783} ensures that $\nabla \widehat \varphi$ is strongly metrically regular around $(\bar{x},0)$. \vspace*{0.03in}
\noindent\textbf{Claim~4:} \emph{The sequences $\{ \varphi(x_k)\}$, $\{w_k\}$, and $\{\nabla g(x_k)\}$ converge at least Q-superlinearly to $\varphi(\bar{x})$, $0$, and $\nabla g(\bar{x})$, respectively.}\\ It follows from the estimates in \eqref{estimationsCOR}, with the replacement of $\varphi$ by $\widehat\varphi$ and with taking into account that $\widehat\varphi(x_k) - \widehat\varphi(\bar x) =\varphi(x_k)-\varphi(\bar x)$ and $ \nabla \widehat \varphi(x_k) = w_k$ due to \eqref{eq_auxvarphi}, that there exist constants $\alpha_1 , \alpha_2 >0$ such that \begin{align*}
\frac{\varphi(x_{k+1}) -\varphi(\bar{x}) }{ \varphi(x_k) -\varphi(\bar{x})} &\leq \alpha_1 \frac{ \| x_{k+1} - \bar{x} \|^2 }{ \| x_{k} -\bar{x} \|^2} \\
\frac{ \|w_{k+1 }\| }{ \|w_k\|} &\leq \alpha_2 \frac{ \| x_{k+1} - \bar{x} \| }{ \| x_{k} - \bar{x} \|} \end{align*} provided that $k$ is sufficiently large. Recalling that $w_k-\nabla g(x_k)=-\nabla g(\bar{x})$ for large $k$ completes the proof of the claim. \vspace*{0.03in}
\noindent\textbf{Claim~5:} \emph{If $g$ is of class $\mathcal{C}^{2,1}$ around $\bar{x}$, then the rate of convergence of the sequences above is at least quadratic.}\\
It is easy to see that the assumed $\mathcal{C}^{2,1}$ property of $g$ yields this property of $\widehat{\varphi}$ around $\bar{x}$. Using estimate
\eqref{eqSLC}, we deduce this claim from the quadratic convergence of the classical Newton method; see, e.g., \cite[Theorem~5.18]{Aragon2019} and \cite[Theorem~2.15]{MR3289054}. This therefore completes the proof of the theorem. \end{proof}\vspace*{-0.25in}
\begin{remark}\label{rem:long} Concerning Theorem~\ref{Cor:max_affine}, observe the following:
(i) It is important to emphasize that the performance of Algorithm~\ref{alg:1} revealed in Theorem~\ref{Cor:max_affine} is mainly due to the usage of the basic subdifferential of the function $-h$ in contrast to that of $h$, which is calculated as \begin{equation}\label{h-sub}
\partial h(x)= \text{co} \left(\bigcup \left\{ x^\ast_i\;\bigg|\;h(x)=\langle x^\ast_i,x\rangle +\alpha_i \right\}\right) \end{equation} by \cite[Theorem~3.46]{MR2191744}. We can see from the proof of Theorem~\ref{Cor:max_affine} that it fails if the evaluation of $\partial(-h)(x)$ in \eqref{FORMSUBD} is replaced by the one of $\partial h(x)$ in \eqref{h-sub}.
(ii) The main assumptions of Theorem~\ref{Cor:max_affine} do not imply the smoothness of $\varphi$ at stationary points. For instance, consider the nonconvex function $\varphi: \mathbb{R}^n \to \mathbb{R}$ defined as in Example~\ref{example3.10} but letting now $h_i(x_i):= |x_i| +| 1- x_i|$.
The function $\varphi$ satisfies the assumptions of Theorem~\ref{Cor:max_affine} at any of its stationary points $\{-2,0,2\}^n$, but $\varphi$ is not differentiable at $\bar{x}=0$; see Figure~\ref{example3.10reviplot2}.
\begin{figure}
\caption{ Plot of function $\varphi_i(x)=\frac{1}{2}x^2 -|x| -| 1- x| $ in Remark~\ref{rem:long}}
\label{example3.10reviplot2}
\end{figure} (iii) The functions $\varphi$, $g$, and $h$ in Example~\ref{example3.10} satisfy the assumptions of Theorem~\ref{Cor:max_affine}. Therefore, the convergence of the sequences generated by Algorithm~\ref{alg:1} is at least quadratic. \end{remark}\vspace*{-0.33in}
\section{Convergence Rates under the Kurdyka--{\L}ojasiewicz Property}\label{sec:4}\vspace*{-0.1in}
In this section, we verify the global convergence of Algorithm~\ref{alg:1} and establish convergence rates in the general setting of Theorem~\ref{The01} without additional assumptions of Theorems~\ref{corSMR} and \ref{Cor:max_affine} while supposing instead that the cost function $\varphi$ satisfies the Kurdyka--{\L}ojasiewicz property. Recall that the \emph{Kurdyka--{\L}ojasiewicz property} holds for $\varphi$ at $ \bar{x}$ if there exist $\eta >0$ and a continuous concave function $ \psi:[0,\eta] \to [0,\infty)$ with $\psi (0)=0$ such that $\psi$ is $\mathcal{C}^1$-smooth on $(0,\eta)$ with the strictly positive derivative $\psi'$ and that \begin{align}\label{Kur-Loj} \psi'\big(\varphi(x) - \varphi(\bar{x})\big)\,{\rm dist}\big(0;\partial \varphi(x)\big)\geq 1 \end{align} for all $x\in \mathbb{B}_\eta(\bar{x})$ with $\varphi(\bar{x}) < \varphi(x) <\varphi( \bar{x} ) + \eta$, where ${\rm dist}(\cdot;\Omega)$ stands for the distance function of a set $\Omega$.
The first theorem of this section establishes the {\em global convergence} of iterative sequence generated by Algorithm~\ref{alg:1} to a {\em stationary point} of \eqref{EQ01}.\vspace*{-0.05in}
\begin{theorem}\label{Teo:Kur-Loj} In addition to the assumptions of Theorem~{\rm\ref{The01}}, suppose that the iterative sequence $\{x_k\}$ generated by Algorithm~{\rm\ref{alg:1}} has an accumulation point $\bar{x}$ at which the Kurdyka--{\L}ojasiewicz property \eqref{Kur-Loj} is satisfied. Then $\{x_k\}$ converges $\bar{x}$ as $k\to\infty$, which is a stationary point of problem~\eqref{EQ01}. \end{theorem}\vspace*{-0.05in} \begin{proof} If Algorithm~\ref{alg:1} stops after a finite number of iterations, there is nothing to prove. Due to the decreasing property of $\{\varphi(x_k)\}$ from Theorem~\ref{The01}(i), we can assume that $\varphi(x_k) > \varphi(x_{k+1}) $ for all $k \in \mathbb{N}$. Let $ \bar{x}$ be the accumulation point of $\{x_k\}$ where $\varphi$ satisfies the Kurdyka--{\L}ojasiewicz inequality \eqref{Kur-Loj}, which by Theorem \ref{The01} is a stationary point of problem~\eqref{EQ01}. Since $\varphi$ is continuous, we have that $\varphi(\bar{x} )= \inf_{k\in\mathbb{N}}\varphi(x_k)$. Taking the constant $\eta>0$ and the function $\psi$ from \eqref{Kur-Loj} and remembering that $g$ is of class $\mathcal{C}^{1,1}$ around $\bar{x}$, suppose without loss of generality that $\nabla g$ is Lipschitz continuous on $\mathbb{B}_{2\eta}( \bar{x})$ with modulus $\kappa$. Let $k_0 \in \mathbb{N}$ be such that $x_{k_0} \in \mathbb{B}_{\eta/2}(\bar{x})$ and that \begin{align}\label{iq:01Ku-Loj} \varphi (\bar{x}) < \varphi(x_{k}) < \varphi(\bar{x}) +\eta, \quad\frac{ \kappa +\rho_{\max} }{\sigma \zeta }\psi\big( \varphi (x_{k}-\varphi(\bar{x})\big) < \eta/2 \end{align}
for all $k \geq k_0$, where $\sigma \in (0,1)$, $\zeta>0$, and $\rho_{\max}>0$ are the constants of Algorithm~\ref{alg:1}. The rest of the proof is split into the following three steps.\vspace*{0.03in}
\noindent\textbf{Claim~1:} \emph{Let $k \geq k_0 $ be such that $x_k \in \mathbb{B}_\eta (\bar{x})$. Then we have the estimate} \begin{align}\label{eq:Kur-Loj}
\| x_k -x_{k+1}\| \leq \frac{ \kappa+\rho_k }{\sigma\zeta }\big( \psi( \varphi(x_{k}) - \varphi(\bar{x})\big) - \psi\big( \varphi(x_{k+1}) - \varphi(\bar{x})\big)\big). \end{align} Indeed, it follows from \eqref{coder:sub}, \eqref{EQALG01}, and \cite[Theorem~1.22]{MR3823783} that \begin{equation}\label{dist:inq} \begin{array}{ll}
{\rm dist}(0;\partial \varphi (x_k)\big) &\leq \| w_k \| \leq \| w_k +\rho_kd_k\|+\rho_k\|d_k\|\\
&\leq (\kappa+\rho_k) \| d_k \| = \displaystyle\frac{ \kappa +\rho_k}{ \tau_k } \| x_{k+1} - x_k\|. \end{array} \end{equation} Then using Step~5 of Algorithm~\ref{alg:1}, Lemma~\ref{lemma1}, the Kurdyka--{\L}ojasiewicz inequality \eqref{Kur-Loj}, the concavity of $\psi$, and estimate \eqref{dist:inq} gives us \begin{align*}
\| x_k - &x_{k+1}\|^2 = \tau^2_k \| d_k\|^2 \leq \frac{ \tau_k }{\sigma\zeta }\big( \varphi(x_k ) - \varphi(x_{k+1})\big) \\ & \leq \frac{ \tau_k }{\sigma\zeta }{\rm dist}\big(0;\partial \varphi (x_k)\big)\,\psi'\big(\varphi(x_k) - \varphi(\bar{x})\big) \big( \varphi(x_k ) - \varphi(x_{k+1})\big)\\ & \leq \frac{ \tau_k }{\sigma\zeta }{\rm dist}\big(0;\partial \varphi (x_k)\big)\big(\psi( \varphi(x_k) - \varphi(\bar{x})\big) - \psi( \varphi(x_{k+1}) - \varphi(\bar{x})\big)\big) \\
&\leq \frac{ \kappa+\rho_k }{\sigma\zeta } \| x_{k+1} - x_k\|\big( \psi\big( \varphi(x_k) - \varphi(\bar{x})\big) - \psi\big(\varphi(x_{k+1}) - \varphi(\bar{x})\big)\big),
\end{align*} which therefore verifies the claimed inequality \eqref{eq:Kur-Loj}.\vspace*{0.03in}
\noindent\textbf{Claim~2:} \emph{For every $k \geq k_0 $, we have the inclusion $x_{k} \in \mathbb{B}_{\eta} (\bar{x})$.}\\
Suppose on the contrary that there exists $k> k_0$ with $x_k \notin \mathbb{B}_\eta (\bar{x})$ and define $\bar{k}:=\min\left\{ k> k_o\;\big|\; x_k \notin \mathbb{B}_\eta (\bar{x}) \right\}$. Since for $k\in\{k_0,\ldots,\bar{k}-1\}$ the estimate in \eqref{eq:Kur-Loj} is satisfied, we get by using~\eqref{iq:01Ku-Loj} that \begin{align*}
\| x_{ \bar{k} } - \bar{x} \|& \leq \| x_{k_0} - \bar{x}\| + \sum_{k=k_0}^{\bar{k}-1}\| x_{k} - x_{k+1}\| \\
&\leq \| x_{k_0} - \bar{x}\| + \frac{\kappa+\rho_{\max}}{\sigma\zeta } \sum_{k=k_0}^{\bar{k}-1}\big( \psi\big( \varphi(x_{k}) - \varphi(\bar{x})\big) - \psi\big( \varphi(x_{k+1}) - \varphi(\bar{x}) \big)\big)\\
&\leq \| x_{k_0} - \bar{x}\| + \frac{\kappa+\rho_{\max}}{\sigma\zeta } \psi\big( \varphi(x_{k_0}) - \varphi(\bar{x})\big) \leq \eta, \end{align*} which contradicts our assumption and thus verifies this claim.\vspace*{0.03in}
\noindent\textbf{Claim~3:} \emph{We have that $\sum_{k=1}^{\infty} \| x_k - x_{k+1}\| < \infty$, and consequently the sequence $\{x_k\}$ converges to $\bar{x}$ as $k\to\infty$.}\\ It follows from Claim~1 and Claim~2 that \eqref{eq:Kur-Loj} holds for all $k \geq k_{0}$. Thus \begin{align*}
\sum_{k=1}^{\infty} \| x_k - x_{k+1}\|& \leq \sum_{k=1}^{k_0-1} \| x_k - x_{k+1}\| + \sum_{k=k_0}^{\infty} \| x_k - x_{k+1}\| \\
& \leq \sum_{k=1}^{k_0-1} \| x_k - x_{k+1}\| + \frac{\kappa+\rho_{\max}}{\sigma\zeta } \psi\big( \varphi(x_{k_0}) - \varphi(\bar{x})\big) <\infty, \end{align*} which therefore completes the proof of the theorem. \end{proof}\vspace*{-0.1in}
The next theorem establishes {\em convergence rates} for iterative sequence $\{x_k\}$ in Algorithm~\ref{alg:1} provided that the function $\psi$ in \eqref{Kur-Loj} is selected in a special way. Since the proof while using Theorem~\ref{Teo:Kur-Loj}, is similar to the corresponding one from \cite[Theorem~4.9]{MR4078808} in a different setting, it is omitted. \vspace*{-0.05in}
\begin{theorem}\label{COR:Kur-Loj}
In addition to the assumptions of Theorem~{\rm\ref{Teo:Kur-Loj}}, suppose that the Kurdyka--{\L}ojasiewicz property \eqref{Kur-Loj} holds at the accumulation point $\bar{x}$ with $\psi(t):= M t^{1-\theta}$ for some $M>0$ and $\theta\in[0,1)$. The following assertions hold:
{\bf(i)} If $\theta =0$, then the sequence $\{x_k\}$ converges in a finite number of steps.
{\bf(ii)} If $\theta\in (0,1/2]$, then the sequence $\{x_k\}$ converges at least linearly.
{\bf(iii)} If $\theta \in (1/2,1)$, then there exist $\mu >0$ and $k_0\in \mathbb{N}$ such that
\begin{align*}
\|x_ k - \bar{x}\| \leq \mu k^{-\frac{1-\theta }{ 2\theta -1 } }\;\text{ for all }\;k \geq k_0.
\end{align*} \end{theorem}\vspace*{-0.1in}
\begin{remark} Together with our main Algorithm~\ref{alg:1}, we can consider its modification with the replacement of $\partial(-h)(x_k)$ by $-\partial h(x_k)$. In this case, the most appropriate version of the Kurdyka--{\L}ojasiewicz inequality \eqref{Kur-Loj}, ensuring the fulfillment the corresponding versions of Theorem~\ref{Teo:Kur-Loj} and \ref{COR:Kur-Loj}, is the one
\begin{align*}
\psi\big(\varphi(x)-\varphi(\bar{x})\big)\,{\rm dist}\big(0;\partial^0\varphi(x)\big)\geq 1
\end{align*} expressed in terms of the {\em symmetric subdifferential} $\partial^0\varphi(x)$ from \eqref{sym}. Note that the latter is surely satisfied where the symmetric subdifferential is replaced by the {\em generalized gradient} $\overline{\partial}\varphi(x)$, which is the convex hull of $\partial^0\varphi(x)$. \end{remark}\vspace*{-0.35in}
\section{Applications to Structured Constrained Optimization}\label{sec:5}\vspace*{-0.05in}
In this section, we present implementations and specifications of our main RCSN Algorithm~\ref{alg:1} for two structured classes of optimization problems. The first class contains functions represented as sums of two nonconvex functions one of which is smooth, while the other is extended-real-valued. The second class concerns minimization of smooth functions over closed constraint sets.\vspace*{-0.25in}
\subsection{Minimization of Structured Sums}\label{subsec1}\vspace*{-0.05in}
Here we consider the following class of structured optimization problems: \begin{equation}\label{ProFBE} \min_{x\in \mathbb{R}^n}\varphi(x):=f(x)+\psi(x), \end{equation} where $f:\mathbb{R}^n\to\mathbb{R}$ is of class $\mathcal{C}^{2,1}$ with the $L_f$-Lipschitzian gradient, and where $\psi: \mathbb{R}^n \to \overline{\mathbb{R}}$ is an extended-real-valued prox-bounded function with the threshold $\lambda_\psi>0$. When both functions $f$ and $\psi$ are convex, problems of type \eqref{ProFBE} have been largely studied under the name of ``convex composite optimization" emphasizing the fact that $f$ and $\psi$ are of completely different structures. In our case, we do not impose any convexity of $f,\psi$ and prefer to label \eqref{ProFBE} as {\em minimization of structured sums} to avoid any confusions with optimization of function compositions, which are typically used in major models of variational analysis and constrained optimization; see, e.g., \cite{MR1491362}.
In contrast to the original class of {\em unconstrained} problems of {\em difference programming} \eqref{EQ01}, the structured {\em sum optimization} form \eqref{ProFBE} covers optimization problems with {\em constraints} given by $x\in\mbox{\rm dom}\,\psi$. Nevertheless, we show in what follows that the general class of problem \eqref{ProFBE} can be reduced under the assumptions imposed above to the difference form \eqref{EQ01} satisfying the conditions for the required performance of Algorithm~\ref{alg:1}.
This is done by using an extended notion of envelopes introduced by Patrinos and Bemporad in \cite{Patrinos2013}, which is now commonly referred as the \emph{forward-backward envelope}; see, e.g., \cite{MR3845278}.\vspace*{-0.1in}
\begin{definition} Given $ \varphi = f + \psi$ and $\lambda >0$, the \emph{forward-backward envelope} (FBE) of the function $\varphi$ with the parameter $\lambda$ is defined by \begin{align}\label{FBE}
\varphi_\lambda(x) :=\inf_{z \in \mathbb{R}^n}\Big\{ f(x) + \langle \nabla f(x),z-x\rangle + \psi(z) +\frac{1}{2\lambda }\| z- x\|^2\Big\}. \end{align} \end{definition}
Remembering the constructions of the Moreau envelope \eqref{moreau} and the Asplund function \eqref{asp} allows us to represent $\varphi_\lambda$ for every $\lambda \in (0, \lambda_\psi)$ as: \begin{equation}\label{rep02} \begin{array}{ll}
\varphi_\lambda(x)&= f(x) -\displaystyle\frac{\lambda}{2}\| \nabla f(x)\|^2 + \MoreauYosida{\psi}{\lambda}\big(x-\lambda \nabla f(x)\big) \\
&=\displaystyle f(x) +\frac{1}{2\lambda} \|x\|^2 -\langle \nabla f(x), x\rangle - \Asp{\lambda}{\psi}\big(x- \lambda \nabla f(x)\big).
\end{array}
\end{equation}
\begin{remark}\label{rem:infimum} It is not difficult to show that whenever $\nabla f$ is $L_f$-Lipschitz on $\mathbb{R}^n$ and $\lambda \in (0,\frac{1}{L_f})$, the optimal values in problems \eqref{EQ01} and \eqref{FBE} are the same \begin{align}\label{eqinf} \inf_{x\in \mathbb{R}^n}\varphi_\lambda(x)= \inf_{x\in \mathbb{R}^n } \varphi(x). \end{align} Indeed, the inequality ``$\leq $'' in \eqref{eqinf} follows directly from the definition of $\varphi_\lambda$. The reverse inequality in \eqref{eqinf} is obtained by \begin{equation*} \begin{array}{ll}
\displaystyle\inf_{x\in \mathbb{R}^n} \varphi_\lambda(x)=\displaystyle\inf_{x\in \mathbb{R}^n}\displaystyle\inf_{ z \in \mathbb{R}^n } \Big\{ f(x) + \langle \nabla f(x),z-x\rangle + \psi(z) +\displaystyle\frac{1}{2\lambda }\|z- x\|^2\Big\}\\
\geq\displaystyle\inf_{x\in \mathbb{R}^n}\displaystyle\inf_{ z \in \mathbb{R}^n}\Big\{ f(z) -\frac{L_f}{2} \|z-x\|^2 + \psi(z) +\displaystyle\frac{1}{2\lambda }\| z- x\|^2\Big\}\\
= \displaystyle\inf_{ z \in \mathbb{R}^n }\displaystyle\inf_{x\in \mathbb{R}^n}\Big\{ f(z) + \psi(z)+\Big(\frac{1}{2\lambda } -\displaystyle\frac{L_f}{2}\Big)\| z- x\|^2\Big\}= \displaystyle\inf_{ z \in \mathbb{R}^n }\varphi(x). \end{array} \end{equation*} Moreover, \eqref{eqinf} does not hold if $\nabla f$ is not Lipschitz continuous on $\mathbb{R}^n$. Indeed, consider $f(x):=\frac{1}{4} x^4$ and $\psi:=0$. Then we have $\inf_{x\in\mathbb{R}^n} \varphi(x) =0$ while $\varphi_\lambda(x)=\frac{1}{4} x^4 - \frac{\lambda}{2} x^6 $, which yields $\inf_{x\in \mathbb{R}^n} \varphi_\lambda(x)=-\infty$, and so \eqref{eqinf} fails. \end{remark}\vspace*{-0.05in}
The next theorem shows that FBE \eqref{FBE} can be written as the difference of a $\mathcal{C}^{1,1}$ function and a Lipschitzian prox-regular function. Furthermore, it establishes relationships between minimizers and critical points of $\varphi$ and $\varphi_\lambda$.\vspace*{-0.1in}
\begin{theorem}\label{diff_repr_varphi} Let $\varphi=f+\psi$, where $f$ is of class $\mathcal{C}^{2,1}$ and where $\psi$ is prox-bounded with threshold $\lambda_\psi>0$. Then for any $\lambda \in (0,\lambda_\psi)$, we have the inclusion \begin{equation}\label{Subbasic} \partial\varphi_\lambda (x)\subseteq \lambda^{-1}\big( I- \lambda \nabla^2 f(x)\big) \big(x -\Prox{\psi}{\lambda}\big(x-\lambda \nabla f(x)\big)\big). \end{equation} Furthermore, the following assertions are satisfied:
{\bf(i)} \label{diff_repr_varphi_a} If $x\in \mathbb{R}^n$ is a stationary point of $\varphi_\lambda$, then $0\in \widehat{\partial} \varphi(x)$ provided that the matrix $ I- \lambda \nabla^2 f(x)$ is nonsingular.
{\bf(ii)} \label{diff_repr_varphi_c} The FBE \eqref{FBE} can be written as $\varphi_\lambda=g-h$, where $g(x):= f(x) +\frac{1}{2\lambda} \|x\|^2 $ is of class $\mathcal{C}^{2,1}$, and where $h(x):= \langle \nabla f(x),x\rangle + \Asp{\lambda}{\psi}(x- \lambda \nabla f(x))$ is locally Lipschitzian and prox-regular on $\mathbb{R}^n$. Moreover, $\nabla^2 g(x)$ is $\xi$-lower-definite for all $x\in \mathbb{R}^n$ with $\xi:=\frac{1}{\lambda} - L_f$.
{\bf(iii)} \label{diff_repr_varphi_d} If $\psi:=\delta_{C}$ for a closed set $C$, then $\partial (-\Asp{\lambda}{\psi})=-\frac{1}{\lambda}\mathtt{P}_C$, where $\mathtt{P}_C$ denotes the $($generally set-valued$)$ projection operator onto $C$. In this case, inclusion \eqref{Subbasic} holds as an equality.
{\bf(iv)} \label{diff_repr_varphi_b} If both $f$ and $\psi$ are convex, we have that $\varphi_\lambda = g - h$, where $g(x):= f(x) + \MoreauYosida{\psi}{\lambda} (x-\lambda \nabla f(x))$ and $h(x):= \frac{\lambda}{2}\| \nabla f(x)\|^2$ are of class $\mathcal{C}^{1,1}$ $($and hence prox-regular$)$ on $\mathbb{R}^n$, and that \begin{align}\label{eqconvexcase}
\big\{x \in \mathbb{R}^n\;\big|\;\nabla \varphi_\lambda(x)=0\big\} =\big\{ x\in \mathbb{R}^n\;\big|\;0 \in \partial \varphi(x)\big\} \end{align} provided that $ I- \lambda \nabla^2 f(x)$ is nonsingular at any stationary point of $\varphi_\lambda$. \end{theorem}\vspace*{-0.05in} \begin{proof} Observe that inclusion \eqref{Subbasic} follows directly by applying the basic subdifferential sum and chain rules from \cite[Theorem~2.19 and Corollary~4.6]{MR3823783}, respectively, the first representation of $\varphi_\lambda$ in \eqref{rep02} with taking into account the results of Lemma~\ref{Lemma5.1}. Now we pick any stationary point $x\in\mathbb{R}^n$ of the FBE $\varphi_\lambda$ and then deduce from $0\in \partial \varphi_\lambda(x)$ and \eqref{Subbasic} that \begin{equation*} x \in\Prox{\psi}{\lambda}\big(x-\lambda \nabla f(x))\big), \end{equation*} which readily implies that $0 \in \nabla f(x) +\widehat{\partial} \psi(x)=\widehat{\partial} \varphi(x)$ and thus verifies (i). Assertion (ii) follows directly from Proposition~\ref{Lemma5.1} and the smoothness of $f$.
To prove (iii), we need to verify the reverse inclusion ``$\supseteq$'' in \eqref{eq_sub_eq01}, for which it suffices to show that the inclusion $v\in\mathtt{P}_C(x)$ yields $v\not\in{\rm co}(\mathtt{P}_C(x)\setminus\{v\})$. On the contrary, if $v\in\mathtt{P}_C(x)\cap{\rm co}(\mathtt{P}_C(x)\setminus\{v\})$, then there exist $c_1,\ldots,c_m\in P_C(x)\setminus\{v\}$ and $\mu_1,\ldots,\mu_m\in (0,1)$ such that $v=\sum_{i=1}^m\mu_i c_i$ with $\sum_{i=1}^m \mu_i=1$. By definition of the projection, we get the equalities \begin{equation*}
|c_1-x\|^2=\ldots=\|c_m-x\|^2=\|v-x\|^2=\Big\|\sum_{i=1}^m\mu_i(c_i-x)\Big\|^2, \end{equation*}
which contradict the strict convexity of $\|\cdot\|^2$ and thus verifies (iii).
The first statement in (iv) follows from the differentiability of $f$ and of the Moreau envelope $\MoreauYosida{\psi}{\lambda}$ by \cite[Theorem~2.26]{MR1491362}. Further, the inclusion ``$\subseteq $'' in \eqref{eqconvexcase} is a consequence of (i). To justify the reverse inclusion in \eqref{eqconvexcase}, observe that any $x$ satisfying $0\in\partial\varphi(x)$ is a global minimizer of the convex function $\varphi$, and so $x = \Prox{\psi}{\lambda} (x-\lambda \nabla f(x))$. The differentiability of $\varphi_\lambda$ and \eqref{Subbasic} (which holds as an equality in this case) tells us that $\nabla\varphi_\lambda(x)=0$, and thus \eqref{eqconvexcase} holds. This completes the proof of the theorem. \end{proof}\vspace*{-0.25in}
\begin{remark}\label{rem:DC repres} Based on Theorem~\ref{diff_repr_varphi}(ii), it is not hard to show that the FBE function $\varphi_\lambda$ can be represented as a difference of convex functions. Indeed, since $\Asp{\lambda}{\psi}$ is a locally Lipschitzian and prox-regular function, we have by \cite[Corollary~3.12]{MR2101873} that $h$ is a lower-$\mathcal{C}^2$ function, and hence by \cite[Theorem~10.33]{MR1491362}, it is locally a DC function. Similarly, $g$ being a $\mathcal{C}^2$ function is a DC function, so the difference $\varphi=g-h$ is also a DC function. However, it is difficult to determine for numerical purposes what is an appropriate representation of $\varphi$ as a difference of convex functions. Moreover, such a representation of the objective in terms of convex functions may generate some theoretical and algorithmic challenges as demonstrated below in Example~\ref{Example5.4}.
\end{remark}\vspace*{-0.35in}
\subsection{Nonconvex Optimization with Geometric Constraints}\label{subsec:5.1}\vspace*{-0.1in}
This subsection addresses the following problem of {\em constrained optimization} with explicit geometric constraints given by: \begin{equation}\label{prob:constrained} \mbox{minimize }\;f(x)\;\mbox{ subject to }\;x\in C, \end{equation} where $f:\mathbb{R}^n\to\mathbb{R}$ is of class $\mathcal{C}^{2,1}$, and where $C\subseteq\mathbb{R}^n$ is an arbitrary closed set. Due to the lack of convexity, most of the available algorithms in the literature are not able to directly handle this problem. Nevertheless, Theorem~\ref{diff_repr_varphi} provides an effective machinery allowing us to reduce \eqref{prob:constrained} to an optimization problem that can be solved by using our developments. Indeed, define $\psi(x): = \delta_{C}(x)$ and observe that $\psi$ is prox-regular with threshold $\lambda_\psi = \infty$. In this setting, FBE \eqref{FBE} reduces to the formula \begin{equation*}
\varphi_\lambda(x)=f(x)-\frac{\lambda}{2}\|\nabla f(x)\|^2+\frac{1}{2\lambda}{\rm dist}^2\big(x-\lambda\nabla f(x);C\big). \end{equation*} Furthermore, it follows from Theorem~\ref{diff_repr_varphi}(iii) that
\begin{align*}
\partial \varphi_\lambda(x)=\lambda^{-1}\big(I-\lambda \nabla^2 f(x)\big)\big(x - \mathtt{P}_C\big( x -\lambda \nabla f(x)\big)\big).
\end{align*} Based on Theorem~\ref{diff_repr_varphi}, we deduce from Algorithm~\ref{alg:1} with $\rho_k=0$ its following version to solve the constrained problem \eqref{prob:constrained}.\vspace*{-0.2in}
\begin{algorithm}[ht!] \begin{algorithmic}[1] \Require{$x_0 \in \mathbb{R}^n$, $\beta \in (0,1)$, $t_{\min}>0 $ and $\sigma\in(0,1)$.} \For{$k=0,1,\ldots$} \State Take $w_k\in \left(\lambda^{-1}I-\nabla^2 f(x_k)\right)\big( x_k - \mathtt{P}_C\big(x-\lambda \nabla f(x_k)\big)\big)$. \State If $ w_k=0$, STOP and return~$x_k$. Otherwise set $d_k$ as the solution to the linear system $(\nabla^2 f(x_k)+\lambda^{-1}I)d_k=w_k$. \State Choose any $\overline{\tau}_k\geq t_{\min}$. Set $\overline{\tau}_k:=\tau_k$. \While{$\varphi_\lambda(x_k + \tau_k d_k) > \varphi_\lambda(x_k) +\sigma \tau_k \langle \nabla w_k , d_k\rangle $} \State $\tau_k = \beta \tau_k$. \EndWhile \State Set $x_{k+1}:=x_k + \tau_kd_k$. \label{step5_2} \EndFor \end{algorithmic} \caption{Projected-like Newton algorithm for constrained optimization}\label{alg:3} \end{algorithm}\vspace*{-0.2in}
To the best of our knowledge, Algorithm~\ref{alg:3} is new even for the case of convex constraint sets $C$. All the results obtained for Algorithm~\ref{alg:1} in Sections~\ref{sec:3} and \ref{sec:4} can be specified for Algorithm~\ref{alg:3} to solve problem \eqref{prob:constrained}. For brevity, we present just the following direct consequence of Theorem~\ref{The01}. \vspace*{-0.05in}
\begin{corollary}\label{Cor:Theo01} Considering problem \eqref{prob:constrained}, suppose that $f: \mathbb{R}^n \to\mathbb{R}$ is of class $\mathcal{C}^{2,1}$, that $C\subset\mathbb{R}^n$ is closed, and that $\inf_{x\in C}f(x) >-\infty$. Pick an initial point $x_0 \in \mathbb{R}^n$ and a parameter $\lambda\in (0, \frac{1}{L_f})$. Then Algorithm~{\rm\ref{alg:3}} either stops at a point $x$ such that $0\in\nabla f(x)+\widehat{N}_C(x)$, or generates infinite sequences $\{x_k\}$, $\{\varphi_\lambda(x_k)\}$, $\{w_k\}$, $\{d_k\}$, and $\{\tau_k\}$ satisfying the assertions:
{\bf(i)} \label{Cor:The01a} The sequence $\{\varphi_\lambda(x_k)\}$ monotonically decreases and converges.
{\bf(ii)} \label{Cor:The01b} If $\{x_{k_j}\}$ is a bounded subsequence of $\{x_k\}$, then $\inf_{j\in\mathbb{N}} \tau_{k_j}>0$ and \begin{align*}
\sum\limits_{j \in \mathbb{N}} \| d_{k_j}\|^2 < \infty,\; \sum\limits_{j\in \mathbb{N} } \| x_{k_j +1} - x_{k_j}\|^2< \infty,\;\sum\limits_{j\in \mathbb{N} } \| w_{k_j}\|^2 < \infty. \end{align*} If, in particular, the entire sequence $\{x_k\}$ is bounded, then the set of its accumulation points is nonempty, closed, and connected.
{\bf(iii)} \label{Cor:The01c} If $x_{k_j} \to \bar{x}$ as $j\to\infty$, then $0\in\nabla f(\bar{x})+\widehat{N}_C(\bar{x})$ and the equality $\varphi_\lambda(\bar{x})=\inf_{k\in \mathbb{N}} \varphi_\lambda(x_k)$ holds.
{\bf(iv)} \label{Cor:The01d} If the sequence $\{x_k\}$ has an isolated accumulation point $\bar{x}$, then it converges to $\bar{x}$ as $k\to\infty$, and we have $0\in\nabla f(\bar{x})+\widehat{N}_C(\bar{x})$. \end{corollary}\vspace*{-0.05in}
The next example illustrates our approach to solve \eqref{prob:constrained} via Algorithm~\ref{alg:3} in contrast to algorithms of the DC type.\vspace*{-0.05in}
\begin{example}\label{Example5.4} Consider the minimization of a quadratic function over a closed (possibly nonconvex) set $C$:
\begin{align}\label{ProblemQUAD_example}
\mbox{minimize }\;\frac{1}{2}x^T Q x + b^T x\; \text{ subject to }\;
x\in C,
\end{align}
where $Q$ is a symmetric matrix, and where $b \in \mathbb{R}^n$. In this setting, FBE \eqref{FBE} can be written as $\varphi_\lambda(x) = g(x) - h(x)$ with
\begin{equation}\label{func_h_Example55}
\begin{array}{ll}
g(x)&:=\displaystyle\frac{1}{2}x^T\big( Q + \lambda^{-1} I\big) x + b^T x,\\ h(x)& := x^T Q x + b^T x+ \Asp{\lambda}{\psi}\big((I- \lambda Q)x-\lambda b\big). \end{array} \end{equation} Our method {\em does not require} a DC decomposition of the objective function $\varphi_\lambda$. Indeed the function $h$ in \eqref{func_h_Example55} is generally nonconvex. Specifically, consider $Q=\begin{bsmallmatrix}0&-1\\-1&0\end{bsmallmatrix}$, $b=(0,0)^T$, and $C$ being the unit sphere centered at the origin. Then $g$ in \eqref{func_h_Example55} is strongly convex for any $\lambda \in (0,1)$, while $h$ therein is not convex whenever $\lambda >0$. More precisely, in this case we have $$ h(x_1,x_2) = -2x_1x_2+ \Asp{\lambda}{\psi}(x_1+\lambda x_2,\lambda x_1+x_2)\;\mbox{ with} $$ \begin{align*}
\Asp{\lambda}{\psi}(x)= \frac{1}{2\lambda}\left(\|x\|^2-d_C^2(x)\right)=\frac{1}{2\lambda}\left(\|x\|^2-(\|x\|-1)^2\right)=\frac{1}{2\lambda}\left(2\|x\|-1\right). \end{align*} This tells us, in particular, that $$ h(-1/2,-1/2)-\frac{1}{2}h(-1,-1)-\frac{1}{2}h(0,0)=\frac{1}{2}, $$ and thus $h$ is not convex regardless of the value of $\lambda$; see Figure~3.\vspace*{-0.2in}
\begin{figure}
\caption{Contour plot of the functions $f$, $\varphi_\lambda$, $g$ and $h $ in \eqref{func_h_Example55} with $\lambda =0.9$}
\end{figure} \end{example}\vspace*{-0.15in}
\section{Further Applications and Numerical Experiments}\label{sec:6}\vspace*{-0.1in}
In this section, we demonstrate the performance of Algorithm~\ref{alg:1} and Algorithm~\ref{alg:3} in two different problems. The first problem is smooth and arises from the study of system biochemical reactions. It can be successfully tackled with DCA-like algorithms, but they require to solve subproblems whose solutions cannot be analytically computed and are thus time-consuming. This is in contrast to Algorithm~\ref{alg:1}, which only requires solving the linear equation \eqref{EQALG01} at each iteration. The second problem is nonsmooth and consists of minimizing a quadratic function under both convex and nonconvex constraints. Employing FBE \eqref{FBE} and Theorem~\ref{diff_repr_varphi}, these two problems can be attacked by using DCA, BDCA, and Algorithm~\ref{alg:3}.
Both Algorithms~\ref{alg:1} and \ref{alg:3} have complete freedom in the choice of the initial value of the stepsizes $\overline{\tau}_k$ in Step~4, as long as they are bounded from below by a positive constant $t_{\min}$, while the choice of $\overline{\tau}_k$ totally determines the performance of the algorithms. On the one hand, a small value would permit the stepsize to get easily accepted in Step~5, but it would imply little progress in the iteration and (likely) in the reduction of the objective function, probably making it more prone to stagnate at local minima. On the other hand, we would expect a large value to ameliorate these issues, while it could result in a significant waste of time in the linesearch Steps~5-7 of both algorithms.
Therefore, it makes sense to consider a choice which sets the trial stepsize $\overline{\tau}_k$ depending on the stepsize $\tau_{k-1}$ accepted in the previous iteration, perhaps increasing it if no reduction of the stepsize was needed. This technique was introduced in~\cite[Section~5]{MR4078808} under the name of \emph{Self-adaptive trial stepsize}, and it was shown there that this accelerates the performance of BDCA in practice. A similar idea is behind the so-called \emph{two-way backtracking} linesearch, which was recently proposed in~\cite{Truong2021} for the gradient descent method, showing good numerical results on deep neural networks. In contrast to BDCA, our theoretical results require $t_{\min}$ to be strictly positive, so the technique should be slightly adapted as shown in Algorithm~\ref{alg:self-adaptive}. Similarly to \cite{MR4078808}, we adopt a conservative rule of only increasing the trial stepsize $\overline{\tau}_k$ when two consecutive trial stepsizes were accepted without decreasing them.\vspace*{0.1in}
\begin{algorithm}[ht!] \begin{algorithmic}[1] \Require{$\gamma>1$, $\overline{\tau}_0>0$}. \State Obtain $\tau_0$ by Steps~5-7 of Algorithms~\ref{alg:1} or \ref{alg:3}. \State Set $\overline{\tau}_1:=\max\{\tau_0,t_{\min}\}$ and obtain $\tau_1$ by Steps~5-7 of Algorithms~\ref{alg:1} or \ref{alg:3}. \For{$k=2,3,\ldots$} \If{$\tau_{k-2}=\overline{\tau}_{k-2}$ \textbf{and} $\tau_{k-1}=\overline{\tau}_{k-1}$} \State $\overline{\tau}_{k}:=\gamma\tau_{k-1}$; \Else \State $\overline{\tau}_{k}:=\max\{\tau_{k-1},t_{\min}\}$. \EndIf \State Obtain $\tau_k$ by Steps~5-7 of Algorithms~\ref{alg:1} or \ref{alg:3}. \EndFor \end{algorithmic} \caption{Self-adaptive trial stepsize}\label{alg:self-adaptive} \end{algorithm} \vspace*{-0.05in}
The codes in the first subsection below were written and ran in MATLAB version R2021b, while for the second subsection we used Python~3.8. The tests were ran on a desktop of Intel Core i7-4770 CPU 3.40GHz with 32GB RAM, under Windows 10 (64-bit).\vspace*{-0.2in}
\subsection{Smooth DC Models in Biochemistry}\vspace*{-0.1in}
Here we consider the problem motivating the development of BDCA in \cite{AragonArtacho2018}, which consists of finding a steady state of a dynamical equation arising in the modeling of {\em biochemical reaction networks}. We ran our experiments on the same 14 biochemical reaction network models tested in \cite{AragonArtacho2018,MR4078808}. The problem can be modeled as finding a zero of the function $$f(x):=\left([F,R]-[R,F]\right)\exp\left(w+[F,R]^T x\right),$$
where $F,R\in\mathbb{Z}_{\geq 0}^{m\times n}$ denote the forward and reverse \emph{stoichiometric matrices}, respectively, where $w\in\mathbb{R}^{2n}$ is the componentwise logarithm of the \emph{kinetic parameters}, where $\exp(\cdot)$ is the componentwise exponential function, and where $[\,\cdot\,,\cdot\,]$ stands for the horizontal concatenation operator. Finding a zero of $f$ is equivalent to minimizing the function $\varphi(x):=\|f(x)\|^2$, which can be expressed as a {\em difference of the convex functions} \begin{equation}\label{eq:bio_DCA}
g(x):=2\left(\|p(x)\|^2+\|c(x)\|^2\right)\quad\text{and}\quad h(x):=\|p(x)+c(x)\|^2, \end{equation} where the functions $p(x)$ and $c(x)$ are given by \begin{equation*} p(x):=[F,R]\exp\left(w+[F,R]^T x\right)\quad\text{and}\quad c(x):=[R,F]\exp\left(w+[F,R]^T x\right). \end{equation*} In addition, it is also possible to write \begin{equation*}
\varphi(x)=\|f(x)\|^2=\|p(x)-c(x)\|^2=\|p(x)\|^2+\|c(x)\|^2-2p(x)c(x), \end{equation*} and so $\varphi(x)$ can be decomposed as the difference of the functions \begin{equation}\label{eq:bio_ours}
g(x):=\|p(x)\|^2+\|c(x)\|^2 \quad\text{and}\quad h(x)=2p(x)c(x) \end{equation} with $g$ being convex. Therefore, $\nabla^2 g(x)$ is $0$-lower definite, and minimizing $\varphi$ can be tackled with Algorithm~\ref{alg:1} by choosing $\rho_k\geq\zeta$ for some fixed $\zeta>0$. As shown in \cite{AragonArtacho2018}, the function $\varphi$ is real analytic and thus satisfies the Kurdyka--{\L}ojasiewicz assumption of Theorem~\ref{COR:Kur-Loj}, but as observed in \cite[Remark~5]{AragonArtacho2018}, a linear convergence rate cannot be guaranteed.
Our first task in the conducted experiments was to decide how to set the parameters $\zeta$ and $\rho_k$. We compared the strategy of taking $\rho_k$ equal to some fixed value for all $k$, setting a decreasing sequence bounded from below by $\zeta$, and choosing $\rho_k=c\|w_k\|+\zeta$ for some constant $c>0$. In spite of Remark~\ref{rem:theorem}(ii), $\zeta$ was added in the last strategy to guarantee both Theorem~\ref{The01}(ii) and Theorem~\ref{COR:Kur-Loj}. We took $\zeta=10^{-8}$ and a constant $c=5$, which worked well in all the models. We tried several options for the decreasing strategy, of which a good choice seemed to be $\rho_k=\frac{\|w_0\|}{10^{\lfloor k/50\rfloor}}+\zeta$, where $\lfloor\cdot\rfloor$ denotes the floor function (i.e., the parameter was initially set to $\|w_0\|$ and then divided by $10$ every 50 iterations). The best option was this decreasing strategy, as can be observed in the two models in Figure~\ref{fig:bio_rhos}, and this was the choice for our subsequent tests.\vspace*{-0.1in}
\begin{figure}
\caption{Comparison of the objective values for three strategies for setting the regularization parameter $\rho_k$: constant (with values $10^6$, $10^5$, $10^3$ and $1$), decreasing, and adaptive with respect to the value of $\|w_k\|$.}
\label{fig:bio_rhos}
\end{figure} \vspace*{-0.3in}
\begin{experiment}\label{exp1} For finding a steady state of each of the 14 biochemical models, we compared the performance of Algorithm~\ref{alg:1} and BDCA with self-adaptive strategy, which was the fastest method tested in~\cite{MR4078808} (on average, 6.7 times faster than DCA). For each model, 5 kinetic parameters were randomly chosen with coordinates uniformly distributed in $(-1,1)$, and 5 random starting points with random coordinates in $(-2,2)$ were picked. BDCA was ran using the same parameters as in~\cite{MR4078808}, while we took $\sigma=\beta=0.2$ for Algorithm~\ref{alg:1}.
We considered two strategies for setting the trial stepsize $\overline{\tau}_k$ in Step~4 of Algorithm~\ref{alg:1}: constantly initially set to 50, and self-adaptive strategy (Algorithm~\ref{alg:self-adaptive}) with $\gamma=2$ and $t_{\min}=10^{-8}$. For each model and each random instance, we computed 500 iterations of BDCA with self-adaptive strategy and then ran Algorithm~\ref{alg:1} until the same value of the target function $\varphi$ was reached. As in~\cite{AragonArtacho2018}, the BDCA subproblems were solved by using the function \texttt{fminunc} with \texttt{optimoptions(\char13 fminunc\char13,} \texttt{\char13 Algorithm\char13,} \texttt{\char13 trust-region\char13,} \texttt{\char13 GradObj\char13,} \texttt{\char13 on\char13,} \texttt{\char13 Hessian\char13,} \texttt{\char13 on\char13,} \texttt{\char13 Display\char13,} \texttt{\char13 off\char13,} \texttt{\char13 TolFun\char13,} \texttt{1e-8,} \texttt{\char13 TolX\char13,} \texttt{1e-8)}.
The results are summarized in Figure~\ref{fig:bio_ratio}, where we plot the ratios of the running times between BDCA with self-adaptive stepsize and Algorithm~\ref{alg:1} with constant trial stepsize against Algorithm~\ref{alg:1} with self-adaptive stepsize. On average, Algorithm~\ref{alg:1} with self-adaptive strategy was $6.69$ times faster than BDCA, and was $1.33$ times faster than Algorithm~\ref{alg:1} with constant strategy. The lowest ratio for the times of self-adaptive Algorithm~\ref{alg:1} and BDCA was $3.17$. Algorithm~\ref{alg:1} with self-adaptive stepsize was only once (out of the 70 instances) slightly slower (a ratio of 0.98) than with the constant strategy.\vspace*{-0.05in}
\begin{figure}
\caption{Ratios of the running times of Algorithm~\ref{alg:1} with constant stepsize and BDCA with self-adaptive stepsize to Algorithm~\ref{alg:1} with self-adaptive stepsize. For each of the models, the algorithms were run using the same random starting points. The overall average ratio is represented with a dashed line}
\label{fig:bio_ratio}
\end{figure}\vspace*{-0.2in}
In Figure~\ref{fig:bio_comparison}, we plot the values of the objective function for each algorithm and also include for comparison the results for DCA and BDCA without self-adaptive strategy. The self-adaptive strategy also accelerates the performance of Algorithm~\ref{alg:1}. We can observe in Figure~\ref{fig:bio_taus} that there is a correspondence between the drops in the objective value and large increases of the stepsizes $\tau_k$ (in a similar way to what was shown for BDCA in~\cite[Fig.~12]{MR4078808}).\vspace*{-0.15in}
\begin{figure}
\caption{Value of the objective function (with logarithmic scale) of Algorithm~\ref{alg:1}, DCA and BDCA for two biochemical models. The value attained after 500 iterations of BDCA with self-adaptive stepsize is shown by a dashed line.}
\label{fig:bio_comparison}
\end{figure}
\begin{figure}
\caption{Comparison of the self-adaptive and the constant (with $\overline{\tau}_k = 50$) choices for the trial stepsizes in Step 4~of Algorithm~\ref{alg:1} for two biochemical models. The plots include two scales, a logarithmic one for the objective function values and a linear one for the stepsizes (which are represented with discontinuous lines).}
\label{fig:bio_taus}
\end{figure} \end{experiment}
\subsection{Solving Constrained Quadratic Optimization Models} \vspace*{-0.1in}
This subsection contains numerical experiments to solve problems of constrained quadratic optimization formalized by \begin{align}\label{ProblemQUAD1} \mbox{minimize }\;\frac{1}{2}x^T Q x + b^T x\; \text{ subject to }\;x\in C:=\bigcup_{i=1}^p C_i, \end{align} where $Q$ is a symmetric matrix (not necessarily positive-semidefinite), $b \in \mathbb{R}^n$, and $C_1,\ldots,C_p\subseteq \mathbb{R}^n$ are nonempty, closed, and convex sets.
When $C=\mathbb{B}_r(0)$ (i.e., $p=1$), this problem is referred as the {\em trust-region subproblem}. If $Q$ is positive-semidefinite, then \eqref{ProblemQUAD1} is a problem of {\em convex quadratic programming}. Even when $Q$ is not positive-semidefinite, Tao and An~\cite{Tao1998} showed that this particular instance of problem \eqref{ProblemQUAD1} could be efficiently addressed with the DCA algorithm by using the following DC decomposition: \begin{equation}\label{eq:Quad_DCA_g_h}
g(x):=\frac{1}{2}\rho\|x\|^2+b^T x+\delta_{\mathbb{B}_r(0)},\quad h(x):=\frac{1}{2}x^T(\rho I-Q)x, \end{equation}
where $\rho\geq\|Q\|_2$. However, this type of decomposition would not be suitable for problem \eqref{ProblemQUAD1} when $C$ is not convex.
As shown in Subsection~\ref{subsec:5.1}, problem \eqref{ProblemQUAD1} for $p\geq 1$ can be reformulated by using FBE \eqref{FBE} to be tackled with Algorithm~\ref{alg:3} with $\lambda\in(0,\frac{1}{\|Q\|_2})$. Although the decomposition in \eqref{func_h_Example55} may not be suitable for DCA when $Q$ is not positive-definite, it can be regularized by adding $\frac{1}{2}\rho\|x\|^2$ to both $g$ and $h$ with $\rho\geq\max\{0,-2\lambda_{\min}(Q)\}$. Such a regularization would guarantee the convexity of the resulting functions $g$ and $h$ given by \begin{align}
g(x)& := \frac{1}{2}x^T \left( Q + (\rho+\lambda^{-1}) I\right) x + b^T x, \label{eq:Quad_FBE_g_h}\\
h(x)& := \frac{1}{2}x^T \left(2Q+\rho I\right) x + b^T x+ \Asp{\lambda}{\delta_C}((I- \lambda Q)x-\lambda b). \label{eq:Quad_FBE_g_h_bis}
\end{align} The function $g$ in \eqref{eq:Quad_DCA_g_h} is not smooth, but the function $g$ in~\eqref{eq:Quad_FBE_g_h} is. Then it is possible to apply BDCA \cite{MR4078808} to formulation \eqref{eq:Quad_FBE_g_h}--\eqref{eq:Quad_FBE_g_h_bis} in order to accelerate the convergence of DCA. Note that it would also be possible to do it with \eqref{eq:Quad_DCA_g_h} if the $\ell_1$ or $\ell_{\infty}$ balls were used; see~\cite{Artacho2019} for more details.\vspace*{0.03in}
Let us describe two numerical experiments to solve problem \eqref{ProblemQUAD1}. \vspace*{-0.05in}
\begin{experiment}\label{exp2}
Consider \eqref{ProblemQUAD1} with $C=\mathbb{B}_r(0)$ and replicate the hardest setting in \cite{Tao1998}, which was originally considered in~\cite{More1983}. Specifically, in this experiment we generated potentially difficult cases by setting $Q:=UDU^T$ for some diagonal matrix $D$ and orthogonal matrix $U:=U_1U_2U_3$ with $U_j:=I-2u_ju_j^T/\|u_j\|^2$, $j=1,2,3$. The components of $u_j$ were random numbers uniformly distributed in $(-1,1)$, while the elements in the diagonal of $D$ were random numbers in $(-5,5)$. We took $b:=Uz$ for some vector $z$ whose elements were random numbers uniformly distributed in $(-1,1)$ except for the component corresponding to the smallest element of $D$, which was set to $0$. The radius $r$ was randomly chosen in the interval $(\|d\|,2\|d\|)$, where $d_i:=z_i/(D_{ii}-\lambda_{\min}(D))$ if $D_{ii}\neq \lambda_{\min(D)}$ and $0$ otherwise.
For each $n\in\{100,200,\ldots,900,1000,1250,1500,\ldots,3750,4000\}$, we generated 10 random instances, took for each instance a random starting point in $\mathbb{B}_r(0)$, and ran from it the four algorithms described above: DCA applied to formulation~\eqref{eq:Quad_DCA_g_h} (without FBE), DCA and BDCA applied to~\eqref{eq:Quad_FBE_g_h}--\eqref{eq:Quad_FBE_g_h_bis}, and Algorithm~\ref{alg:3}. We took $\lambda=0.8/\|Q\|_2$ as the parameter for FBE (both for DCA and Algorithm~\ref{alg:3}). The regularization parameter $\rho$ was chosen as $\max\{0,-2\lambda_{\min}(Q)\}$ for DCA with FBE and $0.1+\max\{0,-2\lambda_{\min}(Q)\}$ for BDCA, as $h$ should be strongly convex. Both Algorithm~\ref{alg:3} and BDCA were ran with the self-adaptive trial stepsize for the backtracking step introduced in~\cite{MR4078808} with parameters $\sigma=\beta=0.2$ and $\gamma=4$, and with $t_{\min}=10^{-6}$. For the shake of fairness, we did not compute function values for the runs of DCA at each iteration, since it is not required by the algorithm. Instead, we used for both versions of DCA the stopping criterion from~\cite{Tao1998} that $er\leq 10^{-4}$, where $$er=\left\{\begin{array}{lc}
\left\|x^{k+1}-x^k\right\| /\left\|x^k\right\| & \text { if }\left\|x^k\right\|>1, \\
\left\|x^{k+1}-x^k\right\|& \text { otherwise.} \end{array}\right.$$ As DCA with FBE was clearly the slowest method, we took the function value of the solution returned by DCA without FBE as the target value for both Algorithm~\ref{alg:3} and BDCA, so these algorithms were stopped when that function value was reached. In Figure~\ref{fig:trust_region_ratio_small}, we plot the time ratio of each algorithm against Algorithm~\ref{alg:3}. On average, Algorithm~\ref{alg:3} was more than 5 times faster than DCA with FBE and more than 2 times faster than DCA without FBE. BDCA greatly accelerated the performance of DCA with FBE, but still Algorithm~\ref{alg:3} was more than 1.5 times faster. Only for size 300, the performance of DCA without FBE was comparable to that of Algorithm~\ref{alg:3}. We observe on the right plot that the advantage of Algorithm~\ref{alg:3} is maintained for larger sizes.
\begin{figure}
\caption{Time ratio for 10 random instances of DCA with FBE, DCA without FBE, and BDCA with respect to Algorithm~\ref{alg:3}. Average ratio within each size is represented with a triangle for DCA with FBE, with a square for DCA without FBE and with a circle for BDCA. The overall average ratio for each pair of algorithms is represented by a dotted line.}
\label{fig:trust_region_ratio_small}
\end{figure} \end{experiment}
\begin{experiment} With the aim of finding the minimum of a quadratic function with integer and box constraints, we modified the setting of Experiment~2 and considered instead a set $C$ composed by $9^n$ balls of various radii centered at $\{-4,-3,-2,-1,0,1,2,3,4\}^n$, with $n\in\{2,10,25,50,100,200,500,1000\}$. As balls of radius $1/2\sqrt{n}$ cover the region $[-4,4]^n$, we ran our tests with balls of radii $c/2\sqrt{n}$ with $c\in\{0.1,0.2,\ldots,0.8, 0.9\}$. This time we considered {\em both convex and nonconvex} objective functions. The nonconvex case was generated as in Experiment~2, while for the convex case, the elements of the diagonal of $D$ were chosen as random numbers uniformly distributed in $(0,5)$.
For each $n$ and~$r$, 100 random instances were generated. For each instance, a starting point was chosen with random coordinates uniformly distributed in $[-5,5]^n$. As the constraint sets are nonconvex, FBE was also needed to run DCA. The results are summarized in Table~\ref{tbl:integer}, where for each $n$ and each radius, we counted the number of instances (out of 100) in which the value of $\varphi_\lambda$ at the rounded output of DCA and BDCA was lower and higher than that of Algorithm~\ref{alg:3} when ran from the same starting point. We used the same parameter settings for the algorithms as in Experiment~2. Finally, we plot in Figure~\ref{fig:integer} two instances in $\mathbb{R}^2$ in which Algorithm~\ref{alg:3} reached a better solution.\vspace*{-0.25in}
\begin{table}[ht!] \begin{subtable}[t]{\textwidth}\centering \scalebox{.85}{\centering \begin{tabular}{c c cc cc cc cc c} \toprule \multicolumn{2}{c}{}&\multicolumn{9}{c}{Radius of the balls}\\ \cmidrule[.7pt]{3-11} & Alg.~\ref{alg:3} vs & $\frac{1}{20}\sqrt{n}$ & $\frac{2}{20}\sqrt{n}$ & $\frac{3}{20}\sqrt{n}$ & $\frac{4}{20}\sqrt{n}$ & $\frac{5}{20}\sqrt{n}$ & $\frac{6}{20}\sqrt{n}$ & $\frac{7}{20}\sqrt{n}$ & $\frac{8}{20}\sqrt{n}$ & $\frac{9}{20}\sqrt{n}$ \\ \midrule[.7pt] \multirow{ 2}{*}{$n=2$}& DCA & 0/34 & 1/20 & 1/26 & 1/19 & 0/19 & 0/18 & 0/3 & 1/1 & 0/2\\ & BDCA & 6/12 & 2/13 & 3/14 & 4/9 & 0/4 & 1/10 & 0/2 & 1/1 & 0/2\\ \midrule[.5pt] \multirow{ 2}{*}{$n=10$}& DCA & 2/89 & 2/83 & 1/66 & 5/53 & 8/28 & 3/7 & 1/1 & 3/1 & 1/0\\ & BDCA & 21/68 & 38/53 & 33/39 & 23/24 & 18/16 & 2/7 & 0/0 & 0/1 & 0/0 \\ \midrule[.5pt] \multirow{ 2}{*}{$n=25$}& DCA & 0/99 & 0/98 & 2/87 & 11/58 & 9/32 & 3/8 & 2/9 & 2/2 & 5/4\\ & BDCA & 16/83 & 29/71 & 40/58 & 37/40 & 13/26 & 2/3 & 0/1 & 0/0 & 1/2\\ \midrule[.5pt] \multirow{ 2}{*}{$n=50$}& DCA & 0/100 & 0/100 & 0/91 & 2/86 & 13/41 & 14/12 & 9/12 & 6/10 & 12/12\\ & BDCA & 8/92 & 6/94 & 31/69 & 36/53 & 16/28 & 8/8 & 6/5 & 3/4 & 5/3\\ \midrule[.5pt] \multirow{ 2}{*}{$n=100$}& DCA & 0/100 & 0/100 & 0/99 & 9/87 & 18/49 & 18/31 & 12/22 & 18/20 & 11/21\\ & BDCA & 2/98 & 6/94 & 39/61 & 36/61 & 23/33 & 16/14 & 9/8 & 9/8 & 13/9\\ \midrule[.5pt] \multirow{ 2}{*}{$n=200$}& DCA & 0/100 & 0/100 & 0/100 & 1/98 & 23/64 & 31/41 & 25/29 & 22/30 & 20/41\\ & BDCA & 3/97 & 2/98 & 38/62 & 37/63 & 33/39 & 27/17 & 18/18 & 14/13 & 16/18\\ \midrule[.5pt] \multirow{ 2}{*}{$n=500$}& DCA & 0/100 & 0/100 & 0/100 & 1/99 & 6/94 & 15/80 & 27/61 & 29/65 & 36/48\\ & BDCA & 0/100 & 1/99 & 41/59 & 44/56 & 33/63 & 25/56 & 34/39 & 32/47 & 17/35\\ \bottomrule \end{tabular}} \caption{Convex case} \end{subtable}
\begin{subtable}[h]{\textwidth}\centering \scalebox{.85}{\centering \begin{tabular}{c c cc cc cc cc c} \toprule \multicolumn{2}{c}{}&\multicolumn{9}{c}{Radius of the balls}\\ \cmidrule[.7pt]{3-11} & Alg.~\ref{alg:3} vs & $\frac{1}{20}\sqrt{n}$ & $\frac{2}{20}\sqrt{n}$ & $\frac{3}{20}\sqrt{n}$ & $\frac{4}{20}\sqrt{n}$ & $\frac{5}{20}\sqrt{n}$ & $\frac{6}{20}\sqrt{n}$ & $\frac{7}{20}\sqrt{n}$ & $\frac{8}{20}\sqrt{n}$ & $\frac{9}{20}\sqrt{n}$ \\ \midrule[.7pt] \multirow{ 2}{*}{$n=2$}& DCA & 1/8 & 1/9 & 1/10 & 1/6 & 0/6 & 0/6 & 0/8 & 3/2 & 1/0\\ & BDCA & 2/4 & 1/4 & 1/3 & 3/4 & 0/5 & 0/4 & 0/6 & 3/2 & 1/0\\ \midrule[.5pt] \multirow{ 2}{*}{$n=10$}& DCA & 9/39 & 4/39 & 7/39 & 4/35 & 10/30 & 3/27 & 5/45 & 2/34 & 8/29\\ & BDCA & 9/31 & 11/33 & 13/29 & 6/31 & 11/29 & 6/25 & 7/38 & 5/29 & 10/30\\ \midrule[.5pt] \multirow{ 2}{*}{$n=25$}& DCA & 6/69 & 13/67 & 7/62 & 5/61 & 10/53 & 3/59 & 6/56 & 3/72 & 3/66\\ & BDCA & 16/58 & 16/63 & 16/55 & 12/48 & 9/52 & 11/52 & 13/52 & 12/57 & 11/58\\ \midrule[.5pt] \multirow{ 2}{*}{$n=50$}& DCA & 11/81 & 10/79 & 8/87 & 5/90 & 3/87 & 4/80 & 2/86 & 5/89 & 8/81\\ & BDCA & 24/68 & 21/64 & 23/70 & 17/73 & 14/75 & 9/73 & 10/75 & 18/74 & 16/71\\ \midrule[.5pt] \multirow{ 2}{*}{$n=100$}& DCA & 4/96 & 6/94 & 4/94 & 5/94 & 4/96 & 3/97 & 2/98 & 7/91 & 9/91\\ & BDCA & 15/85 & 16/83 & 18/80 & 14/84 & 17/83 & 11/89 & 9/91 & 20/79 & 19/80\\ \midrule[.5pt] \multirow{ 2}{*}{$n=200$}& DCA & 4/96 & 4/96 & 4/96 & 2/98 & 1/99 & 2/98 & 4/96 & 3/97 & 0/100\\ & BDCA & 11/89 & 16/84 & 11/89 & 8/92 & 6/94 & 11/89 & 10/90 & 13/87 & 8/92\\ \midrule[.5pt] \multirow{ 2}{*}{$n=500$}& DCA & 1/99 & 2/98 & 0/100 & 0/100 & 0/100 & 1/99 & 1/99 & 2/98 & 1/99\\ & BDCA & 12/88 & 17/83 & 15/85 & 9/91 & 15/85 & 11/89 & 9/91 & 18/82 & 20/80\\ \bottomrule \end{tabular}} \caption{Nonconvex case} \end{subtable} \caption{For different values of $n$ (space dimension) we computed 100 random instances of problem~\eqref{ProblemQUAD1} with $Q$ positive definite and $C$ formed by the union of balls whose centers have integer coordinates between $-4$ and $4$. We counted the number of instances in which DCA and BDCA obtained a lower/upper value than Algorithm~\ref{alg:3}.}\label{tbl:integer} \end{table}\vspace*{-0.15in}
\begin{figure}
\caption{Two instances of problem~\eqref{ProblemQUAD1}. On the left, both line searches of Algorithm~\ref{alg:3} and BDCA help to reach a better solution for a nonconvex quadratic function, while only Algorithm~\ref{alg:3} succeeds on the right for the convex case.}
\label{fig:integer}
\end{figure} \end{experiment}\vspace*{0.15in}
\section{Conclusion and Future Research}\label{sec:7}\vspace*{-0.05in}
This paper proposes and develops a novel RCSN method to solve problems of difference programming whose objectives are represented as differences of generally nonconvex functions. We establish well-posedness of the proposed algorithm and its global convergence under appropriate assumptions. The obtained results exhibit advantages of our algorithm over known algorithms for DC programming when both functions in the difference representations are convex. We also develop specifications of the main algorithm in the case of structured problems of constrained optimization and conduct numerical experiments to confirm the efficiency of our algorithms in solving practical models.
In the future research, we plan to relax assumptions on the program data ensuring the linear, superlinear, and quadratic convergence rates for RCSN and also extend the spectrum of applications to particular classes of constrained optimization problems as well as to practical modeling. \vspace*{-0.25in}
\end{document} | arXiv |
\begin{document}
\title[Rotation sets with nonempty interior and transitivity]{Rotation sets with nonempty interior\\ and transitivity in the universal covering} \author{Nancy Guelman} \address{Nancy Guelman. IMERL, Facultad de Ingenier\'\i a, Universidad de la Rep\'ublica, C.C. 30, Montevideo, Uruguay} \email{[email protected]} \author{Andres Koropecki} \address{Andres Koropecki. Universidade Federal Fluminense, Instituto de Matem\'atica e Estat\'\i stica, Rua M\'ario Santos Braga S/N, 24020-140 Niteroi, RJ, Brasil} \email{[email protected]} \author{Fabio Armando Tal} \address{Fabio Armando Tal. Instituto de Matem\'atica e Estat\'\i stica, Universidade de S\~ao Paulo, Rua do Mat\~ao 1010, Cidade Universit\'aria, 05508-090 S\~ao Paulo, SP, Brazil} \email{[email protected]} \thanks{The first author was supported by Grupo de investigaci\'on de Sistemas Din\'amicos CSIC 618, Universidad de la Rep\'ublica, Uruguay. The second author was supported by CNPq-Brasil. The third author was partially supported by CNPq-Brasil and FAPESP} \keywords{Torus homeomorphisms, rotation set, transitivity}
\begin{abstract} Let $f$ be a transitive homeomorphism of the two-dimensional torus in the homotopy class of the identity. We show that a lift of $f$ to the universal covering is transitive if and only if the rotation set of the lift contains the origin in its interior. \end{abstract}
\maketitle
\section{Introduction}
Given an homeomorphism $f$ of the torus $\T^2=\R^2/\Z^2$ which is in the homotopy class of the identity, and a lift $\widehat{f}\colon \R^2\to \R^2$ of $f$ to the universal covering, one can associate to $\widehat{f}$ its rotation set $\rho(\widehat f)$, a convex and compact subset of the plane consisting of all the limit points of sequences of the form $\lim_{k\to\infty}(\widehat f^{n_k}(x_k)-x_k)/n_k$, with $n_k\to \infty$ and $x_k\in \R^2$. This definition was introduced by Misiurewicz and Ziemian in \cite{MZ89} as a generalization of the rotation number for circle homeomorphisms, and has proved to be a useful tool in the study of the dynamics of these homeomorphisms. In particular, when the rotation set has nonempty interior it has been shown that $f$ exhibits very rich dynamics, with an abundance of periodic points and positive topological entropy \cite{F89, LM91}. But this complex behavior is not restricted to rotation sets with nonempty interior: there are several examples of homeomorphisms with rich dynamical properties such that their rotation set is a singleton.
In this work we are concerned with the interplay between transitivity of $f$, transitivity on the universal covering space and rotation sets. Our motivation in studying transitivity on the universal covering space is to understand when it is possible for the dynamical system to exhibit this form of extreme transitivity, where there exist trajectories that are not only dense on the surface, but also explore all possible loops and directions. This is a strictly more stringent concept, as ergodic rotations of the $2$-torus do not lift to transitive homeomorphisms of the plane. There have been some prior works concerning transitivity of the lifted dynamics of surface homeomorphisms. In \cite{Boyland}, Boyland defines $H_1$-transitivity for a surface homeomorphism as the property of having an iterate which lifts to the universal Abelian covering as a transitive homeomorphism, and he characterized $H_1$-transitivity for rel pseudo-Anosov maps.
In \cite{guelman-et-al} it was proven that a $C^{1+\alpha}$ diffeomorphism of $\T^2$ homotopic to the identity and with positive entropy has a rotation set with nonempty interior if $f$ has a transitive lift to a suitable covering of $\T^2$. In \cite{AZT1} and \cite{AZT2}, homeomorphisms of the annulus with a transitive lift were studied, and it was shown that if there are no fixed points in the boundary then the rotation set of the lift contains the origin in its interior.
More relevant to this work is the result from \cite{Ta1}, where it is shown that if $f$ is a homeomorphism of $\T^2$ homotopic to the identity which has a transitive lift $\widehat{f}$ to $\R^2$, then the origin lies in the interior of $\rho(\widehat f)$. Our main result is the converse of this fact: \begin{theorem}\label{th:transitivoemcima} Let $f$ be a transitive homeomorphism of $\T^2$ homotopic to the identity, and let $\widehat f$ be a lift of $f$ to $\R^2$. If $(0,0)$ belongs to the interior of $\rho(\widehat f)$ then $\widehat f$ is transitive. \end{theorem}
Note that, since the fundamental group of $\T^2$ is abelian, the notion of $H_1$-transitivity (as defined in \cite{Boyland}) in the case of a homeomorphism $f$ of $\T^2$ is equivalent to saying that some iterate of $f$ has a lift to $\R^2$ which is transitive. As an immediate consequence of theorem \ref{th:transitivoemcima} together with the main result of \cite{Ta1}, we have a characterization of $H_1$-transitive homeomorphisms of $\T^2$ in the homotopy class of the identity:
\begin{corollary} A homeomorphism of $\T^2$ in the homotopy class of the identity is $H_1$-transitive if and only if it has (a lift with) a rotation set with nonempty interior. \end{corollary}
The key starting point for the proof of theorem \ref{th:transitivoemcima} is a recent result from \cite{KoTa} that rules out the existence of `unbounded' periodic topological disks. After this, the theorem is reduced to a technical result, theorem \ref{th:maintheorem}, the proof of which is obtained from geometric and combinatorial arguments relying on a well-known theorem of Franks about realization of rational rotation vectors by periodic points \cite{F89} and some classic facts from Brouwer theory. The next section states these preliminary facts and introduces the necessary notation. In section 3, theorem \ref{th:maintheorem} is stated, and its proof is presented after using it to prove theorem \ref{th:transitivoemcima}.
\section{Notations and preliminary results}
We denote by $\pi\colon \R^2\to \T^2$ the universal covering of $\T^2=\R^2/\Z^2$. Given a point $x\in \R^2$, we denote by $(x)_1$ the projection of $x$ onto the first coordinate, by $(x)_2$ its projection onto the second coordinate, and $$\norm{x}_{\infty}=\max\{\abs{(x)_1},\abs{(x)_2}\}.$$ The following definitions will simplify the notation in the proofs ahead. For $v\in\Z^2$ and $L\in\N$, define $$S(v,L)=\{ w\in\Z^2 : \norm{w-v}_{\infty}\le L\},$$ and $$\Sigma(M)=\{(i,v)\in \Z\times\Z^2 : \abs{i} \le M, \norm{v}_{\infty}\le M\}.$$ Given $v\in\R^2$, we denote by $T_v:\R^2\to\R^2$ the translation $x\mapsto x+v$. If $\gamma:[0,1]\to\R^2$ is curve, we denote its image by $[\gamma]$. We say that $\gamma$ is a \emph{$T_v$-translation arc} if it joins a point $z=\gamma(0)$ to its image $T_v(z)=\gamma(1)$, and $[\gamma]\cap T_v([\gamma])=\gamma(1)$.
The following lemma is a direct consequence of lemma 3.1 of \cite{Brown}.
\begin{lemma}\label{lm:disjunto1vezdisjuntosempre} Let $v\in\R^2$ be a nonzero vector and let $K\subset\R^2$ be an arcwise connected set such that $K\cap T_v(K)=\emptyset$. Then $K\cap T^{i}_v(K)=\emptyset$ for all $i\in\Z$ with $i\neq 0$. \end{lemma}
And the next result follows from theorem 4.6 of \cite{Brown}.
\begin{lemma}\label{lm:Brouwerline} Let $v\in\R^2\setminus \{(0,0)\}$ and let $K$ be an arcwise connected subset of $\R^2$ such that $K\cap T_v(K)=\emptyset$, and $\alpha$ a $T_v$-translation arc disjoint from $K$. Then, either $$K\cap\bigcup_{i\in\N}T^{i}_v[\alpha]=\emptyset \quad \text{ or } \quad K\cap\bigcup_{i\in\N}T^{-i}_v[\alpha]=\emptyset.$$ \end{lemma}
We will also use the next theorem from \cite{F89}. \begin{theorem}[Franks] \label{th:Franks} Let $\widehat{f}$ be a lift to $\R^2$ of a homeomorphism of $\T^2$ homotopic to the identity. If $w\in \Z^2$ and $q\in \N$ are such that $w/q$ lies in the interior of $\rho(\widehat{f})$, then there is $\widehat{x}\in \R^2$ such that $\widehat f^q(\widehat x)=\widehat x+w.$ \end{theorem}
Let us recall some terminology used in \cite{KoTa}. An open set $U\subset \T^2$ is \emph{inessential} if every loop contained in $U$ is homotopically trivial in $\T^2$. An arbitrary subset of $\T^2$ is called inessential if it has an inessential neighborhood, and \emph{essential} otherwise. If $A$ is an inessential subset of $\T^2$, and $U$ is an open neighborhood of $\T^2\setminus A$, then $\T^2\setminus U$ is a closed inessential set. This implies that every homotopy class of loops of $\T^2$ is represented by some loop contained in $U$, and for this reason any subset of $\T^2$ with an inessential complement is called a \emph{fully essential} set.
If $f\colon \T^2\to\T^2$ is a homeomorphism, we say that a point $x\in \T^2$ is an \emph{inessential point} for $f$ if there exists $\epsilon>0$ such that the set $\bigcup_{i\in\Z} f^{i}(B_{\epsilon}(x))$ is inessential. The set of all inessential points of $f$ is denoted by $\operatorname{Ine}(f)$. Any point in the set $\operatorname{Ess}(f)=\T^2\setminus \operatorname{Ine}(f)$ is called an \emph{essential point} for $f$. It follows from the definitions that $\operatorname{Ine}(f)$ is open and invariant, while $\operatorname{Ess}(f)$ is closed and invariant.
Given an open connected set $U \subset \T^2$, we denote by $\mathcal{D}(U)$ the diameter in $\R^2$ of any connected component of $\pi^{-1}(U)$. If $\mathcal{D}(U) < \infty$, then we say that $U$ is \emph{bounded}.
Suppose that $f\colon \R^2\to \R^2$ is a homeomorphism homotopic to the identity. We need the following \begin{theorem}[\cite{KoTa}, Theorem C]\label{th:KoTa} If $f$ is nonwandering and has (a lift with) a rotation set with nonempty interior, then $\operatorname{Ine}(f)$ is a disjoint union of periodic simply connected sets which are bounded. \end{theorem}
In particular, when $f$ is transitive one may conclude $\operatorname{Ine}(f)$ is empty:
\begin{lemma}\label{lm:essentialisall} Suppose that $f$ is transitive and has (a lift with) a rotation set with nonempty interior. Then $\operatorname{Ess}(f)=\T^2$. \end{lemma}
\proof Suppose for contradiction that $\operatorname{Ine}(f)$ is nonempty. Then there is some point $x\in \operatorname{Ine}(f)$ with a dense orbit. Let $D$ be the connected component of $\operatorname{Ine}(f)$ that contains $x$. By lemma \ref{th:KoTa}, we have that $D$ is a periodic simply connected open and bounded set. Thus any connected component $\widehat{D}$ of $\pi^{-1}(D)$ is bounded.
Let $k$ be such that $f^k(D)=D$ and let $\widehat{f}$ be a lift of $f$. Then $\widehat{f}^k(\widehat{D})=\widehat{D}+v$ for some $v\in \Z^2$, and so if $\widehat{g}=T_v^{-1}\widehat{f}^k$ we have that $\widehat{g}(\widehat{D})=\widehat{D}$. The set $\widehat{U}=\bigcup_{i=0}^{k-1}\widehat{f}^i(\widehat{D})$ is then bounded $\widehat{g}$-invariant (because $\widehat{g}$ commutes with $\widehat{f}$), and therefore $\cl(\widehat{U})$ is also bounded and $\widehat{g}$-invariant. In particular all points in $\cl(\widehat{U})$ have a bounded $\widehat{g}$-orbit. Since $U=\pi(\widehat{U})$ is the $f$-orbit of $D$, which contains the (dense) orbit of $x$, it follows that $\cl(U)=\T^2$. The boundedness of $\widehat{U}$ then implies that $\pi(\cl(\widehat{U})) = \cl(U)=\T^2$.
It follows from these facts that every point of $\R^2$ has a bounded $g$-orbit, from which we conclude that $\rho(\widehat{g})=\{(0,0)\}$. But it follows from the definition of rotation set that $\rho(\widehat{g})=\rho(T_v^{-1}\widehat{f}^k) = k\rho(\widehat{f})-v$ (see \cite{MZ89}), contradicting the fact that $\rho(\widehat{f})$ has nonempty interior. \endproof
\section{Proof of Theorem \ref{th:transitivoemcima}}
Theorem \ref{th:transitivoemcima} is a consequence of the following technical result. \begin{theorem}\label{th:maintheorem} Let $f\colon \T^2\to \T^2$ be a homeomorphism homotopic to the identity, and $\widehat f$ a lift of $f$ such that $(0,0)$ belongs to the interior of $\rho(\widehat f).$ Let $O\subset \R^2$ be an open connected set such that $\overline{\pi(O)}$ is inessential and $\bigcup_{n\in\Z} f^n(\pi(O))$ is fully essential. Then, for every $w\in\Z^2$ there exists $n\in \N$ such that $\widehat f^n(O)\cap T_w(O)\neq \emptyset$. \end{theorem}
\begin{proof}[Proof of theorem \ref{th:transitivoemcima} assuming theorem \ref{th:maintheorem}] Let $O_1, O_2$ be two open sets of the plane. Since $f$ is transitive, there exists $n_0$ such that $f^{n_0}(\pi(O_1))$ intersects $\pi(O_2),$ which implies that there exists $w\in\Z^2$ such that $\widehat f^{n_0}(O_1)\cap T_w(O_2)\not=\emptyset.$ Let $\widehat x\in\R^2$ and $0<\varepsilon<\frac{1}{2}$ be such that $B_{\varepsilon}(\widehat x)\subset \widehat f^{n_0}(O_1)\cap T_w(O_2)$. Note that, as $\varepsilon<\frac{1}{2}$, the set $\pi(B_{\varepsilon}(\widehat x))$ is inessential.
Since $f$ is transitive and $\rho(\widehat f)$ has interior, lemma \ref{lm:essentialisall} implies that $\operatorname{Ess}(f)=\T^2$ and, in particular, $ U_{\varepsilon}(x)=\bigcup_{i\in\Z} f^{i}(B_{\varepsilon}(x))$ is an essential invariant open set and, since $f$ is nonwandering, the connected component of $\widehat x$ in $U_{\varepsilon}(x)$ is periodic. In $\T^2$, an essential connected periodic open set is either fully essential or contained in a periodic set homeomorphic to an essential annulus, in which case the rotation set is contained in a line segment. Since $\rho(\widehat f)$ has nonempty interior, we conclude that $U_{\varepsilon}(x)$ is fully essential.
Therefore $B_{\varepsilon}(\widehat x)$ satisfies that the hypotheses of theorem \ref{th:maintheorem}. Thus there exists $n_1$ such that $\widehat f^{n_1}(B_{\varepsilon}(\widehat x))$ intersects $T_{-w}(B_{\varepsilon}(\widehat x)).$ But this implies that $\widehat f^{n_0+n_1}(O_1)$ intersects $T_{-w}(T_{w}(O_2))=O_2,$ completing the proof. \end{proof}
\subsection{Proof of theorem \ref{th:maintheorem}}
We begin by fixing both $\overline{w}\in\Z^2$ and an open connected set $O\subset\R^2$ such that $\bigcup_{n\in\Z} f^n(\pi(O))$ is fully essential and $\overline{\pi(O)}$ is inessential. Our aim is to show that $\widehat{f}^n(O)\cap T_{\overline{w}}(O)\neq \emptyset$ for some $n\in \N$.
Since we are assuming that $\rho(\widehat{f})$ contains $(0,0)$ in its interior, we may choose $\delta>0$ (which is fixed from now on) such that $B_{\delta}((0,0))\subset \rho(\widehat f)$.
\begin{proposition}\label{pr:cercadominiofundamental} There exists a positive integer $M$ and a compact set $K$ such that $[0,1]^2$ is contained in a bounded connected component of $\R^2\setminus K$ and $$K \subset\bigcup_{(i,v)\in\Sigma(M)} T_v(\widehat f^{i}(O))$$ \end{proposition}
\proof Since $\bigcup_{n\in\Z} f^n(\pi(O))$ is fully essential, its preimage by $\pi$, which we denote by $\widehat U$, is a connected open set invariant by $\Z^2$ translates. Therefore, given a point $\widehat y$ in $\widehat U,$ there exist two connected arcs $\alpha$ and $\beta$ in $\widehat U$ such that $\alpha$ connects $\widehat y$ to $\widehat y+(1,0)$ and $\beta$ connects $\widehat y$ to $\widehat y+(0,1).$ Let $$\Gamma_{\alpha}=\bigcup_{i=-\infty}^{\infty} T^{i}_{(1,0)}[\alpha],\quad \Gamma_{\beta}=\bigcup_{i=-\infty}^{\infty} T^{i}_{(0,1)}[\beta]$$ and note that, as $\widehat U$ is $\Z^2$ invariant, then all integer translates of $\Gamma_{\alpha}$ and $\Gamma_{\beta}$ are contained in $\widehat U.$
Fix an integer $R> \max \{\norm{x} : x\in [\alpha]\cup[\beta]\}$. Then, since $$\max \{(x)_2 : x \in \Gamma_{\alpha}\}=\max \{(x)_2 : x\in[\alpha]\}<R$$ and $\min \{(x)_2 : x \in \Gamma_{\alpha}\}>-R,$ it follows that $\R^2\setminus \Gamma_{\alpha}$ has at least two connected components, one containing the semi-plane $\{ x : (x)_2\ge R\}$ and another containing $\{x : (x)_2 \le -R\}.$ Likewise, $\R^2\setminus \Gamma_{\beta}$ has at least two connected components, one containing $\{ x : (x)_1\ge R\}$ and another one containing $\{x : (x)_1 \le -R\}.$
Now let $$F= (\Gamma_\alpha-(0,R))\cup (\Gamma_\alpha+(0,R+1))\cup(\Gamma_\beta-(R,0))\cup(\Gamma_\beta+(R+1,0)),$$ and note that $F\subset \widehat U$ and $\R^2\setminus F$ has a connected component $W$ that contains $[0,1]^2$ and is contained in $[-2R, 2R+1]\times [-2R, 2R+1]$ (see figure \ref{fig:1}). \begin{figure}\label{fig:1}
\end{figure}
Let $K = \bd W$. Then $K$ is a compact subset of $\widehat U$. Since $\widehat{U} = \bigcup_{v\in\Z^2}\bigcup_{i\in\Z} T_v(\widehat f^{i}(O))$ is an open cover of $K$, choosing a finite subcover we conclude the existence of $M$. \endproof
\begin{proposition}\label{pr:fundamentaldomainmeets} For every $w\in\Z^2,$ if $n>\frac{\norm{w}}{\delta},$ then $\widehat f^n([0,1]^2)$ intersects $T_w([0,1]^2).$ \end{proposition} \proof By our choice of $\delta$ (at the beginning of this section), $w/n$ is in the interior of $\rho(\widehat{f})$. It follows from theorem \ref{th:Franks} that there is $\widehat y\in \R^2$ such that $\widehat f^n(\widehat y)=\widehat y + w$. Since we may choose $\widehat y \in [0,1]^2$ (by using an appropriate integer translation), the proposition follows. \endproof
\begin{proposition}\label{pr:cadawtemumtransladado} For every $w\in\Z^2,$ and every $n>\frac{\norm{w}}{\delta},$ there exists $(j_w,v_w)$ in $\Sigma(2M)$ such that $\widehat f^{n}(\widehat f^{j_w}(T_{v_w}(O)))\cap T_w(O)\neq \emptyset$. \end{proposition}
\proof By proposition \ref{pr:fundamentaldomainmeets}, if $U$ is the connected component of $\R^2\setminus K$ that contains $[0,1]^2,$ then $\widehat f^n(U)$ intersects $T_w(U)$, and since $U$ is bounded this implies that $\widehat f^n(\partial U)$ intersects $T_w(\partial U).$ Since $\partial U\subset K$, it follows that $\widehat f^{n}(K)\cap T_w(K)\not=\emptyset.$ If $\widehat x$ is a point in this intersection, then there exist integers $j_1, j_2\in [-M,M]$ and $v_1, v_2\in \Z^2$ with $\norm{v_i}_{\infty}<M$, $i\in\{1,2\},$ such that $\widehat x$ belongs to both $\widehat f^{n}(\widehat f^{j_1}(T_{v_1}(O))) $ and $\widehat f^{j_2}(T_{v_2}(O+w)).$ Setting $j_w=j_1-j_2$ and $v_w=v_1-v_2$ yields the result. \endproof
\begin{lemma}\label{lm:inbetween} Let $v\in \R^2$ be a nonzero vector, and $K_1$, $K_2$ be two arcwise connected subsets of $\R^2$ such that $T_v(K_i)\cap K_i=\emptyset$ for $i\in \{1,2\}$. Suppose there are integers $i,j$ with $ i\geq 0$, $j > 0$ such that $T_v^{-i}(K_1)\cap K_2\not=\emptyset$ and $T_v^{j}(K_1)\cap K_2\not=\emptyset$. Then $K_1$ intersects $K_2$. Moreover, there exists a $T_v$-translation arc $\gamma$ contained in $K_1\cup K_2$ and joining a point $x\in K_1$ to $T_v(x)\in K_2$. \end{lemma}
\proof We may assume that $K_1$ is compact by replacing it by a compact arc contained in $K_1$ and joining some point of $T_v^{i}(K_2)$ to a point of $T_v^{-j}(K_2)$.
Let $\alpha:[0,1]\to K_2$ be a simple arc satisfying $\alpha(0)\in T^{-i}_{v}(K_1)$ and $\alpha(1)\in T^j_{v}(K_1).$ Since $[\alpha]$ and $K_1$ are compact and $v$ is not null, there exists an integer $n_0$ such that, if $\abs{n}>n_0,$ then $T^{n}_v(K_1)$ is disjoint from $[\alpha].$
Let \begin{eqnarray}\nonumber s_0 = \max\{t\in[0,1]: \alpha(t)\in \bigcup_{n=0}^{n_0}T^{-n}_{v}(K_1)\},\\
s_1 = \min\{t\in[s_0,1]: \alpha(t)\in \bigcup_{n=1}^{n_0}T^n_{v}(K_1)\},\nonumber \end{eqnarray} and let $i_0$ and $j_0$ be integers such that $\alpha(s_0)\in T^{-i_0}_{v}(K_1)$ and $\alpha(s_1)\in T^{j_0}_{v}(K_1).$ Finally, let $$K_3 = T^{-i_0}_{v}(K_1)\cup\alpha([s_0,s_1])\cup T^{j_0}_{v}(K_1),$$ which is a connected set.
We claim that $i_0=0$ and $j_0=1$. To prove our claim, assume that it does not hold. Then $i_0+j_0>1$. Note that by construction, $\alpha((s_0,s_1))$ is disjoint from $\bigcup_{n\in\Z}T^n_{v}(K_1).$ Since $K_1\cap T_v(K_1)=\emptyset$, it follows from lemma \ref{lm:disjunto1vezdisjuntosempre} that $K_1\cap T_v^n(K_1)=\emptyset$ for any $n \neq 0$, and so $T_v(T^{-i_0}_{v}(K_1))$ is disjoint from $T^{j_0}_{v}(K_1)$ (because $i_0+j_0\neq 1$). From these facts and from the hypotheses follows that $K_3$ is disjoint from $T_v(K_3)$, so again by lemma \ref{lm:disjunto1vezdisjuntosempre} we have that $K_3$ is disjoint from $T_v^n(K_3)$ for all $n\neq 0$. But $i_0+j_0\neq 0$, and clearly $T^{j_0+i_0}_v(K_3)$ intersects $K_3$. This contradiction shows that $i_0+j_0 = 1$, i.e. $i_0=0$ and $j_0=1.$
Since $\alpha(s_0)\in K_2\cap T^{-i_0}_v(K_1) = K_2\cap K_1$, we have shown that $K_1$ intersects $K_2$. Note that $\alpha(s_1)\in T_{v}^{j_0}(K_1) = T_v(K_1)$, and let $\beta:[0,1]\to K_1$ be a simple arc joining $\alpha(s_1)-v$ to $\alpha(s_0)$. Since $[\beta]\subset K_1$ and $\alpha([s_0,s_1))\cap (T_v(K_1)\cup T_{-v}(K_1))=\emptyset,$ it follows that $[\beta]\cap T_v(\alpha([s_0,s_1)))= \emptyset=T_v([\beta])\cap \alpha([s_0,s_1))$. Therefore, letting $\gamma$ be the concatenation of $\beta$ with $\alpha|_{[s_0,s_1]}$ we conclude that $\gamma$ is a $T_v$-translation arc joining $x=\alpha(s_1)-v\in K_1$ to $T_v(x)\in K_2$. \endproof
Define, for each $(j,v)\in \Z\times \Z^2,$ the sets $$I(j,v)=\{w\in\Z^2 : \, \widehat f^{j}(T_v(O))\cap T_w(O)\not=\emptyset\}.$$
\begin{proposition}\label{pr:propertiesofsetofindices} The following properties hold. \begin{enumerate} \item{ If $R>0$ and $n>\frac{R}{\delta}$, then $$\left(\Z^2\cap B_R((0,0))\right)\subset\bigcup_{(j,v)\in \Sigma(2M)} I(j+n,v).$$} \item{For any $w,v\in\Z^2$ and $ j\in\Z,\, T_w(I(j,v))= I(j, T_w(v))$.} \item{ Let $u,v \in \Z^2$ and $j\in\Z.$ If $S(u,C)\subset I(j,v)$, then $S(u,C-\norm{v-w}_{\infty})\subset I(j,w)$.}
\end{enumerate} \end{proposition}
\proof Parts (1) and (2) are direct consequences of definition of $I(j,v)$ and proposition \ref{pr:cadawtemumtransladado}. To prove (3), let $ r \in S(u,C-\norm{v-w}_{\infty})$; since $ r+v-w \in S(u,C)\subset I(j,v)$, it follows from (2) that $r= T_{w-v}(r+v-w)\in I(j,T_{w-v}(v))=I(j,w)$. \endproof
\begin{proposition}\label{pr:pintasegmento} Let $i,k_0,k_1,j$ be integers with $k_0<k_1$, and $u\in\Z^2.$ If both $(i,k_0)$ and $(i,k_1)$ belong to $I(j,u),$ then $(i,k) \in I(j,u)$ for each integer $k\in [k_0,k_1]$. Likewise, if both $(k_0,i)$ and $(k_1,i)$ belong to $I(j,u),$ then $(k,i) \in I(j,u)$ for all integers $k\in [k_0,k_1]$. \end{proposition} \begin{proof} This is a direct consequence of lemma \ref{lm:inbetween}, choosing $K_1 = O +(i,k)$, $K_2=f^{j}(T_u(O))$ and $v=(1,0)$ for the first case, and an analogous choice for the second case. \end{proof}
\begin{lemma}\label{lm:pintasubquadrado} Given $C_1\in\Z$ there is $C_2>0$ such that, for any $w_1\in \Z^2$, $C>C_2$, and $n>(\norm{w_1}+\sqrt{2}C)/\delta$, there exist $w_2\in\Z^2$ and $\overline j \in [-2M,2M]\cap \Z$, such that $$S(w_2,C_1)\subset S(w_1, C)\cap I(n+\overline j, \overline w).$$ \end{lemma}
\proof Define the auxiliary constants $L= (4M+1)^3 +1$, which is larger than the cardinality of the set $\Sigma(2M)$, and $R=2M+\norm{\overline{w}}_{\infty}.$ Let also $D= 2C_1 + 2R+1$, and let $C_2=L^L D.$
Note that if $C>C_2$ and $n$ is chosen as in the statement, proposition \ref{pr:propertiesofsetofindices} (1) implies that $$S(w_1, C)\subset \bigcup_{(j,v)\in \Sigma(2M)}I(n+j,v).$$ Let $i_0,k_0$ be integers such that $(i_0,k_0)= w_1 - (C,C)$. Define, for $1\le s\le L-1, \, k_s = k_0+sD$. For $1\le s \le L$ we will define $i_s$ in $\Z$ and $(j_s,v_s)\in\Sigma(2M)$ recursively satisfying the following properties: \begin{enumerate} \item{$i_{s-1}\le i_{s} \le i_{s-1} +\frac{C_2}{L^{s-1}}-\frac{C_2}{L^{s}}$} \item{For any $ i_{s}\le i \le i_s+ \frac{C_2}{L^s}$, the point $(i,k_{s-1})\in I(n+j_s,v_s)$.} \end{enumerate}
Suppose we have already defined $i_s$ and $(j_s,v_s).$ Consider the $L$ different points of the form $(i_{s,r},k_s)$ with $0\le r \le L-1$, where $i_{s,r}=i_s+r\frac{C_2}{L^{s+1}}$. Note that from the recursion hypothesis (1) follows that $i_{s} \leq i_0+C_2-C_2/L^{s}$, so that $$i_{s,r}\leq i_0+C_2 - C_2/L^s +(L-1)C_2/L^{s+1} = i_0+C_2 - C_2/L^{s+1} \leq i_0+C.$$ In particular, $(i_{s,r},k_s) \in S(w_1, C)$ for each $r\in \{0,\dots, L-1\}$, so there exists $(j_{s,r},v_{s,r})\in \Sigma(2M)$, satisfying $(i_{s,r},k_s)\in I(n+j_{s,r},v_{s,r})$. Since $L$ is greater than the cardinality of $\Sigma(2M)$, by the pigeonhole principle there exists $r_1<r_2$ such that $(j_{s,r_1},v_{s,r_1})=(j_{s,r_2},v_{s,r_2})$. Define $i_{s+1}=i_{s,r_1}$ and $(j_{s+1},v_{s+1})=(j_{s,r_1},v_{s,r_1})$. Since both $(i_{s,r_1},k_s)$ and $(i_{s,r_2},k_s)$ belong to $I(n+ j_{s+1},v_{s+1})$, it follows from proposition \ref{pr:pintasegmento} that $(i,k_s)\in I(n+j_{s+1},v_{s+1})$ whenever $i_{s,r_1}\le i \le i_{s,r_2}$. Note also that $i_{s+1}\ge i_s$, and that $i_{s+1}+\frac{C_2}{L^{s+1}}\le i_{s,r_2}\le i_s+ \frac{C_2}{L^{s}}$. So, $i_{s+1}$ and $(j_{s+1},v_{s+1})$ satisfy properties (1) and (2) (see figure \ref{fig:2}).
\begin{figure}\label{fig:2}
\end{figure}
Having defined $i_s$ and $(j_s,v_s)$ for $1\le s\le L$, we can again use the pigeonhole principle to find integers $1\le s_1<s_2\le L$ such that $(j_{s_1},v_{s_1})=(j_{s_2},v_{s_2}).$
By property (2) in the recursion, for each integer $i \in [i_{s_2}, i_{s_2}+\frac{C_2}{L^{s_2}}]$ the point $(i_{s_2}+i,k_{s_2-1})$ belongs to $I(n+j_{s_2},v_{s_2}).$
As $i_{s_1}\le i_{s_2}$ and $i_{s_2}+\frac{C_2}{L^{s_2}}< i_{s_1}+\frac{C_2}{L^{s_1}}$, then the interval $[i_{s_2},i_{s_2}+\frac{C_2}{L^{s_2}}]$ is contained in $[i_{s_1},i_{s_1}+\frac{C_2}{L^{s_1}}] $, so again by property (2), for each integer $ i \in [i_{s_2}, i_{s_2}+\frac{C_2}{L^{s_2}}]$, the point $(i_{s_2}+i,k_{s_1-1})$ belongs to $I(n+j_{s_1},v_{s_1}) = I(n+j_{s_2},v_{s_2}).$
Hence, for all $0\le i\le \frac{C_2}{L^{s_2}},$ both $(i_{s_2}+i,k_{s_1-1})$ and $(i_{s_2}+i,k_{s_2-1})$ belong to $I(n+j_{s_2},v_{s_2})$. From proposition \ref{pr:pintasegmento} applied to the latter pair of points for each $i$ follows that $$\{(i,k): i_{s_2}\le i \le i_{s_2}+\frac{C_2}{L^{s_2}}, k_{s_1-1}\le k \le k_{s_2-1}\}\subset I(n+j_{s_2},v_{s_2}).$$
Note that $k_{s_2-1}-k_{s_1-1}\ge D,$ and also $\frac{C_2}{L^{s_2}}\ge D.$ Therefore, if we define $w_2=(i_{s_2},k_{s_1-1})+ (R+ C_1+1,R+ C_1+1)$ then $S(w_2, R+C_1)\subset I(n+j_{s_2},v_{s_2}),$ and $S(w_2, R+C_1)\subset S(w_1, C).$
Finally, since $\norm{v_{s_2}-\overline{w}}_{\infty}\le R$, if $\overline{j}=j_{s_2}$ we have, by proposition \ref{pr:propertiesofsetofindices}, $S(w_2, C_1)\subset I(n+\overline{j},\overline{w})$. \endproof
\begin{proposition}\label{pr:intersquadrantes} For every $R>0$ there exists $n_0 \in \N$ such that, if $n>n_0$, then $\widehat f^n(O)$ intersects each of the four sets $$ U_1= \{(y: (y)_1>R, (y)_2>R\},\quad U_2=\{y:(y)_1<-R, (y)_2>R\},$$ $$ U_3= \{y: (y)_1>R, (y)_2<-R\},\quad U_4=\{y:(y)_1<-R, (y)_2<-R\}$$ \end{proposition}
\proof Let $C_1$ be such that $O\subset B_{C_1}((0,0)),$ and let $C_2=\max_{x\in \R^2} \norm{\smash{\widehat f(x)-x}}$, which is finite since $f$ is homotopic to the identity.
Let $C> 2M(C_2+1)+ C_1+ R$ be an integer, and consider $w_1=(C,C), w_2=(C,-C), w_3=(-C,C)$ and $w_4=(-C,-C).$
If $n> \frac{\sqrt{2} C}{\delta}$ then, by proposition \ref{pr:cadawtemumtransladado}, for each $i\in\{1,2,3,4\}$ there exists $(j_i, v_i)\in \Sigma(2M)$ such that $w_i\in I(n+j_i,v_i)$. This implies that $\widehat f^{n+j_i}(T_{v_i}(O))$ intersects $T_{w_i}(O)\subset T_{w_i}(B_{C_1}((0,0)))=B_{C_1}(w_i)$ for each $i\in \N$. Since the definition of $C_2$ implies that, for every $x\in\R^2$, $$\norm{\smash{\widehat f^{n+j_i}(x)-\widehat f^{n}(x)}}\le \abs{j_i} C_2\le 2MC_2,$$ we conclude that $\widehat f^{n}(T_{v_i}(O))$ intersects $B_{C_1+2M C_2}(w_i)$. This means that $\widehat f^{n}(O)$ intersects $T_{v_i}^{-1}(B_{C_1+2M C_2}(w_i))$. Since $\norm{v_i}_{\infty}\le 2M$, it follows that
$$T_{v_i}^{-1}(B_{C_1+2M C_2}(w_i))\subset B_{C_1+2M (C_2+1)}(w_i)\subset U_i,$$ so $\widehat{f}^n(O)\cap U_i\neq \emptyset$. \endproof
\begin{claim}\label{cl:mainclaim} Suppose that $\widehat{f}^n(O)\cap T_{\overline{w}}(O)=\emptyset$ for all $n\in \Z$. Let $k\in \N$ and $v\in\Z^2$ be such that $S(v,1)\subset I(k,\overline{w}).$ Then, for every sufficiently large $n,$ we have $v\in I(n,(0,0)).$ \end{claim}
\proof Assume, by contradiction, that there exists a sequence $(n_i)_{i\in\N}$ such that $\lim_{i\to\infty}n_i=\infty$ and such that, $v\notin I(n_i,(0,0))$ for all $i\in \N$. Then, by proposition \ref{pr:pintasegmento}, we cannot have that both $v-(1,0)$ and $v+(1,0)$ belong to $I(n_i,(0,0))$ simultaneously, and neither can $v-(0,1)$ and $v+(0,1)$. Thus we may assume, with no loss of generality, that for infinitely many values of $i$ neither $v+(1,0)$ nor $v+(0,1)$ belong to $I(n_i,(0,0))$ (since the other cases are analogous). This means that $$\widehat f^{n_i}(O)\cap \left(T_v(O)\cup T_{v+(1,0)}(O)\cup T_{v+(0,1)}(O)\right)=\emptyset.$$ Extracting a subsequence of $(n_i)_{i\in \N}$ we may (and will) assume that the latter holds for \emph{all} $i\in \N$.
Let $K_2=\widehat f^{k}(T_{\overline{w}}(O)).$ Since $S(v,1)\subset I(k,\overline{w}),$ it follows that $K_2$ intersects the open connected sets $T_v(O)$ and $T_{(1,0)}(T_v(O))$ and therefore lemma \ref{lm:inbetween} (with $K_1=T_v(O)$) implies that there exists a $T_{(1,0)}-$translation arc $\alpha$ contained in $T_v(O)\cup K_2$ joining a point $x_1\in T_v(O)$ to $T_{(1,0)}(x_1).$ Note that the assumption that $\widehat{f}^n(O)$ is disjoint of $T_{\overline{w}}(O)$ for all $n\in\Z$ implies that $K_2\cap \widehat f^{n_i}(O)=\emptyset$ for all $i \in \N$. Furthermore, since $v\notin I(n_i,(0,0))$, the set $\widehat f^{n_i}(O)$ is also disjoint from $T_v(O)$. Thus $[\alpha]\cap \widehat f^{n_i}(O)=\emptyset$ for all $i\in \N$.
A similar reasoning shows that there exists a translation arc $\beta\subset T_v(O)\cup K_2,$ joining a point $x_2\in T_v(O)$ to $T_{(0,1)}(x_2)$, such that $[\beta]\cap \widehat f^{n_i}(O)=\emptyset$ for all $i\in \N$.
Let $\gamma\colon [0,1]\to T_v(O)$ be an arc joining $x_1$ to $x_2$, and define $$\alpha^{+}=\bigcup_{i\in\N}T^i_{(1,0)}([\alpha]),\,\, \alpha^{-}=\bigcup_{i\in\N}T^{-i}_{(1,0)}([\alpha]),\,\, \beta^{+}=\bigcup_{i\in\N}T^i_{(0,1)}([\beta]),\,\, \beta^{-}=\bigcup_{i\in\N}T^{-i}_{(0,1)}([\beta]).$$
Since $\widehat f^{n_i}(O)$ is disjoint from $[\alpha] \cup [\beta] \cup [\gamma]$, it follows from lemma \ref{lm:Brouwerline} that $\widehat f^{n_i}(O)$ is disjoint from at least one of the following 4 connected sets: $$F_1=\alpha^{+}\cup\beta^{+}\cup[\gamma], \, F_2=\alpha^{+}\cup\beta^{-}\cup[\gamma],$$ $$F_3=\alpha^{-}\cup\beta^{+}\cup[\gamma], \, F_4=\alpha^{-}\cup\beta^{-}\cup[\gamma].$$
In particular there is $j\in\{1,2,3,4\}$ such that $\widehat{f}^{n_i}(O)$ is disjoint from $F_j$ for infinitely many values of $i\in \N$. We assume that $j=1$, since the other cases are analogous. Extracting again a subsequence of $(n_i)_{i\in \N}$, we may assume that $f^{n_{i}}(O)$ is disjoint from $F_1$ for all $i\in \N$.
Let $R_1=\max_{y\in\alpha} \norm{y}$, $R_2=\max_{y\in\beta}\norm{y}$, $R_3=\max_{y\in\gamma}\norm{y},$ and note that $\max_{y\in\alpha^{+}}(y)_2\le R_1$, and $ \max_{y\in\beta^{+}}(y)_1\le R_2.$
Finally, let $R=\max\{R_1,R_2,R_3\},$ so that $$F_1\subset \{y : \abs{(y)_1}\leq R\}\cup \{y : \abs{(y)_2}\leq R\},$$ and observe that the sets $$ U_1 = \{y : (y)_1>R, (y)_2>R\},\quad U_2=\{y : (y)_1<-R, (y)_2>R\}$$ lie in distinct connected components of $\R^2\setminus F_1$ (see figure \ref{fig:3}).
\begin{figure}\label{fig:3}
\end{figure}
As $\widehat f^{n_i}(O)$ is connected and contained in $\R^2\setminus F_1$, for each $i\in \N$ the set $\widehat{f}^{n_i}(O)$ is disjoint from either $U_1$ or $U_2$. This contradicts proposition \ref{pr:intersquadrantes}, completing the proof of the claim. \endproof
Let us denote by $R_{\overline{w}}(v)= 2\overline{w}-v=T^2_{\overline{w}-v}(v)$ the symmetric point of $v$ with respect to $\overline{w}$.
\begin{claim}\label{cl:pintaquadradoeoposto} There exist $k_1, k_2\in \Z$ and $\overline{v}\in \Z^2$ such that $S(\overline{v},1)\subset I(k_1,\overline{w})$ and $S(R_{\overline{w}}(\overline{v}),1)\subset I(k_2,\overline{w}).$ \end{claim}
\proof
Let $C_2>0$ be such that the conclusion of lemma \ref{lm:pintasubquadrado} holds with $C_1=1$. Using again lemma \ref{lm:pintasubquadrado} but setting $C_1=C_2+1$, and $w_1=(0,0)$, there exists $C_2'>0$, $n_1\in \N$, an integer $j_1\in [-2M,2M]$, and $v_1\in \Z^2$ such that $$S(v_1,C_2+1)\subset S((0,0),C_2'+1) \cap I(n_1+j_1, \overline{w}).$$
Due to our choice of $C_2$, lemma \ref{lm:pintasubquadrado} applied with $C_1=1$ and $w_1=R_{\overline{w}}(v_1)$ implies that there exist $n_2\in \N$, an integer $j_2\in [-2M, 2M]$ and $v_2\in \Z^2$ such that $$S(v_2,1)\subset S(R_{\overline{w}}(v_1),C_2+1)\cap I(n_2+j_2,\overline{w}).$$
Let $\overline{v}=v_2$, $k_1=n_2+j_2$ and $k_2=n_1+j_1$. Then, $S(\overline{v},1)\subset I(k_1,\overline{w})$, while $$S(R_{\overline{w}}(\overline{v}),1) = R_{\overline{w}}(S(\overline{v},1))\subset R_{\overline{w}}(S(R_{\overline{w}}(v_1),C_2+1)) = S(v_1, C_2+1)\subset I(k_2,\overline{w}),$$ so the claim follows (see figure \ref{fig:claim2}). \begin{figure}\label{fig:claim2}
\end{figure} \endproof
To complete the proof of theorem \ref{th:maintheorem}, suppose, suppose that $\widehat f^{n}(O)\cap T_{\overline{w}}(O)= \emptyset$ for all $n\in \Z$. Claims \ref{cl:mainclaim} and \ref{cl:pintaquadradoeoposto} together imply that, for sufficiently large $n,$ both $\overline{v} \in I(n,(0,0))$ and $2\overline{w}- \overline{v} =R_{\overline{w}}(\overline{v}) \in I(n,(0,0))$. This means that $\widehat f^{n}(O) \cap T_{\overline{v}}(O)\neq \emptyset$ and $\widehat f^{n}(O) \cap T_{2\overline{w}-\overline{v}}(O) \neq \emptyset$. Letting $K_1= T_{\overline{w}}(O)$ and $K_2= \widehat f^{n}(O)$, it follows that $T_{\overline{w}-\overline{v}}^{-1}(K_1)\cap K_2\neq \emptyset$ and $T_{\overline{w}-\overline{v}}(K_1)\cap K_2\neq \emptyset$. Thus, lemma \ref{lm:inbetween} (with $v=\overline{w}-\overline{v}$) implies that $K_1$ intersects $K_2$, i.e.\ $\widehat f^{n}(O)\cap T_{\overline{w}}(O)\neq \emptyset$. This contradicts our assumption at the beginning of this paragraph, completing the proof. \qed
\end{document} | arXiv |
Woodin cardinal
In set theory, a Woodin cardinal (named for W. Hugh Woodin) is a cardinal number $\lambda $ such that for all functions
$f:\lambda \to \lambda $
there exists a cardinal $\kappa <\lambda $ with
$\{f(\beta )\mid \beta <\kappa \}\subseteq \kappa $
and an elementary embedding
$j:V\to M$
from the Von Neumann universe $V$ into a transitive inner model $M$ with critical point $\kappa $ and
$V_{j(f)(\kappa )}\subseteq M.$
An equivalent definition is this: $\lambda $ is Woodin if and only if $\lambda $ is strongly inaccessible and for all $A\subseteq V_{\lambda }$ there exists a $\lambda _{A}<\lambda $ which is $<\lambda $-$A$-strong.
$\lambda _{A}$ being $<\lambda $-$A$-strong means that for all ordinals $\alpha <\lambda $, there exist a $j:V\to M$ which is an elementary embedding with critical point $\lambda _{A}$, $j(\lambda _{A})>\alpha $, $V_{\alpha }\subseteq M$ and $j(A)\cap V_{\alpha }=A\cap V_{\alpha }$. (See also strong cardinal.)
A Woodin cardinal is preceded by a stationary set of measurable cardinals, and thus it is a Mahlo cardinal. However, the first Woodin cardinal is not even weakly compact.
Consequences
Woodin cardinals are important in descriptive set theory. By a result[1] of Martin and Steel, existence of infinitely many Woodin cardinals implies projective determinacy, which in turn implies that every projective set is Lebesgue measurable, has the Baire property (differs from an open set by a meager set, that is, a set which is a countable union of nowhere dense sets), and the perfect set property (is either countable or contains a perfect subset).
The consistency of the existence of Woodin cardinals can be proved using determinacy hypotheses. Working in ZF+AD+DC one can prove that $\Theta _{0}$ is Woodin in the class of hereditarily ordinal-definable sets. $\Theta _{0}$ is the first ordinal onto which the continuum cannot be mapped by an ordinal-definable surjection (see Θ (set theory)).
Mitchell and Steel showed that assuming a Woodin cardinal exists, there is an inner model containing a Woodin cardinal in which there is a $\Delta _{4}^{1}$-well-ordering of the reals, ◊ holds, and the generalized continuum hypothesis holds.[2]
Shelah proved that if the existence of a Woodin cardinal is consistent then it is consistent that the nonstationary ideal on $\omega _{1}$ is $\aleph _{2}$-saturated. Woodin also proved the equiconsistency of the existence of infinitely many Woodin cardinals and the existence of an $\aleph _{1}$-dense ideal over $\aleph _{1}$.
Hyper-Woodin cardinals
A cardinal $\kappa $ is called hyper-Woodin if there exists a normal measure $U$ on $\kappa $ such that for every set $S$, the set
$\{\lambda <\kappa \mid \lambda $ is $<\kappa $-$S$-strong$\}$
is in $U$.
$\lambda $ is $<\kappa $-$S$-strong if and only if for each $\delta <\kappa $ there is a transitive class $N$ and an elementary embedding
$j:V\to N$
with
$\lambda ={\text{crit}}(j),$
$j(\lambda )\geq \delta $, and
$j(S)\cap H_{\delta }=S\cap H_{\delta }$.
The name alludes to the classical result that a cardinal is Woodin if and only if for every set $S$, the set
$\{\lambda <\kappa \mid \lambda $ is $<\kappa $-$S$-strong$\}$
is a stationary set.
The measure $U$ will contain the set of all Shelah cardinals below $\kappa $.
Weakly hyper-Woodin cardinals
A cardinal $\kappa $ is called weakly hyper-Woodin if for every set $S$ there exists a normal measure $U$ on $\kappa $ such that the set $\{\lambda <\kappa \mid \lambda $ is $<\kappa $-$S$-strong$\}$ is in $U$. $\lambda $ is $<\kappa $-$S$-strong if and only if for each $\delta <\kappa $ there is a transitive class $N$ and an elementary embedding $j:V\to N$ with $\lambda ={\text{crit}}(j)$, $j(\lambda )\geq \delta $, and $j(S)\cap H_{\delta }=S\cap H_{\delta }.$
The name alludes to the classic result that a cardinal is Woodin if for every set $S$, the set $\{\lambda <\kappa \mid \lambda $ is $<\kappa $-$S$-strong$\}$ is stationary.
The difference between hyper-Woodin cardinals and weakly hyper-Woodin cardinals is that the choice of $U$ does not depend on the choice of the set $S$ for hyper-Woodin cardinals.
Notes and references
1. A Proof of Projective Determinacy
2. W. Mitchell, Inner models for large cardinals (2012, p.32). Accessed 2022-12-08.
Further reading
• Kanamori, Akihiro (2003). The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. ISBN 3-540-00384-3.
• For proofs of the two results listed in consequences see Handbook of Set Theory (Eds. Foreman, Kanamori, Magidor) (to appear). Drafts of some chapters are available.
• Ernest Schimmerling, Woodin cardinals, Shelah cardinals and the Mitchell-Steel core model, Proceedings of the American Mathematical Society 130/11, pp. 3385–3391, 2002, online
• Steel, John R. (October 2007). "What is a Woodin Cardinal?" (PDF). Notices of the American Mathematical Society. 54 (9): 1146–7. Retrieved 2008-01-15.
| Wikipedia |
Recent questions asked in neet 2020 p2
neet 2020 p2
by Mukesh in Physics and Measurement asked Nov 14, 2021 376 views
The angle of 1" (second of arc) in a radian is equal to:
by JeeNeetQnA in Some Basic Principles of Organic Chemistry asked Feb 7, 2021 2,118 views
How many (i) sp2 hybridised carbon atoms and (ii) π bonds are present in the following compound?
by JeeNeetQnA in Solutions asked Feb 7, 2021 54 views
Isotonic solutions have same
Isotonic solutions have same (1) Boiling temperature (2) Vapour pressure (3) Freezing temperature (4) Osmotic pressure
by JeeNeetQnA in Hydrocarbons asked Feb 7, 2021 532 views
CH3CH2CH=CH2 → B2H6 H2O, H2O2, OH− Z. What is Z?
$\require{mhchem}\ce{CH3CH2CH=CH2->[\ce{B2H6}][\ce{H2O,H2O2,OH-}]Z}.$ What is Z? (1) $\ce{CH3CH2CH2CH3}$ (2) $\ce{CH3CH2CH2CH2OH}$ (3) $\begin{array}{c}\ce{CH3CH2CHCH3}\\\quad|\\\quad\ \ \ \ce{OH}\end{array}$ (4) $\ce{CH3CH2CH2CHO}$
by JeeNeetQnA in Classification of Elements and Periodicity in Properties asked Feb 7, 2021 590 views
Match the following aspects with the respective metal. Aspects Metal (a) The metal which reveals a maximum number of
by JeeNeetQnA in Some p - Block Elements asked Feb 7, 2021 128 views
Which of the following oxide is amphoteric in nature?
Which of the following oxide is amphoteric in nature? (1) CO2 (2) SnO2 (3) SiO2 (4) GeO2
carbon-family
by JeeNeetQnA in Some Basic Concepts of Chemistry asked Feb 7, 2021 1,533 views
One mole of carbon atom weighs 12 g, the number of atoms in it is equal to, (Mass of carbon-12 is 1.9926 × 10–23 g)
by JeeNeetQnA in Aldehydes, Ketones and Carboxylic Acids asked Feb 7, 2021 518 views
Which of the following acid will form an (a) Anhydride on heating and (b) Acid imide on strong heating with ammonia?
chemical-reaction-of-carboxylic-acids
by JeeNeetQnA in Electrochemistry asked Feb 7, 2021 1,878 views
Identify the reaction from following having top position in EMF series (Std. red. potential) according to their
standard-reduction-potential
by JeeNeetQnA in s - Block Elements (Alkali and Alkaline Earth Metals) asked Feb 7, 2021 300 views
Identify the correct statement from the following.
Identify the correct statement from the following. (1) Lithium chloride is deliquescent and crystallises as a hydrate, LiCI·H2O. (2) The order of hydration enthalpies of alkaline earth cations Be2+ < Mg2+ < Ca2+ < Sr2+ < Ba2+ (3) Lithium and Magnesium show ...
by JeeNeetQnA in Biomolecules asked Feb 7, 2021 83 views
Which of the following statement is not true about glucose?
Which of the following statement is not true about glucose? (1) It is an aldopentose. (2) It is an aldohexose. (3) It contains five hydroxyl groups. (4) It is a reducing sugar.
by JeeNeetQnA in Chemical Bonding and Molecular Structure asked Feb 7, 2021 351 views
Match the compounds of Xe in column I with the molecular structure in column II. (a) XeF2 (i) Square planar (b) XeF4 | CommonCrawl |
\begin{document}
\title{The peripatric coalescent} \author{\textsc{By Amaury Lambert$^{1,2}$ and Chunhua Ma$^{1,2,3}$} } \date{} \maketitle
\noindent\textsc{$^1$ UPMC Univ Paris 06\\ Laboratoire de Probabilit\'es et Mod\`eles Al\'eatoires CNRS UMR 7599}\\ \noindent\textsc{$^2$ Coll\`ege de France\\ Center for Interdisciplinary Research in Biology CNRS UMR 7241\\ Paris, France}\\\textsc{E-mail: }[email protected]\\ \textsc{URL: }http://www.proba.jussieu.fr/pageperso/amaury/index.htm\\ \\ \noindent\textsc{$^3$Nankai University\\ School of Mathematical Sciences and LPMC \\ Tianjin, P.\,R.\ China}\\ \textsc{E-mail: }[email protected]\\ \textsc{URL: } http://math.nankai.edu.cn/$\sim$mach
\begin{abstract} \noindent We consider a dynamic metapopulation involving one large population of size $N$ surrounded by colonies of size $\varepsilon_{_N}N$, usually called peripheral isolates in ecology, where $N\to\infty$ and $\varepsilon_{_N}\to 0$ in such a way that $\varepsilon_{_N}N\to\infty$. The main population periodically sends propagules to found new colonies (emigration), and each colony eventually merges with the main population (fusion). Our aim is to study the genealogical history of a finite number of lineages sampled at stationarity in such a metapopulation.
We make assumptions on model parameters ensuring that the total outer population has size of the order of $N$ and that each colony has a lifetime of the same order. We prove that under these assumptions, the scaling limit of the genealogical process of a finite sample is a censored coalescent where each lineage can be in one of two states: an inner lineage (belonging to the main population) or an outer lineage (belonging to some peripheral isolate). Lineages change state at constant rate and inner lineages (only) coalesce at constant rate per pair.
This two-state censored coalescent is also shown to converge weakly, as the landscape dynamics accelerate, to a time-changed Kingman coalescent. \end{abstract}
\textit{Running head.} The peripatric coalescent.\\ \textit{Key words and phrases.} Censored coalescent; metapopulation; weak convergence; peripheral isolate; population genetics; peripatric speciation; phylogeny.\\ \textit{AMS 2000 subject classifications.} Primary 60K35; Secondary 60J05; 60G10; 92D10; 92D15; 92D25; 92D40.
\section{Introduction}
\setcounter{equation}{0}
Many plant and animal populations in nature are highly fragmented, and this fragmentation plays a prominent role in the context of adaptation and speciation. Indeed, the emergence of new species is usually thought to be driven by geographical processes \cite{CO04}. First, \emph{allopatric speciation} occurs when various subpopulations belonging to the same initial species are separated by a geographical barrier that prevents hybridization between them (gene flow) and allows them to diverge (genetical differentiation) by local adaptation. Second, \emph{parapatric speciation} is a version of allopatric speciation where local adaptation is mediated by the existence of an environmental gradient (resource availability, environmental conditions). Third, when a species is present in one large, panmictic population surrounded by small colonies, usually called peripheral isolates, it is believed that the combination of founder events and of local adaptation to borderline environmental conditions leads to the formation of new species within the isolates. This phenomenon is called \emph{peripatric speciation}. We aim to study the genealogy of populations embedded in such a spatial context. The present study should then serve as a building brick for future work in the field of speciation modeling.
Population dynamic models specifying explicitly the spatial context are called \emph{metapopulation models} (Hanski and Gilpin \cite{HG97}). Typical such models include: island model, isolation by distance, stepping stone models, extinction-recolonization models. From the point of view of speciation, all these models suffer from the same defect: they assume a given, constant number of subpopulations in the metapopulation, with fixed migration rates between them. As one of the authors of the present paper suggested (Lambert \cite{L10}), an alternative method would consist in considering species as ``spread out on a randomly evolving number of locations, allowing for repeated fragmentations of colonies, colonizations of new locations, as well as secondary contacts between subpopulations''. This author and others have designed such dynamic landscape models \cite{K00, ALC11, ACL13}, but usually in detailed ecological contexts whose study is only possible through numerical simulations (to the exception of \cite{ACL09}).
Here we propose a mathematical study of a dynamic landscape of the peripatric type. More specifically, we consider a dynamic population subdivision which involves one large main population surrounded by a random number of small peripheral isolates, that we will call colonies for simplicity. The size of the main population is constant equal to $N$, the size of each colony is constant equal to $\varepsilon_{_N}$ and the genealogy in each population is given by the Moran model. The number of colonies at time $t$ is denoted by $\xi_N(t)$. The landscape dynamics is as follows (see Figure 1): \begin{itemize} \item At constant rate $\theta_N$, each individual sends independently $\varepsilon_{_N}N$ offspring to found a new colony; \item Each colony independently merges again with the main population at rate $\gamma_N \xi_N^{\alpha -1}$, where $\alpha \ge 1$; at such so-called fusion time, $\varepsilon_{_N} N$ individuals among the new $(1+\varepsilon_{_N}) N$ individuals of the main population are chosen uniformly and simultaneously killed to keep its size constant. \end{itemize} Note that $(\xi_N(t);t\ge 0)$ is a pure-death process with immigration. The parameter $\alpha$ is meant to model the competition for space, since the fusion rate per colony grows with the number of colonies. This density-dependence disappears if $\alpha$ is chosen equal to 1.
The main purpose of this paper is to investigate the genealogy of a finite sample of lineages in the above peripatric metapopulation model. We will show that the history of such a sample, viewed backward in time, can be approximated, as $N\to\infty$ under certain assumptions, by a two-state censored coalescent, where the state of a lineage can be inner (lying in the main population) or outer (lying in a colony). Lineages change state at a constant rate per lineage, but only inner lineages can coalesce, at a constant rate per pair of lineages, as in Kingman coalescent \cite{K82}.
A two-state censored coalescent can be viewed as a new type of structured coalescent. The structured coalescent (see Takahata \cite{T88}, Notohara \cite{N90} and Herbots \cite{H97}) describes the ancestral genealogical process of a sample of lineages in a subdivided population connected by migration. The coalescent on two subpopulations was considered by \cite{T88}; for a finite number of subpopulations by \cite{N90}, and placed in a rigorous framework by \cite{H97}. To date, there have been a number of works dealing with the structured coalescent arising in various special types of metapopulations; see Nordborg and Krone \cite{NK02}, Eldon \cite{E09} and the references therein. Our results show that new types of structured coalescents can arise in some specific dynamic metapopulations.
We now give the heuristics giving rise to the result. We assume that $N\to\infty$ and $\varepsilon_{_N}\to 0$ in such a way that $\varepsilon_{_N}N\to\infty$, so that the size of colonies is large but neglectable compared to the main population (Assumption A). It is known that in a Moran model, inner lineages coalesce at constant rate per pair when time is rescaled by $N$ (Kingman coalescent \cite{K82}). We make assumptions on the parameters ensuring that all events changing the configuration of ancestral lineages occur on this time scale. This can only be done to the exception of coalescences in colonies, which happen instantaneously in the new time scale, leading to outer lineages which always all lie in different colonies. Also, in order to have a total outer population size of the order of $N$, we need to have a number of colonies of the order of $\varepsilon_{_N}^{-1}$. This can be achieved by the following choice of parameters (Assumption B). The \emph{per capita} emigration rate $\theta_N$ is taken equal to $$ \theta_N= \frac{\theta}{\varepsilon_{_N} N^2} , $$ and the fusion rate $\gamma_N$ is taken equal to $$ \gamma_N = \gamma \frac{\varepsilon_{_N}^{\alpha -1}}{N} . $$ Under theses assumptions, the number of colonies is asymptotically deterministic, equal to $\varepsilon_{_N}^{-1}\,(\theta/\gamma)^{1/\alpha}$.
Now the rate at which a single inner lineage changes state is the rate at which a single lineage is taken in a fusion event (backward in time), which happens at rate $$ \frac{\varepsilon_{_N}}{1+\varepsilon_{_N}} \,\gamma_N\, \xi_N^{\alpha} \approx \frac{\varepsilon_{_N}}{1+\varepsilon_{_N}} \,\gamma_N \,\varepsilon_{_N}^{-\alpha}\,\frac{\theta}{\gamma}, $$ which is equivalent to $\theta/N$ as $N\to\infty$. As a consequence, in the new time scale, inner lineages become outer lineages at contant rate $\theta$.
Also note that the probability that two lineages are taken in the same fusion vanishes, so that no two lineages can lie within the same colony. As a consequence, outer lineages are not allowed to coalesce.
Now the lifetime of a colony is approximately exponential with parameter $$ \gamma_N\, \xi_N^{\alpha-1} \approx \gamma_N\,\varepsilon_{_N}^{1-\alpha}\,\left(\frac{\theta}{\gamma}\right)^{1-1/\alpha}, $$ which is equivalent to $(\theta/N)\,(\gamma/\theta)^{1/\alpha}$. As a consequence, in the new time scale, outer lineages become inner lineages at contant rate $\theta\,(\gamma/\theta)^{1/\alpha}$. By making these heuristics rigorous we get the results stated in Theorem \ref{thm}. Namely, the genealogical history of a finite sample of lineages, seen as a process backward in time, converges weakly (except at time 0, where instantaneous coalescences within colonies makes the limiting process not right-continuous) to the following two-state censored coalescent. Inner lineages coalesce at constant rate 1 per pair, and lineages change type at constant rate per lineage: inner lineages become outer lineages at rate $\theta$ and outer lineages become inner lineages at rate $\theta\,(\gamma/\theta)^{1/\alpha}$.
The paper is organized as follows. In Section 2, we give a detailed description of our dynamic metapopulation model in forward and backward time. The main result, Theorem \ref{thm}, is stated in Section 3. In addition, we also prove that under fast landscape dynamics, the censored coalescent converges weakly to a time-changed version of the Kingman coalescent \cite{K82}. Finally, a section is dedicated to the formal proofs of the above results.
\section{Metapopulation model} \setcounter{equation}{0}
\subsection{Forward dynamics}
Let $N\in\mathbb{N}$ with $\mathbb{N}:=\{0,1,2,\cdots\}$ and let $\varepsilon_{_N}$ be any positive number such that $\varepsilon_{_N} N\in\mathbb{N}$. Let $\theta_N$, $\gamma_N$ and $\alpha$ be positive constants. Consider a dynamic metapopulation model involving one large population of size $N$, called main population, and a random number of small populations, called colonies, of size $\varepsilon_{_N}N$. The main population periodically sends propagules (or emigrants) that found new colonies and ultimately each colony merges again with the main population. A further assumption is as follows. See Figure 1 for an illustration.
\noindent{\rm (1)} The number of colonies, denoted by $\{\xi_N(t): t\geq0\}$, evolves as a pure death density-dependent process with immigration and the transition rates are given by \begin{eqnarray}}\def\eeqlb{\end{eqnarray} \begin{array}{lll}
j\rightarrow j+1 &\text{ at rate } & N\theta_N,\\
j\rightarrow j-1 &\text{ at rate } & \gamma_Nj^\alpha.
\end{array}
\label{transition rate}
\eeqlb When $\alpha=1$, the process $\{\xi_N(t)\}$ is reduced to a pure death branching process with immigration. It follows from Kelly \cite{K79} that $\{\xi_N(t)\}$ with any initial value has the stationary distribution $\pi_N$ given by
\begin{eqnarray}}\def\eeqlb{\end{eqnarray}
\pi_N(0)=\Big(1+\sum_{j=1}^\infty\frac{(N\theta_N/\gamma_N)^j}{(j!)^\alpha}\Big)^{-1}
\ \mbox{and}\
\pi_N(k)=\frac{(N\theta_N/\gamma_N)^k}{(k!)^\alpha}
\Big(1+\sum_{j=1}^\infty\frac{(N\theta_N/\gamma_N)^j}{(j!)^\alpha}\Big)^{-1}
\label{stationary distribution}
\eeqlb for $k\geq1$. We assume that $\xi_N(0)$ is distributed as $\pi_N$. Then $\{\xi_N(t)\}$ is a stationary Markov chain. Let $(P_t^N)_{t\geq0}$ be its semigroup. For any finite set $\{t_1< t_2< \cdots< t_n\}\subset \mbb{R}$ define the probability measure on $\mathbb{N}$ by
\begin{eqnarray}}\def\eeqlb{\end{eqnarray}\label{2.16} \eta^N_{t_1,t_2,\cdots,t_n}(j_1,j_2,\cdots,j_n)
= \pi^N(j_1)P^N_{t_2-t_1}(j_1,j_2)\cdots P^N_{t_n-t_{n-1}}(j_{n-1},j_n).
\eeqlb Then $\{\eta^N_{t_1,t_2,\cdots,t_n}: t_1< t_2< \cdots< t_n\in \mbb{R}\}$ is a consistent family. By Kolmogorov's theorem, there is a stochastic process $\{\xi_N(t): t\in \mbb{R}\}$ with finite-dimensional distributions given by (\ref{2.16}). Clearly, $\{\xi_N(t): t\in \mbb{R}\}$ is a stationary Markov chain with one-dimensional marginal distribution $\pi_N$ and transition semigroup $(P^N_t)_{t\ge 0}$.
\noindent{\rm (2)} At the jump times of $\xi_N(t)$ from $j$ to $j+1$, one individual, chosen uniformly at random from the large population, gives birth to $\varepsilon_{_N}N$ emigrant offspring individuals which found a new colony. We refer to such an event as ``emigration'' (of new colonies) or ``fission".
\noindent{\rm (3)} At the jump times of $\xi_N(t)$ from $j$ to $j-1$, one colony is chosen at random from the $j$ current colonies and all the $\varepsilon_{_N}N$ individuals within this colony immediately migrate back into the main population. We refer to such an event as a ``fusion'' (of colonies with the main population). Instead of keeping all those $(1+\varepsilon_{_N})N$ individuals in the main population alive, only $N$ of them survive this fusion event, which are chosen uniformly at random among the $(1+\varepsilon_{_N})N$ previously existing individuals.
\noindent{\rm (4)} Between the jump times of $\xi_N(t)$, the large population and the colonies independently evolve as Moran models, that is, at rate $1$ each individual independently gives birth to a single offspring, and simultaneously a uniformly chosen individual is killed.
\includegraphics[width=0.95\textwidth]{graph.eps}
\subsection{Backward dynamics}
Now we start with a sample of $n$ lineages at time 0 and proceed backward in time. Let ${\mbox{\bf X}}_N(t)=(X_N^0(t),X_N^1(t),\cdots,X_N^n(t))$ be the ancestral process of this sample defined for $t\ge 0$ by \begin{itemize} \item[\;]$X_N^0(t)=$ the number of lineages in the main population at time $-t$,
\item[\;]${X}_N^i(t)=$ the number of colonies containing $i$ lineages at time $-t$ \ ($1\leq i\leq n$). \end{itemize} We set ${\mbox{\bf X}}_N(0)={\mbox{\bf x}}$, where ${\mbox{\bf x}}=(x_0,x_1,\cdots,x_n)\in\mathbb{N}^{n+1}$ with $x_0+\sum_{j=1}^njx_j=n$. The process $\{{\mbox{\bf X}}_N(t): t\geq0 \}$ has state-space
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*} E:=\Big\{(x_0,x_1,\cdots,x_{n})\in \mathbb{N}^{n+1}: 1\leq x_0+\sum_{j=1}^{n}jx_j\leq n \Big\}.
\eeqnn Define the subspace $\Pi$ of $E$ by \begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
\Pi:=\big\{(x_0,x_1,0,\cdots,0)\in \mathbb{N}^{n+1}: 1\leq x_0+x_1\leq n\big\}.
\eeqnn Consider the projection $\Gamma: (x_0,x_1,0,\cdots,0)\mapsto (x_0,x_1)$ from $\Pi$ to $\mathbb{N}^2$. \begin{eqnarray*}}\def\eeqnn{\end{eqnarray*} \Gamma(\Pi)=\big\{(x_0,x_1)\in \mathbb{N}^{2}: 1\leq x_0+x_1\leq n\big\}.
\eeqnn By the action of the homeomorphism $\Gamma$, $\Gamma(\Pi)$ can be regarded as a subspace of $E$, and we thus still denote it by $\Pi$ for simplicity. For ${\mbox{\bf x}}\in E$, let $$ \bar{{\bf x}}:=\Big(x_0,\sum_{j=1}^nx_j\Big). $$ We will use this notation for the following reason. Because in the new time scale lineages lying in the same colony immediately coalesce, the configuration ${\bf x}$ immediately turns into $(x_0,\sum_{j=1}^nx_j, 0,\ldots, 0)$ where all outer lineages are now alone in their respective colonies. Note that ${\mbox{\bf x}}\mapsto\bar{{\mbox{\bf x}}}$ is an injection from $E$ to $\Pi$. We also write ${\mbox{\bf e}}_j=(0,\ldots,0,1,0,\ldots,0)\in\mathbb{N}^{n+1}$ whose $(j+1)$-th component is $1$ for $j=0,\ldots,n$.
Let $\eta_N(t)=\xi_N(-t)$ for $t\geq0$. It follows from \cite[Lemma 1.5, P.9]{K79} that $\{\eta_N(t): t\geq0 \}$ is still a stationary Markov process with the same transition rates as (\ref{transition rate}). Thus, the fission events (fusions seen backward) happen at rate $\theta_NN$ and, conditioned on $\eta_N(t)$, the fusion events (fissions seen backward) happen at rate $\gamma_N\eta_N^{\alpha}(t)$. At any fission time, every lineage independently exits from the main population with probability $\varepsilon_{_N} /(1+\varepsilon_{_N})$. At any fusion time, one colony is chosen at random from the existing colonies and the (say) $i$ lineages in this colony enter the main population, and simultaneously coalesce together (if $i\ge 2$), and coalesce with their ancestor in the main population (if it is also in the sample; but asymptotically, with high probability $i=1$ and the ancestor is not in the sample). Between fission and fusion times, coalescences within the main population or within colonies may happen. We again refer to Figure 1 for an illustration.
Based on the above description, it is not hard to see that $\{({\mbox{\bf X}}_N(t),\eta_N(t)): t\geq0\}$ is a time-homogeneous Markov chain taking values in
$E\times \mathbb{R}_+$. The corresponding generator is given by
\begin{eqnarray}}\def\eeqlb{\end{eqnarray}
\bar{A}_Ng({\mbox{\bf x}},k)=\bar{\psi}_Ng({\mbox{\bf x}},k)+\bar{\phi}_Ng({\mbox{\bf x}},k)+\bar{\Gamma}_Ng({\mbox{\bf x}},k)
\label{bar A_N}
\eeqlb
for any bounded function $g$ on $E\times\mathbb{N}$. Here
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
\bar{\psi}_Ng({\mbox{\bf x}},k)=\sum_{j=2}^{n}x_j\binom{j}{2}\frac{2}{\varepsilon_{_N}N-1}
\Big(g({\mbox{\bf x}}-{\mbox{\bf e}}_j+{\mbox{\bf e}}_{j-1},k)-g({\mbox{\bf x}},k)\Big),
\eeqnn
which corresponds to coalescences within colonies. Note that $\psi_Ng({\mbox{\bf x}},u)\equiv0$ if
${\mbox{\bf x}}\in\Pi$. Then
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
\bar{\phi}_Ng({\mbox{\bf x}},k)
\!\!\!&=\!\!\!&
\binom{x_0}{2}\frac{2}{N-1}\Big(g({\mbox{\bf x}}-{\mbox{\bf e}}_1,k)-g({\mbox{\bf x}},k)\Big)\\
\!\!\!& \!\!\!&
+\,N\theta_{N} \sum_{r=1}^{x_0}\binom{x_0}{r}\Big(\frac{\varepsilon_{_N}}{1+\varepsilon_{_N}}\Big)^r
\Big(\frac{1}{1+\varepsilon_{_N}}\Big)^{x_0-r}
\Big(g({\mbox{\bf x}}-r{\mbox{\bf e}}_0+{\mbox{\bf e}}_{r},k+1)-g({\mbox{\bf x}},k)\Big)\\
\!\!\!& \!\!\!&
+\,\gamma_N k^{\alpha}(x_1/k)(1-(x_0/N))
\Big(g({\mbox{\bf x}}-{\mbox{\bf e}}_1+{\mbox{\bf e}}_0,k-1)-g({\mbox{\bf x}},k)\Big)1_{\{k>0\}}\\
\!\!\!& \!\!\!&
+\,\gamma_Nk^{\alpha}\sum_{j=2}^{N}(x_j/k)(1-(x_0/N))
\Big(g({\mbox{\bf x}}-{\mbox{\bf e}}_j+{\mbox{\bf e}}_0,k-1)-g({\mbox{\bf x}},k)\Big)1_{\{k>0\}}\\
\!\!\!& \!\!\!&
+\,\gamma_Nk^{\alpha}\sum_{j=1}^{N}(x_j/k)(x_0/N)
\Big(g({\mbox{\bf x}}-{\mbox{\bf e}}_j,k-1)-g({\mbox{\bf x}},k)\Big)1_{\{k>0\}}.
\eeqnn In $\bar{\phi}_N$, the first term corresponds to coalescences within the main population, the second term corresponds to exit from the main population, the third term corresponds to entrance into the main population, the last two terms correspond to simultaneous entrance into the main population and coalescence. The fourth term is identically equal to $0$ if ${\mbox{\bf x}}\in\Pi$. Last,
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
\bar{\Gamma}_Ng({\mbox{\bf x}},k)
\!\!\!&=\!\!\!&
N\theta_N\Big(\frac{1}{1+\varepsilon_{_N}}\Big)^{x_0}
\Big(g({\mbox{\bf x}},k+1)-g({\mbox{\bf x}},k)\Big)\\
\!\!\!& \!\!\!&\,
+\gamma_Nk^{\alpha}\Big(1-\sum_{j=1}^{N}x_j/k\Big)
\Big(g({\mbox{\bf x}},k-1)-g({\mbox{\bf x}},k)\Big)1_{\{k>0\}},
\eeqnn
which corresponds to the event that the number of colonies increases or decreases
but the ancestral process does not change.
\section{Convergence to the two-state censored coalescent}
\setcounter{equation}{0}
\subsection{Main results}
Let $D([0,\infty), S)$ be the space of all c\`{a}dl\`{a}g functions $x: [0,\infty) \rightarrow S$ endowed with the Skorokhod topology for any separable and complete metric space $S$; see Ethier and Kurtz \cite[p.116]{EK86} for details. For $N\in\mathbb{N}$, we consider the sequence of processes $\{({\mbox{\bf X}}_N(\cdot), \eta_N(\cdot))\}$. Define
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*} {\mbox{\bf Y}}_N(t)={\mbox{\bf X}}_N(Nt)\quad\mbox{and}\quad\tilde{\eta}_N(t)=\varepsilon_{_N}\eta_N(Nt).
\eeqnn Let $\theta>0$ and $\gamma>0$ be constants. We further assume the following conditions: \begin{itemize}
\item[(A)]\;$\varepsilon=\varepsilon_{_N}$ satisfying $\varepsilon_{_N}\rightarrow0$
and $\varepsilon_{_N} N\rightarrow\infty$ as $N\rightarrow\infty$;
\item[(B)]\;$\theta_N=\theta/(\varepsilon_N N^2)$ and $\gamma_N=\gamma\varepsilon_{_N}^{\alpha-1}/N$. \end{itemize} Recall that $ {\mbox{\bf y}}\in E$ and the corresponding $\bar{{\mbox{\bf y}}}\in\Pi$. The main result of the paper follows. \begin{theorem}}\def\etheorem{\end{theorem}\label{thm}\;Let ${\mbox{\bf x}}\in E$. Under conditions (A) and (B), the finite-dimensional distributions of the ancestral process $\{{\mbox{\bf Y}}_N(t),\,t\geq0\}$ starting at ${\mbox{\bf y}}$ converges to those of a $\Pi$-valued continuous time Markov chain $\{{\mbox{\bf Y}}(t),\ t\geq0\}$ starting at $\bar{{\mbox{\bf y}}}$, except at time $0$. The corresponding infinitesimal generator ${\bf Q}=(q_{{\mbox{\bf r}},{\mbox{\bf r}}'})_{{\mbox{\bf r}},{\mbox{\bf r}}'\in\Pi}$ is given by
\begin{eqnarray}}\def\eeqlb{\end{eqnarray} q_{{\mbox{\bf r}},{\mbox{\bf r}}'} = \left\{ \begin{array}{ll}
-(\theta r_0+\theta(\gamma/\theta)^{1/\alpha}r_1+r_0(r_0-1)), &
\text{ if } {\mbox{\bf r}}'={\mbox{\bf r}},\\
\theta r_0, &
\text{ if } r_0\neq 0\text{ and }{\mbox{\bf r}}'={\mbox{\bf r}}+(-1,1), \\
\theta(\gamma/\theta)^{1/\alpha}r_1, &
\text{ if } r_1\neq0\text{ and } {\mbox{\bf r}}'={\mbox{\bf r}}+(1,-1),\\
r_0(r_0-1), &
\text{ if } {\mbox{\bf r}}'={\mbox{\bf r}}+(-1,0),\\
0, &
\text{ otherwise.}
\end{array} \right.
\label{Y}
\eeqlb where ${\mbox{\bf r}}=(r_0,r_1)\in\Pi$. Furthermore, if the initial value ${\mbox{\bf y}}\in\Pi$, weak convergence on $D([0,\infty), \Pi)$ to $\{{\mbox{\bf Y}}(t)\}$ holds. \etheorem The previous statement describes the asymptotic genealogical history of a finite sample of lineages, seen as a process backward in time. Except at time 0, where instantaneous coalescences within colonies makes the limiting process not right-continuous, this process converges weakly to a two-state censored coalescent, where type 0 corresponds to inner lineages (lying in the main population) and type 1 to outer lineages (lying in pairwise distinct colonies). Inner lineages coalesce at constant rate 1 per (ordered) pair, and lineages change type at constant rate per lineage: inner lineages become outer lineages at rate $\theta$ and outer lineages become inner lineages at rate $\theta\,(\gamma/\theta)^{1/\alpha}$.\\ \\ Now consider a sequence of censored coalescent processes $\{{\mbox{\bf Y}}_k(t)\}$ defined by (\ref{Y}) with parameters $\theta$ and $\gamma$ replaced by $\theta_k$ and $\gamma_k$, and the initial value ${\mbox{\bf Y}}_k(0)={\mbox{\bf y}}\in\Pi$ with $y_0+y_1=n$. Let $Y_k(t)=Y_k^0(t)+Y_k^1(t)$ and let $I_n=\{0,1,2,\cdots,n\}$. We assume that
\begin{itemize}
\item[(C)]\;As $k\rightarrow\infty$, $\theta_k\rightarrow\infty$, $\gamma_k\rightarrow\infty$ and
$\theta_k/\gamma_k\rightarrow p$ for some constant $p>0$.
\end{itemize} The above condition corresponds to the acceleration of the landscape dynamics (emigration and fusion). The following theorem states that such an acceleration gives rise to a single state coalescent process, where coalescence rates are obtained by averaging over the probability of presence in the main population.
\begin{theorem}}\def\etheorem{\end{theorem}\label{thm2}\; Under condition (C), the process $\{Y_k(t),\,t\geq0\}$ starting at $n$ converges weakly to the time-changed $n$-Kingman coalescent process $\{K(t),\,t\geq0\}$ on $D([0,\infty), I_n)$. When $K=l$, the coalescence rate is given by
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
c_l=\sum_{j=1}^lj(j-1)\binom{l}{j}\Big(\frac{p^{1/\alpha}}{1+p^{1/\alpha}}\Big)^j
\Big(\frac{1}{1+p^{1/\alpha}}\Big)^{l-j}.
\eeqnn
\etheorem \begin{remark}}\def\eremark{\end{remark} It is easy to see that if $p=0$ which corresponds to predominant emigrations, $\{Y_k(t),\,t\geq0\}$ converges weakly to the constant process $\{K(t)\equiv n, t\geq0\}$; if $p=\infty$ which corresponds to predominant fusions, $\{Y_k(t),\,t\geq0\}$ converges weakly to the standard Kingman coalescent $\{K(t), t\geq0\}$ (i.e., $c_l=l(l-1)$). \eremark \subsection{Proofs}
To prove Theorem \ref{thm}, we start by proving the following lemmas.
\begin{lemma}}\def\elemma{\end{lemma}\label{lemma 1}\;Under conditions (A) and (B), as $N\rightarrow\infty$,
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
\tilde{\eta}_N(\cdot)\overset{p}{\longrightarrow}(\theta/\gamma)^{1/\alpha}
\eeqnn in $D([0,\infty), \mathbb{R}_+)$. \elemma
\noindent{\it Proof.~~}}\def\qed{
$\Box$
Recall that the number $\xi_N(Nt)$ of colonies of size $\varepsilon_{_N}N$ is a pure death density-dependent
process with immigration with transition rates given by \begin{eqnarray*}}\def\eeqnn{\end{eqnarray*} \begin{array}{lll}
j\rightarrow j+1 &\text{ at rate } & \theta/\varepsilon_{_N},\\
j\rightarrow j-1 &\text{ at rate } & \gamma\varepsilon_{_N}^{\alpha-1}j^\alpha.
\end{array}
\eeqnn By (\ref{stationary distribution}), it has the stationary distribution $\pi_N$ given by
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
\pi_N(0)=\Big(1+\sum_{j=1}^\infty\frac{(\theta/(\gamma\varepsilon_{_N}^\alpha))^j}{(j!)^\alpha}\Big)^{-1}
\ \mbox{and}\
\pi_N(k)=\frac{(\theta/(\gamma\varepsilon_{_N}^\alpha))^k}{( k!)^\alpha}
\Big(1+\sum_{j=1}^\infty\frac{(\theta/(\gamma\varepsilon_{_N}^\alpha))^j}{(j!)^\alpha}\Big)^{-1}.
\eeqnn Recall that $\xi_N(0)$ is distributed as $\pi_N$. Let $M_{N}=[(\theta/\gamma)^{\frac{1}{\alpha}}/\varepsilon_{_N}]$. For $k>M_N$,
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
\!\!\!&\!\!\!&\pi_N(k)\\
\!\!\!&=\!\!\!&\frac{(\theta/(\gamma\varepsilon_{_N}^\alpha))^{M_N}(\theta/(\gamma\varepsilon_{_N}^\alpha))^{k-M_N}} {(M_N!\prod_{j=M_N+1}^k j)^\alpha}
\bigg(1+\Big(\sum_{j=1}^{M_N}+\sum_{j=M_N+1}^\infty\Big)\frac{(\theta/(\gamma\varepsilon_{_N}^\alpha))^j}{(j!)^\alpha}\bigg)^{-1}\\
\!\!\!&=\!\!\!&
\prod_{j=M_N+1}^k\frac{\theta/(\gamma\varepsilon_{_N}^\alpha)}{j^\alpha}
\bigg(\frac{(M_N!)^\alpha}{(\theta/(\gamma\varepsilon_{_N}^\alpha))^{M_N}}
+\sum_{j=1}^{M_N-1}\prod_{i=j+1}^{M_N}\frac{i^\alpha}{\theta/(\gamma\varepsilon_{_N}^\alpha)}
+1
+\sum_{j=M_N+1}^\infty\prod_{i=M_N+1}^j\frac{\theta/(\gamma\varepsilon_{_N}^\alpha)}{i^\alpha}\bigg)^{-1}.
\eeqnn Note that if $j>M_N$, $j^\alpha>\theta/(\gamma\varepsilon_{_N}^\alpha)$ and if $j>3M_N$, $\frac{\theta/(\gamma\varepsilon_{_N}^\alpha)}{j^\alpha}\leq \frac{\theta/(\gamma\varepsilon_{_N}^\alpha)}{(3M_N)^\alpha}<\frac{1}{2^\alpha}$ for sufficiently large $N$. Then $\prod_{j=M_N+1}^{4M_N}\frac{\theta/(\gamma\varepsilon_{_N}^\alpha)}{j^\alpha}\leq 2^{-\alpha M_N}$ and
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*} \sum_{k=4M_N+1}^\infty\prod_{j=M_N+1}^k\frac{\theta/(\gamma\varepsilon_{_N}^\alpha)}{j^\alpha}
\leq \prod_{j=M_N+1}^{4M_N}\frac{\theta/(\gamma\varepsilon_{_N}^\alpha)}{j^\alpha}\sum_{k=1}^\infty2^{-\alpha k}\leq O(2^{-\alpha M_N}).
\eeqnn If follows that $\pi_N([4M_N,\infty))\leq O(2^{-\alpha M_N})$. Thus the sequence $\{\varepsilon_{_N}\xi_N(0)\}$ is tight. On the other hand, $\{\varepsilon_{_N}\xi_N(Nt)\}$ takes values in $\{i\varepsilon_{_N}: i\in\mathbb{N}\}$ and its generator is given by
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
L_Nf(z)=\gamma\varepsilon_{_N}^{\alpha-1}(z/\varepsilon_{_N})^\alpha(f(z-\varepsilon_{_N})-f(z))
+(\theta/\varepsilon_{_N})(f(z+\varepsilon_{_N})-f(z)),
\eeqnn for any continuous bounded function $f$ on $\mathbb{R}_+$. Let $C_c^2(\mathbb{R}_+)$ be the set of
twice differentiable functions with compact support on $\mathbb{R}_+$.
It is not hard to see that as $N\rightarrow\infty$, for $f\in C_c^2(\mathbb{R}_+)$,
\begin{eqnarray}}\def\eeqlb{\end{eqnarray}
\|L_Nf-Lf\|\rightarrow0 \mbox{ and } Lf(z)=(\theta-\gamma z^\alpha)f'(z),
\label{xi}
\eeqlb where $\|f\|=\sup_{x\in\mathbb{R}_+}|f(x)|$. The Markov process $\xi$ with generator $L$ is actually deterministic and satisfies the ODE:
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
\xi'(t)=\theta-\gamma\xi^\alpha(t),
\eeqnn which has the unique equilibrium point $(\theta/\gamma)^{1/\alpha}$. It follows from (\ref{xi}), \cite[Theorem 6.1, P.28]{EK86} and \cite[Theorem 9.10, P.244]{EK86} that $\varepsilon_{_N}\xi_N(0)\overset{w}{\longrightarrow} (\theta/\gamma)^{1/\alpha}$ as $N\rightarrow\infty$. Again by (\ref{xi}), \cite[Corollary 8.7, p.231]{EK86} shows that $\{\xi_N(t):t\geq0\}$ converges weakly to the constant function $\{\xi(t)\equiv(\theta/\gamma)^{1/\alpha},\ t\geq0\}$ on $D([0,\infty), \mathbb{R}_+)$. Since $\xi_N(\cdot)$ is stationary and time-reversible, the same weak convergence holds for $\tilde{\eta}_N(\cdot)$. Then this lemma is proved. \qed
As in Section 2 it is easy to see that
$({\mbox{\bf Y}}_N(\cdot), \tilde{\eta}_N(\cdot))$ is a continuous time Markov chain taking values in
$E\times\mathbb{R}_+$. Based on (\ref{bar A_N}) and Conditions (A) and (B), a simple calculation shows that
the corresponding generator is given by
\begin{eqnarray}}\def\eeqlb{\end{eqnarray}
A_Ng({\mbox{\bf y}},u)=\psi_Ng({\mbox{\bf y}},u)+\phi_Ng({\mbox{\bf y}},u)+\Gamma_Ng({\mbox{\bf y}},u)
\label{A_N}
\eeqlb
for any bounded function $g$ on $\mathbb{R}_+\times E$. Here
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
\psi_Ng({\mbox{\bf y}},u)=2\sum_{j=2}^{n}y_j\binom{j}{2}\frac{1}{\varepsilon_{_N}}
\Big(1+\frac{1}{\varepsilon_{_N}N-1}\Big)
\Big(g({\mbox{\bf y}}-{\mbox{\bf e}}_j+{\mbox{\bf e}}_{j-1},u)-g({\mbox{\bf y}},u)\Big).
\eeqnn
Note that $1/(\varepsilon_{_N}N-1)\rightarrow0$ as $N\rightarrow\infty$ by Condition (A). We also have
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
\phi_Ng({\mbox{\bf y}},u)
\!\!\!&=\!\!\!&
2\binom{y_0}{2}\Big(g({\mbox{\bf y}}-{\mbox{\bf e}}_0,u)-g({\mbox{\bf y}},u)\Big)\\
\!\!\!& \!\!\!&
+\,\theta y_0
\Big(g({\mbox{\bf y}}-{\mbox{\bf e}}_0+{\mbox{\bf e}}_1,u+\varepsilon_{_N})-g({\mbox{\bf y}},u)\Big)\\
\!\!\!& \!\!\!&
+\,\gamma u^{\alpha-1}y_1
\Big(g({\mbox{\bf y}}-{\mbox{\bf e}}_1+{\mbox{\bf e}}_0,u-\varepsilon_{_N})-g({\mbox{\bf y}},u)\Big)1_{\{u>0\}}\\
\!\!\!& \!\!\!&
+\,\gamma u^{\alpha-1}\sum_{j=2}^{N}y_j
\Big(g({\mbox{\bf y}}-{\mbox{\bf e}}_j+{\mbox{\bf e}}_0,u-\varepsilon_{_N})-g({\mbox{\bf y}},u)\Big)1_{\{u>0\}}\\
\!\!\!& \!\!\!&
+\,\Big(\varepsilon_{_N}R_{1,N}g({\mbox{\bf y}},u)+\frac{1}{N}u^{\alpha-1}1_{\{u>0\}}
R_{2,N}g({\mbox{\bf y}},u)\Big).
\eeqnn Here the fourth term is identically equal to $0$ if ${\mbox{\bf y}}\in\Pi$. In the last term, $R_{1,N}$ and $R_{2,N}$
are bounded linear operators satisfying $\|R_{i,N}\|\leq C$ for some constant $C$. This last term includes the simultaneous entrance into the main population and coalescence and it vanishes if $c_1\leq u\leq c_2$ for positive numbers $c_1$ and $c_2$. Last, we have
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
\Gamma_Ng({\mbox{\bf y}},u)
\!\!\!&=\!\!\!&
\theta\varepsilon_{_N}^{-1}(1-y_0\varepsilon_{_N})
\Big(g({\mbox{\bf y}},u+\varepsilon_{_N})-g({\mbox{\bf y}},u)\Big)\\
\!\!\!& \!\!\!&\,
+\gamma u^{\alpha}\varepsilon_{_N}^{-1}\Big(1-\varepsilon_{_N}u^{-1}\sum_{j=1}^Ny_j\Big)
\Big(g({\mbox{\bf y}},u-\varepsilon_{_N})-g({\mbox{\bf y}},u)\Big)1_{\{u>0\}}.
\eeqnn Let us write $c_{\psi}^N({\mbox{\bf y}})$ (resp. $c_{\phi}^N({\mbox{\bf y}},u), c_{\Gamma}^N({\mbox{\bf y}},u))$ the total rate of the events generated by $\psi_N$ (resp. $\phi_N$, $\Gamma_N$) when $A_Ng$ is applied to $({\mbox{\bf y}},u)$. Then
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
c_{\psi}^N({\mbox{\bf y}})
=2\sum_{j=2}^{n}y_j\binom{j}{2}\frac{1}{\varepsilon_{_N}}
\Big(1+\frac{1}{\varepsilon_{_N}N-1}\Big),
\eeqnn
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
c_{\phi}^N({\mbox{\bf y}},u)
=
2\binom{y_0}{2}+\theta y_0
+\gamma u^{\alpha-1}1_{\{u>0\}}\sum_{j=1}^{N}y_j+\varepsilon_{_N}(1+u^{\alpha-1}1_{\{u>0\}})
\eeqnn
and
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
c_{\Gamma}^N({\mbox{\bf y}},u)
=
\theta\varepsilon_{_N}^{-1}(1-y_1\varepsilon_{_N})
+\gamma u^{\alpha}\varepsilon_{_N}^{-1}\Big(1-\varepsilon_{_N}u^{-1}\sum_{j=1}^Ny_j\Big)1_{\{u>0\}}.
\eeqnn
Let us introduce the following notation,
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
\sigma^N_0=\inf\{t\geq0: {\mbox{\bf Y}}_N(t)\in\Pi\}
\eeqnn
and
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
\sigma_1^N=\inf\{t\geq0:\ \mbox{a}\ \phi_N\mbox{-event occurs at }t\ \}.
\eeqnn \begin{lemma}}\def\elemma{\end{lemma}\label{lemma 2} $\sigma^N_0\overset{p}{\longrightarrow}0$ as $N\rightarrow\infty$. \elemma
\noindent{\it Proof.~~}}\def\qed{
$\Box$
By Lemma 1.3, we have for any $T$ and $0<\delta<(\theta/\gamma)^{1/\alpha}$, as $N\rightarrow\infty$,
\begin{eqnarray}}\def\eeqlb{\end{eqnarray}
{\mbox{\bf P}}\Big(\sup_{0\leq t\leq T}|\tilde{\eta}_N(t)-(\theta/\gamma)^{1/\alpha}|>\delta\Big)\rightarrow0.
\label{constant}
\eeqlb Fix above $\delta$. Let $c_1=(\theta/\gamma)^{1/\alpha}-\delta$ and $c_2=(\theta/\gamma)^{1/\alpha}+\delta$. Conditioned on $({\mbox{\bf Y}}_N(t),\tilde{\eta}_N(t))=({\mbox{\bf y}},u)$ with $({\mbox{\bf y}},u)\in(E\setminus\Pi)\times[c_1,c_2]$ at the current time $t$,
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
{\mbox{\bf P}}(\mbox{the next event is a } \phi^N\mbox{-event} )=\frac{c_{\phi}^N({\mbox{\bf y}},u)}
{c_{\psi}^N({\mbox{\bf y}})+c_{\phi}^N({\mbox{\bf y}},u)+c_{\Gamma}^N({\mbox{\bf y}},u)}\leq C\varepsilon_{_N},
\eeqnn for some positive constant $C$; \begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
{\mbox{\bf P}}(\mbox{the next event is a } \psi^N\mbox{-event} )=\frac{c_{\psi}^N({\mbox{\bf y}})}
{c_{\psi}^N({\mbox{\bf y}})+c_{\phi}^N({\mbox{\bf y}},u)+c_{\Gamma}^N({\mbox{\bf y}},u)}\leq
\frac{2n^3}{2n^3+\theta+\gamma c_1^\alpha},
\eeqnn for sufficiently large $N$;
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
{\mbox{\bf P}}(\mbox{the next event is a } \Gamma^N\mbox{-event} )=\frac{c_{\Gamma}^N({\mbox{\bf y}},u)}
{c_{\psi}^N({\mbox{\bf y}})+c_{\phi}^N({\mbox{\bf y}},u)+c_{\Gamma}^N({\mbox{\bf y}},y)}\leq
\frac{\theta+\gamma c_2^\alpha}{2+\theta+\gamma c_2^\alpha},
\eeqnn for sufficiently large $N$. Inspired by Taylor and V\'{e}ber \cite[Lemma 3.1]{TV09}, we fix some $s>0$ and consider
$$
{\mbox{\bf P}}(\sigma_0^N>s)= {\mbox{\bf P}}(D)+o(1),
$$ where $$
D = \{\sigma_0^N>s,\ \sup_{0\leq t\leq s}|\tilde{\eta}_N(t)-(\theta/\gamma)^{1/\alpha}|\leq\delta\}.
$$ Then
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
{\mbox{\bf P}}(D)
\!\!\!&=\!\!\!& {\mbox{\bf P}}(\{\mbox{at most } n\ \psi^N\mbox{-events occur in} [0,s]\}\cap D)\\
\!\!\!&=\!\!\!&{\mbox{\bf P}}(\{\mbox{at most } n\ \psi^N\mbox{-}\mbox{ and at least a } \phi^N\mbox{-events occur in} [0,s]\}\cap D)\\
\!\!\!& \!\!\!&
+\,{\mbox{\bf P}}(\{\mbox{at most } n\ \psi^N\mbox{-}\mbox{ and no } \phi^N\mbox{-events occur in} [0,s]\}\cap D)\\
\!\!\!&=:\!\!\!& I_1+I_2.
\eeqnn Note that we have $Y^N(t)\in E\setminus\Pi$ for $t\in[0,s]$ if $\sigma^N_0>s$. Let $p=\frac{2n^3}{2n^3+\theta+\gamma c_1^\alpha}\vee \frac{\theta+\gamma c_2^\alpha}{2+\theta+\gamma c_2^\alpha}$. Then
\begin{eqnarray}}\def\eeqlb{\end{eqnarray}
I_1\!\!\!&\leq\!\!\!&\sum_{k=0}^n {\mbox{\bf P}}(
\{\mbox{exactly } k\ \psi^N\mbox{-events}
\mbox{ before a }
\phi^N\mbox{-event occur in } [0,s]\}\cap D)\nonumber\\
\!\!\!&=\!\!\!&
\sum_{k=0}^n\sum_{l=0}^\infty{\mbox{\bf P}}(
\{\mbox{exactly } k\ \psi^N\mbox{- and }l\ \Gamma^N\mbox{-events }\mbox{before a }
\phi^N\mbox{-event in } [0,s]\}\cap D)
\nonumber\\
\!\!\!&\leq\!\!\!&
\sum_{k=0}^n\sum_{l=0}^\infty\binom{k+l}{k}p^{k+l}
(C\varepsilon_{_N}),
\label{I_1}
\eeqlb Since $0<p<1$, $\sum_{k=0}^n\sum_{l=0}^\infty\binom{k+l}{k}p^{k+l}<\infty$. Then $I_1\rightarrow0$ as $N\rightarrow\infty$. Let $U_j^N$ be the arrival time
of the $j$'th event occurring to $({\mbox{\bf Y}}_N,\tilde{\eta}_N)$. For $I_2$,
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
I_2
\!\!\!&=\!\!\!& \sum_{k=0}^n\sum_{l=0}^\infty{\mbox{\bf P}}(
\{\mbox{exactly } k\ \psi^N\mbox{-events, }l\ \Gamma^N\mbox{-events }\mbox{and no } \phi^N\mbox{-events occur in } [0,s]\}\cap D)\\
\!\!\!&\leq\!\!\!& \sum_{k=0}^n\sum_{l=0}^\infty\binom{k+l}{k}p^{k+l}{\mbox{\bf P}}(\{U_{k+l}^N<s, U^N_{k+l+1}>s\}\cap D).
\eeqnn Conditioned on $({\mbox{\bf Y}}_N(t),\tilde{\eta}_N(t))=({\mbox{\bf y}},u)$ with ${\mbox{\bf y}}\in(E\setminus\Pi)$, the rate for the event occurring to $({\mbox{\bf Y}}_N,\tilde{\eta}_N)$ at time $t$ is $c_{\psi}^N({\mbox{\bf y}})+c_{\phi}^N({\mbox{\bf y}},u)+c_{\Gamma}^N({\mbox{\bf y}},u)$ and $c^N_{\psi}({\mbox{\bf y}})\geq 2/\varepsilon_{_N}$. Then $U^N_{k+l+1}$ is stochastically bounded by the sum of $k+l+1$ i.i.d. exponential variables with parameter $2/\varepsilon_{_N}$ whose distribution becomes concentrated close to $0$ as $N\rightarrow\infty$. Thus as $N\rightarrow\infty$, ${\mbox{\bf P}}(\{U_{k+l}^N<s, U^N_{k+l+1}>s\}\cap D)\rightarrow0$ and by the dominated convergence theorem, $I_2\rightarrow0$.\qed
\begin{lemma}}\def\elemma{\end{lemma}\label{lemma 3} There exist positive constants $M$ and $K_1$ such that for any $s>0$,
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
\limsup_{N\rightarrow\infty}{\mbox{\bf P}}(\sigma^N_1\leq s)\leq M(1-e^{-K_1 s}).
\eeqnn \elemma
\noindent{\it Proof.~~}}\def\qed{
$\Box$
By the proof of $(\ref{I_1})$, ${\mbox{\bf P}}(\mbox{at least one }\phi_{N}\mbox{-event occurs before } \sigma_0^N)\rightarrow0$ as $N\rightarrow\infty$. Then by (\ref{constant}), we have $$ {\mbox{\bf P}}(\sigma_1^N\leq s) = {\mbox{\bf P}}(G)+o(1), $$ where
$$
G= \{\mbox{only }\psi_N\mbox{- or }\Gamma_N\mbox{- events before } \sigma_0^N,
\sup_{0\leq t\leq s}|\tilde{\eta}_N(t)-(\theta/\gamma)^{1/\alpha}|
\leq\delta\mbox{ and } \sigma_1^N\leq s\}. $$ Recall that ${\mbox{\bf Y}}_N(0)={\mbox{\bf y}}$. If only $\psi_N$- or $\Gamma_N$- events occur before $\sigma_0^N$, $\sigma_0^N<\sigma_1^N$ and ${\mbox{\bf Y}}_N(t)=\bar{{\mbox{\bf y}}}$ for $t\in[\sigma_0^N,\sigma_1^N]$. Furthermore $Y_N(t)\in\Pi$ and $c_{\psi}^N\equiv0$ for $t\geq\sigma_0^N$. Conditioned on $({\mbox{\bf Y}}_N(t),\tilde{\eta}_N(t))=({\mbox{\bf y}},u)$ with $({\mbox{\bf y}},u)\in\Pi\times[c_1,c_2]$, \begin{eqnarray*}}\def\eeqnn{\end{eqnarray*} K_1\leq c_{\phi}^N({\mbox{\bf y}},u)\leq K_2,\quad \varepsilon^{-1}_{_N}(\theta+\gamma c_1^{\alpha})/2\leq c_{\Gamma}^N({\mbox{\bf y}},u)\leq\varepsilon_{_N}^{-1}(\theta+\gamma c_2^\alpha), \eeqnn for sufficiently large $N$, where $K_1=[\gamma(c_1^{\alpha-1}\wedge c_2^{\alpha-1})]\wedge\theta$ and $K_2=n^2+n\theta+\gamma n(c_1^{\alpha-1}\vee c_2^{\alpha-1})$. Then
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
\frac{c_{\phi}^N({\mbox{\bf y}},u)}
{c_{\phi}^N({\mbox{\bf y}},u)+c_{\Gamma}^N({\mbox{\bf y}},u)}
\leq \frac{2K_2\varepsilon_{_N}} {2K_2\varepsilon_{_N}+\theta+\gamma c_1^\alpha},\quad \frac{c_{\Gamma}^N({\mbox{\bf y}},u)}
{c_{\phi}^N({\mbox{\bf y}},u)+c_{\Gamma}^N({\mbox{\bf y}},u)}
\leq \frac{\theta+\gamma c_2^\alpha} {K_1\varepsilon_{_N}+\theta+\gamma c_2^\alpha}.
\eeqnn For $({\mbox{\bf Y}}_N(\cdot), \tilde{\eta}_N(\cdot))$ with initial value $({\mbox{\bf y}},u)\in\Pi\times[c_1,c_2]$, recall that $U_j^N$ denotes the arrival time of the $j$'th event occurring to $({\mbox{\bf Y}}_N,\tilde{\eta}_N)$ and $U_0^N=0$. It is not hard to see that $U^N_j$ is stochastically larger than the sum of $j$ i.i.d. exponential variables with parameter $\varepsilon_{_N}^{-1}(\theta+\gamma c_2^\alpha)+K_2$. We have
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
{\mbox{\bf P}}(G)
\!\!\!&=\!\!\!& \sum_{k=0}^\infty {\mbox{\bf P}}(\{\mbox{exactly } k\ \Gamma_N\mbox{-events occur in } [\sigma_0^N,\sigma_1^N]\}\cap G)\\
\!\!\!&\leq\!\!\!&
\sum_{k=0}^\infty \Big(\frac{\theta+\gamma c_2^\alpha} {K_1\varepsilon_{_N}+\theta+\gamma c_2^\alpha}\Big)^k \frac{2K_2\varepsilon_{_N}} {2K_2\varepsilon_{_N}+\theta+\gamma c_1^\alpha} {\mbox{\bf P}}\Big(\sigma_0^N+U_k^N\leq s\Big)\\
\!\!\!&\leq\!\!\!&
\sum_{k=0}^\infty \Big(\frac{\theta+\gamma c_2^\alpha} {K_1\varepsilon_{_N}+\theta+\gamma c_2^\alpha}\Big)^k \frac{2K_2\varepsilon_{_N}} {2K_2\varepsilon_{_N}+\theta+\gamma c_1^\alpha} {\mbox{\bf P}}\Big(\sigma_0^N+\sum_{j=1}^k \tilde{V}^N_j\leq s\Big)\\
\!\!\!&\leq\!\!\!& M{\mbox{\bf P}}\Big(\sigma_0^N+\sum_{j=1}^{T_N}\tilde{V}_j^N\leq s\Big),
\eeqnn for some positive constant $M$ and sufficiently large $N$, where $\{\tilde{V}_j^N\}$ are i.i.d. exponential variables with parameter $\varepsilon_{_N}^{-1}(\theta+\gamma c_2^\alpha)+K_2$, and $T_N$ is a geometric variable with parameter $\frac{K_1\varepsilon_{_N}}{K_1\varepsilon_{_N}+\theta+\gamma c_2^\alpha}$ independent of $\{\tilde{V}_j^N\}$. Since $\sigma_0^N\overset{p}{\rightarrow}0$, a simple calculation shows that $\sigma_0^N+\sum_{j=0}^{M_N}V_j^N$ converges weakly to an exponential variable with parameter $K_1$. The lemma is proved.\qed
\begin{lemma}}\def\elemma{\end{lemma}\label{lemma 4} \;Under conditions (A) and (B), the ancestral process $\{{\mbox{\bf Y}}_N(t),\ t\geq0\}$ starting at ${\mbox{\bf y}}$ with ${\mbox{\bf y}}\in\Pi$ converges weakly on $D([0,\infty), \Pi)$ to $\{{\mbox{\bf Y}}(t),\ t\geq0\}$ given by (\ref{Y}) starting at ${{\mbox{\bf y}}}$. \elemma
\noindent{\it Proof.~~}}\def\qed{
$\Box$
If the process $Y_N(t)$ stays in the space of $\Pi$, $\psi_N$ and the fourth term in $\phi_N$ vanishes. Then for any bounded function $g$ on $E\times\mathbb{R}_+$ define $B_Ng=\tilde{\phi}_Ng+\Gamma_Ng$, where $\Gamma_N$ is given in (\ref{A_N}) and
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
\tilde{\phi}_Ng({\mbox{\bf y}},u)
\!\!\!&=\!\!\!&
2\binom{y_0}{2}\Big(g({\mbox{\bf y}}-{\mbox{\bf e}}_0,u)-g({\mbox{\bf y}},u)\Big)\\
\!\!\!& \!\!\!&
+\,\theta y_0
\Big(g({\mbox{\bf y}}-{\mbox{\bf e}}_0+{\mbox{\bf e}}_1,u+\varepsilon_{_N})-g({\mbox{\bf y}},u)\Big)\\
\!\!\!& \!\!\!&
+\,\gamma u^{\alpha-1}y_1
\Big(g({\mbox{\bf y}}-{\mbox{\bf e}}_1+{\mbox{\bf e}}_0,u-\varepsilon_{_N})-g({\mbox{\bf y}},u)\Big)1_{\{u>0\}}\\
\!\!\!& \!\!\!&
+\,\Big(\varepsilon_{_N}R_{1,N}g({\mbox{\bf y}},u)+\frac{1}{N}u^{\alpha-1}1_{\{u>0\}}
R_{2,N}g({\mbox{\bf y}},u)\Big).
\eeqnn Let $\mathcal{F}^N_t= \{({\mbox{\bf Y}}_N(s),\tilde{\eta}_N(s)):\ 0\leq s\leq t\}$. Because of the Markov property of $({\mbox{\bf Y}}_N(\cdot),\tilde{\eta}_N(\cdot))$,
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
g({\mbox{\bf Y}}_N(t),\tilde{\eta}_N(t))-g({\mbox{\bf y}},\tilde{\eta}_N(0))-\int_0^t(B_Ng)({\mbox{\bf Y}}_N(s),\tilde{\eta}_N(s))ds
\eeqnn is a local $(\mathcal{F}^N_t)$-martingale. Let
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
\tau_N=\inf\{
t\geq0,\ |\tilde{\eta}_N(t-)-(\theta/\gamma)^{1/\alpha}|\geq\delta\mbox{ or } |\tilde{\eta}_N(t)-(\theta/\gamma)^{1/\alpha}|\geq\delta
\},
\eeqnn where $\delta$ is a positive constant satisfying $0<\delta<(\theta/\gamma)^{1/\alpha}$. For sufficiently large $N$, there exist $c_1>0$ and $c_2>0$ such that $c_1\leq \tilde{\eta}_N(t\wedge \tau_N)\leq c_2$ for any $t\geq0$. Then for any function $f$ on $E$,
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
\zeta_N(t)=f({\mbox{\bf Y}}_N(t\wedge\tau_N))-f({\mbox{\bf y}})
-\int_0^{t\wedge\tau_N}(B_Nf)({\mbox{\bf Y}}_N(s\wedge\tau_N),\tilde{\eta}_N(s\wedge\tau_N))ds
\eeqnn is a bounded martingale. Indeed, for sufficiently large $N$,
$|(B_Nf)({\mbox{\bf Y}}_N(s\wedge\tau_N),\tilde{\eta}_N(s\wedge\tau_N))|\leq 2\big(n^2+n\theta+n\gamma(c_1^{\alpha-1}\vee c_2^{\alpha-1})\big)\|f\|$ for any $s\geq0$. Note that $E$ is a finite set, so the discrete topology on $E$ makes it a complete and compact metric space and any real valued function $f$ on $E$ is bounded and continuous. By Ethier and Kurtz \cite[Theorem 9.1 and 9.4, p.142]{EK86}, the process $\{Y_N(t\wedge\tau_N)\}$ is relatively compact. On the other hand, for any $t\geq0$,
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
{\mbox{\bf P}}(\tau_N<t)\leq{\mbox{\bf P}}\Big(\sup_{0\leq s\leq t}|\tilde{\eta}_N(s)-(\theta/\gamma)^{1/\alpha}|\geq\delta\Big)\rightarrow0,
\eeqnn which shows that $\tau_N\overset{p}{\rightarrow}\infty$. Let $\{{\mbox{\bf Y}}(t)\}$ be any limit point of $\{{\mbox{\bf Y}}_N(t\wedge\tau_N)\}$. By Skorokhod's representation theorem we may assume that on some Skorokhod space $(\tau_N,{\mbox{\bf Y}}_N(t\wedge\tau_N),\tilde{\eta}_N(t\wedge\tau_N),) \overset{a.s.}{\rightarrow}(\infty,{\mbox{\bf Y}}(t),(\theta/\gamma)^{1/\alpha})$ in the topology of $\mathbb{R}_+\times D([0,\infty),E\times\mathbb{R}_+)$. Thus $\zeta_N(t)\overset{a.s.}{\rightarrow}\zeta(t)$ and $\zeta(t)$ is given by
\begin{eqnarray}}\def\eeqlb{\end{eqnarray}
\zeta(t)=f({\mbox{\bf Y}}(t))-f({\mbox{\bf x}})
-\int_0^{t}(Bf)({\mbox{\bf Y}}(s))ds,\label{martingale problem}
\eeqlb where
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
Bf(u,{\mbox{\bf y}})\!\!\!&=\ar2\binom{y_0}{2}\Big(f({\mbox{\bf y}}-{\mbox{\bf e}}_0)-f({\mbox{\bf y}})\Big)
+\theta y_0
\Big(f({\mbox{\bf y}}-{\mbox{\bf e}}_0+{\mbox{\bf e}}_{1})-f({\mbox{\bf y}})\Big)\\
\!\!\!& \!\!\!&
+\,\theta(\gamma/\theta)^{1/\alpha}y_1
\Big(f({\mbox{\bf y}}-{\mbox{\bf e}}_1+{\mbox{\bf e}}_0)-f({\mbox{\bf y}})\Big).
\eeqnn Since $\sup_N|\zeta_N(t)|<\infty$, $\zeta_N(t)\overset{L_1}{\rightarrow}\zeta(t)$ for any $t\geq0$. Thus $\zeta(t)$ is a martingale. Since $\{{\mbox{\bf Y}}(t)\}$ given by (\ref{Y}) is the unique solution to the martingale problem (\ref{martingale problem}), We have that $\{{\mbox{\bf Y}}_N(t\wedge\tau_N)\}$ converges weakly to $\{{\mbox{\bf Y}}(t)\}$ given by (\ref{Y}) in $D([0,\infty),E)$. Furthermore, for any $\epsilon>0$ and any $t\geq0$,
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
{\mbox{\bf P}}\Big(\sup_{0\leq s\leq t} |Y_N(s\wedge\tau_N)-Y_N(s)|>\epsilon\Big) \leq{\mbox{\bf P}}(\tau^N<t)\rightarrow0,
\eeqnn as $N\rightarrow\infty$. The lemma follows from the above limit.\qed
{\it Proof of Theorem \ref{thm}\;} Let ${\mbox{\bf P}}_{{\mbox{\bf y}}}(\cdot)$ be the distribution of $({\mbox{\bf Y}}_N(\cdot),\tilde{\eta}_N(\cdot))$ with initial value $({\mbox{\bf y}},\tilde{\eta}_N(0))$, where $\tilde{\eta}_N(\cdot)$ is distributed as $\pi_N$ given in Section 2. Let $f_1,\cdots, f_k$ be real-valued functions on $E$. Choose $0<s<t_1<\cdots<t_k$. Let $Q_N=\{\sigma_0^N<s<\sigma_1^N\}$. Then
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
\!\!\!&\!\!\!&{\mbox{\bf E}}_{{\mbox{\bf y}}}\Big[\Pi_{i=1}^kf_i({\mbox{\bf Y}}_N(t_i))1_{Q_N}\Big]\\
\!\!\!&=\!\!\!&{\mbox{\bf E}}_{{\mbox{\bf y}}}\Big[1_{Q_N}{\mbox{\bf E}}\Big[\Pi_{i=1}^kf_i({\mbox{\bf Y}}_N(t_i))|\mathcal{F}^N_s\Big]\Big]\\
\!\!\!&=\!\!\!& {\mbox{\bf E}}_{{\mbox{\bf y}}}\Big[1_{Q_N}{\mbox{\bf E}}_{(\bar{{\mbox{\bf y}}},\,\tilde{\eta}_N(s))}\Big[\Pi_{i=1}^kf_i({\mbox{\bf Y}}_N(t_i-s))\Big]\Big]\\
\!\!\!&=\!\!\!& {\mbox{\bf E}}_{{\mbox{\bf y}}}\Big[{\mbox{\bf E}}_{(\bar{{\mbox{\bf y}}},\,\tilde{\eta}_N(s))}\Big[\Pi_{i=1}^kf_i({\mbox{\bf Y}}_N(t_i-s))\Big]\Big] -{\mbox{\bf E}}_{{\mbox{\bf y}}}\Big[1_{\bar{Q}_N}{\mbox{\bf E}}_{((\bar{{\mbox{\bf y}}},\,\tilde{\eta}_N(s))}\Big[\Pi_{i=1}^kf_i({\mbox{\bf Y}}_N(t_i-s))\Big]\Big]\\ \!\!\!&=\!\!\!& {\mbox{\bf E}}_{\bar{{\mbox{\bf y}}}}\Big[\Pi_{i=1}^kf_i({\mbox{\bf Y}}_N(t_i-s))\Big] -{\mbox{\bf E}}_{{\mbox{\bf y}}}\Big[1_{\bar{Q}_N}{\mbox{\bf E}}_{(\bar{{\mbox{\bf y}}},\,\tilde{\eta}_N(s))}\Big[\Pi_{i=1}^kf_i({\mbox{\bf Y}}_N(t_i-s))\Big]\Big].
\eeqnn where $\bar{Q}_N$ is the complement of the set $Q_N$. The last equality follows from the fact that $\tilde{\eta}_N(\cdot)$ is stationary. Then by Lemmas \ref{lemma 2}, \ref{lemma 3} and \ref{lemma 4},
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*} \!\!\!& \!\!\!&\limsup_{N\rightarrow\infty}
\bigg|{\mbox{\bf E}}_{{\mbox{\bf y}}}\Big[\Pi_{i=1}^kf_i({\mbox{\bf Y}}_N(t_i))\Big]
-{\mbox{\bf E}}_{\bar{{\mbox{\bf y}}}}\Big[\Pi_{i=1}^kf_i({\mbox{\bf Y}}(t_i-s))\Big]\bigg|\\ \!\!\!&\leq\!\!\!&
2\max_i\|f_i\|\limsup_{N\rightarrow\infty}{\mbox{\bf P}}(\bar{Q}_N)+ \limsup_{N\rightarrow\infty}
\bigg|{\mbox{\bf E}}_{\bar{{\mbox{\bf y}}}}\Big[\Pi_{i=1}^kf_i({\mbox{\bf Y}}_N(t_i-s))\Big]
-{\mbox{\bf E}}_{\bar{{\mbox{\bf y}}}}\Big[\Pi_{i=1}^kf_i({\mbox{\bf Y}}(t_i-s))\Big]\bigg|\\
\!\!\!&\leq\!\!\!& 2M\max_i\|f_i\|(1-e^{-K_1 s}),
\eeqnn which goes to $0$ as $s\rightarrow0$. Note that ${\mbox{\bf Y}}(t)$ is stochastically continuous. ${\mbox{\bf E}}_{\bar{{\mbox{\bf y}}}}\Big[\Pi_{i=1}^kf_i({\mbox{\bf Y}}(t_i-s))\Big]$ converges to ${\mbox{\bf E}}_{\bar{{\mbox{\bf y}}}}\Big[\Pi_{i=1}^kf_i({\mbox{\bf Y}}(t_i))\Big]$ as $s\rightarrow0$. Then we have that $\lim_{N\rightarrow\infty} {\mbox{\bf E}}_{{\mbox{\bf y}}}\Big[\Pi_{i=1}^kf_i({\mbox{\bf Y}}^N(t_i))\Big]= {\mbox{\bf E}}_{\bar{{\mbox{\bf y}}}}\Big[\Pi_{i=1}^kf_i({\mbox{\bf Y}}(t_i))\Big]$.\qed
{\it Proof of Theorem \ref{thm2}\;} Step 1: recall the notation in Section 2.2. Under the homeomorphism, the subspace $\Pi$ can be regarded as $\Gamma(\Pi)$ for simplicity.
It follows from (\ref{martingale problem}) that for any function $f$ on $\Pi$,
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*} f({\mbox{\bf Y}}_k(t))-f({\mbox{\bf y}})
-\int_0^{t}(B_kf)({\mbox{\bf Y}}_k(s))ds
\
\eeqnn is a martingale, where
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
B_kf({\mbox{\bf y}})
\!\!\!&=\!\!\!&
2\binom{y_0}{2}\Big(f({\mbox{\bf y}}+(-1,0))-f({\mbox{\bf y}})\Big)
+\theta_k y_0
\Big(f({\mbox{\bf y}}+(-1,1))-f({\mbox{\bf y}})\Big)\\
\!\!\!& \!\!\!&
+\,\theta_k(\gamma_k/\theta_k)^{1/\alpha}y_1
\Big(f({\mbox{\bf y}}+(1,-1))-f({\mbox{\bf y}})\Big).
\eeqnn Recall that ${\mbox{\bf Y}}_k(t)=(Y_k^0(t),Y_k^1(t))$ and $Y_k(t)=Y_k^0(t)+Y_k^1(t)$. For any function $g$ on $I_n$, let $f({\mbox{\bf y}})=g(y_0+y_1)$ for ${\mbox{\bf y}}\in\Pi$. Then \begin{eqnarray*}}\def\eeqnn{\end{eqnarray*} g(Y_k(t))-g(n)
-\int_0^{t}(\tilde{B}_kg)(Y_k^0(s),Y_k^1(s))ds
\eeqnn is also a martingale, where $\tilde{B}_kg(y)=2\binom{y_0}{2}(g(y-1)-g(y))$. Note that $I_n$ is a finite set, so the discrete topology on $I_n$ makes it a complete and compact metric space. Any real valued function $g$ on $I_n$ is bounded and continuous. Then $Y_k(\cdot)$ satisfies the compact containment condition. For each $T>0$,
$\sup_k\int_0^T|\tilde{B}_kg(Y_k^0(s),Y_k(s))|ds\leq 2n^2T\|g\|$, where $\|g\|=\sup_{y\in I_n}
|g(y)|$. By Ethier and Kurtz \cite[Theorem 9.1 and 9.4, p.142]{EK86}, $Y_k(\cdot)$ is relatively compact in $D([0,\infty), I_n)$.
Step 2: suppose that $\{\xi^k_j(\cdot)\}_{j=1}^n$ is the sequence of i.i.d.\,Markov chains taking values in $\{0,1\}$ and whose transition rate matrix is given by
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
\left( \begin{array}{ll}
-1 &
1\\
(\frac{\gamma_k}{\theta_k})^{\frac{1}{\alpha}} & -(\frac{\gamma_k}{\theta_k})^{\frac{1}{\alpha}} \\
\end{array} \right).
\eeqnn Let $P^k_{ij}(t)={\mbox{\bf P}}(\xi^k_1(t)=j|\xi^k_1(0)=i)$. A simple calculation shows that
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
P_{00}^k(t)=1-P^k_{01}(t)\!\!\!&=\!\!\!& \frac{(\theta_k/\gamma_k)^{1/\alpha}}{1+(\theta_k/\gamma_k)^{1/\alpha}} +\frac{1}{1+(\theta_k/\gamma_k)^{1/\alpha}}e^{-(1+(\theta_k/\gamma_k)^{1/\alpha})t},\\
P_{10}^k(t)=1-P_{11}^k(t)\!\!\!&=\!\!\!& \frac{(\theta_k/\gamma_k)^{1/\alpha}}{1+(\theta_k/\gamma_k)^{1/\alpha}} -\frac{(\theta_k/\gamma_k)^{1/\alpha}}{1+(\theta_k/\gamma_k)^{1/\alpha}}e^{-(1+(\theta_k/\gamma_k)^{1/\alpha})t}.
\eeqnn Let $\zeta_n^k(t)=\sum_{j=1}^n1_{\{\xi_j^k(t)=0\}}$. Since $\{\xi_i^k(t)\}_{i=1}^n$ are independent of each other, it is not hard to see that for any $g$ on $I_n$,
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
\sup_{x,y \in I_n}\Big|{\mbox{\bf E}}_x[g(\zeta_n^k(t))]
-{\mbox{\bf E}}_y[g(\zeta_n^k(t))]\Big|\leq2n\|g\|e^{-(1+(\theta_k/\gamma_k)^{1/\alpha})t},\ t\geq0.
\eeqnn This implies $\zeta_n^k(t)$ satisfies the $\phi$-mixing condition (see \cite[P.111]{B05}). By (1.13) of \cite[p.109]{B05},
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
\!\!\!& \!\!\!& \sup_{y\in I_n}\big|{\mbox{\bf E}}_y[g(\zeta_n^k(t_2))g(\zeta_n^k(t_1))]-{\mbox{\bf E}}_y[g(\zeta_n^k(t_2))]{\mbox{\bf E}}_y[g(\zeta_n^k(t_1))]\big|\\
\!\!\!&\leq\!\!\!& 2\sqrt{2n}\|g\|^2e^{-(1+(\theta_k/\gamma_k)^{1/\alpha})(t_2-t_1)/2}
\eeqnn for any $t_2\geq t_1\geq0$. Then
\begin{eqnarray}}\def\eeqlb{\end{eqnarray}
\!\!\!&\!\!\!&{\mbox{\bf E}}_y\Big[\Big(\int_0^t\Big(g(\zeta_n^k(\theta_ks))-{\mbox{\bf E}}_y[g(\zeta_n^k(\theta_ks))]\Big)ds\Big)^2\Big]\nonumber\\
\!\!\!&=\!\!\!&{\mbox{\bf E}}_y\bigg[\int_0^t\int_0^tds_1ds_2\Big(g(\zeta_n^k(\theta_ks_1))-{\mbox{\bf E}}_y[g(\zeta_n^k(\theta_ks_1))]\Big)
\Big(g(\zeta_n^k(\theta_ks_2))-{\mbox{\bf E}}_y[g(\zeta_n^k(\theta_ks_2))]\Big)\bigg]\nonumber\\
\!\!\!&=\!\!\!& \int_0^t\int_0^tds_1ds_2
\Big({\mbox{\bf E}}_y[g(\zeta_n^k(\theta_ks_2))
g(\zeta_n^k(\theta_ks_1))]-{\mbox{\bf E}}_y[g(\zeta_n^k(\theta_ks_2))]{\mbox{\bf E}}_y[g(\zeta_n^k(\theta_ks_1))]\Big)\nonumber\\
\!\!\!&\leq\!\!\!&
C(n)\|g\|^2\int_0^tds_2\int_0^{t}e^{-\theta_k(1+(\theta_k/\gamma_k)^{1/\alpha})|s_2-s_1|/2}ds_1\nonumber\\
\!\!\!&\leq\!\!\!& C(n)\|g\|^2t/\theta_k,
\label{inequility}
\eeqlb where $C(n)$ is a constant only depending $n$. Since $P_{00}^k(\theta_kt)\rightarrow\frac{p^{1/\alpha}}{1+p^{1/\alpha}}$ and $P_{01}^k(\theta_kt)\rightarrow\frac{1}{1+p^{1/\alpha}}$ as $k\rightarrow\infty$, it is easy to see for any $t\geq0$, $\zeta_n^k(\theta_kt)\overset{d}{\rightarrow}\zeta_n$ as $k\rightarrow\infty$, where $\zeta_n$ follows the Binomial distribution, i.e., $\zeta_n\sim Bn(n,\frac{p^{1/\alpha}}{1+p^{1/\alpha}})$. Note that $I_n$ is finite. The dominated convergence theorem shows that
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
\sup_{y\in I_n}\int_0^t\Big|{\mbox{\bf E}}_y[g(\zeta_n^k(\theta_ks))]-{\mbox{\bf E}}[g(\zeta_n)]\Big|ds\rightarrow0,
\eeqnn as $k\rightarrow\infty$. Combined with (\ref{inequility}), we have as $k\rightarrow\infty$,
\begin{eqnarray}}\def\eeqlb{\end{eqnarray}
\sup_{y\in I_n}{\mbox{\bf E}}_y\Big[\Big(\int_0^t\Big(g(\zeta_n^k(\theta_ks))-{\mbox{\bf E}}[g(\zeta_n)]\Big)ds\Big)^2\Big]\rightarrow0.
\label{convergence}
\eeqlb
Step 3: $(Y^0_k(t),Y_k(t))$ is a Markov process as in Step 1. $Y^0_k(0)=y\in I_n$ and $Y_k(0)=n$. Let $\mathcal{F}^k_t=\sigma\{(Y^0_k(s),Y_k(s)):\ 0\leq s\leq t\}$. Define $T^k_j=\inf\{t\geq0: Y_k(t)=n-j\}$ with $T^k_0=0$ and $\tau_j^k=T_j^k-T_{j-1}^k$. Set $h(y)=y(y-1)$ for $y\in I_n$. By (\ref{convergence}), we have
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
{\mbox{\bf P}}(\tau^k_{j+1}>t)\!\!\!&=\!\!\!&{\mbox{\bf E}}\Big[{\mbox{\bf P}}\Big(\tau^k_{j+1}>t\Big|\mathcal{F}^k_{T^k_{j}}\Big)\Big]\nonumber\\
\!\!\!&=\!\!\!&{\mbox{\bf E}}\Big[{\mbox{\bf P}}_{(Y_k^0(T^k_{j}),n-j)}(\tau^k_{j+1}>t)\Big]\nonumber\\
\!\!\!&=\!\!\!&{\mbox{\bf E}}\Big[{\mbox{\bf E}}_{Y_k^0(T^k_{j})}\Big(\exp\Big\{-\int_0^th(\zeta^k_{n-j}(\theta_ks))ds\Big\}\Big)\Big]\nonumber\\
\!\!\!&\rightarrow\!\!\!& e^{-{\mbox{\bf E}}[h(\zeta_{n-j})]t},
\label{convergence2}
\eeqnn as $k\rightarrow\infty$. Similarly
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*} {\mbox{\bf P}}(\tau^k_1>t,\tau^k_2>s)= {\mbox{\bf E}}\Big[1_{\{\tau^k_1>t\}}{\mbox{\bf E}}_{Y_k^0(T^k_1)}\Big(\exp\Big\{-\int_0^th(\zeta^k_{n-1}(\theta_ks))ds\Big\}\Big)\Big].
\eeqnn Then
\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}
\!\!\!&\!\!\!&|{\mbox{\bf P}}(\tau^k_1>s,\tau^k_2>t)-e^{-{\mbox{\bf E}}[h(\zeta_n)]s-{\mbox{\bf E}}[h(\zeta_{n-1})]t}|\\
\!\!\!&\leq\!\!\!&|{\mbox{\bf P}}(\tau^k_1>s)-e^{-{\mbox{\bf E}}[h(\zeta_n)]s}|+
\sup_{y\in I_n}{\mbox{\bf E}}_y\Big[\Big|\int_0^t\Big(h(\zeta_{n-1}^k(\theta_ks))-{\mbox{\bf E}}[h(\zeta_{n-1})]\Big)ds\Big|\Big].
\eeqnn By (\ref{convergence}) and (\ref{convergence2}) we have that the second term in the right-hand side of the above inequality goes to $0$ as $k\rightarrow\infty$. By induction, $(\tau_1^k,\cdots,\tau_{n-1}^k)\overset{d}{\rightarrow}(\tau_1,\cdots,\tau_{n-1})$, where $\{\tau_j\}_{j=1}^{n-1}$ is independent of each other and $\tau_j$ follows the exponential distribution with parameter $c_{n-j+1}$. It follows that $\{Y_k(t),\,t\geq0\}$ converges in the sense of finite-dimensional distributions to the $n$-Kingman coalescent process $\{K(t),\,t\geq0\}$. Since $\{Y_k(t)\}$ is relatively compact, the theorem is proved.\qed
\paragraph{Acknowledgments.} AL and CM were financially supported by grant MANEGE `Mod\`eles Al\'eatoires en \'Ecologie, G\'en\'etique et \'Evolution' 09-BLAN-0215 of ANR (French national research agency). AL also thanks the {\em Center for Interdisciplinary Research in Biology} (Coll\`ege de France) for funding. CM also thanks the financial support from the National Natural Science Foundation of China (NSFC) (No.11001137 and No.11271204) and the China Scholarship Council (CSC).
\noindent{\Large\bf References}
\small
\begin{enumerate}
\renewcommand{[\arabic{enumi}]}{[\arabic{enumi}]}
\bibitem{ACL09} Aguil\'ee, R., Claessen, D., Lambert, A. (2009) Allele fixation in a dynamic metapopulation: Founder effects vs refuge effects \emph{Theoretical Population Biology} 76(2) 105--117.
\bibitem{ALC11} Aguil\'ee, A., Lambert, A., Claessen, D. (2011) Ecological speciation in dynamic landscapes. \emph{Journal of Evolutionary Biology} 24(12) 2663--2677.
\bibitem{ACL13} Aguil\'ee, R., Claessen, D., Lambert, A. (2013) Adaptive radiation driven by the interplay of eco-evolutionary and landscape dynamics. \emph{Evolution}. In press.
\bibitem{B05} Bradley, R.C. (2005) Basic properties of strong mixing
conditions. A survey and some open questions. \textit{Probability
Surveys} \textbf{2} 107--144.
\bibitem{CO04} Coyne, J.A., Orr, H.A. (2004) \emph{Speciation}. Sinauer Associates Sunderland, MA.
\bibitem{E09} Eldon, B. (2009) Structured coalescent processes from a modified Moran model with large offspring numbers. \textit{Theoretical Population Biology} \textbf{76} 92--104.
\bibitem{EK86} Ethier, S.N. and Kurtz, T.G. (1986) \textit{Markov processes: Characterization and Convergence.} John Wiley and Sons Inc., New York.
\bibitem{HG97} Hanski, I.A. and Gilpin, M.E. (1997) \textit{Metapopulation biology. Ecology, genetics and evolution.} Academic Press, San Diego.
\bibitem{H97} Herbots, H.M. (1997) The structured coalescent, in \textit{Progress in Population Genetics and Human Evolution}. Springer, New York, 231--255.
\bibitem{K79} Kelly, F.P. (1979) \textit{Reversibility and Stochastic Networks.} John Wiley and Sons Inc, New York.
\bibitem{K00} Keymer, J., Marquet, P., Velasco-Hern\'andez, J., Levin, S. (2000) Extinction thresholds and metapopulation persistence in dynamic landscapes. \emph{American Naturalist} {\bf 156}(5) 478--494.
\bibitem{K82} Kingman, J.F.C. (1982) The coalescent. \emph{Stochastic Processes and their Applications} {\bf 13} 235--248.
\bibitem{L10} Lambert, A. (2010) Population genetics, ecology and the size of populations \textit{Journal of Mathematical Biology} \textbf{60} 469--472.
\bibitem{NK02} Nordborg, M. and Krone, S.M. (2002) Separation of time scales and convergence to the coalescent in structured populations. \textit{Modern Developments in Theoretical Population Genetics.} Oxford University Press, Oxford, UK.
\bibitem{N90} Notohara, M. (1990) The coalescent and the genealogical process in geographically structured population. \textit{Journal of Mathematical Biology} \textbf{29} 59--75.
\bibitem{T88} Takahata, N. (1988) The coalescent in two partially isolated diffusion populations. \textit{Genetical Research} \textbf{52} 213--222.
\bibitem{TV09} Taylor, J.E. and V\'{e}ber, A. (2009) Coalescent processes in subdivided populations subject to recurrent mass extinctions. \textit{Electronic Journal of Probability} \textbf{14} 242--288.
\end{enumerate}
\end{document} | arXiv |
\begin{document}
\title{PoD-BIN: A Probability of Decision Bayesian Interval Design for Time-to-Event Dose-Finding Trials with Multiple Toxicity Grades }
\section*{Abstract} We consider a Bayesian framework based on ``probability of decision'' for dose-finding trial designs. The proposed PoD-BIN design evaluates the posterior predictive probabilities of up-and-down decisions. In PoD-BIN, multiple grades of toxicity, categorized as the mild toxicity (MT) and dose-limiting toxicity (DLT), are modeled simultaneously, and the primary outcome of interests is time-to-toxicity for both MT and DLT. This allows the possibility of enrolling new patients when previously enrolled patients are still being followed for toxicity, thus potentially shortening trial length. The Bayesian decision rules in PoD-BIN utilize the probability of decisions to balance the need to speed up the trial and the risk of exposing patients to overly toxic doses. We demonstrate via numerical examples the resulting balance of speed and safety of PoD-BIN and compare to existing designs.
\section{Introduction}\label{intro} \subsection{Overview} Phase I clinical trials are the first-in-human studies evaluating the safety and tolerability of a new treatment. Given a series of ascending dose levels, the goal of phase I clinical trial is to identify a maximum tolerated dose (MTD) defined as the highest dose having a dose-limiting toxicity (DLT) probability close to or lower than a target toxicity rate, say 0.3. Under this goal, previous designs for phase I trials can generally be categorized into two classes: $(i)$ algorithm-based designs, such as the 3+3 \citep{storer1989} design and the i3+3 design \citep{Liu2020}, and $(ii)$ model-based designs, such as the continual reassessment method (CRM) \citep{o1990continual}, the Bayesian logistic regression method (BLRM) \citep{neuenschwander2008critical}, and the interval-based designs like mTPI \citep{ji2010modified}, mTPI-2 \citep{guo2017bayesian}, and keyboard \citep{yan2017keyboard}.
In conventional phase I trials, patients are enrolled sequentially, and doses are assigned for successive patients based on outcomes observed within a certain assessment window. With recent advances in targeted therapy and immunotherapy in oncology, the problem of late-onset toxicity has been a major concern in phase I trials as the toxicities associated with these new treatments can take longer to observe than traditional cytotoxic therapies \citep{kanjanapan2019delayed}. In addition, clinically it is important to distinguish different grades of toxicities instead of a binary DLT outcome, usually referring to grade 3 or higher grade toxicities \citep{CTCAE}. For example, in the phase I trial that tests the toxicity of certinib \citep{cho2017ascend}, grades 1 and 2 adverse events (AEs) at the 750mg dose have led to concomitant medication, study drug intervention, or dose reduction. These AEs would not be categorized as a DLT, although they still lead to a dose reduction to 450mg in the trial, which shows comparable PK exposure and a more favorable safety profile. It is, therefore, desirable to consider the safety of a dose based on its toxicity beyond the conventional definition of DLT -- defined by dichotomizing severity of toxicities into a binary variable with a significant loss of toxicity information obtained from each patient.
\subsection{Existing approaches on time-to-toxicity and multi-grade toxicity models} We attempt to shorten the trial duration and better characterize the toxicity profile of a dose by modeling time-to-event measurements for toxicity outcomes of multiple toxicity grades. To shorten the duration of phase I trials, several designs have proposed rolling enrollment of future patients while some existing patients in the trial are still pending for DLT assessment. \cite{cheung2000sequential} introduce the time-to-event CRM (TITE-CRM) design which considers a weighted likelihood approach, where a patient with complete outcomes receives full weight, and a patient with pending outcome receives a partial weight based on the time of follow-up. Both the expectation maximization CRM (EM-CRM) \citep{yuan2011robust} and data augmentation CRM (DA-CRM) \citep{liu2013bayesian} adopt the idea of missing data modeling, where the missing/pending outcomes are explicitly considered. \cite{yuan2018time} propose the time-to-event BOIN (TITE-BOIN) design to accommodate late-onset toxicity by imputing the unobserved pending toxicity outcome using the single mean imputation method. The rolling TPI (R-TPI) design is an extension of the mTPI-2 design with a fixed set of decision rules to accommodate pending outcomes \citep{guo2019r}. Recently, in the probability-of-decision toxicity probability interval design (PoD-TPI) by \cite{zhou2020pod}, the dose assignment decision is treated as a random variable, and the posterior distribution of the decisions is computed to accommodate uncertainties in pending outcomes.
Separately, several methods have been proposed to model different toxicity grades in the estimation of MTD based on non-time-to-event observations. \cite{bekele2004dose} first introduce a concept of total toxicity burden (TTB), which is a weighted sum of the toxicity grades across different toxicity types. \cite{yuan2007continual} propose the quasi-CRM method, in which toxicity grades are first converted to the equivalent toxicity score, and then incorporated into the CRM using the quasi-Bernoulli likelihood to find MTD. \cite{lee2010continual} propose the continual reassessment method with multiple constraints (CRM-MC), which allows for the specification of various toxicity thresholds. \cite{iasonos2011incorporating} introduce a two-stage CRM that incorporates lower-grade toxicities in the first stage and separately models the rates of DLT and lower-grade toxicities in the second stage. \cite{van2012dose} propose the proportional odds CRM, incorporating toxicity severity with proportional odds models. Later, \cite{ezzalfani2013dose} define the total toxicity profile (TTP) as the Euclidean norm of the weights of toxicities experienced by a patient and characterize the overall severity of multiple types and grades of toxicities.
Recently, \cite{Lee2019} develop the TITE-CRMMC (Time to Event Continual Reassessment Method with Multiple Constraints) design that incorporates time-to-event information with multiple toxicity grades, combining the TITE-CRM \citep{cheung2000sequential} and CRM-MC \citep{lee2010continual} designs. It takes into account partial information on both moderate toxicities (MTs) and DLTs observed during a trial. However, TITE-CRMMC assumes that once a patient experiences a DLT, having additional moderate toxicity would no longer be counted as a toxicity outcome. In other words, patients with DLT only and patients with both moderate toxicity and DLT are not differentiated.
In this paper, we propose a Bayesian design, namely the probability-of-decision Bayesian interval design (PoD-BIN), to accelerate phase I trials by modeling time to different toxicity grades. Specifically, time-to-toxicity for both DLT and MT are modeled using a latent probit regression, and a summary statistic called the ``toxicity burden" is used to summarize the total toxicity impact to a patient. Importantly, dose-finding decisions are treated as a random variable and statistical inference is based on the posterior predictive distribution of pending toxicity outcomes. Innovatively, we compute and threshold probability of decisions to control the risk of making aggressive decisions. The remainder of this paper is organized as follows. In Sections \ref{met} and \ref{des}, we describe the probability model and propose the PoD-BIN design, respectively. In Section \ref{sim}, we present simulation studies and results. The paper ends with a discussion in Section \ref{dis}.
\section{Probability model and Inference}\label{met} \subsection{Probability model}\label{prob} Consider estimating the probabilities of DLT and MT among a set of $D$ dose levels, $d \in \{1,2,\dots, D\}$. Based on the Common Terminology Criteria for Adverse Events (CTCAE) by \cite{CTCAE}, general guidelines for all possible adverse events describe grade 1 as ``mild,'' 2 as ``moderate,'' 3 as ``severe,'' 4 as ``life-threatening or disabling,'' and 5 as ``death.'' We define MT as mild or moderate toxicities that are usually not considered as DLTs (i.e., grade $1-2$ based on CTCAE criteria), and DLT as the pre-specified severe adverse events usually with grade $3-5$ by CTCAE. In practice, individual trials may extend CTCAE to define their own MTs and DLTs.
Let $W$ denote the length of the pre-specified assessment window. In oncology, $W$ is usually set to 21 to 28 days, corresponding to the length of one treatment cycle. Now assume a sample size of $N$ patients is to be enrolled in a trial. We denote $v_i$ the follow-up time for patient $i$, for $i=1, \dotsc ,N$, and $x_i$ the dose assigned to patient $i$. Let $T_{MT,i}$ be the time to the manifestation of the first MT event for patient $i$, and $T_{DLT,i}$ the time to the manifestation of the first DLT event. We introduce $Y_i$ as an ordinal variable with four levels, summarizing the MT and DLT outcomes patient $i$. Suppose patient $i$ has been followed for time $v_i$, Define an outcome variable as \begin{equation}
Y_i^{(v_i)} =
\left\{ \begin{array}{lr}
1, & \text{if \ } T_{MT,i}>v_i, T_{DLT,i}>v_i,\\
2, & \text{if \ } T_{MT,i} \leq v_i, T_{DLT,i}>v_i,\\
3, & \text{if \ } T_{MT,i}>v_i, T_{DLT,i}\leq v_i,\\
4, & \text{if \ } T_{MT,i}\leq v_i, T_{DLT,i}\leq v_i.
\end{array} \right. \end{equation} In words, $Y_i$ takes a value of $1$ if no toxicity is observed by time $v_i$, $2$ if only MT is observed, $3$ if only DLT is observed, and $4$ if both MT and DLT are observed by time $v_i$. By definition, $Y_i$ is ordinal and the higher its value the more severe the toxicity. Also, $Y_i$ is a function of follow-up time $v_i$, and therefore it is time-varying as patient $i$ continues to be followed. When patient $i$ completes follow-up, define $Y_i^*$ as the ordinal outcome at the end of assessment window, i.e., \begin{equation}\label{eq0}
Y^*_i =
\left\{ \begin{array}{lr}
1, & \text{if \ } T_{MT,i}>W, T_{DLT,i}>W,\\
2, & \text{if \ } T_{MT,i} \leq W, T_{DLT,i}>W,\\
3, & \text{if \ } T_{MT,i}>W, T_{DLT,i}\leq W,\\
4, & \text{if \ } T_{MT,i}\leq W, T_{DLT,i}\leq W,
\end{array} \right. \end{equation} where $W$ is the duration of the assessment window. Since $W$ is a fixed value, such as 21 days, $Y_i^*$ is not time-varying.
In a traditional phase I trial, DLT$=\mathbbm{1}\{Y^*_i=3 \text{ or } 4\}$ and no DLT$=\mathbbm{1}\{Y^*_i=1 \text{ or } 2\}$. We first model the relationship between the dose level $x_i$ and the time-to-toxicity $Y^*_i,$ and then the relationship between $Y^{(v_i)}_i$ and $Y^*_i$. We assume a latent Gaussian variable $Z$ that underlies the generation of the ordinal response $Y_i^*$, given by a regression of $Z_i$ on dose $x_i$ as \begin{equation}\label{eq:probit}
Z_i=x_i\beta+\epsilon, \end{equation} where $\epsilon$ is the error term drawn from a standard Gaussian distribution, i.e., $Z_i\sim N(x_i\beta, 1)$. We assume the effect of doses, $\beta \in \mathbb{R}^+$, is positive across the four ordered toxicity categories to enforce monotone toxicity over doses. Therefore, let $\Phi(\cdot)$ denote the distribution function of a standard Gaussian random variable, and by Equation~(\ref{eq0}) we define the probability of toxicity outcome $Y_i^*=c$ at dose $d$ as \begin{equation}\label{pdj}
p_{d,c} \triangleq Pr(Y_i^*=c|x_i=d) = Pr(g_{c-1} \leq Z_i < g_c) = \Phi (g_c - d\beta) - \Phi (g_{c-1} - d\beta), \end{equation} where $c= 1,\dotsc,4$, the parameter vector $\boldsymbol{g} = (g_0, g_1, \dotsc, g_4)$ are the cutoff values, and $-\infty = g_0 < g_1 < \dotsb < g_4 = + \infty$. We also require $g_1\equiv 0$ to ensure identifiability \citep{johnson2006ordinal}.
Suppose a total of $n$ patients have been enrolled, and denote $\{Y_i^{(v_i)}\}$ as the currently observed data. The likelihood function can be written as
\begin{equation}\label{eq1}
L_n(\beta,\boldsymbol{g},\boldsymbol{Z} | \boldsymbol{y, v, x})=\prod_{i=1}^n \prod_{ k=1}^{4} [Pr(Y_i^{(v_i)}=k|x_i)]^{\mathbbm{1}(Y_i^{(v_i)}=k)}.
\end{equation} To proceed with Bayesian modeling, we express $Pr(Y_i^{(v_i)}=k \mid x_i)$ in Equation~(\ref{eq1}) as a function of model parameters, $\boldsymbol{g}$ and $\beta$, using the latent probit regression in Equation~(\ref{pdj}). Specifically, write \begin{equation}\label{eq2} \begin{aligned}
Pr(Y_i^{(v_i)}= k | x_i=d)
&= \sum_{c=1}^4Pr(Y_i^{(v_i)}= k |Y_i^{*}= c )Pr(Y_i^{*}= c|x_i =d) \\
& = \sum_{c=1}^4 w_{k, c} \cdot p_{d, c} \\ &= \sum_{c=1}^4 w_{k, c} \cdot \{\Phi (g_c - d \beta) - \Phi (g_{c-1} - d \beta) \}, \end{aligned} \end{equation}
where $w_{k, c}=Pr(Y_i^{(v_i)}=k |Y_i^{*}=c)$ is a conditional probability and usually modeled as a weight function in time-to-event models \citep{cheung2000sequential, Lee2019}. Denote $\boldsymbol{W}=[w_{k,c}; k,c=1,\dots,4]$ as a $4\times 4$ matrix given by \[ \boldsymbol{W}=\begin{bmatrix}
w_{1,1} & \dots & w_{1,4} \\
\vdots & \ddots & \vdots \\
w_{4,1} & \dots & w_{4,4}
\end{bmatrix}. \] Therefore, Equation~(\ref{eq2}) can be rewritten as
\begin{align}\label{eq3} \begin{bmatrix}
Pr(Y_i^{(v_i)}=1|x_i) \\
\vdots \\
Pr(Y_i^{(v_i)}=4|x_i)
\end{bmatrix} = \boldsymbol{W}
\begin{bmatrix}
Pr(Y_i^{*}=1|x_i) \\
\vdots \\
Pr(Y_i^{*}=4|x_i)
\end{bmatrix}. \end{align}
In $\boldsymbol{W}$, $w_{k, c}=0$ if $k> c$ since it is impossible to observe a toxicity outcome by an early time point but not later. By the same argument, $w_{1,1}=1$. Following the recommendation in TITE-CRM \citep{cheung2000sequential}, the linear weight function yielded similar operating characteristics compared to other complicated weight functions, and therefore we assume the $w_{k,c}$'s are linear in time, i.e., $Pr(T_{MT,i}\leq v_i|T_{MT,i}\leq W)= \frac{v_i}{W}$. We derive the entire $\boldsymbol{W}$ matrix based on the linearity assumption given by \begin{align}\label{eq3} \boldsymbol{W}=
\begin{bmatrix}
1 & 1-\frac{v_i}{W} & 1-\frac{v_i}{W} & (1-\frac{v_i}{W})^2 \\
0 & \frac{v_i}{W} & 0 & \frac{v_i}{W}(1-\frac{v_i}{W}) \\
0 & 0 & \frac{v_i}{W} & \frac{v_i}{W}(1-\frac{v_i}{W}) \\
0 & 0 & 0 & \frac{v_i^2}{W^2}
\end{bmatrix}. \end{align} See details in Web Appendix A.
\subsection{Inference}\label{inference}
Treating $\boldsymbol{Z}$ as augmented data, posterior inference is conducted by sampling the model parameters $(\beta, \boldsymbol{g}, \boldsymbol{ Z})$. We fix the first cutoff parameter $g_1$ at $0$, and assume a flat prior $f(\boldsymbol{g})\propto 1$ on $\boldsymbol{g}=( g_2, g_3)$ and a truncated flat prior $f(\beta) \propto \mathbbm{1}(\beta>0)$ on $\beta$ over the positive real values. With the likelihood function in Equation~(\ref{eq1}) and the prior models, we conduct posterior inference on the parameters $(\beta, \boldsymbol{g})$ and the latent variable $\boldsymbol{Z}$ using the Metropolis-Within-Gibbs algorithm \citep{johnson2006ordinal}, a technique of \mbox{Markov-chain} Monte Carlo (MCMC) simulation (Web Appendix B). The ordering constraints on the cutoffs $\boldsymbol{g}=(g_2, g_3)$ are imposed through the sampling step in MCMC.
One drawback of the augmented data approach for ordinal data is that MCMC has poor numeric stability, especially when the numbers observed at each ordinal category are small. As such, we implement a two-stage design -- in the first stage, only complete follow-up data will be used until sufficient information from patients has been acquired. This means no time-to-event modeling, and we will introduce a special algorithm for the first stage. The second stage allows for rolling enrollment and dose assignments while toxicity outcomes of enrolled patients are still pending. The dose assignment decisions are made based on the posterior predictive distribution of the toxicity outcomes for the pending patients. The two-stage design will become clear later on.
\subsection{Posterior Predictive Imputation}\label{imputation}
We follow the idea of the PoD-TPI design \citep{zhou2020pod} for decision-making. Specifically, the observed time-to-event data can be used to predict the probability of a pending patient experiencing MT or DLT
at the end of follow-up, under which we can compute the probability of possible dose allocation decisions. These probabilities of decisions (PoDs) are used to guide dose allocation decisions. We discuss details next.
At a given moment of the trial, suppose a total of $n$ patients have been enrolled, and the current dose used for treating patients in the trial is $d$. Recall that $Y^{(v_i)}_i$ denotes the observed toxicity outcome by follow-up time $v_i$ and $x_i$ denotes the assigned dose level for patient $i$. Let $M_i = 1$ if patient $i$ is still being followed without definitive toxicity outcome, i.e., $v_i < W$; and let $M_i = 0$ if the patient finishes the follow-up, i.e., $v_i=W$. In missing data literature, $M_i$ is the missingness indicator. Denote $\mathcal{I}_d = \{i : x_i = d, M_i = 1\}$ the index set of the pending patients treated at dose $d$. Using this notation, we define the outcomes of the patients as follows. At dose $d$, let $S_d = \{Y^*_i : i \in \mathcal{I}_d\}$ be the set of future unobserved outcomes of pending patients when they complete the follow-up. And let $C_d = \{Y^*_i : i \notin \mathcal{I}_d\}$ be the set of observed outcomes of patients who have completed follow-up. If $S_d$ were known, then given a statistical design, the dose-finding decision would be given by $A_d = \mathcal{A}(S_d,C_d),$ where $\mathcal{A}$ represents a statistical design using complete data $(S_d,C_d)$. However, since $S_d$ is not observed, it is treated as random, which in turn makes $A_d$ a random variable. With proper probability modeling of $S_d$, we can then calculate the probability of $A_d$, which is the PoD.
Recall $p_{d,c} = Pr(Y_i^{*}= c | x_i =d)$ is the marginal probability of experiencing toxicity outcome $c\in\{1,\dots,4\}$ at dose level $d$ when a patient completes the follow-up.
Denote $\boldsymbol{p_d}=(p_{d,1},p_{d,2},p_{d,3},p_{d,4})$. The proposed model relies on the estimation of $\boldsymbol{p_d}$ using the observed data $\{Y_i^{(v_i)}, v_i\}$, which can be connected by the following predictive probability. In particular, for patient $i$ who has been followed for time $v_i$ with outcome $\{Y_i^{(v_i)}\}$, the predictive probability for future outcome $Y_i^*$ at the end of follow-up is given by
\begin{align}\label{eq4}
Pr(Y_i^*=c | Y_i^{(v_i)}=k, x_i=d)
= &\int \underbrace{ Pr(Y_i^*=c | Y_i^{(v_i)}=k,x_i=d, \boldsymbol{p_d})}_{I} \pi(\boldsymbol{p_d} | \text{data} )
d\boldsymbol{p_d}. \end{align}
\noindent where $\pi(\boldsymbol{p_d} | \text{data} )$ is the posterior distribution of $\boldsymbol{p_d}$. For $I$, using Bayes' theorem, we express it as a function of $\boldsymbol{p_d} $ as follows:
\begin{align*}
I
=& \frac{Pr(Y_i^{(v_i)}= k |Y^{*}_i=c) Pr(Y^{*}_i=c|x_i=d,\boldsymbol{p_d})}{\sum_{h=1}^4Pr(Y_i^{(v_i)}= k | Y^{*}_i=h)Pr(Y^{*}_i=h|x_i=d, \boldsymbol{p_d}) }\\
=&\frac{w_{k, c} p_{d,k}}{\sum_{h=1}^4 w_{k, h} p_{d,h}}. \end{align*}
Therefore, Equation (\ref{eq4}) can be estimated by posterior inference using the latent probit model. Specifically, we obtain MCMC samples of $\boldsymbol{g}$ and $\beta$, denoted as $\{\boldsymbol{g}^{(b)}, \beta^{(b)}, b=1,\dots,B\}$. Therefore, the posterior predictive probability in Equation (\ref{eq4}) can be approximated as, \begin{equation}
Pr(Y_i^{*}=c| Y_i^{(v_i)}= k, x_i=d)
= \frac{1}{B} \sum_{b=1}^B \frac{ w_{k, c} \hat{p}_{d,k}^{(b)}}{\sum_{h=1}^4 w_{k, h} \hat{p}^{(b)}_{d,h}}, \end{equation} where $\hat{p}^{(b)}_{d,c} = \Phi(\hat{q}_c^{(b)} - d\hat{\beta}^{(b)}) - \Phi(\hat{q}_{c-1}^{(b)} - d\hat{\beta}^{(b)}).$
Given the predictive probability of the $i$th patient, we can easily derive the probability of observing $S_d = s$ at dose $d$, where $s \in \{(s_1,\dots,s_J); s_j \in \{1,2,3,4\} \}$ is the set of possible future outcomes for the pending patients and $J_d=|\mathcal{I}_d|$ is the number of pending patients at dose $d$. Assuming independence across patients' outcomes, we have \begin{equation}
Pr(S_d=s|Y_i^{(v_i)},x_i) = \prod_{i\in \mathcal{I}_d} Pr(Y^*_i = s_j|Y_i^{(v_i)},x_i) , \ j=1,\dots,J_d. \end{equation} \noindent Finally, we define the PoD as \begin{equation}\label{eq9}
Pr(A_d=a|Y_i^{(v_i)},x_i) = \sum_{s:\mathcal{A}(s,C_d)=a}Pr(S_d=s|Y_i^{(v_i)},x_i), \end{equation} where decision $a$ denotes the dose-finding decision based on a dose-finding algorithm $\mathcal{A}$ for complete data. An algorithm $\mathcal{A}$ is introduced in Section \ref{algorithm} next. Based on PoDs, the proposed PoD-BIN design assigns patients to a dose according to the decision $A^*_d$ with the largest PoD, i.e.,
\begin{equation}\label{eq10}A^*_d = \argmax_{a} Pr(A_d=a|Y_i^{(v_i)},x_i).\end{equation}
\section{Trial Design}\label{des} \subsection{ Target toxicity }
\noindent The goal of the proposed PoD-BIN design is to incorporate the time-to-event of both MT and DLT outcomes into the dose-escalation decisions. We first combine the MT and DLT outcomes in the form of a weighted average, representing the toxicity burden of each patient and dose. The concept of toxicity burden was first introduced by \cite{bekele2004dose}. Toxicity burden requires elicitation of positive-valued numerical weights with a higher weight corresponding to greater severity, where the weights characterize the relative extent of harm that is associated with experiencing the toxicity at the given severity level in relation to the other severity levels. These weights are elicited based on physicians' consensus at the design stage of a trial. In this case, we denote $r_M$ as the \textit{relative severity weight} for MT relative to DLT. For example, $r_M=0.2$ is interpreted as that the impact of experiencing five MT events on patients' health is equivalent to that of experiencing one DLT event. We define the toxicity burden (TB) at dose $d$ as \begin{equation} TB_d= r_M{p}_{MT,d} + {p}_{DLT,d}, \end{equation}
where ${p}_{MT,d},{p}_{DLT,d}$ are the probabilities of MT and DLT at dose $d$, defined as \begin{equation} \begin{aligned}
p_{MT, d} &= Pr(Y^*_i = 2 |x_i=d) + Pr(Y^*_i = 4 |x_i=d)=p_{d,2}+p_{d,4} , \ \text{and} \\ p_{DLT, d} &= Pr(Y^*_i = 3 |x_i=d) + Pr(Y^*_i = 4 |x_i=d)=p_{d,3}+p_{d,4},
\end{aligned} \end{equation}
respectively. Let $p^*_{DLT}$ be the target DLT probability as in conventional phase I trials, for example, $p^*_{DLT}=0.3$ . We further define the target toxicity burden as
\begin{equation}
TTB=p^*_{DLT} + r_Mp^*_{MT},
\end{equation}
where $p^*_{MT}$ is the target MT probability. For simplicity, we set $p^*_{MT}=0$ in subsequent numerical examples, although it needs not to be $0$. Similar to $p^*_{DLT}$, $p^*_{MT}$ should be elicited with trial physicians based on the specific context of a trial.
\subsection{Dose assignment algorithms}\label{algorithm} The proposed PoD-BIN design consists of two stages. The first stage is a simple interval-based design that ends until sufficient information from patients has been collected. Stage I requires that enrolled patients are fully followed with no pending outcomes, i.e., no rolling enrollment. At the second stage, the Bayesian time-to-event probability model introduced in Section \ref{prob} is applied, allowing new patients to be enrolled when previously enrolled patients have pending outcomes. In addition, the posterior predictive imputation of pending outcomes is performed, and patients are allocated to available doses according to the optimal PoD.
\paragraph{Stage I }
The proposed Stage I algorithm follows and extends the idea in the i3+3 design \citep{Liu2020} (Web Appendix C). Specifically, consider decisions based on an estimated toxicity burden $\widehat{TB_d}$ and an equivalence interval ($EI$) of the $TTB$. We define the $EI$ of $TTB$ as $(TTB-\epsilon_1, TTB+\epsilon_2)$ \noindent where $\epsilon_1, \epsilon_2$ are small constants that reflect investigators' tolerance of the deviation from $TTB$ for the MTD. In other words, $(TTB-\epsilon_1)$ is the lowest toxicity burden for which a dose can be considered as the MTD, and $(TTB+\epsilon_2)$ is the highest burden. Suppose dose $d$ is the current dose used to treat patients. Consider an estimated toxicity burden, $ \widehat{TB}_d= r_M\frac{n_{MT,d}}{n_d} + \frac{n_{DLT,d}}{n_d},$ where $n_{d}$ is the total number of patients, and $n_{MT,d}\leq n_d$ and $n_{DLT,d}\leq n_d$ are the numbers of patients who experience MT and DLT at dose $d$, respectively. Define $\widehat{TB}_{d, -1}$ the value of $\widehat{TB_d}$ by assuming that the patient with the least severe toxicity outcome enrolled at the current dose $d$ had no toxicity at all. Mathematically, $\widehat{TB}_{d,-1}= r_M\frac{n_{MT,d,-1}}{n_d} + \frac{n_{DLT,d,-1}}{n_d},$ where $n_{MT,d,-1}=n_{MT,d}-1$ if the least severe patient experiences MT, $n_{MT,d,-1}=n_{MT,d}$ otherwise. Similarly, $n_{M=DLT,d,-1}=n_{DLT,d}-1$ if the least severe patient experiences DLT, $n_{DLT,d,-1}=n_{DLT,d}$ otherwise. Table \ref{tab:stage1} provides an example for the calculation of $\widehat{TB}_{d}$ and $\widehat{TB}_{d, -1}$. In the example, the patient with the least toxicity treated at the current dose $d$ is the first patient with only MT. Then the MT is removed and replaced by no toxicity in order to calculate $\widehat{TB}_{d, -1}$. In the case when the patient with the least toxicity has no toxicity, then $\widehat{TB}_{-1,d} =\widehat{TB}_{d}$.
\begin{table} \caption{An example of $\widehat{TB}_d$ and $\widehat{TB}_{d, -1}$ for $n_d=3$ patients, with $n_{MT,d}=2$ MT events and $n_{DLT,d}=2$ DLT events. Assume $r_M=0.2,$ which means a DLT event is the same as $1/0.2=5$ MT events in terms of harm to health. Top panel: the observed data and the corresponding toxicity burden $\widehat{TB}_d$. Bottom panel: the data with one MT (for patient 1) deleted and the corresponding toxicity burden $\widehat{TB}_{d, -1}$.} \label{tab:stage1} \begin{center} \small \begin{tabular}{ccccc} \hline \multirow{2}{*}{Patient $\#$}& \multicolumn{3}{c}{Observed data} & \multirow{2}{*}{$\widehat{TB}_{d} =r_M\frac{n_{MT,d}}{n_d} + \frac{n_{DLT,d}}{n_d}$ } \\
& & MT & DLT & \\ \cline{1-5} 1& & \bf{1} & 0 & \\
2 & &1 & 1 & $ 0.2 \times \frac{2}{3} + \frac{2}{3} = 0.8$ \\
3 & & 0 & 1 & \\ \hline $n_d=3$& & $n_{MT,d}= 2$ & $n_{DLT,d}=2$\\ \hline \end{tabular}
\begin{tabular}{ccccc} \hline \multirow{2}{*}{Patient $\#$}& \multicolumn{3}{c}{Delete-1-MT from Pat $\#1$} & \multirow{2}{*}{$\widehat{TB}_{d,-1} =r_M\frac{n_{MT,d,-1}}{n_d} + \frac{n_{DLT,d,-1}}{n_d}$ } \\
& & MT & DLT & \\ \cline{1-5}
1 & & \st{1} \bf{0} & 0 &\\
2 & & 1 & 1 & $0.2 \times \frac{1}{3} + \frac{2}{3} = 0.73 $ \\
3 & & 0 & 1 & \\
\hline $n_d=3$& &$n_{MT,d,-1}= \bf{1}$ & $n_{DLT,d,-1}=2$ \\ \hline
\end{tabular}
\end{center} \end{table} Based on $\widehat{TB}_d$ and $\widehat{TB}_{d,-1}$, PoD-BIN uses a fixed algorithm for Stage I dose-finding summarized in Algorithm~\ref{stageIrules}. The algorithm follows the idea in i3+3 to simplify dose-finding decisions.
\begin{algorithm}[tbh] \caption{Stage I dose assignment rules $\mathcal{A}(S_d=\emptyset, C_d)$} \label{stageIrules} \begin{algorithmic} \STATE Suppose dose $d$ is currently administered for patients enrolled in the trial. \IF{$\widehat{TB}_d$ is below the $EI$} \STATE Escalate and enroll patients at the next higher dose $\mathcal{A}(\emptyset, C_d)=(d + 1)$;
\ELSIF{$\widehat{TB}_d$ is inside the $EI$}
\STATE Stay and enroll patients at the current dose $\mathcal{A}(\emptyset, C_d)=d$;
\ELSIF{$\widehat{TB}_d$ is above the $EI$} \IF{$\widehat{TB}_{d, -1}$ is below the $EI$} \STATE Stay and enroll patients at the current dose $\mathcal{A}(\emptyset, C_d)=d$; \ELSE \STATE De-escalate and enroll patients at the next lower dose $\mathcal{A}(\emptyset, C_d)=(d-1)$. \ENDIF \ENDIF \end{algorithmic} \end{algorithm}
To summarize, in Stage I at dose $d$, the unobserved data $S_d=\emptyset$ is an empty set since Stage I only makes a decision when all the patients at dose $d$ complete follow-up and record outcomes. Therefore, the Stage I decision can be described as $A^*_d=\mathcal{A}(S_d=\emptyset, C_d)\in \{d-1,d,d+1\}$, where $\mathcal{A}(\emptyset, C_d)$ represents Algorithm \ref{stageIrules}. This decision not only guides dose-finding in Stage I, but it will also be used in Stage II as well, discussed next.
\paragraph{Stage II } During the first stage, patients are enrolled in cohorts, say, every three patients and newly enrolled patients are assigned to doses following the set of simple dose assignment rules. After each enrollment, check if all four outcomes are observed in the trial or if the number of patients reaches a certain sample size threshold denoted as $n^*$. If the answer is no, the trial continues as Stage I; otherwise, the trial proceeds to Stage II. We recommend using a sample size threshold $n^*$ no less than 12 patients. In Stage II, PoD-BIN switches to full model-based rolling enrollment guided by PoDs. Algorithm~\ref{stageIIrules} provides details.
The suspension rules in Algorithm~\ref{stageIIrules} are needed to account for the variability of the pending outcomes and protect patient safety. In practice, the values $\lambda_E$ and $\lambda_D$ should be chosen according to the desired extent of safety and calibrated based on simulations. To ensure safety, we recommend choosing $\lambda_E\geq 0.8$ and setting $\lambda_D\leq0.25$. For example, in the simulation studies (Section \ref{Sim}), we use $\lambda_E=1$ and $\lambda_E=0$ , which minimizes the chance of risky decisions while previously enrolled patients have not completed follow-up.
\begin{algorithm}[H] \caption{Stage II dose assignment rules} \label{stageIIrules} \begin{algorithmic} \STATE Suppose dose $d$ is currently administered for patients enrolled in the trial. Denote $S_d$ the pending data and $C_d$ the observed data at dose $d$. \IF{there are patients at dose $d$ with pending outcomes} \STATE 1) Apply the inference based on the Bayesian models in Sections \ref{prob}-\ref{imputation}. \STATE 2) Compute the PoD in Equation (\ref{eq9}), in which the decision $\mathcal{A}(S_d, C_d)$ is based on Algorithm \ref{stageIrules}. And obtain the optimal decision $A^*_d \in \{d-1, d, d+1\}$. \STATE 3) To ensure patient safety, we adopt the following suspension rules \citep{zhou2020pod}. If none of the following suspension rules are invoked, the next cohort of patients is assigned to the dose indicated by $A^*_d$. Otherwise, suspend the trial until none of the suspension rules are invoked.
\begin{itemize}
\item {\bf Suspension rule 1}: If the current dose $d$ has not been tested before and if the number of patients with pending outcomes exceeds or equals to 3 ($J_d\geq3$), suspend the enrollment.
\item {\bf Suspension rule 2}: If $A^*_d=d+1$ (i.e., escalate): the enrollment is suspended if (1) $ Pr(A_d= d+1 | Y_i^{(v_i)},x_i) < \lambda_E$ for some pre-determined threshold $\lambda_E \in [0.33, 1]$ or (2) if the number of patients who have completed follow-up without DLT is 0.\\
Condition (1) suggests that dose escalation is only allowed if the confidence of escalation is higher than $\lambda_E$, and a larger $\lambda_E$ represents more conservative escalation decisions. Condition (2) states that escalation is allowed only if at least one patient has finished follow-up with no DLT at the current dose.
\item {\bf Suspension rule 3}: If $A^*_d=d$ (i.e., stay): the enrollment is suspended if $ Pr(A_d= d-1 | Y_i^{(v_i)},x_i) > \lambda_D$ for some pre-determined threshold $\lambda_D \in [0, 0.5]$. This means that stay is allowed only if there is a relatively low chance of de-escalation. A smaller $\lambda_D$ represents more conservative stay decisions.
\end{itemize} \ELSE \STATE Assign the next cohort of patients according to $\mathcal{A}(\emptyset, C_d)$ in Algorithm \ref{stageIrules}. \ENDIF \end{algorithmic} \end{algorithm}
\subsection{Safety rules}\label{safety} In addition to the suspension rules, we include the following safety rules throughout the trial. Recall that $n_d$ is the number of patients treated at dose $d$, $p_{DLT,d}$ is the DLT probability at dose $d$, and $p^*_{DLT}$ is the target DLT rate.
\begin{itemize}
\item Safety rule 1 (\textit{early termination}): At any moment in the trial, if $n_1\geq 3$ and $Pr( p_{DLT,1} > p^*_{DLT}|\text{data}) > 0.95$, terminate the trial due to excessive toxicity.
\item Safety rule 2 (\textit{dose exclusion}): At any moment in the trial, if $n_d \geq 3$ and $Pr( p_{DLT,d} > p^*_{DLT}|\text{data})>0.95$, remove dose $d$ and its higher doses from the trial. \end{itemize}
The posterior probability $Pr(p_{DLT,d}>p^*_{DLT}| \text{data)}$ is calculated using the observed binary DLT data at dose $d$ with a prior distribution $p_{DLT,d}\sim Beta(1,1)$. If Safety rule 1 is triggered in Stages I or II, then the trial is terminated. Safety rule 2 has an exception. Removed doses may be put back to trials if the rule is no longer violated after patients with pending data complete follow-up.
\subsection{MTD selection} The trial is completed if the number of enrolled patients reaches the pre-specified maximum sample size or Safety rule 1 is triggered. If Safety rule 1 is triggered, no MTD is selected since all doses are considered overly toxic. Otherwise, let $\widetilde{TB}_d=r_M\tilde{p}_{MT,d}+\tilde{p}_{DLT,d}$ where $\tilde{p}_{MT,d}$ and $\tilde{p}_{DLT,d}$ are the posterior means of $p_{MT,d}$ and $p_{DLT,d}$. Let $\{ \widetilde{TB}'_d\}$ be the isotonically transformed $\{ \widetilde{TB}_d\}$ and
PoD-BIN recommends dose $d^\ast$ as the MTD, defined as
\begin{equation}
d^\ast= \argmin_{d\in D, n_d>0 } |\widetilde{TB}'_d - TTB|.
\end{equation} among all doses $d$ for which $n_d>0$. Figure \ref{fig:flowchart} illustrates the PoD-BIN design as a flowchart.
\begin{figure}
\caption{ A flowchart illustration of the proposed PoD-BIN design. The first cohort is treated at the lowest dose. For subsequent cohorts in Stage I, dose assignments are made based on simple decision rules with complete follow-up data. Once the number of enrolled patients reaches sample size threshold $n^*$ or all the ordinal outcomes are observed, whichever comes first, Stage II of the design is invoked. This process will be repeated until the stopping criteria are satisfied.}
\label{fig:flowchart}
\end{figure}
\section{ Numerical Studies }\label{sim} \subsection{Illustration of a single trial in the simulation}
\noindent Through a hypothetical dose-finding trial in Figure \ref{fig:hypo}, we illustrate how PoD-BIN is used as a design to guide decision-making. The x-axis is the study time in days, and the y-axis is the dose levels at which patients are treated. Each circle represents a patient.
Assume $TTB$ is $0.25$ with $ EI=(0.15, 0.35)$, the assessment window $W=50$ days, $r_M=0.15,$ $\lambda_E = 0.95,$ and $\lambda_D = 0.15 $. Patients are enrolled in cohort sizes of three with a total sample size of $n=24$. The inter-patient arrival time is generated from an exponential distribution with a mean of 10 days, i.e., on average, a new patient is enrolled every 10 days. In addition, the sample size threshold for starting Stage II is set to be $n^*=12$, which is half of the total sample size. The first 12 patients are enrolled in cohorts. This means that the patients in the previously enrolled cohorts must complete the 50-day observation period before the next cohort of patients can be enrolled. Therefore, no rolling enrollment is allowed in the first stage. Stage II is invoked after the assignment of patient 12 as Stage I ends. Patients $13-15$ are placed at dose level $5$. Patient $13$ experiences DLT, and patient $15$ experiences MT. Hence, dose level 5 is deemed unsafe, and the next cohort (patients $16-18$) is enrolled at dose level 4. Later, patients 16, 17, and 18 experience DLT and MT outcomes, which renders dose level 4 unsafe. Subsequently, the trial de-escalates to dose level 3, at which patients 19 through 24 are enrolled
according to PoD-BIN.
At the end of the trial, dose level 3 is recommended as the MTD by PoD-BIN, with 2 MT and 0 DLT events
out of 9 patients. The trial lasts
563 days in total.
\begin{figure}
\caption{ A hypothetical dose-finding trial using the PoD-BIN design.}
\label{fig:hypo}
\end{figure}
\subsection{ Simulations}\label{Sim}
We conduct simulation studies to evaluate the operating characteristics of the PoD-BIN design. The implementation of PoD-BIN requires the specification of a few parameters. We set $TTB = p^*_{DLT}=0.25,$ $ EI=(0.15, 0.35)$, $n^*=12$ and $r_M=0.15$ (so that a DLT event is as severe as more than six MT events). We let $\lambda_E = 1$ and $\lambda_D = 0$ for the suspension criteria in Stage II to minimize the chance of risky decisions. These two values are the most conservative choices that greatly reduce the possibility of making aggressive decisions when there are pending patients at the current dose. Less conservative choices will be discussed in the sensitivity studies next.
We compare the proposed PoD-BIN design to the TITE-CRMMC design \citep{Lee2019}, an extension of the TITE-CRM, which allows for the specification of a toxicity target defined based on both DLT and MT constraints. Importantly, TITE-CRMMC incorporates the follow-up times of the pending patients using a weighted likelihood approach. For TITE-CRMMC, we apply the recommended model configuration with a $N(0, 1.34^2)$ prior for the model parameter, and a skeleton of $(0.06, 0.14, 0.25 ,0.38, 0.50)$. Note that in TITE-CRMMC, the toxicity outcome $Y$ has three levels, $Y = 0$ if no toxicity, $Y=1$ if MTs occur, and $Y=2$ if DLTs occur. There are two toxicity constraints based on the tail probability of $Y$ under dose $x$, $Pr(Y
\geq 1 | x) = 0.5$ and $Pr(Y = 2 | x) = 0.25$. The next dose level for TITE-CRMMC is selected based on the marginal posterior of the dose levels that minimizes the distance to both pre-specified toxicity constraints, and the MTD recommendation is based on the same definition. In addition, we add Safety rules 1 \& 2 in Section \ref{safety} to the TITE-CRMMC design for a fair comparison.
Furthermore, we compare the performance of the PoD-BIN design against a benchmark version of the design, in which Stage I of PoD-BIN is implemented throughout a trial. In other words, no rolling enrollment is allowed, and all enrolled patients must be followed for the full duration with known toxicity outcomes before future cohorts may be enrolled.
We consider a total of six different scenarios and simulate trials investigating five dose levels with a maximum sample size of $n=30$ patients and a cohort size of three. The starting dose level is dose 1. As in a typical oncology phase I trial, we set the length of the assessment window at $W=28$ days. A conditional uniform model is used to generate the time-to-toxicity in simulations for the proposed method: we first determine if a patient has a toxic response based on the true toxicity probability of MT and DLT; and if toxicity is generated, we generate time-to-toxicity from Uniform$(0,W)$. Moreover, the arrival time between two consecutive patients follows an exponential distribution with rate 0.14, meaning on average, one patient is enrolled per week. For each scenario, we simulate 1,000 trials.
We also conduct the same simulation with different sample sizes 24 and 36 as a sensitivity study, the results of which are reported in Web Appendix D.
\subsection{ Operating Characteristics }
The simulation results with sample size 30 are summarized in Table \ref{tab:comp1}, which shows the percentage of trials that select each dose as the MTD and the percentage of patients allocated to each dose. The true MTD under different designs is bolded. In Scenarios 1 to 3, PoD-BIN outperforms other methods in terms of recommending the correct MTD (PCS). In Scenario 4, the Benchmark method yields slightly better PCS ($80\%$) compared to PoD-BIN($78\%$), with TITE-CRMMC falling behind ($65\%$). In Scenario 5, TITE-CRMMC yields higher PCS compared to PoD-BIN and Benchmark. TITE-CRMMC recommends the correct MTD in $66\%$ of the trials, while PoD-BIN and Benchmark in $61\%$. In Scenario 6, dose levels 4 and 5 are both considered as the true MTD based on PoD-BIN, whereas only dose 5 is considered as the true MTD based on TITE-CRMMC. PoD-BIN and the Benchmark design produce high PCS if dose 5 is considered as the MTD. In terms of allocating patients to the correct dose, PoD-BIN and Benchmark yield similar results and perform well in Scenarios 1,2,5,6. Overall, the performance of PoD-BIN with $\lambda_E = 1, \lambda_D = 0$ is comparable to that of the Benchmark method, in which dose assignments are performed with complete follow-up data. Across all studied scenarios, the PoD-BIN design results in the best performance in terms of the percentage of trials concluding a dose above the true MTD (POS) and the percentage of patients allocated to doses higher than the MTD (POA), two important safety metrics.
\begin{table} \scriptsize \caption{Performance of the PoD-BIN method compared with existing methods. The target toxicity burden is $0.25$ and the $EI$ is $(0.15, 0.35)$.} \label{tab:comp1} \centering \tiny \begin{tabular}{llllllllllll} \hline
& \multicolumn{5}{c}{Recommendation percentage} & & \multicolumn{5}{c}{Allocated dose percentage} \\\hline \multicolumn{1}{c}{Scenario 1} & 1 & 2 & 3 & 4 & 5 & & 1 & 2 & 3 & 4 & 5 \\\hline Pr(MT) & \textbf{0.31} & 0.40 & 0.50 & 0.60 & 0.69 & & \textbf{0.31} & 0.40 & 0.50 & 0.60 & 0.69 \\ Pr(DLT) & \textbf{0.26} & 0.38 & 0.50 & 0.62 & 0.74 & & \textbf{0.26} & 0.38 & 0.50 & 0.62 & 0.74 \\ TB & \textbf{0.31} & 0.44 & 0.58 & 0.71 & 0.84 & & \textbf{0.31} & 0.44 & 0.58 & 0.71 & 0.84 \\\hline PoD-BIN & \textbf{73.0 } & 13.0 & 1.0 & 0.0 & 0.0 & & \textbf{85.0 } & 13.0 & 2.0 & 0.0 & 0.0 \\ Benchmark & \textbf{66.7 } & 16.8 & 1.2 & 0.0 & 0.0 & & \textbf{82.4 } & 15.2 & 2.3 & 0.2 & 0.0 \\ TITE-CRMMC & \textbf{65.0 } & 24.0 & 1.0 & 0.0 & 0.0 & & \textbf{57.0 } & 32.0 & 9.0 & 2.0 & 0.0 \\\hline
& & & & & & & & & & & \\\hline \multicolumn{1}{c}{Scenario 2} & 1 & 2 & 3 & 4 & 5 & & 1 & 2 & 3 & 4 & 5 \\\hline Pr(MT) & 0.11 & \textbf{0.26} & 0.47 & 0.68 & 0.85 & & 0.11 & \textbf{0.26} & 0.47 & 0.68 & 0.85 \\ Pr(DLT) & 0.07 & \textbf{0.20} & 0.42 & 0.67 & 0.86 & & 0.07 & \textbf{0.20} & 0.42 & 0.67 & 0.86 \\ TB & 0.09 & \textbf{0.24} & 0.49 & 0.77 & 0.98 & & 0.09 & \textbf{0.24} & 0.49 & 0.77 & 0.98 \\\hline PoD-BIN & 6.0 & \textbf{86.0 } & 7.0 & 0.0 & 0.0 & & 37.0 & \textbf{50.0 } & 12.0 & 1.0 & 0.0 \\ Benchmark & 3.7 & \textbf{85.9 } & 10.2 & 0.0 & 0.0 & & 31.4 & \textbf{52.2 } & 15.2 & 1.2 & 0.1 \\ TITE-CRMMC & 7.0 & \textbf{69.0 } & 24.0 & 0.0 & 0.0 & & 15.0 & \textbf{39.0 } & 35.0 & 9.0 & 1.0 \\\hline
& & & & & & & & & & & \\\hline \multicolumn{1}{c}{Scenario 3} & 1 & 2 & 3 & 4 & 5 & & 1 & 2 & 3 & 4 & 5 \\\hline Pr(MT) & 0.09 & 0.21 & \textbf{0.38} & 0.56 & 0.70 & & 0.09 & 0.21 & \textbf{0.38} & 0.56 & 0.70 \\ Pr(DLT) & 0.02 & 0.08 & \textbf{0.21} & 0.43 & 0.68 & & 0.02 & 0.08 & \textbf{0.21} & 0.43 & 0.68 \\ TB & 0.03 & 0.11 & \textbf{0.27} & 0.52 & 0.78 & & 0.03 & 0.11 & \textbf{0.27} & 0.52 & 0.78 \\\hline PoD-BIN & 0.0 & 18.0 & \textbf{81.0 } & 1.0 & 0.0 & & 14.0 & 36.0 & \textbf{39.0 } & 9.0 & 1.0 \\ Benchmark & 0.1 & 19.7 & \textbf{79.1 } & 1.1 & 0.0 & & 13.9 & 33.1 & \textbf{42.2 } & 10.2 & 0.7 \\ TITE-CRMMC & 0.0 & 13.0 & \textbf{73.0 } & 14.0 & 0.0 & & 12.0 & 19.0 & \textbf{42.0 } & 23.0 & 4.0 \\\hline
& & & & & & & & & & & \\\hline
& \multicolumn{5}{c}{Recommendation percentage} & & \multicolumn{5}{c}{Allocated dose percentage} \\\hline \multicolumn{1}{c}{Scenario 4} & 1 & 2 & 3 & 4 & 5 & & 1 & 2 & 3 & 4 & 5 \\\hline Pr(MT) & 0.11 & 0.21 & \textbf{0.31} & 0.40 & 0.48 & & 0.11 & 0.21 & \textbf{0.31} & 0.40 & 0.48 \\ Pr(DLT) & 0.03 & 0.09 & \textbf{0.21} & 0.37 & 0.57 & & 0.03 & 0.09 & \textbf{0.21} & 0.37 & 0.57 \\ TB & 0.05 & 0.13 & \textbf{0.25} & 0.43 & 0.64 & & 0.05 & 0.13 & \textbf{0.25} & 0.43 & 0.64 \\ PoD-BIN & 1.0 & 18.0 & \textbf{78.0 } & 3.0 & 0.0 & & 17.0 & 35.0 & \textbf{36.0 } & 11.0 & 1.0 \\ Benchmark & 0.3 & 16.0 & \textbf{79.9 } & 3.7 & 0.1 & & 15.5 & 32.5 & \textbf{38.5 } & 12.1 & 1.5 \\ TITE-CRMMC & 0.0 & 12.0 & \textbf{65.0 } & 22.0 & 0.0 & & 13.0 & 20.0 & \textbf{40.0 } & 22.0 & 4.0 \\
& & & & & & & & & & & \\\hline \multicolumn{1}{c}{Scenario 5} & 1 & 2 & 3 & 4 & 5 & & 1 & 2 & 3 & 4 & 5 \\\hline Pr(MT) & 0.02 & 0.07 & 0.20 & \textbf{0.37} & 0.48 & & 0.02 & 0.07 & 0.20 & \textbf{0.37} & 0.48 \\ Pr(DLT) & 0.00 & 0.00 & 0.03 & \textbf{0.11} & 0.30 & & 0.00 & 0.00 & 0.03 & \textbf{0.11} & 0.30 \\ TB & 0.00 & 0.01 & 0.06 & \textbf{0.17} & 0.38 & & 0.00 & 0.01 & 0.06 & \textbf{0.17} & 0.38 \\\hline PoD-BIN & 0.0 & 0.0 & 25.0 & \textbf{61.0 } & 13.0 & & 10.0 & 11.0 & 19.0 & \textbf{37.0 } & 23.0 \\ Benchmark & 0.0 & 1.6 & 23.5 & \textbf{61.4 } & 13.5 & & 10.1 & 10.6 & 16.9 & \textbf{37.2 } & 25.3 \\ TITE-CRMMC & 0.0 & 0.0 & 6.0 & \textbf{66.0 } & 28.0 & & 11.0 & 10.0 & 13.0 & \textbf{33.0 } & 33.0 \\\hline
& & & & & & & & & & & \\\hline \multicolumn{1}{c}{Scenario 6} & 1 & 2 & 3 & 4 & 5 & & 1 & 2 & 3 & 4 & 5 \\\hline Pr(MT) & 0.19 & 0.24 & 0.28 & \textbf{0.32} & \textbf{0.35} & & 0.19 & 0.24 & 0.28 & \textbf{0.32} & \textbf{0.35} \\ Pr(DLT) & 0.04 & 0.06 & 0.08 & \textbf{0.11} & \textbf{0.15} & & 0.04 & 0.06 & 0.08 & \textbf{0.11} & \textbf{0.15} \\ TB & 0.07 & 0.09 & 0.12 & \textbf{0.16} & \textbf{0.20} & & 0.07 & 0.09 & 0.12 & \textbf{0.16} & \textbf{0.20} \\\hline PoD-BIN & 0.0 & 4.0 & 26.0 & \textbf{29.0 } & \textbf{41.0 } & & 15.0 & 18.0 & 23.0 & \textbf{21.0 } & \textbf{23.0 } \\ Benchmark & 0.6 & 6.2 & 22.2 & \textbf{26.3 } & \textbf{44.7 } & & 15.2 & 18.4 & 20.2 & \textbf{20.7 } & \textbf{25.6 } \\ TITE-CRMMC & 0.0 & 3.0 & 23.0 & 40.0 & \textbf{34.0 } & & 15.0 & 18.0 & 26.0 & 25.0 & \textbf{16.0 } \\\hline \end{tabular}
\footnotesize Note: The true MTD under each scenario is bolded. \end{table}
\subsection{ trade-off between Safety and Speed }\label{trade-off}
While rolling enrollment can speed up a dose-finding trial, a major issue for designs allowing rolling enrollment is the possibility of making an inconsistent decision. Here, an inconsistent decision refers to a decision by the rolling design that is different from its corresponding non-rolling design. In other words, an inconsistent decision by the rolling design is different from the decision that would have been made if all patients had been followed for the full duration with complete toxicity outcomes. For PoD-TPI and other rolling designs, there can be six types of inconsistent decisions
denoted as AB, where A is the decision that should have been made if complete toxicity data were available, and B is the decision made by the rolling design allowing pending toxicity outcomes. The six types are: (1) should de-escalate but stayed (DS), (2) should de-escalate but escalated (DE), (3) should stay but escalated (SE), (4) should stay but de-escalated (SD), (5) should escalate but de-escalated (ED), and (6) should escalate but stayed (ES). For example, SE refers to an inconsistent decision of E, escalate, made by the rolling design while the corresponding non-rolling design based on complete data would make the decision S, stay at the current dose. Apparently, SE is a risky decision since the rolling design wrongly exposes patients to a high dose.
Three inconsistent decisions DS, DE and ES are considered as {\bf risky decisions}, as they can expose patients to overly toxic doses, and SD, ED and ES are conservative decisions, which could allocate patients to sub-therapeutic doses. In general, the more liberal a rolling design allows incomplete data to inform doses for future patients, the higher risk the design has in putting patients at overly toxic doses. To balance the trade-off of speed and safety, PoD-BIN uses the suspension thresholds $(\lambda_E,\lambda_D )$ to control the chance of making these inconsistent decisions.
We report simulation results of PoD-BIN with different combinations of the suspension threshold values $(\lambda_E,\lambda_D)$. For each scenario and suspension threshold, we simulate 1,000 trials using PoD-BIN.
Table \ref{Sen1} reports the mean and standard deviation of different metrics over the six scenarios with 1,000 simulated trials for each scenario. The scenario-specific operating characteristics are reported in Web Appendix E. We calculate seven different metrics to summarize the operating characteristics of the PoD-BIN with different thresholds, shown in Panel A of Table \ref{Sen1}. In addition, we report the percentage of inconsistent decisions to evaluate the reliability and safety of PoD-BIN in Panel B.
Table \ref{Sen1} Panel A shows comparable PCS and POS of PoD-BIN to those of Benchmark since the selection of MTD is always based on complete data. The POA, the average percentage of patients who experience MT (POMT), and the average percentage of patients who experience DLT (PODLT) of PoD-BIN are slightly better compared to Benchmark. Panel B shows the effect of $\lambda_E$ and $\lambda_D$ on limiting aggressive decisions. The closer to 1 is for $\lambda_E$ and to 0 for $\lambda_D$, the safer the PoD-BIN but also the longer the trial. Importantly, the trial duration can be greatly reduced by using the PoD-BIN method in all cases. On average, the trial duration is shortened by at least 50 days using PoD-BIN compared to Benchmark.
An important feature of PoD-BIN is the flexibility in calibrating the values of $\lambda_E$ and $\lambda_D$ to achieve the desired trade-off between speed and safety. As noted in \cite{zhou2020pod}, the threshold $\lambda_E$ controls the inconsistent of DE and SE decisions, and $\lambda_D$ controls the rate of SD decisions. They both affect the speed of the trial through suspension rules. With $\lambda_E=1, \lambda_D=0$, PoD-BIN achieves the lowest POA, POMT and PODLT, and the numbers of risky decisions (DS, DE, SE) are minimized. On the contrary, by setting $\lambda_E=0.85, \lambda_D=0.25$, the design produces higher risks of making aggressive decisions. However, the average trial duration is further shortened with less strict threshold values, which is a result of less frequent and shorter periods of suspension during a trial. The trade-off between trial speed and safety can be carefully adjusted by choosing an appropriate threshold for suspension in practice, a distinctive and important feature of PoD-BIN.\\
\begin{table}[hbtp] \caption{ Performance of PoD-BIN with different choices of $\lambda_E, \lambda_D$. $\mbox{Mean}_{\mbox{sd}}$ values are shown in all the entries. }\label{Sen1} \centering \small \subcaption*{Panel A: PCS: the average percentage of correct selection of the true MTD; PCA: the average percentage of patients correctly allocated to the true MTD; POS: the average percentage of selecting a dose above the true MTD; POA: the average percentage of patients allocated to a dose above the true MTD; POMT: the average percentage of patients experiencing moderate toxicities; PODLT: the average percentage of patients experiencing dose-limiting toxicities. The unit of PCS, PCA, POA, POMT, and PODLT is $\% $.} \begin{tabular}{lcccccc} \hline
& PCS & PCA & POA & POS & POMT & PODLT \\\hline
Benchmark & $74.00_{9.20}$ & $49.77_{16.89}$ & $13.96_{8,39}$ & $7.77_{7.27}$& $28.39_{2.51}$& $16.68_{5.64}$ \\ $ \lambda_E = 1,\lambda_D = 0$ & $\bf{74.90_{8.95}}$ & $48.62_{18.74}$ & $\bf{12.18_{7.34}}$ & $\bf{6.37_{6.22}}$ & $27.75_{3.15}$ & $16.44_{5.77}$ \\ $ \lambda_E = 0.95,\lambda_D = 0$ & $74.10_{10.27} $& $48.93_{18.30}$ & $12.54_{7.41}$ & $6.78_{6.51}$ & $\bf{27.65_{2.78}}$& $\bf{16.38_{5.73}}$ \\
$\lambda_E = 1,\lambda_D = 0 .15$ & $74.02_{9.09}$ & $49.29_{18.26}$ & $12.55_{7.85}$ & $6.67_{6.25}$ & $27.66_{2.62}$ & $16.62_{5.86}$ \\
$\lambda_E = 0.95,\lambda_D = 0 .15$ & $74.60_{9.13}$ & $49.02_{18.02}$ & $12.78_{7.63}$ & $6.47_{5.95}$ & $27.84_{2.90}$ & $16.60_{5.82}$ \\
$\lambda_E = 0.85,\lambda_D = 0 .25$ & $73.53_{9.34}$ & $\bf{49.80_{17.46}} $ & $13.75_{8.22}$ & $7.38_{6.97} $& $28.11_{2.67}$ & $16.92_{5.86}$\\\hline \end{tabular}
\subcaption*{Panel B: Frequencies of inconsistent decisions and duration. The unit of inconsistent decisions (DS, DE, SE, SD, ED, ES) is $1/100$, and the unit of duration (Dur) is day. }
\begin{tabular}{lccccccc} \hline
& DS & DE & SE & SD & ED & ES & Dur \\\hline
Benchmark & - & - & - & - & - & - & $476_{20}$ \\
$\lambda_E = 1,\lambda_D = 0 $ & $\bf{0.37_{0.57}}$ & $\bf{0.01_{0.01}}$& $\bf{0.01_{0.01}}$ & $3.87_{2.12}$ & $0.27_{0.29}$ & $\bf{1.60_{0.76}}$ & $423_{24}$ \\
$\lambda_E = 0.95,\lambda_D = 0 $ & $0.46_{0.79}$ & $0.06_{0.04}$ & $0.81_{0.41}$ & $3.97_{2.18}$ & $0.32_{0.23}$ & $1.28_{0.58}$ & $411_{25}$ \\
$\lambda_E = 1,\lambda_D = 0 .15$ & $2.38_{0.53}$ & $\bf{0.01_{0.01}}$ & $\bf{0.01_{0.01}}$ & $3.05_{1.60}$ & $0.18_{0.15}$ & $3.90_{1.84}$ & $392_{28}$ \\ $ \lambda_E = 0.95,\lambda_D = 0 .15$ & $2.43_{0.70}$ & $0.07_{0.04}$ & $0.87_{0.44}$ & $2.93_{1.55}$ & $0.20_{0.15}$ & $3.44_{1.69}$ & $383_{26}$ \\ $ \lambda_E = 0.85,\lambda_D = 0 .25$ & $3.18_{0.70}$ & $0.21_{0.19}$ & $1.82_{0.66}$ & $\bf{2.82_{1.56}}$ & $\bf{0.17_{0.15}}$ & $2.67_{1.30}$ & $\bf{376_{20}} $\\\hline \end{tabular}
\footnotesize Note: Bold values indicate the optimal performance among all cases. \end{table}
\subsection{Sensitivity Analysis} Additionally, we conduct sensitivity analysis to demonstrate the performance of PoD-BIN with different generative models to simulate patients' time-to-toxicity profiles and trial accrual rates. First, assuming patient accrual time follows $Exp(1/7)$ and assessment window length $W=28$ days, we simulate time to MT/DLT for a patient from a Weibull distribution with shape and scale parameters uniquely identified based on the following four settings. See Web Appendix G for details.\\ \indent $\bullet$ Setting 1: $80\%$ of DLTs occur in the first half of the assessment window $W=28$ days, and $80\%$ of MTs occur in the first half of the assessment window days;\\
\indent $\bullet$ Setting 2: $20\%$ of DLTs occur in the first half of the assessment window $W=28$ days, and $20\%$ of MTs occur in the first half of the assessment window days;\\ \indent $\bullet$ Setting 3: $80\%$ of DLTs occur in the first half of the assessment window $W=28$ days, and $20\%$ of MTs occur in the first half of the assessment window days;\\ \indent $\bullet$ Setting 4: $20\%$ of DLTs occur in the first half of the assessment window $W=28$ days, and $80\%$ of MTs occur in the first half of the assessment window days. \\ For each scenario and time-to-toxicity setting, we simulate 1,000 trials. The sensitivity results with four time-to-toxicity settings are summarized in Table \ref{Sen2}. Setting 1 suggests that both DLT and MT occur early during the follow-up period, and it achieves the shortest trial duration with the lowest POMT and PODLT. On the other hand, setting 2 indicates that both DLT and MT occur late during the follow-up period, and it achieves the longest trial duration with the highest POMT and PODLT. In terms of inconsistent decisions, both settings 2 and 4 with late DLT occurrence tend to have higher frequencies of risky decisions DS, DE and SE comparing to the other two settings with early DLT occurrence.
\begin{table}[H] \caption{ Performance of PoD-BIN under different time-to-toxicity
profiles. $\mbox{Mean}_{\mbox{sd}}$ alues are shown in all the entries.
}\label{Sen2} \centering \small \subcaption*{Panel A: PCS: the average percentage of correct selection of the true MTD; PCA: the average percentage of patients correctly allocated to the true MTD; POS: the average percentage of selecting a dose above the true MTD; POA: the average percentage of patients allocated to a dose above the true MTD; POMT: the average percentage of patients experiencing moderate toxicities; PODLT: the average percentage of patients experiencing dose-limiting toxicities. The unit of PCS, PCA, POA, POMT, and PODLT is $\% $.}
\begin{tabular}{lllllll}\hline
& PCS & PCA & POA & POS & POMT & PODLT \\\hline
Setting 1 & $74.42_{8.23}$ &$48.46_{18.74}$& $12.22_{7.17}$& $6.80_{6.47}$ & $\bf{27.40_{2.93}}$& $\bf{16.21_{5.82}}$ \\ Setting 2 & $\bf{75.32_{10.14}}$ &$48.74_{18.22}$ &$12.39_{7.35}$& $\bf{6.38_{6.40}}$&$27.96_{3.32}$&$16.57_{5.94}$\\ Setting 3 & $74.73_{10.95}$& $\bf{49.32_{19.19}}$& $\bf{12.18_{7.60}}$& $6.68_{6.11}$ &$27.69_{2.84}$& $16.43_{5.87}$ \\ Setting 4 &$ 74.24_{8.82}$ &$48.70_{18.68}$& $12.49_{7.62}$& $6.77_{6.54}$ &$27.74_{2.75}$& $16.44_{5.76}$ \\\hline \end{tabular}
\subcaption*{Panel B: Frequencies of inconsistent decisions and duration. The unit of inconsistent decisions (DS, DE, SE, SD, ED, ES) is $1/100$, and the unit of duration (Dur) is day. }
\begin{tabular}{llllllll}\hline &DS & DE & SE & SD & ED & ES & Dur \\\hline
Setting 1 & $\bf{0.26_{0.53}}$& $0.01_{0.01}$ &$\bf{0.01_{0.02}}$& $4.56_{2.68}$& $0.38_{0.34}$&$1.99_{0.95}$ &$\bf{415_{27}}$\\ Setting 2 & $0.46_{0.83}$ &$0.01_{0.01}$& $0.02_{0.01}$ &$\bf{3.43_{1.88}}$& $0.21_{0.19}$ &$\bf{1.29_{0.68}}$ &$429_{21}$\\ Setting 3 & $0.27_{0.56}$& $\bf{0.00_{0.01}}$& $\bf{0.01_{0.01}}$&$3.95_{2.30}$&$\bf{0.20_{0.15}}$&$1.41_{0.81}$ &$423_{26}$\\ Setting 4 & $0.55_{0.73}$& $0.02_{0.02}$&$0.05_{0.03}$ &$3.97_{2.34}$& $0.43_{0.44}$&$1.67_{0.83}$ &$422_{24}$\\\hline \end{tabular}
\footnotesize Note: Bold values indicate the optimal performance among all settings. \end{table}
\section{Discussion}\label{dis}
In this article, we have proposed the two-stage PoD-BIN design to incorporate time-to-toxicity data of multiple toxicities to speed-up phase I trials. The proposed design defines the concept of toxicity burden to summarize multiple toxicity outcomes information simultaneously and adopts a decision algorithm similar to the i3+3 design \citep{Liu2020} in Stage I. Furthermore, the proposed PoD-BIN design evaluates the posterior predictive probabilities of dose escalation decisions by using a latent probit model in Stage II. Flexible suspension rules based on the risk of different dosing decisions are added to further ensure trial safety and control the chance of making inconsistent decisions. The thresholds for suspension can be adjusted to balance the trade-off between speed and safety, as demonstrated in Section~\ref{trade-off}.
We considered a two-stage design, in which Stage I acquires complete follow-up data from patients, and Stage II allows rolling enrollment based on model inference. There are two criteria to invoke Stage II: 1) once the sample size reaches a pre-specified threshold $n^*$ , or 2) once all 4 toxicity outcomes are observed. We recommend $n^*$ no less than 12 patients. In practice, the sample size threshold $n^*$ can be determined through sensitivity analysis to evaluate its effect on the design performance. Additionally, the severity weight that quantifies the relative severities of MT and DLT must be elicited from the physicians planning the trial. These decisions are inherently subjective and require close collaboration between the statisticians and the clinical team. Multiple physicians are recommended to quantify this severity weight, and additional sensitivity analysis can be implemented to evaluating its impact on the operating characteristics of the design.
PoD-BIN delivers sound operating characteristics that are comparable to existing designs. Comparing to the TITE-CRMMC design, the proposed design tends to reduce the probability of selecting a dose having excessive toxicities and the number of patients allocated to dose above the MTD. Using the information on moderate toxicities to guide the estimation of MTD seems to result in conservative dose-finding decisions in comparison with standard phase I designs.
In a real trial, it is desirable to account for both times to toxicities as well as the number of occurrences of different toxicities. Future research is needed to extend the model to handle count data.
\end{document} | arXiv |
\begin{document}
\title{Constructions for Quantum \Indistinguishability Obfuscation
}
\begin{abstract} An \emph{indistinguishability obfuscator} is a probabilistic polynomial-time algorithm that takes a circuit as input and outputs a new circuit that has the same functionality as the input circuit, such that for any two circuits of the same size that compute the \emph{same} function, the outputs of the indistinguishability obfuscator are indistinguishable. Here, we study schemes for indistinguishability obfuscation for \emph{quantum} circuits. We present two definitions for indistinguishability obfuscation: in our first definition ($qi\mathcal{O}$) the outputs of the obfuscator are required to be indistinguishable if the input circuits are perfectly equivalent, while in our second definition ($qi\mathcal{O}_{\bf D}$), the outputs are required to be indistinguishable as long as the input circuits are approximately equivalent with respect to a pseudo-distance~{\bf D}. Our main results provide (1) a computationally-secure scheme for $qi\mathcal{O}$ where the size of the output of the obfuscator is exponential in the number of non-Clifford (${\sf T}$ gates), which means that the construction is efficient as long as the number of ${\sf T}$ gates is logarithmic in the circuit size and (2) a statistically-secure $qi\mathcal{O}_{\bf D},$ for circuits that are close to the $k$th level of the Gottesman-Chuang hierarchy (with respect to {\bf D}); this construction is efficient as long as $k$ is small and fixed.
\end{abstract}
\section{Introduction} At the intuitive level, an \emph{obfuscator} is a probabilistic polynomial-time algorithm that transforms a circuit $C$ into another circuit $C’$ that has the same functionality as $C$ but that does not reveal anything about $C$, except its functionality \emph{i.e.}, anything that can be learned from $C’$ about $C$ can also be learned from black-box access to the input-output functionality of $C$. This concept is formalized in terms of \emph{virtual black-box obfuscation}, and was shown~\cite{BGI+12} to be unachievable in general. Motivated by this impossibility result, the same work proposed a weaker notion called \emph{indistinguishability obfuscation} ($i\mathcal{O}$).
In the classical case, an \emph{indistinguishability obfuscator} is a probabilistic polynomial-time algorithm that takes a circuit~$C$ as input and outputs a circuit~$i\mathcal{O}(C)$ such that $i\mathcal{O}(C)(x)=C(x)$ for all inputs~$x$ and the size of $i\mathcal{O}(C)$ is at most polynomial in the size of~$C$. Moreover, it must be that for any two circuits $C_1$ and~$C_2$ of the same size and that compute the same function, their obfuscations are computationally indistinguishable. It is known that $i\mathcal{O}$ achieves the notion of \emph{best possible obfuscation}, which states that any information that is not hidden by the obfuscated circuit is also not hidden by any circuit of similar size computing the same functionality~\cite{GR14}. Indistinguishability obfuscation is a very powerful cryptographic tool which is known to enable, among others: digital signatures, public key encryption~\cite{SW14}, multiparty key agreement, broadcast encryption~\cite{BZ14}, fully homomorphic encryption~\cite{CLTV15} and witness-indistinguishable proofs~\cite{BP15}. Notable in the context of these applications is the \emph{punctured programming technique}~\cite{SW14} which manages to render an $i\mathcal{O}(C)$ into an intriguing cryptographic building block, and this, despite that fact that
the security guarantees of $i\mathcal{O}(C)$ appear quite weak as they are applicable only if the two original circuits have \emph{exactly} the same functionality.
The first candidate construction of $i\mathcal{O}$ was published in~\cite{GGH+13}, with security relying on the presumed hardness of multilinear maps~\cite{CLT13,LSS14,GGH15}. Unfortunately, there have been many quantum attacks on multilinear maps~\cite{ABD16,CDPR16,CGH17}. Recently, new $i\mathcal{O}$ schemes were proposed under different assumptions ~\cite{AJL+19, JLS20,GJLS21}. Whether or not these schemes are resistant against quantum attacks remains to be determined.
Indistinguishability obfuscation has been studied for \emph{quantum} circuits in~\cite{AJJ14,AF16arxiv}. In a nutshell (see \Cref{sec:obf-quantum} for more details), ~\cite{AJJ14} shows a type of obfuscation for quantum circuits, but without a security reduction. On the other hand, the focus of ~\cite{AF16arxiv} is on impossibility of obfuscation for quantum circuits in a variety of scenarios. Thus, despite these works, until now, the achievability of indistinguishability obfuscation for quantum circuits has remained wide open.
\subsection{Overview of Results and Techniques} \label{sec:techniques}
Our contribution establishes indistinguishability obfuscation for certain families of quantum circuits. We now overview each of our two main definitions, and methods to achieve them (\Cref{sec:intro-summary-results-first-definition} and \Cref{sec:intro-summary-results-second-definition}). We then compare the two approaches (\Cref{sec:intro-comparison}).
\subsubsection{Indistinguishability obfuscation for quantum circuits} \label{sec:intro-summary-results-first-definition}
First, we define indistinguishability obfuscation for quantum circuits ($qi\mathcal{O}$) (\Cref{sec:definitions}) as an extension of the conventional classical definition. This definition specifies that on input a classical description of a quantum circuit~$C_q$, the obfuscator outputs a \emph{pair} $(\ket{\phi}, C_q^\prime)$, where $\ket{\phi}$ is an auxiliary quantum state and $C_q^\prime$ is a quantum circuit. For correctness, we require that $ ||C_q^\prime(\ket{\phi}, \cdot)-C_{q}(\cdot) ||_\diamond=0$, whereas for security, we require that, on input two functionally equivalent quantum circuits, the outputs of $qi\mathcal{O}$ are indistinguishable. As a straightforward extension of the classical results, we then argue that \emph{inefficient} indistinguishability obfuscation exists.
In terms of constructing $qi\mathcal{O}$, we first focus on the family of \emph{Clifford} circuits and show two methods of obfuscation: one straightforward method based on the canonical representation of Cliffords, and another based on the principle of gate teleportation~\cite{GC99}. Clifford circuits are quantum circuits that are built from the gate-set $\{{\sf X}, {\sf Z}, {\sf P}, {\sf CNOT}, {\sf H}\}$. They are known not to be universal for quantum computation and are, in a certain sense, the quantum equivalent of classical \emph{linear circuits}. It is known that Clifford circuits can be efficiently simulated on a classical computer~\cite{Got98}; however, note that this simulation is with respect to a \emph{classical} distribution, hence for a purely quantum computation, quantum circuits are required, which motivates the obfuscation of this circuit class. Furthermore, Clifford circuits are an important building block for fault-tolerant quantum computing, for instance, due to the fact that Cliffords admit transversal computations in many fault-tolerant codes. We provide two methods to achieve $qi\mathcal{O}$ for Clifford circuits.
\paragraph{Obfuscating Cliffords using a canonical form.} Our first construction of~$qi\mathcal{O}$ for Clifford circuits starts with the well-known fact that a canonical form is an $i\mathcal{O}$. We point out that a canonical form for Clifford circuits was presented in~\cite{AG04}; this completes this construction (we also note that an alternative canonical form was also presented in~\cite{Sel13arxiv}). This canonical form technique does not require any computational assumptions. Moreover, the obfuscated circuits are classical, and hence can be easily communicated, stored, used and copied.
\paragraph{Obfuscating Cliffords using gate teleportation.} Our second construction of $qi\mathcal{O}$ for Clifford circuits takes a very different approach. We start with the gate teleportation scheme~\cite{GC99}: according to this, it is possible to \emph{encode} a quantum computation~$C_q$ into a quantum state (specifically, by preparing a collection of entangled qubit pairs, and applying~$C_q$ to half of this preparation). Then, in order to perform a quantum computation on a target input $\ket{\psi}$, we \emph{teleport}~$\ket{\psi}$ \emph{into} the prepared entangled state. This causes the state $\ket{\psi}$ to undergo the evolution of~$C_q$, \emph{up to some corrections}, based on the teleportation outcome. If~$C_q$ is chosen from the Clifford circuits, these corrections are relatively simple\footnote{The correction is a tensor products of \emph{Pauli} operators, which is computed as a function of $C_q$ and of the teleportation outcome.} and thus we can use a classical $i\mathcal{O}$ to provide the correction function. In contrast to the previous scheme, the gate teleportation scheme requires the assumption of quantum-secure classical $i\mathcal{O}$ for a certain family of functions (\cref{update function}) and the obfuscated circuits include a quantum system. While this presents a technological challenge to communication, storage and also usage, there could be advantages to storing quantum programs into quantum states, for instance to take advantage of their \emph{uncloneability} ~\cite{Aar09,BL19arxiv}.
\paragraph{Obfuscating Beyond Cliffords.} Next, in our main result for \Cref{QiO:Clifford+T:family}, we generalize the gate teleportation scheme for Clifford circuits, and show a $qi\mathcal{O}$ obfuscator for all quantum circuits where the number of non-Clifford gates is at most logarithmic in the circuit size. For this, we consider the commonly-used Clifford+${\sf T}$ gate-set, and we note that the ${\sf T}$ relates to the ${\sf X}, {\sf Z}$ as: ${\sf T} {\sf X}^b {\sf Z}^a={\sf X}^b {\sf Z}^{a\oplus b}{\sf P}^b {\sf T}$. This means that, if we implement a circuit $C$ with ${\sf T}$ gates as in the gate teleportation scheme above, then the \emph{correction} function is no longer a simple Pauli update (as in the case for Cliffords). However, this is only partially true: since the Paulis form a basis, there is always a way to represent an update as a complex, linear combination of Pauli matrices. In particular, for the case of a~${\sf T}$, we note that ${\sf P}=(\frac{1+i}{2}) {\sf I} + (\frac{1-i}{2}){\sf Z}$. Hence, it \emph{is} possible to produce an update function for general quantum circuits that are encoded via gate teleportation. To illustrate this, we first analyze the case of a general Clifford+${\sf T}$ quantum circuit on a \emph{single} qubit (\Cref{sec:1-qubit}). Here, we are able to provide $qi\mathcal{O}$ for all circuits. Next, for general quantum circuits, (\Cref{sec:n-qubit-circuits}), we note that the update function exists for all circuits, but becomes more and more complex as the number of ${\sf T}$ gates increases. We show that if we limit the number of ${\sf T}$ gates to be logarithmic in the circuit size, we can reach an efficient construction. Both of these constructions assume a quantum-secure, classical indistinguishability obfuscation.
To the best of our knowledge, our gate teleportation provides the first method for indistinguishability obfuscation that is efficient for a large class of quantum circuits, beyond Clifford circuits. Note, however that canonical forms (also called \emph{normal} forms) are known for \emph{single}-qubits universal quantum circuits~\cite{MA08arxiv,GS19arxiv}. We note that, for many other quantum cryptographic primitives, it is the case that the ${\sf T}$-gate is the bottleneck (somewhat akin to a \emph{multiplication} in the classical case). This has been observed, \emph{e.g.}, in the context of \emph{homomorphic quantum encryption}~\cite{BJ15,DSS16}, and instantaneous quantum computation~\cite{Spe16}. Because of these applications, and since the ${\sf T}$ is also typically also the bottleneck for fault-tolerant quantum computing, techniques exist to reduce the number of ${\sf T}$ gates in quantum circuits~\cite{AMMR13,AMM14,DMM16} (see \Cref{sec:intro-related-work} for more on this topic).
\subsubsection{Indistinguishability obfuscation for quantum circuits, with respect to a pseudo-distance} \label{sec:intro-summary-results-second-definition} Next in \Cref{sec:quantum:iO:approx:circuits}, we define indistinguishability obfuscation for quantum circuits with respect to some pseudo-norm {\bf D}, which we call~$qi\mathcal{O}_{\bf D}$. This definition specifies that on input a classical description of a quantum circuit~$C_q$, the obfuscator outputs a \emph{pair} $(\ket{\phi}, C_q^\prime)$, where~$\ket{\phi}$ is an auxiliary quantum state and $C_q^\prime$ is a quantum circuit. For correctness, we require that ${\bf D}(C_q^\prime(\ket{\phi}, \cdot),C_{q}(\cdot))\leq {\tt negl}(n)$, whereas for security, we require that, on input two \emph{approximately} equivalent quantum circuits (\Cref{def:aqec}), the outputs of $qi\mathcal{O}_{\bf D}$ are statistically indistinguishable. This definition is more in line with~\cite{AF16arxiv}.
We show how to construct a statistically-secure quantum indistinguishability obfuscation with respect to the pseudo-distance ${\bf D}$ (see \Cref{QiO:gottesman-chuang}) for quantum circuits that are very close to $k$th level of the Gottesman-Chuang hierarchy~\cite{GC99}, for some fixed $k$ (see \Cref{sec:gottesman-chuang}). The construction takes a circuit~$U_q$ as an input with a promise that the distance ${\bf D}(U_q, C)\leq \epsilon <\frac{1}{2^{k+1/2}}$ for some $C\in \mathcal{C}_k.$ It computes the conjugate circuit $U_q^\dagger$ and then runs Low's learning algorithm as a subroutine on inputs $U_q$ and $U_q^\dagger$ \cite{Low09}. The algorithm outputs whatever Low's learning algorithm outputs. Note that Low's learning algorithm runs in time super-polynomial in $k,$ therefore for our construction to remain efficient the parameter $k$ is some small fixed integer (say $k=5$). Note that for $k>2$, the set $\mathcal{C}_k$ includes all Clifford unitaries as well as some non-Clifford unitaries~\cite{Low09}.
\subsubsection{Comparison of the two Approaches} \label{sec:intro-comparison} Our notions of $qi\mathcal{O}$ and $qi\mathcal{O}_{{\bf D}}$ are incomparable.
To see this, on one hand, note that the basic instantiation of an indistinguishability obfuscator that outputs a canonical form is no longer secure in the definition of indistinguishability with respect to a pseudo-norm.\footnote{If two different circuits are close in functionality but not identical, then we have no guarantee that their canonical forms are close.} On the other hand, the construction for $qi\mathcal{O}_{{\bf D}}$ that we give in \Cref{QiO:gottesman-chuang} does not satisfy the definition of $qi\mathcal{O}$, because the functionality is not perfectly preserved, which is a requirement for $qi\mathcal{O}$.
We recall that in the classical case, it is generally considered an \emph{advantage} that $i\mathcal{O}$ is a relatively weak notion (since it is more easily attained) and that, despite this, a host of uses of $i\mathcal{O}$ are known. We thus take $qi\mathcal{O}$ as the more natural extension of classical indistinguishability obfuscation to the quantum case, but we note that issues related to the continuity of quantum mechanics and the inherent approximation in any universal quantum gateset justify the relevance for our approach to $qi\mathcal{O}_{{\bf D}}$.
We now compare the schemes that we achieve. The most general scheme that we give as a construct for $qi\mathcal{O}$ (\Cref{QiO:qcircuit-teleportation}) allows to obfuscate any polynomial-size quantum circuit (with at most $O(\log)$ non-Clifford gates). While this is a restricted class, it is well-understood and we believe that this technique may be amenable to an extension that would result into a full $qi\mathcal{O}$.
In comparison, the scheme that we give for $qi\mathcal{O}_{{\bf D}}$, based on Low's learning algorithm \cite{Low09}
has some advantages over the teleportation-based constructions. Firstly, the circuits to be obfuscated don't need to be of equal size or perfectly equivalent and the outputs of the obfuscator remain statistically indistinguishable as long as the circuits are approximately equivalent (with respect to the pseudo-distance {\bf D}). Secondly, \Cref{QiO:gottesman-chuang} does not require any computational assumptions, whereas the teleportation-based constructions require a quantum-secure classical indistinguishability obfuscator. However, beyond the fact that~$\mathcal{C}_k$ contains all Clifford circuits, it is not clear how powerful unitaries are in the $k$th level of the Gottesman-Chuang hierarchy (especially for a fixed small~$k$). Even when $k\rightarrow \infty$, the hierarchy does not include all unitaries. In terms of extending this technique, Low's learning algorithm exploits the structure of the Gottesman-Chuang hierarchy and it not obvious how one can apply this technique to arbitrary quantum circuits.
\subsection{More on Related Work} \label{sec:intro-related-work}
\paragraph{Quantum Obfuscation.} \label{sec:obf-quantum} Quantum obfuscation was first studied in ~\cite{AJJ14}, where a notion called $(G,\Gamma)$-{\em indistinguishability obfuscation} was proposed, where $G$ is a set of gates and $\Gamma$ is a set of relations satisfied by the elements of~$G.$ In this notion, any two circuits over the set of gates $G$ are perfectly indistinguishable if they differ by some sequence of applications of the relations in~$\Gamma.$
Since perfect indistinguishability obfuscation is known to be impossible under the assumption that $\mathsf{P} \neq \mathsf{NP}$ \cite{GR14}, one of the motivations of this work was to provide a weaker definition of perfectly indistinguishable obfuscation, along with possibility results. However, to the best of our knowledge, $(G,\Gamma)$-{\em indistinguishability obfuscation} is incomparable with computational indistinguishability obfuscation \cite{BGI+12,GGH+13}, which is the main focus of our work.
Quantum obfuscation is studied in ~\cite{AF16arxiv}, where the various notions of quantum obfuscation are defined (including quantum black-box obfuscation, quantum indistinguishability obfuscation, and quantum best-possible obfuscation). A contribution of ~\cite{AF16arxiv} is to extend the classical impossibility results to the quantum setting, including \emph{e.g.} showing that
each of the three variants of quantum indistinguishability obfuscation is equivalent to the analogous variant of quantum best-possible obfuscation, so long as the obfuscator is efficient. This work shows that the existence of a computational quantum indistinguishability obfuscation implies a witness encryption scheme for all languages in~\textsf{QMA}. Various impossibiliity results are also shown: that efficient statistical indistinguishability obfuscation is impossible unless \textsf{PSPACE} is contained in \textsf{QSZK}\footnote{\textsf{PSPACE} is the class of decision problems solvable by a Turing machine in polynomial space and \textsf{QSZK} is the class of decision problems that admit a quantum statistical zero-knowledge proof system.} (for the case of circuits that include measurements), or unless \textsf{coQMA}\footnote{\textsf{coQMA} is the \emph{complement} of \textsf{QMA}, which is the class of decision problems that can be verified by a one-message quantum interactive proof.} is contained in \textsf{QSZK} (for the case of unitary circuits). Notable here is that \cite{AF16arxiv}
defines a notion of indistinguishability obfuscation where security must hold for circuits that are \emph{close} in functionality (this is similar to our definition of $qi\mathcal{O}_{\bf D}$); it is however unclear if their impossibility results hold for a notion of quantum indistinguishability along the lines of our definition of $qi\mathcal{O}$. See \Cref{sec:defs-qiO} for further discussion of the links between this definition and ours. We note that \cite{AF16arxiv} does not provide any concrete instantiation of obfuscation.
Recently it has been shown that virtual black-box obfuscation of classical circuits via quantum mechanical means is also impossible ~\cite{ABDS20,AP20}.
\paragraph{Quantum Homomorphic Encryption.} In \emph{quantum homomorphic encryption}, a computationally-weak client is able to send a ciphertext to a quantum server, such that the quantum server can perform a quantum computation on the encrypted data, thus producing an encrypted output which the client can decrypt, and obtaining the result of the quantum computation.
This primitive was formally defined in~\cite{BJ15} (see also ~\cite{DSS16,Bra18}), where it was shown how to achieve homomorphic quantum computation for quantum circuits of low ${\sf T}$-depth, by assuming quantum-secure classical fully homomorphic encryption. We note that even the simplest scheme in~\cite{BJ15} (which allows the homomorphic evaluation of \emph{any} Clifford circuit), requires computational assumptions in order for the server to update homomorphically the classical portion of the ciphertext, based on the choice of Clifford. In contrast, here we are able to give information-theoretic constructions for this class of circuits (essentially, because the choice of Clifford is chosen by the obfuscator, not by the evaluator). We thus emphasize that in $i\mathcal{O}$, we want to hide the \emph{circuit}, whereas in homomorphic encryption, we want to hide the \emph{plaintext} (and allow remote computations on the ciphertext). Since the evaluator in homomorphic encryption has control of the circuit, but not of the data, the evaluator knows which types of gates are applied, and the main obstacle is to perform a correction after a ${\sf T}$-gate, controlled on a classical value that is held only in an encrypted form by the evaluator. In contrast to this, in $i\mathcal{O}$, we want to hide the inner workings of the circuit. By using gate teleportation, we end up in a situation where the evaluator \emph{knows} some classical values that have affected the quantum computation in some undesirable way, and then we want to hide the inner workings of \emph{how} the evaluator should compensate for these undesirable effects. Thus, the techniques of quantum homomorphic encryption do not seem directly applicable, although we leave as an open question if they could be used in some indirect way, perhaps towards efficient $qi\mathcal{O}$ for a larger family of circuits.
\subsection{Open Questions} The main open question is efficient quantum indistinguishability obfuscation for quantum circuits with super-logarithmic number of ${\sf T}$-gates. Another open question is about the applications of quantum indistinguishability obfuscation. While we expect that many of the uses of classical $i\mathcal{O}$ carry over to the quantum case, we leave as future work the formal study of these techniques.
\paragraph{Outline.} The remainder of this paper is structured as follows. \Cref{sec:prelims} overviews basic notions required in this work. In \Cref{sec:definitions}, we formally define indistinguishability obfuscation for quantum circuits. In \Cref{QiO:Clifford-Circuits}, we provide the construction for Clifford circuits. In \Cref{QiO:Clifford+T:family}, we give our main result which shows quantum indistinguishability obfuscation for quantum circuits, which is efficient for circuits having at most a logarithmic number of ${\sf T}$ gates. Finally in \Cref{sec:quantum:iO:approx:circuits}, we consider the notion of quantum indistinguishability obfuscation with respect to a pseudo-distance, and show how to instantiate it for a family of circuits close to the Gottesman-Chuang hierarchy.
\section{Preliminaries} \label{sec:prelims} \subsection{Basic Classical Cryptographic Notions} \label{sec:classical-prelims} Let $\mathbb{N}$ be the set of positive integers. For $n \in \mathbb{N}$, we set $[n] = \{1, \cdots, n\}.$ We denote the set of all binary strings of length $n$ by $\bit{n}.$
An element $s \in \bit{n}$ is called a bitstring, and $|s|=n$ denotes its length. Given two bit strings $x$ and~$y$ of equal length, we denote their bitwise XOR by $x \oplus y$. For a finite set $X$, the notation $x \xleftarrow{\text{\$}} X$ indicates that $x$ is selected uniformly at random from~$X$. We denote the set of all $d \times d$ unitary matrices by $\mathcal{U}(d)=\{U \in \mathbb{C}^{d\times d} \mid UU^\dagger={\bf I}\}$, where $U^\dagger$ denotes the conjugate transpose of $U.$
A function $ {\tt negl}:\mathbb{N}\rightarrow\mathbb{R}^{+}\cup \{0\}$ is \emph{negligible} if for every positive polynomial~$p(n)$, there exists a positive integer $n_0$ such that for all $n>n_0,$ $ {\tt negl}(n) < 1/ p(n).$ A typical use of negligible functions is to indicate that the probability of success of some algorithm is too small to be amplified to a constant by a feasible (\emph{i.e.}, polynomial) number of repetitions.
\subsection{Classical Circuits and Algorithms} \label{sec:cir:alg}
A deterministic polynomial-time (or {\bf PT}) algorithm $\mathcal{C}$ is defined by a polynomial-time uniform\footnote{Recall that polynomial-time uniformity means that there exists a polynomial-time Turing machine which, on input~$n$ in unary, prints a description of the $n$th circuit in the family.} family $\mathcal{C}=\{C_{n}\mid n\in \mathbb{N} \}$ of classical Boolean circuits over some gate set, with one circuit for each possible input size $n\in\mathbb{N}.$ For a bitstring $x$, we define $\mathcal{C}(x) := \mathcal{C}_{|x|}(x)$. We say that a function family $f:\{0,1\}^n \rightarrow \{0,1\}^m$ is {\bf PT}-computable if there exists a polynomial-time $\mathcal{C}$ such that $\mathcal{C}(x) = f(x)$ for all~$x$; it is implicit that $m$ is a function of $n$ which is bounded by some polynomial, \emph{e.g.}, the same one that bounds the running time of $\mathcal{C}.$ Note that in the literature, circuits that compute functions whose range is $\{0,1\}^m$ are often called multi-output Boolean circuits \cite{GMOR15}, but in this paper we simply called them Boolean circuits \cite{Sip12}.
A probabilistic polynomial-time algorithm (or {\bf PPT}) is again a polynomial-time uniform family of classical Boolean circuits, one for each possible input size~$n.$ The $n$th circuit still accepts $n$ bits of input, but now also has an additional ``coins'' register of $p(n)$ input wires. Note that uniformity enforces that the function $p$ is bounded by some polynomial. For a {\bf PPT} algorithm~$\mathcal{C},$ $n$-bit input $x$ and $p(n)$-bit coin string $r$, we set $\mathcal{C}(x; r) := \mathcal{C}_n(x; r).$ In contrast with the PT case, the notation algorithm $\mathcal{C}(x)$ will now refer to the random variable algorithm $\mathcal{C}(x; r)$ where $r \xleftarrow{\text{\$}} \{0,1\}^{p(n)}.$
\subsection{Classical Indistinguishability}
Here, we define indistinguishability for classical random variables, against a quantum distinguisher (\Cref{def:classical-indis}).
\begin{definition} (Statistical Distance) Let $X$ and $Y$ be two random variables over some countable set $\Omega$. The statistical distance between $X$ and~$Y$ is
\begin{center}
$\Delta(X,Y)=\frac{1}{2}\left\{\sum_{\omega\in \Omega} \left |Pr[X(\omega)]- Pr[Y(\omega)]\right | \right\}.$ \end{center} \end{definition}
\begin{definition} (Indistinguishability) \label{def:classical-indis} Let $\mathcal{X}=\{X_n\}_{n\in\mathbb{N}}$ and $\mathcal{Y}=\{Y_n\}_{n\in\mathbb{N}}$ be two distribution ensembles indexed by a parameter $n.$ We say \begin{enumerate} \item $\mathcal{X}$ and $\mathcal{Y}$ are \emph{perfectly indistinguishable} if for all $n,$ $$\Delta(X_n,Y_n)=0.$$ \item $\mathcal{X}$ and $\mathcal{Y}$ are \emph{statistically indistinguishable} if there exists a negligible function {\tt negl} such that for all sufficiently large $n$: $$\Delta(X_n,Y_n)\leq {\tt negl}(n).$$ \item $\{X_n\}_{n\in\mathbb{N}}$ and $\{Y_n\}_{n\in\mathbb{N}}$ are \emph{computationally indistinguishable} if for any polynomial-time quantum distinguisher $\mathcal{D}_q$, there exists a negligible function {\tt negl} such that:
$$\Big |{\rm Pr}[\mathcal{D}_q(X_n)=1]-{\rm Pr}[\mathcal{D}_q(Y_n)=1] \Big |\leq {\tt negl}(n).$$ \end{enumerate} \end{definition}
\subsection{Classical Indistinguishability Obfuscation} \label{def:iO}
Let $\mathcal{C}$ be a family of probabilistic polynomial-time circuits. For $n\in\mathbb{N},$ let~$C_n$ be the circuits in $\mathcal{C}$ of input length~$n.$ We now provide a definition of classical indistinguishability obfuscation ($i\mathcal{O}$) as defined in \cite{GR14}, but where we make a few minor modifications.\footnote{We make a few design choices that are more appropriate for our situation, where we show the \emph{possibility} of $i\mathcal{O}$ against quantum adversaries: our adversary is a probabilistic polynomial-time quantum algorithm, we dispense with the mention of the random oracle, and note that our indistinguishability notions are defined to hold for all inputs.}
\begin{definition}\label{def:quantum-secureiO} {\rm({\bf Indistinguishability Obfuscation}, $i\mathcal{O}$)} A probabilistic polynomial-time algorithm is a \emph{quantum-secure indistinguishability obfuscator} ($i\mathcal{O}$) for a class of circuits ${\mathcal C},$ if the following conditions hold: \begin{enumerate} \item {\tt Preserving Functionality:} For any $C\in C_n:$
$$i\mathcal{O}(x)=C(x), \mbox{ for all } x \in \{0,1\}^n$$ The probability is taken over the $i\mathcal{O}$'s coins.
\item {\tt Polynomial Slowdown:} There exists a polynomial $p(n)$ such that for all input lengths, for any $C\in C_n,$ the obfuscator $i\mathcal{O}$ only enlarges $C$ by a factor of
$p(|C|):$
$$ |i\mathcal{O}(C)| \leq p(|C|).$$
\item {\tt Indistinguishability:} An $i\mathcal{O}$ is said to be a computational/statistical/\\perfect indistinguishability obfuscation for the family $\mathcal{C},$ if for all large enough input lengths, for any circuit $C_1\in C_n$ and for any $C_2\in C_n$ that computes the same function as $C_1$ and such that $|C_1|=|C_2|,$ the distributions $i\mathcal{O}(C_1))$ and $i\mathcal{O}(C_2)$ are (respectively) computationally/statistically/perfectly indistinguishable. \end{enumerate} \end{definition}
\subsection{Basic Quantum Notions} \label{sec:quantum-prelims} Given an $n$-bit string $x$, the corresponding $n$-qubit quantum computational basis state is denoted~$\ket{x}$. The $2^n$-dimensional Hilbert space spanned by $n$-qubit basis states is denoted: \begin{equation} \label{eq:hilbert-space} \mathcal{H}_n := \textbf{span} \left\{ \ket{x} : x \in \bit{n} \right\}\,. \end{equation} We denote by $\mathcal{D}(\mathcal{H}_n)$ the set of density operators (\emph{i.e.}, valid quantum states) on~$\mathcal{H}_n$. These are linear operators on $\mathcal{D}(\mathcal{H}_n)$ which are positive-semidefinite and have trace equal to $1$.
\subsection{Norms and Pseudo-Distance} \label{sec:norms} The trace distance between two quantum states $\rho, \sigma\in \mathcal{D}(\mathcal{H}_n)$ is given by:
$$||\rho-\sigma||_{tr}:=\frac{1}{2}\Tr\left(\left\lvert\sqrt{(\rho-\sigma)^\dagger (\rho-\sigma)}\right\rvert\right),$$
where $\lvert \cdot\rvert$ denotes the positive square root of the matrix $\sqrt{(\rho-\sigma)^\dagger (\rho-\sigma)}.$
Let $\Phi$ and $\Psi$ be two admissible operators of type $(n,m)$\footnote{An operator is admissible if its action on density matrices is linear, trace-preserving, and completely positive. A operator's type is $(n,m)$ if it maps $n$-qubit states to $m$-qubit states.}. The \emph{diamond norm} between two quantum operators is
$$||\Phi-\Psi||_\diamond :=\underset{\rho\in \mathcal{D}(\mathcal{H}_{2n}) }{max} ||(\Phi \otimes I_n) \rho-(\Psi \otimes I_n) \rho||_{tr}$$
The Frobenius norm of a matrix $A\in\mathbb{C}^{n \times m}$ is defined as $||A||_F=\sqrt{\Tr (AA^\dagger)}.$ Let $U_1, U_2 \in \mathcal{U}(d)$ be two $d \times d$ unitary matrices. The phase invariant distance between $U_1$ and $U_2$ is
$${\bf D}(U_1,U_2)=\frac{1}{\sqrt{2d^2}} ||U_1\otimes U_1^*- U_2\otimes U_2^*||_F$$
$$=\sqrt{1-\left|\frac{ {\rm Tr}(U_1U_2^\dagger)}{d}\right|^2}\,,$$
where $U_i^*$ denotes the matrix with only complex conjugated entries and no transposition and $| z |$ denotes the norm of the complex number $z$. Note that~${\bf D}$ is a pseudo-distance since ${\bf D}(U_1,U_2) =0$ does not imply $U_1=U_2,$ but that $U_1$ and $U_2$ are equivalent up to a phase so the difference is unobservable. It is easy to see that {\bf D} satisfies the axioms of symmetry (${\bf D}(U_1,U_2)={\bf D}(U_2,U_1)$), the triangle inequality (${\bf D}(U_1,U_2)\leq{\bf D}(U_1,U)+{\bf D}(U,U_2)$) and non-negativity (${\bf D}(U_1,U_2)\geq 0$).
\subsection{Bell Basis and Measurement} \label{sec:bell:basis} The four states $\{\ket{\beta_{00}}, \ket{\beta_{01}}, \ket{\beta_{10}}, \ket{\beta_{11}}\}$ are called \emph{Bell States} or \emph{EPR pairs} and form an orthonormal basis of $\mathcal{H}_2.$
$$\ket{\beta_{00}}=\frac{1}{\sqrt2}\left(\ket{00}+\ket{11}\right) \hspace{1cm} {\beta_{01}}=\frac{1}{\sqrt2}\left(\ket{01}+\ket{10}\right)$$ $$\ket{\beta_{10}}=\frac{1}{\sqrt2}\left(\ket{00}-\ket{11}\right) \hspace{1cm} \ket{\beta_{11}}=\frac{1}{\sqrt2}\left(\ket{01}-\ket{10}\right)$$ We define a generalized Bell state as a tensor product of $n$ Bell states
$$\ket{\beta_{s}}=\ket{\beta_{a_i,b_i}}^{\otimes_{i=1}^{n}},$$
where $s=a_1b_1,\ldots,a_nb_n\in\{0,1\}^{2n}.$ The set of generalized Bell States $\{\ket{\beta_{s}} \mid s\in \{0,1\}^{2n}\}$ forms an orthonormal basis of $\mathcal{H}_n.$ Given a quantum state $$\ket{\psi}=\sum_{s\in\{0,1\}^{2n}} \alpha_{s}\ket{\beta_{s}},$$
a Bell measurement in the (generalized) Bell basis on the state $\ket{\psi}$ outputs the string $s$ with probability $|\alpha_{s}|^2$ and leaves the system in the state $\ket{\beta_{s}}.$
\subsection{Quantum Gates} We will work with the following set of unitary gates $$ {\sf I} = \left[\begin{array}{cc} 1 & 0\\ 0 & 1\end{array}\right], \quad {\sf X} = \left[\begin{array}{cc} 0 & 1\\ 1 & 0\end{array}\right], \quad{\sf Y} = \left[\begin{array}{cc} 0 & i\\ -i & 0\end{array}\right], \quad{\sf Z} = \left[\begin{array}{cc} 1 & 0\\ 0 & -1\end{array}\right], $$ $$ \quad{\sf H} = \frac{1}{\sqrt{2}}\left[\begin{array}{cc}1 & 1\\1 & -1\end{array}\right], \quad{\sf CNOT} = \left[\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\end{array}\right], \text{ and} \quad{\sf T} = \left[\begin{array}{cc} 1 & 0\\ 0 & e^{i\pi/4}\end{array}\right].$$
For any single-qubit density operator $\rho \in \mathcal{D} (\mathcal{H}_1)$, we can encrypt it via the \emph{quantum one-time pad} by sampling uniform bits $s$ and $t$, and producing
${\sf X}^s {\sf Z}^t \rho {\sf Z}^t {\sf X}^s$. To an observer that has no knowledge of $s$ and $t$, this system is information-theoretically indistinguishable from the state $\mathds{1}_1/2$ (where $\mathds{1}_1$ is the 2 by 2 identity matrix)\cite{AMTW00}.
\subsection{Gottesman-Chuang Hierarchy} \label{sec:gottesman-chuang} The $n$-qubit Pauli group $\mathcal{P}_n$ is a multiplicative group of order $4^{n+1}$, defined as: $$\mathcal{P}_n=\{\alpha_1 P_1 \otimes \cdots \otimes \alpha_n P_n \mid \alpha_i \in \{\pm 1, \pm i\}, P_i\in\{{\sf I}, {\sf X}, {\sf Z}, {\sf Y} \}\}\,.$$
Let $C_1$ be the Pauli group $\mathcal{P}_n.$ Then the level $\mathcal{C}_k$ of the \emph{Gottesman-Chuang} hierarchy is defined recursively~\cite{GC99}: $$\mathcal{C}_k=\{U\in \mathcal{U}(2^n): U\mathcal{P}_nU^\dagger \subseteq \mathcal{C}_{k-1}\}.$$ Note that $\mathcal{C}_2$ is the Clifford group and for $k>2$, $\mathcal{C}_k$ is no longer a group but contains unitaries that contains a universal gate set.
The set of gates $\{{\sf X}, {\sf Z}, {\sf P}, {\sf CNOT},{\sf H}\}$ applied to arbitrary wires redundantly generates the \emph{Clifford group}. We note the following relations between these gates (these relations hold up to \emph{global phase}; in this work, we use the convention that equal signs for pure states and unitaries hold up to global phase.) $${\sf X}{\sf Z} = - {\sf Z}{\sf X},\quad {\sf T}^2={\sf P},\quad{\sf P}^2={\sf Z},\quad{\sf H}{\sf X}{\sf H}={\sf Z},\quad {\sf T}{\sf P}={\sf P}{\sf T},\quad{\sf P}{\sf Z}={\sf Z}{\sf P}.$$ Also, for any $a,b\in\{0,1\}$ we have ${\sf H}{\sf X}^b{\sf Z}^a={\sf X}^a{\sf Z}^b {\sf H}$\,.
\subsection{Quantum Circuits and Algorithms} \label{sec:quantum-algo} A quantum circuit is an acyclic network of quantum gates connected by wires. The quantum gates represent quantum operations and wires represent the qubits on which gates act. In general, a quantum circuit can have $n$-input qubits and $m$-output qubits for any integer $n, m\geq 0.$ The \emph{${\sf T}$-count} is the total number of ${\sf T}$-gates in a quantum circuit.
A quantum circuit that computes a unitary matrix is called a \emph{reversible quantum circuit}, \emph{i.e.}, it always possible to uniquely recover the input, given the output. A set of gates is said to be \emph{universal} if for any integer $n \geq 1,$ any $n$-qubit unitary operator can be approximated to arbitrary accuracy by a quantum circuit using only gates from that set \cite{KLM07}. It is a well-known fact that Clifford gates are not universal, but adding any non-Clifford gate, such as ${\sf T}$, gives a universal set of gates \cite{KLM07}\footnote{In this work, we assume circuits are given in the Clifford + ${\sf T}$ gateset}. \emph{Generalized quantum circuits} (which implement \emph{superoperators}) are composed of the unitary gates, together with trace-out and measurement operations. It is well-known that a generalized quantum circuit can be implemented by adding auxiliary states to the original system, applying a unitary operation on the joint system, and then tracing out some subsystem~\cite{KLM07}.
A family of generalized quantum circuits $\mathcal{C}=\{C_{q_n}\mid n\in \mathbb{N}\}$, one for each input size $n\in \mathbb{N}$, is called \emph{polynomial-time uniform} if there exists a deterministic Turing machine~$M$ such that:
\begin{inlinelist}
\item for each $n\in\mathbb{N},$ $M$ outputs a description of $C_{q_n}\in \mathcal{C}$ on input $1^n$; and
\item for each $n\in\mathbb{N},$ $M$ runs in $poly(n).$
\end{inlinelist} We define a \emph{quantum polynomial-time algorithm} (or QPT) to be a polynomial-time uniform family of generalized quantum circuits.
\subsection{Quantum Indistinguishability} \label{sec:comp-stat} Here, we define indistinguishability for indistinguishability for quantum states (\Cref{def:indis-quantum}).
\begin{definition} (Indistinguishability of Quantum States) \label{def:indis-quantum}
Let $\mathcal{R}=\{\rho_n\}_{n\in \mathbb{N}}$ and $\mathcal{S}=\{\sigma_n\}_{n\in \mathbb{N}}$ be two ensembles of quantum states such that $\rho_n$ and $\sigma_n$ are $n$-qubit states. We say
\begin{enumerate}
\item $\mathcal{R}$ and $\mathcal{S}$ are \emph{perfectly indistinguishable} if for all $n,$
$$\rho_n=\sigma_n.$$
\item $\mathcal{R}$ and $\mathcal{S}$ are \emph{statistically indistinguishable} if there exists a negligible function {\tt negl} such that for all sufficiently large $n$:
$$||\rho_n-\sigma_n||_{tr}\leq {\tt negl}(n).$$
\item $\mathcal{R}$ and $\mathcal{S}$ are \emph{computationally indistinguishable} if there exists a negligible function {\tt negl} such that for every state $\rho_n\in \mathcal{R}$, $\sigma_n\in \mathcal{S}$ and for all polynomial-time quantum distinguisher $\mathcal{D}_q$, we have:
$$\Big |{\rm Pr}[\mathcal{D}_q(\rho_n)=1]-{\rm Pr}[\mathcal{D}_q(\sigma_n)=1] \Big |\leq {\tt negl}(n).$$ \end{enumerate} \end{definition}
\subsection{Quantum Teleportation} Here we provide a high-level description of quantum teleportation; for a more rigorous treatment see \cite{BBC+93}. Suppose Alice has a quantum state $\ket{\psi}=\alpha\ket{0}+\beta\ket{1}$\footnote{For simplicity we assume that $\ket{\psi}$ is single-qubit pure state.} that she wants to send to Bob who is located far away from Alice. One way for Alice to send her qubit to Bob is via the quantum teleportation protocol. For teleportation to work, Alice prepares a $2$-qubit Bell state $$\ket{\beta_{00}}_{AB}=\frac{1}{\sqrt2}\left(\ket{00}_{AB}+\ket{11}\right)_{AB},$$ and sends physically one of the qubit to Bob and keeps the other to herself (this is what subscript $AB$ means). We can now write the 3-qubit system as \begin{equation} \label{eq:sys}
\ket{\psi} \otimes \ket{\beta_{00}}_{AB} \end{equation} Alice now performs a joint measurement on $\ket{\psi}$ and her part of the EPR pair in the Bell basis and obtains the output of the measurement (classical bits $a,b$). After this step, Bob's part of EPR pair has been transformed into the state $${\sf X}^b {\sf Z}^a \ket{\psi}.$$ Alice sends the two classical bits $(a,b)$ to Bob, who performs the correction unitary $Z^aX^b$ to the state he possesses and obtains the state~$\ket{\psi}.$
\subsection{Gate Teleportation} One of the main applications of quantum teleportation is in fault-tolerant quantum computation~\cite{GC99}. To construct unitary quantum circuits, we need to have access to some universal set of quantum gates\footnote{$\{{\sf H}, {\sf T}, {\sf CNOT}\}$ is a universal set of quantum gates \cite{KLM07}.}. Suppose we want to evaluate a single-qubit gate on some quantum state $\ket{\psi}.$ If we directly apply $U$ on $\ket{\psi}$ and~$U$ fails, then it may also destroy the state. Quantum teleportation gives a way of solving this problem. Instead of applying $U$ directly to $\ket{\psi}$, we can apply $U$ to the system $B$ in \Cref{eq:sys} and then follow the gate teleportation protocol and obtain $U(\ket{\psi}).$ If $U$ fails, then the Bell state might be destroyed, but there is no harm done, since we can create another EPR pair and try again. The gate teleportation can easily be generalized to evaluate any $n$-qubit Clifford circuit (\Cref{algo:gate-teleport}).
\begin{remark}
In this section, we only discuss how to evaluate Clifford gates using gate teleportation. Note that we can evaluate any unitary circuit using gate teleportation but the correction unitary becomes more complicated (it is no longer a tensor product of Paulis). This is discussed in \Cref{QiO:Clifford+T:family}. \end{remark}
\begin{algorithm}[] {\bf Input}: A $n$-qubit Clifford Circuit $C_q$ and $n$-qubit quantum state $\ket{\psi}$ \caption{Gate Teleportation Protocol.} \label{algo:gate-teleport} \begin{enumerate}
\item Prepare a tensor product of $n$ Bell states: $\ket{\beta^{ 2n}}=\ket{\beta_{00}}\otimes \cdots \otimes \ket{\beta_{00}}.$ \item Write the joint system as $\ket{\psi}_C \ket{\beta^{ 2n}}_{AB}.$ \item Apply the circuit $C_q$ on the subsystem $B.$ \item Perform a measurement in the generalized Bell Basis (generalized Bell measurement) on the system $CA$ and obtain a binary string $a_1b_1,\ldots, a_nb_n.$ The remaining system after the measurement is \begin{equation}
\label{prl:eq2:gate-teleport}
C_q \left({{\sf X}^{b_i} {\sf Z}^{a_i}}\right)^{\otimes_{i=1}^{n}}\ket{\psi}. \end{equation} \item Compute the correction bits using the update function $F_{C_q}$ (\Cref{update function}). \begin{equation}\label{algo:update:func}
F_{C_q}(a_1b_1,\dots,a_nb_n)= a_1^\prime b_1^\prime,\dots,a_n^\prime b_n^\prime \in\{0,1\}^{2n}. \end{equation} \item Compute the correction unitary $U_{F_{C_q}}=\left({{\sf Z}^{a_i^\prime} {\sf X}^{b_i^\prime}}\right)^{\otimes_{i=1}^{n}}$ \item Apply $U_{F_{C_q}}$ to the system (\Cref{prl:eq2:gate-teleport}).
\begin{equation}
\label{prl:eq4:gate-teleport}
\begin{aligned}
&U_{F_{C_q}} \cdot C_q \left({{\sf X}^{b_i} {\sf Z}^{a_i}}\right)^{\otimes_{i=1}^{n}}\ket{\psi}=\left({{\sf Z}^{a_i^\prime} {\sf X}^{b_i^\prime}}\right)^{\otimes_{i=1}^{n}} C_q \left({{\sf X}^{b_i} {\sf Z}^{a_i}}\right)^{\otimes_{i=1}^{n}}\ket{\psi}\\
&=\left({{\sf Z}^{a_i^\prime} {\sf X}^{b_i^\prime}}\right)^{\otimes_{i=1}^{n}} \left({{\sf X}^{b_i^\prime} {\sf Z}^{a_i^\prime}}\right)^{\otimes_{i=1}^{n}}C_q(\ket{\psi})\\
& =C_q( \ket{\psi}).
\end{aligned}
\end{equation} \end{enumerate} \end{algorithm}
\subsection{Update Functions for Quantum Gates}\label{update function}
Let $C_q$ be an $n$-qubit circuit consisting of a sequence of Clifford gates $g_1,\ldots, g_{|C_q|}.$ Then the update function for $C_q$ is a map from $\{0,1\}^{2n}$ to $\{0,1\}^{2n}$ and is constructed by composing the update functions for each gate in $C_q$\footnote{This composition implicitly assumes that when an update function is applied, it acts non-trivially on the appropropriate bits, as indicated by the original circuit, and as the identity elsewhere.}
\begin{equation} \begin{aligned} &F_{C_q}=\{0,1\}^{2n} \longrightarrow \{0,1\}^{2n}\\
& F_{C_q}= f_{g_{{|C_q|}}}\circ \cdots \circ f_{g_2} \circ f_{g_1} \end{aligned} \end{equation}
For each Clifford gate $g,$ the update function $f_g$ is defined below. Note how~$g$ relates to the ${\sf X}$ and ${\sf Z}$ gates. \begin{equation*} \begin{aligned} &{\sf X} ({\sf X}^b{\sf Z}^a)\psi=({\sf X}^b{\sf Z}^a) {\sf X} \ket{\psi} \mbox{ (update function) } f_{\sf X}(a,b)=(a,b)\\ &{\sf Z} ({\sf X}^b{\sf Z}^a){\sf Z}=({\sf X}^b{\sf Z}^a) {\sf Z}\ket{\psi} \mbox{ (update function) } f_{\sf Z}(a,b)=(a,b)\\ &{\sf H} ({\sf X}^b{\sf X}^a){\sf Z}=({\sf X}^b{\sf X}^a) {\sf H}\ket{\psi} \mbox{ (update function) } f_{\sf H}(a,b)=(b,a)\\ &{\sf P} ({\sf X}^b{\sf X}^a){\sf Z}=({\sf X}^b{\sf X}^a) {\sf P}\ket{\psi} \mbox{ (update function) } f_ {\sf P}(a,b)=(a,a\oplus b)\\ &{\sf CNOT}({\sf X}^{a_1}{\sf Z}^{b_1}\otimes {\sf X}^{a_2}{\sf Z}^{b_2})\ket{\psi}=({\sf X}^{b_1}{\sf Z}^{a_1\oplus a_2}\otimes {\sf X}^{b_1\oplus b_2} {\sf Z}^{ b_2}){\sf CNOT}(\ket{\psi}) \mbox{ (update function) }\\ & f_{{\sf CNOT}}(a_1,b_1,a_2,b_2)=(a_1\oplus a_2,b_1,a_2, b_1\oplus b_2). \end{aligned} \end{equation*}
\section{Definitions} \label{sec:definitions} In this section, we provide a definition of perfectly equivalent quantum circuits (see \Cref{sec:perfectly-equivalent}), and define our notion of quantum indistinguishability obfuscation for equivalent circuits (\Cref{sec:defs-qiO}). At the end of the section, we also make an observation about the existence of inefficient quantum indistinguishability obfuscation. Note that in~\Cref{sec:quantum:iO:approx:circuits}, we present our alternative definition for quantum indistinguishability obfuscation, applicable to the case where the circuits are approximately equivalent.
\subsection{Perfectly Equivalent Quantum Circuits} \label{sec:perfectly-equivalent}
\begin{definition}({\em Perfectly Equivalent Quantum Circuits}): \label{def:qec} Let $C_{q_0}$ and $C_{q_1}$ be two $n$-qubit quantum circuits. We say $C_{q_0}$ and $C_{q_1}$ are \emph{perfectly equivalent} if
$$||C_{q_0}-C_{q_1}||_\diamond=0.$$ \end{definition}
\subsection{Indistinguishability Obfuscation for Quantum Circuits} \label{sec:defs-qiO} \begin{definition} ({\em Quantum Indistinguishability Obfuscation for Perfectly Equivalent Quantum Circuits}): \label{def:QiO} Let $\mathcal{C}_Q$ be a polynomial-time family of reversible quantum circuits. For $n\in\mathbb{N}$, let $C_{q^n}$ be the circuits in $\mathcal{C}_Q$ of input length $n.$ A polynomial-time quantum algorithm for~$\mathcal{C}_Q$ is a \emph{Computational/Statistical/Perfect } \emph{quantum indistinguishability obfuscator} ($qi\mathcal{O}$) if the following conditions hold: \begin{enumerate}
\item {\tt Functionality:} There exists a negligible function ${\tt negl}(n)$ such that for every $C_q\in C_{q^n}$
$$(\ket{\phi}, C_q^\prime)\leftarrow qi\mathcal{O}(C_q) \; \mbox{ and }\; ||C_q^\prime(\ket{\phi}, \cdot)-C_{q}(\cdot) ||_\diamond=0.$$ Where $\ket{\phi}$ is an $\ell$-qubit state, the circuits $C_q$ and $C_q^\prime$ are of type $(n,n)$ and $(m,n)$ respectively ($m= \ell +n$).\footnote{A circuit is of type $(i,j)$ if it maps $i$ qubits to $j$ qubits.}
\item {\tt Polynomial Slowdown:} There exists a polynomial $p(n)$ such that for any $C_{q}\in C_{q^n},$ \begin{itemize}
\item $\ell\leq p(|C_{q}|)$
\item $m \leq p(|C_{q}|)$
\item $|C_{q}^\prime| \leq p(|C_{q}|).$ \end{itemize}
\item {\tt Computational/Statistical/Perfect Indistinguishability:} For any two perfectly equivalent quantum circuits $C_{q_1},C_{q_2}\in C_{q^n},$ of the same size, the two distributions $qi\mathcal{O}(C_{q_1})$ and $qi\mathcal{O}(C_{q_2})$ are (respectively) computationally/statistically/perfectly indistinguishable. \end{enumerate} \end{definition}
\begin{remark}\label{re:ktime} A subtlety that is specific to the quantum case is that \Cref{def:QiO} only requires that $(\ket{\phi}, C_q^\prime)$ enable a \emph{single} evaluation of $C_q$. We could instead require a $k$-time functionality, which can be easily achieved by executing the single-evaluation scheme $k$ times in parallel. This justifies our focus here on the single-evaluation scheme. \end{remark}
\begin{note} \label{note:differences-AF16}
As described in \Cref{sec:obf-quantum}, our \Cref{def:QiO} differs from \cite{AF16arxiv} as it requires security only in the case of equivalent quantum circuits (see \Cref{def:aQiO} for a definition that addresses this). Compared to~\cite{AF16arxiv}, we note that in this work we focus on unitary circuits only.\footnote{This is without loss of generality, since a $qi\mathcal{O}$ for a generalized quantum circuit can be obtained from a $qi\mathcal{O}$ for a reversible version of the circuit, followed by a trace-out operation (see~\Cref{sec:quantum-algo}).} Another difference is that the notion of indistinguishability (computational or statistical) in \cite{AF16arxiv} is more generous than ours, since it allows a finite number of inputs that violate the indistinguishability inequality. Since our work focuses on \emph{possibility} of obfuscations, our choice leads to the strongest results; equally, since \cite{AF16arxiv} focuses on impossibility, their results are strongest in their model. We also note that that \cite{AF16arxiv} defines the efficiency of the obfuscator in terms of the number of qubits. We believe that our definition, which bounds the size of the output of the obfucation by a polynomial in the \emph{size} of the input circuit, is more appropriate\footnote{It would be unreasonable to allow an obfuscator that outputs a circuit on $n$ qubits, but of depth super-polynomial in~$n$.} and follows the lines of the classical definitions.
As far as we are aware, further differences in our definition are purely a choice of style. For instance, we do not include an \emph{interpreter} as in \cite{AF16arxiv}, but instead we let the obfuscator output a quantum circuit together with a quantum state; we chose this presentation since it provides a clear separation between the quantum circuit output by the $qi\mathcal{O}$ and the ``quantum advice state''. \end{note}
\subsubsection{Inefficient Quantum Indistinguishability Obfuscators Exist} Finally, we show a simple extension of a result in \cite{BGI+12}, which shows that if we relax the requirement that the obfuscator be efficient, then information-theoretic indistinguishability obfuscation exists.
\begin{claim}
\label{claim:inefficient-qiO}
Inefficient indistinguishability obfuscators exist for all circuits.
\end{claim}
\begin{proof}
Let $qi\mathcal{O}(C)_q$ be the lexicographically first circuit of size $|C_q|$ that computes the same quantum map as $C_q$.
\end{proof}
\section{Quantum Indistinguishability Obfuscation for Clifford Circuits} \label{QiO:Clifford-Circuits}
Here, we show how to construct $qi\mathcal{O}$ for Clifford circuits with respect to definition \Cref{def:QiO}. The first construction (\Cref{sec:Clifford-iO-canonical}) is based on a canonical form, and the second is based on gate teleportation (\Cref{sec:Clifford-iO-teleportaion}).
\subsection{$qi\mathcal{O}$ for Clifford Circuits via a Canonical Form} \label{sec:Clifford-iO-canonical} Aaronson and Gottesman developped a polynomial-time algorithm that takes a Clifford circuit $C_q$ and outputs its canonical form (see~\cite{AG04}, section~VI), which is invariant for any two equivalent $n$-qubit circuits\footnote{Their algorithm outputs a canonical form (unique form) provided it runs on the standard initial tableau see pages 8-10 of \cite{AG04}.}. Moreover the size of the canonical form remains polynomial in the size of the input circuit. Based on this canonical form, we define a $qi\mathcal{O}$ in \Cref{QiO:Canonical-Clifford}. \begin{algorithm}[]
\caption{$qi\mathcal{O}$-Canonical}
\label{QiO:Canonical-Clifford}
\begin{itemize} \item Input: An $n$-qubit Clifford Circuit $C_q.$
\begin{enumerate}
\item Using the Aaronson and Gottesman algorithm~\cite{AG04}, compute the canonical form of $C_q$
\begin{equation*}
C_q^\prime \xleftarrow{\mbox{canonical form}}C_q
\end{equation*}
\item Let $\ket{\phi}$ be an empty register.
\item Output $\left(\ket{\phi},C_q^\prime \right)$.
\end{enumerate}
\end{itemize} \end{algorithm}
\begin{lemma} \Cref{QiO:Canonical-Clifford} is a Perfect Quantum Indistinguishability Obfuscation for all Clifford Circuits. \end{lemma}
\begin{proof}
We have to show that \Cref{QiO:Canonical-Clifford} satisfies the definition of a perfect quantum indistinguishability obfuscation (\Cref{def:QiO}) for all Clifford circuits.
\begin{enumerate}
\item {\tt Functionality:} Since $\ket{\phi}$ is an empty register, it is a 0 qubit state ($\ell=0.$) The circuit $C_q^\prime$ is the canonical form of $C_q,$ therefore, it is also of type $(n,n)$ and has the same functionality as $C_q.$ We have $ ||C_q^\prime(\ket{\phi}, \cdot)-C_{q}(\cdot)||_\diamond =0\leq {\tt negl}(n)$ for any negligible function ${\tt negl}(n).$
\item {\tt Polynomial Slowdown:} Note $C_q^\prime$ is constructed using Aaronson and Gottesman algorithm~\cite{AG04}. Therefore, there exists a polynomial $q(\cdot)$ such that $|C_q^\prime|\leq q(|C_q|).$ Let $p(n)=q(n)+n$ then we clearly have \begin{itemize}
\item $\ell \leq p(|C_{q}|)$
\item $n \leq p(|C_{q}|)$
\item $|C_{q}^\prime| \leq p(|C_{q}|).$ \end{itemize} \item {\tt Perfectly Indistinguishability:} Let $C_{q_1},C_{q_2}\in C_{q^n},$ be any two equivalent Clifford circuits of the same size. Let $$(\ket{\phi_1}, C_{q_1}^\prime)\leftarrow qi\mathcal{O}\mbox{-Canonical}(C_{q_1}) \mbox{ and } (\ket{\phi_2}, C_{q_2}^\prime)\leftarrow qi\mathcal{O}\mbox{-Canonical}(C_{q_2}).$$ Since the canonical form of any two equivalent Clifford circuits are exactly the same, we have $C_{q_1^\prime}=C_{q_2^\prime}.$ Moreover both $\ket{\phi_1}$ and $\ket{\phi_2}$ are empty registers we have $\ket{\phi_1}=\ket{\phi_2}.$ Then we have $qi\mathcal{O}\mbox{-Canonical}(C_{q_1})=qi\mathcal{O}\mbox{-Canonical}(C_{q_2}).$ Therefore, \Cref{QiO:Canonical-Clifford} is a perfect quantum indistinguishability obfuscation for all Clifford circuits.\qedhere
\end{enumerate} \end{proof}
\subsection{$qi\mathcal{O}$ for Clifford Circuits via Gate Teleportation} \label{sec:Clifford-iO-teleportaion} In this section, we show how gate teleportation (see \Cref{algo:gate-teleport}) can be used to construct a quantum indistinguishability obfuscation for Clifford circuits. Our construction, given in \Cref{QiO:Clifford-teleportation}, relies on the existence of a quantum-secure~$i\mathcal{O}$ for classical circuits; however, upon closer inspection,
our construction relies on the assumption that a quantum-secure classical $i\mathcal{O}$ exists for a very specific class of classical circuits\footnote{Circuits that compute update functions for Clifford circuits, see \Cref{update function}.}. In fact, it is easy to construct a perfectly secure $i\mathcal{O}$ for this class of circuits: like Clifford circuits, the circuits that compute the update functions also have a canonical form. Then the $i\mathcal{O}$ takes as input a Clifford circuit and outputs a canonical form of a classical circuit that computes the update function for~$C_q.$ The $i\mathcal{O}$ is described formally in \Cref{alg:classical-iO-Clifford}.
\begin{algorithm}[h!]
\caption{$qi\mathcal{O}$ via Gate Teleportation for Clifford}
\label{QiO:Clifford-teleportation}
\begin{itemize}
\item Input: An $n$-qubit Clifford Circuit $C_q.$
\begin{enumerate}
\item Prepare a tensor product of $n$ Bell states: $\ket{\beta^{ 2n}}=\ket{\beta_{00}}\otimes \cdots \otimes \ket{\beta_{00}}.$
\item Apply the circuit $C_q$ on the right-most $n$ qubits to obtain a system $\ket{\phi}$:
$$\ket{\phi}=({\sf I}_n\otimes C_q) \ket{\beta^{2n}}.$$
\item Compute a classical circuit $C$ that computes the update function $F_{C_q}.$ The classical circuit $C$ can be computed in polynomial-time by \Cref{lemma: iO-clifford-functions}. \item Set $C^\prime\leftarrow i\mathcal{O}(C)$, where $i\mathcal{O}(C)$ is a perfectly secure indistinguishability obfuscation defined in \Cref{sec: iO-clifford-functions}.
\item Description of the circuit $C_q^\prime:$
\begin{enumerate}
\item Perform a general Bell measurement on the leftmost $2n$-qubits on the system $\ket{\phi}\otimes \ket{\psi}$, where $\ket{\phi}$ is an auxiliary state and $\ket{\psi}$ is an input state. Obtain classical bits $(a_1,b_1\ldots,a_n,b_n)$ and the state
\begin{equation}
\label{bld:eq1:QiO-Clifford}
C_q({\sf X}^{\otimes_{i=1}^{n} b_{i}} \cdot {\sf Z}^{\otimes_{i=1}^{n} a_{i}})\ket{\psi}.
\end{equation}
\item Compute the correction bits
\begin{equation}
\label{bld:eq2:QiO-Clifford}
(a_1^\prime, b_1^\prime,\ldots, a_n^\prime, b_n^\prime)=C^\prime(a_1,b_1\ldots,a_n,b_n).
\end{equation}
\item Using the above, the correction unitary is
$U^\prime=({\sf X}^{\otimes_{i=1}^{n} b_{i}^\prime} \cdot {\sf Z}^{\otimes_{i=1}^{n} a_{i}^\prime}).$
\item Apply $U^\prime$ to the system $C_q({\sf X}^{\otimes_{i=1}^{n} b_{i}} \cdot {\sf Z}^{\otimes_{i=1}^{n} a_{i}})\ket{\psi}$ to obtain the state $C_q(\ket{\psi}).$
\end{enumerate}
\item Output $\left(\ket{\phi},C_q^\prime \right).$
\end{enumerate}
\end{itemize} \end{algorithm} \pagebreak
\begin{theorem}\label{th:qio:cliff}
\Cref{QiO:Clifford-teleportation} is a perfect quantum indistinguishability obfuscation for all Clifford Circuits. \end{theorem} \begin{proof} We have to show that \Cref{QiO:Clifford-teleportation} satisfies \Cref{def:QiO}. \begin{enumerate} \item {\tt Functionality:} Let $C_q$ be an $n$-qubit Clifford Circuit and $\left(\ket{\phi},C_q^\prime \right)$ be the output of the \Cref{QiO:Clifford-teleportation} on input $C_q.$ On input $({\sf I}_n\otimes C_q) \ket{\beta^{2n}}$ and $\ket{\psi}$ the circuit $C_q^\prime$ outputs the state $C_q(\ket{\psi})$ (this follows from the principle of gate teleportation). Therefore $C_q^\prime(\ket{\phi}, \cdot)=C_q(\psi)$, which implies that
$||C_q^\prime(\ket{\phi}, \cdot)-C_{q}(\cdot) ||_\diamond=0\leq {\tt negl}(n)$ for any negligible function ${\tt negl}(n).$
\item {\tt Polynomial Slowdown:} Note $\ell=2n$ (the number of qubits in $\ket{\phi}$) and $C_q^\prime$ is a circuit of type $(3n,n).$ The size of the circuit $|C_q^\prime|= |i\mathcal{O}(C)|+|\mbox{Bell measurement}|$, where $C$ is the classical circuit that computes the update function corresponding to $C_q.$ The size of a Bell measurement circuit for an $O(n)$-qubit state is $O(n).$ Therefore, there exists a polynomial $q(|C|)$ such that $|\mbox{Bell measurement}|\leq q(|C|)|.$ The size of $|i\mathcal{O}(C)|$ is at most $r(|C|)$ for some polynomial $r(\cdot)$ (\Cref{lemma: iO-clifford-functions}). Further, the size of $|C|$ is at most $s(|C_q|)$ for some polynomial $|C_q|$ (\Cref{lemma: iO-clifford-functions}). By setting $p(|C|)=2|C_q|+q(|C|)+r(s(|C|)),$ we have
\begin{itemize}
\item $\ell\leq p(|C|)$
\item $m=3n \leq p(|C|)$
\item $|C_q^\prime|= |i\mathcal{O}(C)|+|\mbox{Bell measurement}|\leq p(|C|).$ \end{itemize}
\item {\tt Perfect Indistinguishability:} Let $C_{q_1}$ and $C_{q_2}$ be two $n$-qubit equivalent Clifford circuits of the same size. Let $\left(\ket{\phi_1},C_{q_1}^\prime \right)$ and $\left(\ket{\phi_2},C_{q_2}^\prime \right)$ be the outputs of \Cref{QiO:Clifford-teleportation} on inputs $C_{q_1}$ and $C_{q_2}$ respectively. Since $C_{q_1}(\ket{\tau})=C_{q_2}(\ket{\tau})$ for every quantum state $\ket{\tau}$ we have,
\begin{equation} \ket{\phi_1}=(I\otimes C_{q_1}) \ket{\beta^{2n}}=(I\otimes C_{q_2}) \ket{\beta^{2n}}=\ket{\phi_2}. \end{equation}
The update functions for any two equivalent Clifford circuits are equivalent (\Cref{lem:clifford-functions}), further the classical $i\mathcal{O}$ that obfuscates the update functions (circuits) is perfectly indistinguishable for any two equivalent Clifford circuits (not necessarily of the same size) (\Cref{lemma: iO-clifford-functions}). Therefore, $C_{q_1}^\prime$ and $C_{q_2}^\prime$ are perfectly indistinguishable. \qedhere \end{enumerate} \end{proof}
\begin{lemma}\label{lem:clifford-functions} Let $C_{q_1}$ and $C_{q_2}$ be two equivalent $n$-qubit Clifford circuits. Then their corresponding update functions are also equivalent. \end{lemma}
\begin{proof} Let $F_{C_{q_1}}$ and $F_{C_{q_2}}$ be the update functions for two $n$-qubit Clifford circuits $C_{q_1}$ and $C_{q_2}$ respectively. Suppose $F_{C_{q_1}}\neq F_{C_{q_2}},$ then there must exist at least one binary string ${\bf s}=a_1b_1 \ldots a_n b_n \in \{0,1\}^{2n} $ such that \begin{equation} \label{bld:equ1:QiO-Clifford} F_{C_{q_1}}({\bf s})\neq F_{C_{q_2}}({\bf s}) \end{equation}
Since $C_{q_1}$ and $C_{q_2}$ are equivalent circuit we must have that for every quantum state $\ket{\psi}$: \begin{equation} \label{bld:equ2:QiO-Clifford} C_{q_1} ({\sf X}^{\otimes_{i=1}^{n} b_{i}} \cdot {\sf Z}^{\otimes_{i=1}^{n}a_{i}})\ket{\psi}=C_{q_2} ({\sf X}^{\otimes_{i=1}^{n} b_{i}} \cdot {\sf Z}^{\otimes_{i=1}^{n}a_{i}})\ket{\psi} \end{equation} Let $F_{C_{q_1}}({\bf s})=(a_1^\prime b_1^\prime,\dots a_n^\prime b_n^\prime)$ and $F_{C_{q_2}}({\bf s})=(d_1^\prime e_1^\prime, \dots, d_n^\prime e_n^\prime),$ then we can rewrite \Cref{bld:equ2:QiO-Clifford} as
\begin{equation} \label{bld:eq3:QiO-Clifford}
({\sf X}^{\otimes_{i=1}^{n} b_{i}^\prime} \cdot {\sf Z}^{\otimes_{i=1}^{n}a_{i}^\prime})C_{q_1} (\ket{\psi})=({\sf X}^{\otimes_{i=1}^{n} e_{i}^\prime} \cdot {\sf Z}^{\otimes_{i=1}^{n}d_{i}^\prime})C_{q_2}(\ket{\psi}). \end{equation}
We can replace $C_{q_2} (\ket{\psi})$ with $C_{q_1} (\ket{\psi})$ in \Cref{bld:eq3:QiO-Clifford} \begin{equation} \label{bld:eq4:QiO-Clifford} ({\sf X}^{\otimes_{i=1}^{n} b_{i}^\prime} \cdot {\sf Z}^{\otimes_{i=1}^{n}a_{i}^\prime})C_{q_1} (\ket{\psi})=({\sf X}^{\otimes_{i=1}^{n} e_{i}^\prime} \cdot {\sf Z}^{\otimes_{i=1}^{n}d_{i}^\prime})C_{q_1}(\ket{\psi}). \end{equation}
Now if there exists a $j$ such that $a_j^\prime \neq d_j^\prime$ or $b_j^\prime \neq e_j^\prime,$ then \Cref{bld:eq4:QiO-Clifford} does not hold. This contradicts the assumption that $C_{q_1}$ and $C_{q_2}$ are equivalent Clifford circuits. Therefore $F_{C_{q_1}}$ and $F_{C_{q_2}}$ are equivalent functions.\footnote{The \Cref{bld:eq3:QiO-Clifford} is derived from the assumption that $C_{q_1}$ and $C_{q_2}$ are equivalent Clifford circuits.} \end{proof}
\subsubsection{Indistinguishability Obfuscator for Clifford: Update Functions}
\label{sec: iO-clifford-functions}
Here, we describe a perfect indistinguishability obfuscator $i\mathcal{O}$ for the update functions corresponding to the Clifford circuits. The algorithm takes an $n$-qubit Clifford circuit $C_q$ and output a classical circuit $C$ that computes the update function~$F_{C_q}.$ The circuit $C$ is invariant for any two equivalent Clifford circuits. The main idea here is to compute the canonical form for the Clifford circuit, and then compute the update function for the canonical form.
\begin{algorithm}[] \caption{$i\mathcal{O}$ for Clifford: Update Functions.} \label{alg:classical-iO-Clifford}
\begin{enumerate}
\item Compute the canonical form $C_q$ using the algorithm presented in ~\cite{AG04} (section VI). Denote the canonical form as $\widehat{C}_q.$
\item Let $g_1,g_2, \dots, g_m$ be a topological ordering of the gates in $\widehat{C}_q,$ where $m=|{\widehat{C}_q}|.$
\item Construct the classical circuit $\hat{C}$ that computes the update function $F_{\widehat{C}_q}$ as follows. For $i=1$ to $m$, implement the update rule for each gate $g_i$ (\Cref{update function}).
\item Output the classical circuit $\hat{C}.$
\end{enumerate} \end{algorithm}
\begin{lemma}\label{lemma: iO-clifford-functions} \Cref{alg:classical-iO-Clifford} is a perfect classical indistinguishability obfuscator for the Clifford update functions. \end{lemma}
\begin{proof} We have to show that \Cref{alg:classical-iO-Clifford} satisfies \Cref{def:quantum-secureiO}. \begin{enumerate} \item {\tt Functionality:} Let $C_q$ be an $n$-qubit Clifford circuit and $C$ be the circuit that computes $F_{C_q}$ (the update function for $C_q).$ Let $\widehat{C}_q$ be the canonical form of $C_q.$ Since $C_q$ and ${C_q}^\prime$ are equivalent circuits, it follows from \cref{lem:clifford-functions} that $F_{C_q}$ and $F_{C_q}^\prime$ are equivalent functions, therefore any circuit that computes $F_{C_q}^\prime$ also computes $F_{C_q}.$ From the construction above (\Cref{alg:classical-iO-Clifford}) it follows that the circuit $\hat{C}$ computes $F_{C_q}^\prime$ therefore $\hat{C}(x)= C(x)$ for all inputs~$x.$
\item {\tt Polynomial Slowdown:} For each gate $g_i$ in $\widehat{C}_q$, the classical circuit $\widehat{C}$ has to implement one of the following operations (\Cref{update function}):
\begin{itemize}
\label{cost}
\item[] $(a,b)\xrightarrow{{\sf X}, {\sf Z}}(a,b).$
\item[] $(a,b)\xrightarrow{{\sf H}}(b,a)$ (one swap).
\item[] $(a,b)\xrightarrow{{\sf P}}(a,a\oplus b)$ (one $\oplus$ operation).
\item[] $(a_1,b_1,a_2,b_2)\xrightarrow{\mbox{{\sf CNOT}}}(a_1\oplus a_2,b_1,a_2, b_1\oplus b_2)$ (two $\oplus$ operations).
\end{itemize}
Therefore the size of $|\widehat{C}|$ can be at most be $O(|\widehat{C}_q|).$ The $\widehat{C}_q$ is a canonical form of $C_q$ and of at most $q(|C_q|)$ for some polynomial $q(\cdot)$ \cite{AG04}. Therefore, there exists a polynomial $p(\cdot)$ such that $|\widehat{C}|\leq p(|C_q|).$
\item {\tt Perfectly Indistinguishability:} Let $C_{q_1}, C_{q_2}$ be two equivalent $n$-qubit Clifford circuits (not necessarily of the same size) and $\widehat{C_{q}}$ be their canonical form. Note that the output of \cref{alg:classical-iO-Clifford} only depends on the canonical form of the input Clifford circuit. Since, $C_{q_1}$ and $C_{q_2}$ have the same canonical form we have
$$\hat{C}\leftarrow\Cref{alg:classical-iO-Clifford}(\widehat{C_{q}})=\Cref{alg:classical-iO-Clifford}(C_{q_1})=\Cref{alg:classical-iO-Clifford}(C_{q_2}),$$ Therefore, \Cref{alg:classical-iO-Clifford} is a perfect indistinguishability obfuscation for the Clifford update functions.\qedhere \end{enumerate} \end{proof} \begin{remark} \label{re:iO:sizes}
What is convenient about the $i\mathcal{O}$ of \Cref{alg:classical-iO-Clifford} is that it works for any two equivalent Clifford circuits (regardless of their relative sizes) (see \Cref{lem:clifford-functions}). However, we can use any perfectly secure $i\mathcal{O}$ in our construction with some care. Suppose $i\mathcal{O}$ is some perfectly secure indistinguishability obfuscator for classical circuits (for Clifford update functions) of the same size. Suppose we want to obfuscate an update circuit corresponding to some Clifford~$C_q$. The classical circuit~$C$ is constructed by going through each gate in~$C_{q}.$ Some gates are more costly than others (for \emph{e.g.}, ${\sf CNOT}$ vs. ${\sf Z},$ see proof of \Cref{th:qio:cliff} or \Cref{update function}). Since we assume all Clifford circuits are of the same size, we can obtain an upper bound on all the classical circuits (for the update functions) by replacing each gate in $C_q$ with the most costly gate and then computing the classical circuit for the resulting quantum gate. Now suppose $m$ is the upper bound on the size of classical circuits, then for any circuit $C_q,$ we first calculate the circuit $C$ that computes $F_{C_q}$ and then pad $C$ with $m-|C|$ identity gates. This will ensure that if $|C_{q_1}|=|C_{q_2}|,$ then $|C_1|=|C_2|.$ \end{remark}
\section{Obfuscating Beyond Clifford Circuits} \label{QiO:Clifford+T:family} In this section, we extend the gate teleportation technique to show how we can construct $qi\mathcal{O}$ for \emph{any} quantum circuit. Our construction is efficient as long as the circuit has ${\sf T}$-count at most logarithmic in the circuit size. For the sake of simplicity, we first construct a $qi\mathcal{O}$ for an arbitrary 1-qubit quantum circuit (\Cref{sec:1-qubit}), then extend the 1-qubit construction to any $n$-qubit quantum circuit (\Cref{sec:n-qubit-circuits}).
We first start with some general observations on quantum circuits which are relevant to this section. Consider the application of the ${\sf T}$-gate on an encrypted system using the quantum one-time pad. The following equation relates the ${\sf T}$-gate to the ${\sf X}$- and ${\sf Z}$-gates: \begin{equation} \label{eq:pgate} {\sf T} {\sf X}^b {\sf Z}^a={\sf X}^b {\sf Z}^{a\oplus b}{\sf P}^b {\sf T}\,. \end{equation} If $b=0,$ then ${\sf P}^b$ is the identity; otherwise we have a ${\sf P}$-gate correction. This is undesirable as ${\sf P}$ does not commute with ${\sf X}$, making the update of the encryption key $(a,b)$ complicated (since it is no longer a tensor product of Paulis). Note that we can write ${\sf P}=\left(\frac{1+i}{2}\right) {\sf I} + \left(\frac{1-i}{2}\right){\sf Z}$, therefore \Cref{eq:pgate} can be rewritten as: \begin{equation} \label{eq:pgate1} {\sf T} {\sf X}^b {\sf Z}^a={\sf X}^b {\sf Z}^{a\oplus b}\left[\left(\frac{1+i}{2}\right) {\sf I} + \left(\frac{1-i}{2}\right){\sf Z}\right]^b {\sf T} \end{equation} Since $\left[\left(\frac{1+i}{2}\right) {\sf I} + \left(\frac{1-i}{2}\right){\sf Z}\right]^b=\left(\frac{1+i}{2}\right) {\sf I} + \left(\frac{1-i}{2}\right){\sf Z}^b$ for $b \in \{0,1\},$ we can rewrite \Cref{eq:pgate1} as, \begin{equation} \label{eq:pgate2} \begin{aligned} {\sf T} {\sf X}^b {\sf Z}^a={\sf X}^b {\sf Z}^{a\oplus b}\left[\left(\frac{1+i}{2}\right) {\sf I} + \left(\frac{1-i}{2}\right){\sf Z}^b\right] {\sf T} \\ =\left[\left(\frac{1+i}{2}\right) {\sf X}^b {\sf Z}^{a\oplus b} + \left(\frac{1-i}{2}\right){\sf X}^b {\sf Z}^a\right] {\sf T}. \\ \end{aligned} \end{equation}
It follows from \Cref{eq:pgate2} that for any $a,b\in\{0,1\},$ we can represent ${\sf T} {\sf X}^b {\sf Z}^a$ as a linear combination of ${\sf X}$ and ${\sf Z}.$ \begin{equation} \label{eq:pgate3} \begin{aligned} {\sf T} {\sf X}^b {\sf Z}^a= (\alpha_1 {\sf I} + \alpha_2 {\sf X} + \alpha_3 {\sf Z} + \alpha_4 {\sf X}{\sf Z}){\sf T} \\ \end{aligned} \end{equation} where $\alpha_j \in\left\{0,1, \frac{1+i}{2}, \frac{1-i}{2}\right\},$ for $j\in[4].$
We further note that for a general $n$-qubit quantum unitary $U$ and $n$-qubit Pauli~$P$, there exists a Clifford $C$ such that $UP\ket{\psi} = CU \ket{\psi}$. This is due to the \emph{Clifford hierarchy} \cite{GC99}. We also mention that if an $n$-qubit Clifford operation is given in matrix form, an efficient procedure exists in order to produce a circuit that executes this Clifford\cite{NWD14}. This is a special case of the general problem of \emph{synthesis} of quantum circuits, which aims to produce quantum circuits, based on an initial description of a unitary operation.
\subsection{Single-Qubit Circuits} \label{sec:1-qubit}
Here, we show an indistinguishability obfuscation for single-qubit circuits. As previously mentionned, we note that for the single-qubit case, an efficient indistinguishability obfuscation can also be built using the Matsumoto-Amano normal form~\cite{MA08arxiv,GS19arxiv}. Here, we give an alternate construction based on gate teleportation. Let $C_q$ be a $1$-qubit circuit we want to obfuscate and $\ket{\psi}$ be the quantum state on which we want to evaluate $C_q.$ Note that we can write any $1$-qubit circuit as a sequence of gates from the set $\{{\sf H}, {\sf T}\}$\footnote{The set $\{{\sf H}, {\sf T}\}$ is universal for 1-qubit unitaries \cite{KLM07}.}
$$C_q=(g_{|C_q|},\ldots,g_2,g_1 ),\; g_i\in\{{\sf H}, {\sf T}\}\,.$$ For the indistinguishability obfuscation of a single-qubit circuit, we use the gate teleportation protocol (\Cref{algo:gate-teleport}), which leaves us (after the teleportation) with a subsystem of the form $C_q {\sf X}^b{\sf Z}^a (\ket{\psi})$ \begin{equation} \label{eq:pgate4}
C_q {\sf X}^b{\sf Z}^a (\ket{\psi})=(g_{|C_q|},\ldots,g_2,g_1 ){\sf X}^b{\sf Z}^a (\ket{\psi}), \end{equation} and to evaluate the circuit on $\ket{\psi}$, we have to apply a correction unitary. Now suppose we apply the gate $g_1$. We can write the system in \Cref{eq:pgate4} as \begin{equation} \begin{aligned} \label{eq:pgate5}
C_q {\sf X}^b{\sf Z}^a (\ket{\psi})=(g_{|C_q|},\ldots,g_2)(\alpha_0 {\sf I} + \alpha_1 {\sf X} + \alpha_2 {\sf Z} + \alpha_3 {\sf X} {\sf Z})g_1 (\ket{\psi}) \end{aligned} \end{equation}
where $\alpha_i \in\left\{0,1, \frac{1+i}{2}, \frac{1-i}{2}\right\}.$ Since $\{{\sf I}, {\sf X}, {\sf Z}, {\sf X} {\sf Z}\},$ forms a basis, after applying the remaining gates in the sequence $(g_{|C_q|},\ldots,g_3,g_2),$ we can write \Cref{eq:pgate5} as \begin{equation} \begin{aligned} \label{eq:pgate8}
C_q {\sf X}^b{\sf Z}^a (\ket{\psi})=(\beta_1 {\sf I} + \beta_2 {\sf X} + \beta_3 {\sf Z} + \beta_4{\sf X} {\sf Z})(g_{|C_q|},\ldots,g_2,g_1 )(\ket{\psi}) \end{aligned} \end{equation} where each $\beta_i\in\mathbb{C}$ and is computed by multiplying and adding numbers from the set $\{0,1,\frac{1+i}{2},\frac{1-i}{2}\}.$ We show in \Cref{coeff:size} that the size of the coefficients~$\beta_i$ grows at most as a polynomial in the number of ${\sf T}$-gates. Therefore it follows from \Cref{eq:pgate8} that the update function for any $1$-qubit circuit $C_q$ can be defined as the following map, \begin{equation*} F_{C_q}:\{0,1\}^2\rightarrow \mathbb{C}^4,\; (a,b)\mapsto (\beta_1, \beta_2,\beta_3,\beta_4), \end{equation*} and is in one-to-one correspondence with the correction unitary $\beta_1 {\sf I} + \beta_2 {\sf X} + \beta_3 {\sf Z} + \beta_4{\sf X} {\sf Z}.$ As indicated, our construction for 1-qubit circuits is nearly the same as the gate teleportation scheme for Clifford circuits (\Cref{QiO:Clifford-teleportation}). The proof that this is a $qi\mathcal{O}$ scheme is also very similar to the proof for the Clifford construction (\Cref{sec:Clifford-iO-teleportaion}); we thus omit the formal proof here (it can also be seen as a special case of the proof of \Cref{theorem:main}). Some subtleties, however are addressed below: the equivalence of the update functions (\Cref{lemma:1qubit}) and the circuit synthesis (\Cref{lem:1-qubit-synthesis}).
\begin{lemma} \label{lemma:1qubit} Let $C_{q_1}$ and $C_{q_2}$ be two equivalent $1$-qubit circuits. Then their corresponding update functions in the gate teleportation protocol are also equivalent. \end{lemma}
\begin{proof} Suppose $F_{C_{q_1}}(a,b)=(\beta_1, \beta_2 , \beta_3 , \beta_4,)$ and $F_{C_{q_2}}(a,b)=(\gamma_1, \gamma_2, \gamma_3, \gamma_4)$ are the corresponding update functions for $a,b\in\{0,1\}.$ Since $C_{q_1}$ and $C_{q_2}$ are equivalent circuits, for every quantum state $\ket{\psi}$ and any $a,b \in\{0,1\},$
$$C_{q_1} {\sf X}^b {\sf Z}^a(\ket{\psi})=C_{q_2} {\sf X}^b {\sf Z}^a(\ket{\psi})$$ $$\Leftrightarrow (\beta_1 {\sf I} + \beta_2 {\sf X} + \beta_3 {\sf Z} + \beta_4{\sf X} {\sf Z})C_{q_1}(\ket{\psi})= (\gamma_1 {\sf I} + \gamma_2 {\sf X} + \gamma_3 {\sf Z} + \gamma_4{\sf X} {\sf Z})C_{q_2}(\ket{\psi})$$ $$\Leftrightarrow (\beta_1 {\sf I} + \beta_2 {\sf X} + \beta_3 {\sf Z} + \beta_4{\sf X} {\sf Z})C_{q_1}(\ket{\psi})= (\gamma_1 {\sf I} + \gamma_2 {\sf X} + \gamma_3 {\sf Z} + \gamma_4{\sf X} {\sf Z})C_{q_1}(\ket{\psi})$$ $$\Leftrightarrow ((\beta_1- \gamma_1){\sf I} + (\beta_2 -\gamma_2) {\sf X} + (\beta_3 - \gamma_3){\sf Z} + (\beta_4 -\gamma_4){\sf X} {\sf Z})C_{q_1}(\ket{\psi})= {\bf 0}.$$ $$\Rightarrow (\beta_1- \gamma_1){\sf I} + (\beta_2 -\gamma_2) {\sf X} + (\beta_3 -\gamma_3){\sf Z} + (\beta_4 -\gamma_4){\sf X} {\sf Z}= {\bf 0}.$$ $$\Rightarrow \beta_1= \gamma_1,\, \beta_2 =\gamma_2,\, \beta_3 =\gamma_3,\, \beta_4 =\gamma_4.$$
Therefore, $F_{C_{q_1}}$ and $F_{C_{q_2}}$ are equivalent functions. \end{proof}
We note that, on top of being equal, the circuits that compute the update functions $F_{C_{q_1}},$ $F_{C_{q_2}}$ can be assumed to be of the same size. This follows by an argument very similar to the one in \Cref{re:iO:sizes}.\\
\begin{lemma} \label{lem:1-qubit-synthesis} Based on the classical $i\mathcal{O}$ that computes the coefficients in \Cref{eq:pgate8}, it is possible to build a quantum circuit that performs the correction efficiently. \end{lemma}
\begin{proof} Given a $2 \times 2$ unitary matrix that represents a Clifford operation as in \Cref{eq:pgate8}, it is simple to efficiently derive the Clifford circuit that implements the unitary. This is a special case of the general efficient synthesis for Clifford circuits as presented in~\cite{NWD14}. \end{proof}
\subsection{$qi\mathcal{O}$ via Gate Teleportation for all Quantum Circuits} \label{sec:n-qubit-circuits}
In this section, we construct a $qi\mathcal{O}$ for all quantum circuits. The construction is efficient whenever the number of ${\sf T}$-gates is at most logarithmic in the circuit size (see \Cref{QiO:qcircuit-teleportation}). The reason for this limitation is that the update function blows up once the number of ${\sf T}$-gates is greater than logarithmic in the circuit size. The construction is very similar to the gate teleportation for Clifford circuits (\Cref{sec:Clifford-iO-teleportaion}) and assumes the existence of a quantum-secure $i\mathcal{O}$ for classical circuits.
\begin{algorithm}[]
\caption{$qi\mathcal{O}$ via Gate Teleportation for Quantum Circuits}
\label{QiO:qcircuit-teleportation}
\begin{itemize}
\item Input: A $n$-qubit quantum Circuit $C_q$ with ${\sf T}$-count $\in O(\log(|C_q|)).$
\begin{enumerate}
\item Prepare a tensor product of $n$ Bell states: $\ket{\beta^{ 2n}}=\ket{\beta_{00}}\otimes \cdots \otimes \ket{\beta_{00}}.$
\item Apply the circuit $C_q$ on the right-most $n$ qubits to obtain a system $\ket{\phi}$:
$$\ket{\phi}=({\sf I}_n\otimes C_q) \ket{\beta^{2n}}.$$
\item Set $\hat{C}\leftarrow i\mathcal{O}(C).$ Where $C$ is a circuit that computes the update function $F_{C_q}$ as in \Cref{sec:sizeofupdate}. Note the size of $C$ is at most a polynomial in $|C_q|$ (\Cref{lemma:nqubit:cost}).
\item Description of the circuit $C_q^\prime:$
\begin{enumerate}
\item Perform a general Bell measurement on the leftmost $2n$-qubits on the system $\ket{\phi}\otimes \ket{\psi}$, where $\ket{\phi}$ is an auxiliary state and $\ket{\psi}$ is an input state. Obtain classical bits $(a_1,b_1\ldots,a_n,b_n)$ and the state
$$C_q({\sf X}^{\otimes_{i=1}^{n} b_{i}} \cdot {\sf Z}^{\otimes_{i=1}^{n} a_{i}})\ket{\psi}.$$
\item Compute the correction using the obfuscated circuit
$$((\beta_1, {\bf s}_1),\ldots, (\beta_n, {\bf s}_k))=\hat{C}(a_1,b_1\ldots,a_n,b_n).$$
\item Using the above, the correction unitary is $$U_{F_{C_q}}=\sum_{i=1}^{4^k} \beta_i {\sf X}^{b_{i_1}} {\sf Z}^{a_{i_1}}\otimes \cdots \otimes {\sf X}^{b{i_n}} {\sf Z}^{a_{i_n}}.$$
Compute a quantum circuit that applies $U_{F_{C_q}}$, using the circuit synthesis method of \cite{NWD14}.
\item Apply the quantum circuit for $U_{F_{C_q}}$ to the system $C_q({\sf X}^{\otimes_{i=1}^{n} b_{i}} \cdot {\sf Z}^{\otimes_{i=1}^{n} a_{i}})\ket{\psi}$ to obtain the state $C_q(\ket{\psi}).$
\end{enumerate}
\end{enumerate}
\end{itemize} \end{algorithm}
We are now ready to present our main theorem (\Cref{theorem:main}). For ease of presentation, the proof relies on three auxiliary lemmas that are presented in the following section: \Cref{lemma:nqubit} (which shows that equivalent circuits have equivalent update functions), \Cref{lem1:mainTh} (which bounds the number of terms of the update function), and \Cref{lemma:nqubit:cost} (which shows that update functions can be computed by a polynomial-size circuits).
\begin{theorem} (Main Theorem)\label{theorem:main}
If $i\mathcal{O}$ is a perfect/statistical/computational quantum-secure indistinguishability obfuscation for classical circuits, then \Cref{QiO:qcircuit-teleportation} is a perfect/statistical/computational quantum indistinguishability obfuscator for any quantum circuit~$C_q$ with ${\sf T}$-count $\in O(\log|C_q|).$ \end{theorem}
\begin{proof} We have to show that \Cref{QiO:qcircuit-teleportation} satisfies \Cref{def:QiO}. Throughout the proof, we assume that the quantum circuits have a logarithmic ${\sf T}$-count in the circuit size.
\begin{enumerate} \item {\tt Functionality:} The proof of functionality follows from the principle of gate teleportation and is very similar to the proof in \Cref{th:qio:cliff}. Since the Clifford circuit synthesis has perfect correctness\cite{NWD14}, we have
$||C_q^\prime(\ket{\phi}, \cdot)-C_{q}(\cdot) ||_\diamond=0\leq {\tt negl}(n)$ for any negligible function ${\tt negl}(n).$
\item {\tt Polynomial Slowdown:} Note that $\ket{\phi}$ is a $2n$-qubit state and $C_q^\prime$ is of type $(m,n),$ where $m=3n,$ therefore both $\ket{\phi}$ and $m$ have size in $O(|C_q|).$ The size of $C_q^\prime$ is equal to the size of $\hat{C}$ plus the size of the circuit that performs the general Bell measurement ($GBM$) and the size of the circuit that computes the circuit for $U_{F_{C_q}}$.
Since the size of $|GBM|$ is in $O(n)$, the size of $|\hat{C}|$ is polynomial in $|C_q|$ (\Cref{lem1:mainTh}, \Cref{lemma:nqubit:cost}, \Cref{lem:beta:size} and the definition of $i\mathcal{O}$). Moreover, efficient Clifford synthesis implies that the size of the circuit for $U_{F_{C_q}}$ is polynomial~\cite{NWD14}.
Therefore, there exists a polynomial $p(\cdot)$ such that \begin{itemize}
\item $|\ket{\phi}|\leq p(|C_q|)$
\item $m\leq p(|C_q|)$
\item $|C_q^\prime|\leq p(|C_q|).$ \end{itemize}
\item {\tt Perfect/Statistical/Computational Indistinguishability:} Let $C_{q_1}$ and $C_{q_2}$ be two $n$-qubit circuits of the same size. Let $\left(\ket{\phi_1},C_{q_1}^\prime \right)$ and $\left(\ket{\phi_2},C_{q_2}^\prime \right)$ be the outputs of \Cref{QiO:Clifford-teleportation} on inputs $C_{q_1}$ and $C_{q_2}$ respectively. Since $C_{q_1}(\ket{\tau})=C_{q_2}(\ket{\tau})$ for every quantum state $\ket{\tau}$ we have, \looseness=-1
\begin{equation} \ket{\phi_1}=(I\otimes C_{q_1}) \ket{\beta^{2n}}=(I\otimes C_{q_2}) \ket{\beta^{2n}}=\ket{\phi_2}, \end{equation} The update functions for any two equivalent quantum circuits are equivalent (\Cref{lemma:nqubit}). If the classical $i\mathcal{O}$ that obfuscates the circuits for the update functions is perfectly/statistically/computationally indistinguishable, then states $C_{q_1}^\prime$ and $C_{q_2}^\prime$ are perfectly/statistically/computationally indistinguishable.\footnote{Note that circuits that compute update functions (for equivalent quantum circuits) may have different sizes. However, that can be managed as discussed in \Cref{re:iO:sizes}.} Therefore, \Cref{QiO:qcircuit-teleportation} is a perfectly/statistically/ computationally indistinguishable quantum obfuscator for the quantum circuits. \qedhere \end{enumerate} \end{proof}
\subsection{Equivalent Update Functions} \label{sec:Update-equivalent}
Here, we provide a generalization of \Cref{lem:clifford-functions}, applicable to the case of general circuits.
\begin{lemma} \label{lemma:nqubit} Let $C_{q_1}$ and $C_{q_2}$ be two equivalent $n$-qubit circuits. Then their corresponding update functions are also equivalent. \end{lemma}
\begin{proof} Suppose $C_{q_1}$ and $C_{q_2}$ are two equivalent $n$-qubit quantum circuits, then for any quantum state $\ket{\psi}$ and for any binary string ${\bf r}\in\{0,1\}^{2n}.$ \begin{equation} \label{equ1:nqubit} C_{q_1} ({\sf X}^{a_{i}}{\sf Z}^{b_{i}})^{\otimes_{i=1}^{n}}\ket{\psi} =C_{q_2} ({\sf X}^{a_{i}}{\sf Z}^{b_{i}})^{\otimes_{i=1}^{n}}\ket{\psi}. \end{equation} Then the corresponding update functions are (see \Cref{nqubit:eq0})
\begin{center} $F_{C_{q_1}}({\bf r})=((\beta_1, {\bf s}_1),\ldots, (\beta_n, {\bf s}_k))$ \\ $F_{C_{q_1}}({\bf r})=((\beta_1^\prime, {\bf s}_1^\prime),\ldots, (\beta_{\ell}^\prime, {\bf s}_{\ell}^\prime))$ \end{center} where ${\bf s}_i=a_{i_1}b_{i_1}, \ldots, a_{i_n}b_{i_n} \in\{0,1\}^{2n},$ and ${\bf s}_j^\prime=a_{j_1}b_{j_1}, \ldots, a_{j_n}b_{j_n} \in\{0,1\}^{2n}.$ Without loss of generality, we can assume that $\beta_i\neq 0, \beta_j^\prime\neq 0$ for $i\in[k], j\in[\ell].$ Using the update functions, we can rewrite \Cref{equ1:nqubit} as
\begin{equation}\label{equ2:nqubit} \left(\sum_{i=1}^{4^k} \beta_i {\sf X}^{b_{i_1}} {\sf Z}^{a_{i_1}}\otimes \cdots \otimes {\sf X}^{b{i_n}} {\sf Z}^{a_{i_n}}\right)C_{q_1}\ket{\psi} =\left(\sum_{j=1}^{4^{\ell}} \beta_j^\prime {\sf X}^{b_{j_1}^\prime} {\sf Z}^{a_{j_1}^\prime}\otimes \cdots \otimes {\sf X}^{b_{j_n}^\prime} \right)C_{q_2}\ket{\psi}. \end{equation} Since $C_{q_1}$ and $C_{q_2}$ are equivalent, we can replace $C_{q_2}$ with $C_{q_1}$ in \Cref{equ2:nqubit}
\begin{equation}\label{equ3:nqubit} \left(\sum_{i=1}^{4^k} \beta_i {\sf X}^{b_{i_1}} {\sf Z}^{a_{i_1}}\otimes \cdots \otimes {\sf X}^{b{i_n}} {\sf Z}^{a_{i_n}}\right)C_{q_1}\ket{\psi} =\left(\sum_{j=1}^{4^{\ell}} \beta_j^\prime {\sf X}^{b_{j_1}^\prime} {\sf Z}^{a_{j_1}^\prime}\otimes \cdots \otimes {\sf X}^{b_{j_n}^\prime} \right)C_{q_1}\ket{\psi} \end{equation} Then \Cref{equ3:nqubit} can only hold if
\begin{equation}\label{equ4:nqubit} \left(\sum_{i=1}^{4^k} \beta_i {\sf X}^{b_{i_1}} {\sf Z}^{a_{i_1}}\otimes \cdots \otimes {\sf X}^{b{i_n}} {\sf Z}^{a_{i_n}}\right)=\left(\sum_{j=1}^{4^{\ell}} \beta_j^\prime{\sf X}^{b_{j_1}^\prime} {\sf Z}^{a_{j_1}^\prime}\otimes \cdots \otimes {\sf X}^{b_{j_n}^\prime} \right) \end{equation}
Note the update functions $F_{C_{q_1}}$ and $F_{C_{q_2}}$ are in one-to-one mapping with the unitaries on the left- and right-hand side of \Cref{equ4:nqubit} respectively. Since, the unitaries are equivalent, the corresponding update functions are also equivalent. \end{proof}
\subsection{Complexity of Computing the Update Function} \label{sec:sizeofupdate}
Let $C_q$ be an $n$-qubit circuit consisting of a sequence of gates $g_1\ldots,g_{|C_q|}.$ The update function $F_{C_q}$ is computed by composing update functions for each gate in $C_q.$
$$ F_{C_q}= f_{g_{{|C_q|}}}\circ \cdots \circ f_{g_2} \circ f_{g_1}.$$ Therefore the update function for any $n$-qubit quantum circuit $C_q$ with $k$ ${\sf T}$-gates can be defined as the following map. \begin{equation}\label{nqubit:eq0} \begin{aligned} &F_{C_q}: \{0,1\}^{2n}\longrightarrow (\mathbb{C} \times \{0,1\}^{2n})^{min(k,n)},\\
&
(a_1,b_1,\ldots, a_n,b_n) \mapsto \left((\beta_1, {\bf s}_1),\dots, (\beta_{4^k}, {\bf s}_{4^k})\right).
\end{aligned}
\end{equation} which corresponds to the following correction unitary \begin{equation} \label{exp:nqubit-correction} \sum_{i=1}^{4^k} \beta_i {\sf X}^{b_{i_1}} {\sf Z}^{a_{i_1}}\otimes \cdots \otimes {\sf X}^{b{i_n}} {\sf Z}^{a_{i_n}}. \end{equation}
where $\beta_i\in\mathbb{C}$ and ${\bf s}_i=a_{i_1}b_{i_1}, \ldots, a_{i_n}b_{i_n} \in\{0,1\}^{2n},$ $i\in[4^k].$ Therefore, in order to satisfy the efficiency requirement (polynomial-slowdown), we must have $k\in O(\log(|C_q|))$. Note that the range of the update function can increase exponentially in the number of ${\sf T}$-gates as long as $k\leq n$ (\Cref{nqubit:eq0}). This is because there are at most $2^{2n}$ binary strings of length $2n,$ therefore for any $n$-qubit circuit, the correction unitary \Cref{exp:nqubit-correction} can be written as a summation of at most $2^{2n}$ terms. We will first prove that as long as the ${\sf T}$-count in $O(\log(|C_q|),$ the number of terms in $F_{C_q}$ has at most $O(|C_q|)$ terms.
\begin{lemma}\label{lem1:mainTh}
If $C_q$ is an $n$-qubit quantum circuit with ${\sf T}$-count in $O(\log(|C_q|),$ then the update function $F_{C_q}$ (\Cref{nqubit:eq0}) has at most $O(|C_q|)$ terms. \end{lemma} \begin{proof}
Let $C_q$ be an $n$-qubit circuit with ${\sf T}$-count in $O(\log(|C_q|).$ Suppose we want to evaluate $C_q$ on some $n$-qubit state $\ket{\psi},$ then after the step 4a of \Cref{QiO:qcircuit-teleportation}, we will obtain a state \begin{equation}\label{nqubit:eq1} C_q ({\sf X}^{a_{1}}{\sf Z}^{b_{1}}\otimes {\sf X}^{a_{2}}{\sf Z}^{b_{2}}\otimes \cdots \otimes {\sf X}^{a_{n}}{\sf Z}^{b_{n}})\ket{\psi}. \end{equation} In order to recover $C_q(\ket{\psi})$ from the above expression, we multiply the correction unitary $U_{F_{C_q}}$ to the left hand side of expression \Cref{nqubit:eq1}. To compute $U_{F_{C_q}}$ we first compute the update function $F_{C_q}$ on the input $a_1b_1,\ldots,a_nb_n$
$$F_{C_q}(a_1b_1,\ldots,a_nb_n)=\left((\beta_1, {\bf s}_1),\dots, (\beta_{4^k}, {\bf s}_{4^k})\right)$$ where $\beta_i \in \mathbb{C},$ ${\bf s}_i\in\{0,1\}^{2n},$ $k\in \mathbb{N}$ $$U_{F_{C_q}}=\sum_{i=1}^{4^k} \beta_i {\sf X}^{b_{i_1}} {\sf Z}^{a_{i_1}}\otimes \cdots \otimes {\sf X}^{b{i_n}} {\sf Z}^{a_{i_n}}.$$
We will show that if the ${\sf T}$-count is in $O(\log(|C_q|),$ then $k\in O(|C_q|)$ (number of terms). Recall that \begin{equation}\label{nqubit:eq2} {\sf T} {\sf X}^b {\sf Z}^a= (\alpha_1 {\sf I} + \alpha_2 {\sf X} + \alpha_3 {\sf Z} + \alpha_4 {\sf X}{\sf Z}){\sf T} \end{equation}
\begin{itemize} \item {\bf Case 0}: Suppose $C_q$ has no ${\sf T}$-gates, then $C_q$ is a Clifford and there is only one term in $F_{C_q}.$ Therefore $k=0.$ \item {\bf Case 1}: Suppose $C_q$ has one ${\sf T}$-gate (acting on some $\ell$-th wire). \begin{equation}\label{nqubit:eq3} \begin{aligned} &C_q ({\sf X}^{a_{1}}{\sf Z}^{b_{1}}\otimes \cdots \otimes {\sf X}^{a_{n}}{\sf Z}^{b_{n}})\\ &=({\sf X}^{\otimes_{i=1}^{\ell-1} b_{i}^\prime} {\sf Z}^{\otimes_{i=1}^{\ell-1}a_{i}^\prime})\otimes \left(\sum_{j=1}^{4} \beta_j {\sf X}^{b_j^\prime}{\sf Z}^{a_j^\prime}\right) \otimes ({\sf X}^{\otimes_{i=\ell+1}^{n} b_{i}^\prime} {\sf Z}^{\otimes_{i=\ell+1}^{n}a_{i}^\prime}) C_q\\ &=\sum_{i=1}^{4} \beta_i {\sf X}^{b_{i_1}^\prime} {\sf Z}^{a_{i_1}^\prime}\otimes \cdots \otimes {\sf X}^{b_{i_n}^\prime} {\sf Z}^{a_{i_n}^\prime}. \end{aligned} \end{equation} Therefore $k\leq 4.$ It is important to realize that $4$ is the maximum number of terms a circuit with ${\sf T}$ can have. No Clifford gate including ${\sf CNOT}$ can increase the number of terms beyond 4. This is because only a ${\sf T}$ gate can contribute a 4-term expression and then we expand all the terms in the correction unitary to maximum (\Cref{nqubit:eq3}). Now any Clifford acting on the correction unitary will act linearly on $U_{F_{C_q}}$ and only affect the bits $a_i^\prime$ and~$b_i^\prime$ (\Cref{nqubit:eq3}). \item {\bf Case 2}: Similarly, if $C_q$ has two ${\sf T}$ gates, then each will contribute at most one expression of the form $(\beta_1 {\sf I} + \beta_2 {\sf X} + \beta_3 {\sf Z} + \beta_4 {\sf X}{\sf Z}).$ This will give us a correction unitary $U_{F_{C_q}}=\sum_{i=1}^{4^2} \beta_i {\sf X}^{b_{i_1}} {\sf Z}^{a_{i_1}}\otimes \cdots \otimes {\sf X}^{b{i_n}} {\sf Z}^{a_{i_n}},$ therefore $k\leq 16.$ Note that it makes no difference whether ${\sf T}$ gates are acting on the same wire or on different wires for the worst-case analysis. If they are on the same wire, then the first ${\sf T}$ will contribute 4 terms and the second ${\sf T}$ will expand each term into 4 more terms, resulting in 16 terms. If they are acting on different wires, say $i$ and $j$ and suppose ${\sf CNOT}$s are acting on the $i$-th to $j$-th wire, then we may have to expand all terms in the unitary to apply ${\sf CNOT}$s. This again can contribute at most 16 terms.
\item[] {\bf General Case}: Suppose $C_q$ has $O(\log(|C_q|)$ ${\sf T}$ gates, then each~${\sf T}$-gate will contribute at most a linear combination of 4 terms and in total at most $4^{O(\log(|C_q|)}$ terms.
therefore $k\in O(|C_q|).$\qedhere \end{itemize} \end{proof}
\begin{lemma}\label{lemma:nqubit:cost}
If $C_q$ be an $n$-qubit quantum circuit with ${\sf T}$-count $\in O(\log|C_q|),$ then there exists a classical circuit $C$ and a polynomial $p(\cdot)$ such that \begin{itemize} \item $C$ computes the update function $F_{C_q},$
\item $|C|\leq p(|C_q|).$ \end{itemize} \end{lemma} \begin{proof}
Let $C_q$ be a $n$-qubit quantum circuit with ${\sf T}$-count $\in O(\log|C_q|).$ Recall from \Cref{update function} that the corresponding update functions for the Clifford + ${\sf T}$ gate set are: \begin{itemize} \item $f_{\sf X}(a_1,b_1)=(a_1,b_1)$ \item $f_{\sf Z}(a_1,b_1)=(a_1,b_1)$ \item $f_{\sf H}(a_1,b_1)=(b_1,a_1)$ \item $f_ {\sf P}(a_1,b_1)=(a_1,a_1\oplus b_1)$ \item $f_{{\sf CNOT}}(a_1,b_1,a_2,b_2)= (a_1\oplus a_2,b_1,a_2, b_1\oplus b_2)$ \item $f_{{\sf T}}(a,b)=(\alpha_1,\alpha_2,\alpha_3,\alpha_4)$ \end{itemize}
Let $C_{\sf X},$ $C_{\sf Z},$ $C_{\sf H},$ $C_{\sf P},$ $C_{\sf CNOT}$ and $C_{{\sf T}}$ denote the classical circuits (called \emph{subcircuits}) that compute $f_{\sf X}, f_{\sf Z}, f_{\sf H} f_ {\sf P}, f_{{\sf CNOT}}$ and $f_{{\sf T}}$ respectively. Clearly, these subcircuits are of constant size. Recall that all subcircuits for Cliffords map $k$-bit strings to $k$-bit strings ($k\in\{0,1\}$), but $C_{{\sf T}}$ expands its 2-bit input into a $4\ell$-bit strings (where $\ell =\max\{\left|\frac{1+ i}{2}\right|, \left|\frac{1-i}{2}\right|\}$\footnote{The notation $|a+bi|$ denotes the number of bits to represent the complex number $a+bi.$}. Let $C$ be a circuit that computes~$F_{C_q}$. Then $C$ can be expressed in terms of these gadgets. To construct $C$, we go gate-by-gate in $C_q$ and employ the corresponding subcircuit. If $C_q$ is a Clifford circuit, then $C$ only consists of Clifford subcircuit (as mentioned earlier they map $k$ bits to $k$ bits $k\in\{2,4\}$ there are $O(|C_q|)$ gadgets in the circuit $C$). Otherwise if $C_q$ has $k$ ${\sf T}$-gates, then each $C_{{\sf T}}$ subcircuit will map a 2-bit string to a $4\ell$-bit string, potentially increasing the size of $C$ to $O({(4\ell)}^k),$ but since $k\in O(\log(|C_q|),$ we have $O({(4\ell)}^k)\in O(|C_q|).$ Therefore, there exists a polynomial $p(\cdot)$ such that $|C|\leq p(n).$ \end{proof}
\section{Quantum Indistinguishability Obfuscation with Respect to a Pseudo-Distance} \label{sec:quantum:iO:approx:circuits}
\label{sec:approx:circuits} In this section, we provide a definition for circuits that are approximately equivalent (with respect to a pseudo-distance) (\Cref{def:aqec}). In \Cref{sec:qiO:approx:circuits}, we present a definition of quantum indistinguishability obfuscation with respect to a pseudo-distance, and in \Cref{sec:Gottesman-Chuang}, we present a scheme that satisfies this definition, for circuits close to a fixed level of the Gottesman-Chuang hierarchy.
\subsection{Approximately Equivalent Quantum Circuits} \begin{definition}{\rm (Approximately Equivalent Quantum Circuits):} \label{def:aqec} Let $C_{q_0}$ and~$C_{q_1}$ be two $n$-qubit quantum circuits and {\bf D} be a pseudo-distance. We say $C_{q_0}$ and $C_{q_1}$ are \emph{approximately equivalent} with respect to {\bf D} if there exists a negligible function ${\tt negl}(n)$ such that $${\bf D}(C_{q_0}, C_{q_1})\leq {\tt negl}(n).$$ \end{definition}
\subsection{Indistinguishability Obfuscation for Approximately Equivalent Quantum Circuits} \label{sec:qiO:approx:circuits} In this section, we provide a definition of quantum indistinguishability obfuscation for approximately equivalent circuits, $qi\mathcal{O}_{\bf D}.$ To be consistent with \Cref{def:QiO}, we require that the obfuscator, on input a quantum circuit $C_q,$ outputs an auxiliary quantum state $\ket{\phi}$ and a quantum circuit $C_q^\prime$, but note in the actual construction (\Cref{QiO:gottesman-chuang}), the state $\ket{\phi}$ is an empty register. Here, we consider only the case of \emph{statistical} security. Notable here is the indistinguishability property is required to hold not only for equivalent quantum circuits, but also for \emph{approximately} equivalent quantum circuits. Also, contrary to \Cref{def:QiO}, we only require the indistinguishability for large values of~$n$.
\begin{definition} \label{def:aQiO} Let $\mathcal{C}_Q$ be a polynomial-time family of reversible quantum circuits and let {\bf D} be a pseudo-distance. For $n\in\mathbb{N}$, let $C_{q^n}$ be the circuits in $\mathcal{C}_Q$ of input length $n.$ A polynomial-time quantum algorithm for~$\mathcal{C}_Q$ is a \emph{statistically secure quantum indistinguishability obfuscator} ($qi\mathcal{O}_{\bf D}$) for $\mathcal{C}_Q$ \emph{with respect to {\bf D}} if the following conditions hold:
\begin{enumerate} \item {\tt Functionality:} There exists a negligible function ${\tt negl}(n)$ such that for every $C_q\in C_{q^n}$ $$(\ket{\phi}, C_q^\prime)\leftarrow qi\mathcal{O}_{\bf D}(C_q) \; \mbox{ and }\; \mathbf{ D}(C_q^\prime(\ket{\phi}, \cdot),C_{q}(\cdot))\leq {\tt negl}(n).$$ Where $\ket{\phi}$ is an $\ell$-qubit state, the circuits $C_q$ and $C_q^\prime$ are of type $(n,n)$ and $(m,n)$ respectively ($m= \ell +n$).\footnote{A circuit is of type $(i,j)$ if it maps $i$ qubits to $j$ qubits.}
\item {\tt Polynomial Slowdown:} There exists a polynomial $p(n)$ such that for any $C_{q}\in C_{q^n},$ \begin{itemize}
\item $\ell\leq p(|C_{q}|)$
\item $m \leq p(|C_{q}|)$
\item $|C_{q}^\prime| \leq p(|C_{q}|).$ \end{itemize}
\item {\tt Statistically Secure Indistinguishability:} For any two \textbf{approximately equivalent} quantum circuits $C_{q_0},C_{q_1}\in C_{q^n},$ of the same size \textbf{and for large enough $n,$} the two distributions $qi\mathcal{O}_{\bf D}(C_{q_0})$ and $qi\mathcal{O}_{\bf D}(C_{q_1})$ are statistically indistinguishable. \end{enumerate} \end{definition}
\subsection{$qi\mathcal{O}_{{\bf D}}$ for Circuits Close to the Gottesman-Chuang Hierarchy } \label{sec:Gottesman-Chuang}
Here, we present a quantum indistinguishability obfuscation (\cref{def:QiO}) for a family of circuits that are approximately equivalent (\Cref{def:aqec}) with respect to the pseudo-distance ${\bf D}(U_1,U_2)=\frac{1}{\sqrt{2d^2}} ||U_1\otimes U_1^*- U_2\otimes U_2^*||_F \;$ (see \Cref{sec:norms}). There are two main ingredient in our construction, one is Low's learning algorithm~\cite{Low09} (described below)
and the second is~\Cref{lem:approx}.
In \cite{Low09} Low presents a learning algorithm that, given oracle access to a unitary~$U$ and its conjugate $U^\dagger$ with the promise that the distance ${\bf D}(U, C)\leq \epsilon <\frac{1}{2^{k-1/2}}$ for some $C\in \mathcal{C}_k$ (\Cref{sec:gottesman-chuang}), outputs a circuit $C_q$ for computing $C$ with probability at least $1-\delta$ with $$O\left(\frac{1}{{\epsilon^\prime}^2} (2n)^{k-1} \log\left(\frac{(2n+1)^{k-1})}{\delta}\right)\right)$$ queries. Where $\epsilon^\prime:=\sqrt{2(1-(2^{k-1}\epsilon)^2}-1>0$ and ${\bf D}(U_1,U_2)=\frac{1}{\sqrt{2d^2}} ||U_1\otimes U_1^*- U_2\otimes U_2^*||_F$ is the pseudo-distance defined in \Cref{sec:norms}.
Based on Low's work, we construct an quantum indistinguishability obfuscation $qi\mathcal{O}_{{\bf D}}$ with respect to this pseudo-distance ${\bf D}$ for circuits that are very close to $\mathcal{C}_k.$ Note that the run-time of Low's algorithm is exponential in $k.$ Moreover, the algorithm becomes infeasible if $\epsilon^\prime$ is very small. Therefore, to ensure that our construction in \Cref{QiO:gottesman-chuang} runs in polynomial-time we set $k$ to be some fixed positive integer and $\epsilon \leq {\tt negl}(n)<\frac{1}{2^{k-1/2}}$ for all $n.$ Note if $\epsilon<\frac{1}{2^{k-1/2}},$ then $ \epsilon^\prime \geq \frac{\sqrt{7}}{2}-1.$
\begin{lemma} \label{lem:approx} Let $U$ and $C$ be unitaries. If the distance ${\bf D}(U,C)<\frac{1}{2^{k-1/2}}$ for some $C\in \mathcal{C}_k$, then $C$ is unique up to phase. \end{lemma} \begin{proof} See~\cite{Low09}. \end{proof}
\begin{theorem}\label{th:approx} Let $\mathcal{C}_Q=\{U_{q^{n,k}} \mid n \in \mathbb{N} \mbox{ and } k \mbox{ is fixed positive integer} \},$ be a polynomial-time family of reversible quantum circuits. Here, $U_{q^{n,k}}$ denotes the $n$-qubit circuits for which there exists a negligible function ${\texttt{negl}}(n)$ such that for any $U_q\in U_{q^{n,k}},$ there exists a $C_q\in\mathcal{C}_k$ that satisfies ${\bf D}(U_q, C_q)< {\tt negl}(n)<\frac{1}{2^{k+1/2}}.$ Then \Cref{QiO:gottesman-chuang} is a statistically-secure quantum indistinguishability obfuscation for $\mathcal{C}_Q$ with respect to~${\bf D}.$ \end{theorem}
\begin{algorithm}[]
\caption{$qi\mathcal{O}$-Gottesman-Chuang}
\label{QiO:gottesman-chuang}
\begin{itemize} \item Input: An $n$-qubit circuit $U_q\in U_{q^{n,k}},$ $k$ and $\delta={\tt negl}(n)).$
\begin{enumerate}
\item From $U_q$ compute the circuit $U_q^\dagger.$
\item Using Low's approximate learning algorithm on inputs $U_q$ and ${U_q}^\dagger$ compute the circuit $C_q $~\cite{Low09}.
\item Output the circuit $C_q.$ \end{enumerate}
\end{itemize} \end{algorithm}
\begin{proof} We have to show that \Cref{QiO:gottesman-chuang} satisfies \Cref{def:aQiO}. \begin{enumerate}
\item {\tt Functionality:} On input $U_q\in U_{q^{n,k}}$, let $C_q$ be the output of \Cref{QiO:gottesman-chuang}. By assumption (\Cref{th:approx}) there exists a unitary $C \in \mathcal{C}_k$ such that ${\bf D}(U_q, C)< {\tt negl}(n)<\frac{1}{2^{k+1/2}}.$ From \Cref{lem:approx}, $C$ is unique up to a global phase. Therefore with overwhelming probability, Low's approximate learning algorithm will output a circuit $C_q$ that computes~$C$. We have ${\bf D}(U_q, C_q)={\bf D}(U_q, C_q)< {\tt negl}(n),$ therefore $U_q$ and $C_q$ are approximately equivalent.
\item {\tt Polynomial Slowdown:} The total cost of \Cref{QiO:gottesman-chuang} is the cost of computing the circuit $U_q^\dagger$ from $U_q$ plus the cost of Low's learning algorithm. Clearly, we can compute $U_q^\dagger$ from $U_q^\dagger$ in polynomial-time. Using Low's algorithm (for parameters defined in \Cref{th:approx} and setting $\delta={\tt negl}(n)$) we can learn $C_q$ with probability at least $1-{\tt negl}(n)$ in time at most $$O\left(n^{k-1} \left[(k-1)\log((2n+1))-\log({\tt negl}(n))\right]\right).$$ Which is at most a polynomial in $n$ (since $k$ is a constant). Therefore, \Cref{QiO:gottesman-chuang} runs in polynomial-time.
\item {\tt Statistically Indistinguishability:} Let $U_q, U_q^\prime\in U_{q^{n,k}}$ be two circuits such that $${\bf D}(U_q, U_q^\prime)<{\tt negl}(n).$$ By assumption (\Cref{th:approx}), there exist unitaries $C,C^\prime\in \mathcal{C}_k$ such that ${\bf D}(U_q, C)<{\tt negl}(n)<\frac{1}{2^{k+1/2}}$ and ${\bf D}(U_q^\prime, C^\prime)<{\tt negl}(n)<\frac{1}{2^{k+1/2}}.$ Using the triangle inequality we can easily show that $C$ and $C^\prime$ are equivalent circuit (up to a global phase) for large $n.$
$${\bf D}(U_q, C^\prime)\leq {\bf D}(U_q, U_q^\prime)+{\bf D}(U_q^\prime, C^\prime)\leq {\tt negl}(n)+\frac{1}{2^{k+1/2}}$$ $$\Longrightarrow {\bf D}(U_q, C^\prime)<\frac{1}{2^{k+1/2}}+\frac{1}{2^{k+1/2}}<\frac{1}{2^{k-1/2}}$$
According \Cref{lem:approx} $C^\prime$ is unique (up to a global phase) such that ${\bf D}(U_q, C^\prime)<\frac{1}{2^{k-1/2}},$ But $C$ also satisfies ${\bf D}(U_q, C)<\frac{1}{2^{k-1/2}}.$ It follows that $C$ and $C^\prime$ are equivalent circuits (up to a global phase). Moreover, the outputs of Low's algorithm $C_q$ and $C_q^\prime$ on any two equivalent unitaries (up to global phase) is statistically indistinguishable~\cite{Low09}. Therefore, for any $U_q, U_q^\prime\in U_{q^{n,k}},$ the two distributions $qi\mathcal{O}_{\bf D}(U_q)$ and $qi\mathcal{O}_{\bf D}(U_q^\prime)$ are statistically indistinguishable. \qedhere \end{enumerate} \end{proof}
\section{Size of Coefficients in the Update Functions} \label{sec:appendix}
\label{coeff:size} Here, we prove a Lemma that is used in \Cref{sec:1-qubit}. \begin{lemma}\label{lem:beta:size} Let $C_q$ be an $n$-qubit $poly(n)$ size quantum circuit with $O(\log(n))$ number of gates and $F_{C_q}$ be the corresponding update function \begin{equation*} \begin{aligned} &F_{C_q}: \{0,1\}^{2n}\longrightarrow (\mathbb{C} \times \{0,1\}^{2n})^{min(k,n)},\\
&
(a_1,b_1,\ldots, a_n,b_n) \mapsto \left((\beta_1, {\bf s}_1),\dots, (\beta_{4^k}, {\bf s}_{4^k})\right).
\end{aligned}
\end{equation*}
then there exists a polynomial $p(\cdot)$ such $\beta_i \in O(p(n))$ for all $i\in[4^k].$ \end{lemma}
\begin{proof} The following map is an isomorphism between $\mathbb{C}$ and $\mathbb{R}^2$
\begin{equation}
\label{size:map:real-complex)}
f:\mathbb{C}\leftarrow \mathbb{R}^2, \; (a+b_i)\mapsto (a,b) \end{equation}
Note that there is a one-to-one map between $F_{C_q}$ and the corresponding unitary $U_{F_{C_q}}=\sum_{i=1}^{4^k} \beta_i {\sf X}^{b_{i_1}} {\sf Z}^{a_{i_1}}\otimes \cdots \otimes {\sf X}^{b{i_n}} {\sf Z}^{a_{i_n}},\; \; k\leq n.$ Note that any Clifford gate can only affect the correction bits in unitaries of type $U_{F_{C_q}},$ but will have no effect on the coefficients $\beta_i.$ So the coefficients can be affected by ${\sf T}$ gates. Therefore, to estimate the size of the coefficients we can ignore other gates. Each $\beta_i$ is constructed by adding and multiplying numbers from the set $\left\{0,1, \frac{1+i}{2},\frac{1-i}{2}\right\}.$ We note the following relationships between $\frac{1+i}{2},\frac{1-i}{2}$ \begin{equation*}
\left(\frac{1+i}{2}\right)\pm \left(\frac{1-i}{2}\right)=\pm1. \end{equation*} If $ m=2\ell+1$ and $\ell\in\mathbb{N},$ then
\begin{equation*}
\hspace{4cm}\left(\frac{1\pm i}{2}\right)^m=\left(\frac{a}{2}\right)^\ell,\; a\in\{\pm 1,\pm i\}. \end{equation*} Else if $m=2\ell$ and $\ell\in\mathbb{N}\cup \{0\},$ then \begin{equation*}
\left(\frac{1\pm i}{2}\right)^m=\left(\frac{a}{2}\right)^\ell \left(\frac{1\pm i}{2}\right),\; a\in\{\pm 1,\pm i\} \end{equation*}
Therefore, we can represent $\left(\frac{1\pm i}{2}\right)^m$ in $O(m)$ bits and $\left(\frac{1+ i}{2}\right)^{m} \left(\frac{1- i}{2}\right)^{m}$ in $O(m)$ bits. Of course $1^m=1$ and adding $1$ to itself is $m.$ For each application of ${\sf T}$, a coefficient will multiply and add at most polynomial time in the circuit size and there are $O(\log(n))$ such gates, therefore there exists a polynomial $p(\cdot)$ such that $|\beta_i|\in O(p(n)),$ for every $i\in[4^k].$ \end{proof}
\addcontentsline{toc}{section}{References}
\end{document} | arXiv |
pdgLive Home > New Heavy Bosons (${{\mathit W}^{\,'}}$, ${{\mathit Z}^{\,'}}$, leptoquarks, etc.), Searches for > Limits for other ${{\mathit Z}^{\,'}}$
Limits for other ${{\boldsymbol Z}^{\,'}}$ INSPIRE search
VALUE (GeV)
CL%
$\text{none 580 - 3100}$ 95 1
AABOUD
ATLS ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit t}}{{\overline{\mathit t}}}$
$\text{none 1300 - 3100}$ 95 2
ATLS ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit W}}{{\mathit W}}$
$> 3800$ 95 3
SIRUNYAN
CMS ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit t}}{{\overline{\mathit t}}}$
CMS ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit W}}{{\mathit W}}$ , ${{\mathit H}}{{\mathit Z}}$ , ${{\mathit \ell}^{+}}{{\mathit \ell}^{-}}$
CMS ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit H}}{{\mathit Z}}$
ATLS ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit b}}{{\overline{\mathit b}}}$
2018 AI
ATLS ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit H}}{{\mathit Z}}$
$\text{none 400 - 3000}$ 95 10
2018 BI
$\text{none 1200 - 2800}$ 95 11
$> 2300$ 95 12
2018 ED
CMS ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit W}}{{\mathit W}}$
$\bf{>2900}$ 95 14
ATLS ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit q}}{{\overline{\mathit q}}}$
2017 AO
$>2300$ 95 16
CMS ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit W}}{{\mathit W}}$ , ${{\mathit H}}{{\mathit Z}}$
$\bf{> 2500}$ 95 17
2019 AJ
ATLS DM simplified ${{\mathit Z}^{\,'}}$
ATLS ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit e}^{+}}{{\mathit e}^{-}}$ , ${{\mathit \mu}^{+}}{{\mathit \mu}^{-}}$
RVUE Electroweak
PANDEY
RVUE neutrino NSI
CMS ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit t}}{{\mathit T}}$ , ${{\mathit T}}$ $\rightarrow$ ${{\mathit H}}{{\mathit t}}$ , ${{\mathit Z}}{{\mathit t}}$ , ${{\mathit W}}{{\mathit b}}$
CMS DM simplified ${{\mathit Z}^{\,'}}$
2019 CB
CMS ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit q}}{{\overline{\mathit q}}}$
2019 CD
CMS ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit H}}{{\mathit \gamma}}$
ATLS ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit H}}{{\mathit \gamma}}$
ATLS ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit W}}{{\mathit W}}$ , ${{\mathit H}}{{\mathit Z}}$ , ${{\mathit \ell}^{+}}{{\mathit \ell}^{-}}$
AAIJ
2018 AQ
LHCB ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit \mu}^{+}}{{\mathit \mu}^{-}}$
2018 DR
CMS ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit \mu}^{+}}{{\mathit \mu}^{-}}$
CMS ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit b}}{{\overline{\mathit b}}}$
KHACHATRYAN
CMS ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit \ell}}{{\mathit \ell}}{{\mathit \ell}}{{\mathit \ell}}$
2017 U
CMS ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit H}}{{\mathit A}}$
CMS ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit T}}{{\mathit t}}$
ATLS ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit a}}{{\mathit \gamma}}$ , ${{\mathit a}}$ $\rightarrow$ ${{\mathit \gamma}}{{\mathit \gamma}}$
$\text{none 1000 - 1100, none 1300 - 1500}$ 95 47
ATLS monotop
ATLS ${{\mathit H}}$ $\rightarrow$ ${{\mathit Z}}{{\mathit Z}^{\,'}}$ , ${{\mathit Z}^{\,'}}{{\mathit Z}^{\,'}}$ ; ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit \ell}^{+}}{{\mathit \ell}^{-}}$
CMS monotop
ATLS ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit Z}}{{\mathit \gamma}}$
CMS ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit V}}{{\mathit V}}$
$\text{>1320 or 1000 - 1280}$ 95 58
$>915$ 95 58
CDF ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit t}}{{\overline{\mathit t}}}$
CHATRCHYAN
2013 BM
2012 BV
2012 AR
CDF Chromophilic
CDF ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\overline{\mathit t}}}{{\mathit u}}$
$> 835$ 95 64
ABAZOV
D0 ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit t}}{{\overline{\mathit t}}}$
CMS ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit t}}{{\overline{\mathit u}}}$
2011 AE
CMS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit t}}{{\mathit t}}$
D0 Repl. by ABAZOV 2008AA
COSM Nucleosynthesis; light ${{\mathit \nu}_{{R}}}$
RVUE $\mathit E_{6}$-motivated
CDF ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\overline{\mathit q}}}{{\mathit q}}$
1 AABOUD 2019AS search for a resonance decaying to ${{\mathit t}}{{\overline{\mathit t}}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV. The quoted limit is for a top-color ${{\mathit Z}^{\,'}}$ with $\Gamma _{{{\mathit Z}^{\,'}}}/\mathit M_{{{\mathit Z}^{\,'}}}$ = 0.01. Limits are also set on ${{\mathit Z}^{\,'}}$ masses in simplified Dark Matter models.
2 AAD 2019D search for resonances decaying to ${{\mathit W}}{{\mathit W}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV. The quoted limit is for heavy-vector-triplet ${{\mathit Z}^{\,'}}$ with ${{\mathit g}_{{V}}}$ = 3. The limit becomes $\mathit M_{{{\mathit Z}^{\,'}}}$ $>$ 2900 GeV for ${{\mathit g}_{{V}}}$ = 1. If we assume $\mathit M_{{{\mathit Z}^{\,'}}}$ = $\mathit M_{{{\mathit W}^{\,'}}}$, the limit increases $\mathit M_{{{\mathit Z}^{\,'}}}$ $>$ 3800 GeV and $\mathit M_{{{\mathit Z}^{\,'}}}$ $>$ 3500 GeV for ${{\mathit g}_{{V}}}$ = 3 and ${{\mathit g}_{{V}}}$ = 1, respectively. See their Fig. 9 for limits on $\sigma \cdot{}B$.
3 SIRUNYAN 2019AA search for a resonance decaying to ${{\mathit t}}{{\overline{\mathit t}}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV. The quoted limit is for a leptophobic top-color ${{\mathit Z}^{\,'}}$ with $\Gamma _{{{\mathit Z}^{\,'}}}/\mathit M_{{{\mathit Z}^{\,'}}}$ = 0.01.
4 SIRUNYAN 2019CP present a statistical combinations of searches for ${{\mathit Z}^{\,'}}$ decaying to pairs of bosons or leptons in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV. The quoted limit is for heavy-vector-triplet ${{\mathit Z}^{\,'}}$ with ${{\mathit g}_{{V}}}$ = 3. If we assume ${{\mathit M}}_{{{\mathit Z}^{\,'}}}$ = ${{\mathit M}}_{{{\mathit W}^{\,'}}}$, the limit becomes ${{\mathit M}}_{{{\mathit Z}^{\,'}}}$ $>$ 4500 GeV for ${{\mathit g}_{{V}}}$ = 3 and ${{\mathit M}}_{{{\mathit Z}^{\,'}}}$ $>$ 5000 GeV for ${{\mathit g}_{{V}}}$ = 1. See their Figs. 2 and 3 for limits on $\sigma \cdot{}B$.
5 SIRUNYAN 2019I search for resonances decaying to ${{\mathit Z}}{{\mathit W}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV. The quoted limit is for heavy-vector-triplet ${{\mathit Z}^{\,'}}$ with ${{\mathit g}_{{V}}}$ = 3. The limit becomes $\mathit M_{{{\mathit Z}^{\,'}}}$ $>$ 2800 GeV if we assume $\mathit M_{{{\mathit Z}^{\,'}}}$ = $\mathit M_{{{\mathit W}^{\,'}}}$.
6 AABOUD 2018AB search for resonances decaying to ${{\mathit b}}{{\overline{\mathit b}}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV. The limit quoted above is for a leptophobic ${{\mathit Z}^{\,'}}$ with SM-like couplings to quarks. See their Fig. 6 for limits on $\sigma \cdot{}$B.
7 AABOUD 2018AI search for resonances decaying to ${{\mathit H}}{{\mathit Z}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV. The quoted limit is for heavy-vector-triplet ${{\mathit Z}^{\,'}}$ with ${{\mathit g}_{{V}}}$ = 3. The limit becomes $\mathit M_{{{\mathit Z}^{\,'}}}$ $>$ 2650 GeV for ${{\mathit g}_{{V}}}$ = 1. If we assume $\mathit M_{{{\mathit W}^{\,'}}}$ = $\mathit M_{{{\mathit Z}^{\,'}}}$, the limit increases $\mathit M_{{{\mathit Z}^{\,'}}}$ $>$ 2930 GeV and $\mathit M_{{{\mathit Z}^{\,'}}}$ $>$ 2800 GeV for ${{\mathit g}_{{V}}}$ = 3 and ${{\mathit g}_{{V}}}$ = 1, respectively. See their Fig. 5 for limits on $\sigma \cdot{}\mathit B$.
8 AABOUD 2018AK search for resonances decaying to ${{\mathit W}}{{\mathit W}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ =1 3 TeV. The limit quoted above is for heavy-vector-triplet ${{\mathit Z}^{\,'}}$ with ${{\mathit g}_{{V}}}$ = 3. The limit becomes $\mathit M_{{{\mathit Z}^{\,'}}}$ $>$ 2750 GeV for ${{\mathit g}_{{V}}}$ = 1.
9 AABOUD 2018B search for resonances decaying to ${{\mathit W}}{{\mathit W}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV. The quoted limit is for heavy-vector-triplet ${{\mathit Z}^{\,'}}$ with ${{\mathit g}_{{V}}}$ = 1. See their Fig.11 for limits on $\sigma \cdot{}{{\mathit B}}$.
10 AABOUD 2018BI search for a resonance decaying to ${{\mathit t}}{{\overline{\mathit t}}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV. The quoted limit is for a top-color assisted TC ${{\mathit Z}^{\,'}}$ with $\Gamma _{{{\mathit Z}^{\,'}}}/\mathit M_{{{\mathit Z}^{\,'}}}$ = 0.01. The limits for wider resonances are available. See their Fig. 14 for limits on $\sigma \cdot{}{{\mathit B}}$.
11 AABOUD 2018F search for resonances decaying to ${{\mathit W}}{{\mathit W}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV. The quoted limit is for heavy-vector-triplet ${{\mathit Z}^{\,'}}$ with ${{\mathit g}_{{V}}}$ = 3. The limit becomes $\mathit M_{{{\mathit Z}^{\,'}}}$ $>$ 2200 GeV for ${{\mathit g}_{{V}}}$ = 1. If we assume $\mathit M_{{{\mathit Z}^{\,'}}}$ = $\mathit M_{{{\mathit W}^{\,'}}}$, the limit increases $\mathit M_{{{\mathit Z}^{\,'}}}$ $>$ 3500 GeV and $\mathit M_{{{\mathit Z}^{\,'}}}$ $>$ 3100 GeV for ${{\mathit g}_{{V}}}$ = 3 and ${{\mathit g}_{{V}}}$ = 1, respectively. See their Fig.5 for limits on $\sigma \cdot{}{{\mathit B}}$.
12 SIRUNYAN 2018ED search for resonances decaying to ${{\mathit H}}{{\mathit Z}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV. The limit above is for heavy-vector-triplet ${{\mathit Z}^{\,'}}$ with ${{\mathit g}_{{V}}}$ = 3. If we assume $\mathit M_{{{\mathit Z}^{\,'}}}$ = $\mathit M_{{{\mathit W}^{\,'}}}$, the limit increases $\mathit M_{{{\mathit Z}^{\,'}}}$ $>$ 2900 GeV and $\mathit M_{{{\mathit Z}^{\,'}}}$ $>$ 2800 GeV for ${{\mathit g}_{{V}}}$ = 3 and ${{\mathit g}_{{V}}}$ = 1, respectively.
13 SIRUNYAN 2018P give this limit for a heavy-vector-triplet ${{\mathit Z}^{\,'}}$ with ${{\mathit g}_{{V}}}$ = 3. If they assume $\mathit M_{{{\mathit Z}^{\,'}}}$ = $\mathit M_{{{\mathit W}^{\,'}}}$, the limit increases to $\mathit M_{{{\mathit Z}^{\,'}}}$ $>$ 3800 GeV.
14 AABOUD 2017AK search for a new resonance decaying to dijets in $pp$ collisions at $\sqrt {s }$ = 13 TeV. The limit quoted above is for a leptophobic ${{\mathit Z}^{\,'}}$ boson having axial-vector coupling strength with quarks ${{\mathit g}_{{q}}}$ = 0.2. The limit is 2100 GeV if ${{\mathit g}_{{q}}}$ = 0.1.
15 AABOUD 2017AO search for resonances decaying to ${{\mathit H}}{{\mathit Z}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV. The limit quoted above is for a ${{\mathit Z}^{\,'}}$ in the heavy-vector-triplet model with ${{\mathit g}_{{V}}}$ = 3. See their Fig.4 for limits on $\sigma \cdot{}{{\mathit B}}$.
16 SIRUNYAN 2017AK search for resonances decaying to ${{\mathit W}}{{\mathit W}}$ or ${{\mathit H}}{{\mathit Z}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 and 13 TeV. The quoted limit is for heavy-vector-triplet ${{\mathit Z}^{\,'}}$ with ${{\mathit g}_{{V}}}$ = 3. The limit becomes $\mathit M_{{{\mathit Z}^{\,'}}}$ $>$ 2200 GeV for ${{\mathit g}_{{V}}}$ =1. If we assume $\mathit M_{{{\mathit Z}^{\,'}}}$ = $\mathit M_{{{\mathit W}^{\,'}}}$, the limit increases $\mathit M_{{{\mathit Z}^{\,'}}}$ $>$ 2400 GeV for both ${{\mathit g}_{{V}}}$ = 3 and ${{\mathit g}_{{V}}}$ = 1. See their Fig.1 and 2 for limits on ${{\mathit \sigma}}\cdot{}{{\mathit B}}$.
17 SIRUNYAN 2017Q search for a resonance decaying to ${{\mathit t}}{{\overline{\mathit t}}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV. The limit quoted above is for a resonance with relative width $\Gamma _{{{\mathit Z}^{\,'}}}$ $/$ $\mathit M_{{{\mathit Z}^{\,'}}}$ = 0.01. Limits for wider resonances are available. See their Fig.6 for limits on $\sigma \cdot{}\mathit B$.
18 SIRUNYAN 2017R search for resonances decaying to ${{\mathit H}}{{\mathit Z}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV. The quoted limit is for heavy-vector-triplet ${{\mathit Z}^{\,'}}$ with ${{\mathit g}_{{V}}}$ = 3. Mass regions $\mathit M_{{{\mathit Z}^{\,'}}}$ $<$ 1150 GeV and 1250 GeV $<$ $\mathit M_{{{\mathit Z}^{\,'}}}$ $<$ 1670 GeV are excluded for ${{\mathit g}_{{V}}}$ = 1. If we assume $\mathit M_{{{\mathit Z}^{\,'}}}$ = $\mathit M_{{{\mathit W}^{\,'}}}$, the excluded mass regions are 1000 $<$ $\mathit M_{{{\mathit Z}^{\,'}}}$ $<$ 2500 GeV and 2760 $<$ $\mathit M_{{{\mathit Z}^{\,'}}}$ $<$ 3300 GeV for ${{\mathit g}_{{V}}}$ = 3; 1000 $<$ $\mathit M_{{{\mathit Z}^{\,'}}}$ $<$ 2430 GeV and 2810 $<$ $\mathit M_{{{\mathit Z}^{\,'}}}$ $<$ 3130 GeV for ${{\mathit g}_{{V}}}$ = 1. See their Fig.5 for limits on ${{\mathit \sigma}}\cdot{}{{\mathit B}}$.
19 AABOUD 2019AJ search in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for a new resonance decaying to ${{\mathit q}}{{\overline{\mathit q}}}$ and produced in association with a high $p_T$ photon. For a leptophobic axial-vector ${{\mathit Z}^{\,'}}$ in the mass region 250 GeV $<$ $\mathit M_{{{\mathit Z}^{\,'}}}$ $<$ 950 GeV, the ${{\mathit Z}^{\,'}}$ coupling with quarks ${{\mathit g}_{{q}}}$ is constrained below 0.18. See their Fig.2 for limits in $\mathit M_{{{\mathit Z}^{\,'}}}−{{\mathit g}_{{q}}}$ plane.
20 AABOUD 2019D search in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for a new resonance decaying to ${{\mathit q}}{{\overline{\mathit q}}}$ and produced in association with a high-$p_T$ photon or jet. For a leptophobic axial-vector ${{\mathit Z}^{\,'}}$ in the mass region 100 GeV $<$ $\mathit M_{{{\mathit Z}^{\,'}}}$ $<$ 220 GeV, the ${{\mathit Z}^{\,'}}$ coupling with quarks ${{\mathit g}_{{q}}}$ is constrained below 0.23. See their Fig. 6 for limits in $\mathit M_{{{\mathit Z}^{\,'}}}−{{\mathit g}_{{q}}}$ plane.
21 AABOUD 2019V search for Dark Matter simplified ${{\mathit Z}^{\,'}}$ decaying invisibly or decaying to fermion pair in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV.
22 AAD 2019L search for resonances decaying to ${{\mathit \ell}^{+}}{{\mathit \ell}^{-}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV. See their Fig. 4 for limits in the heavy vector triplet model couplings.
23 LONG 2019 uses the weak charge data of Cesium and proton to constrain mass of ${{\mathit Z}^{\,'}}$ in the 3-3-1 models.
24 PANDEY 2019 obtain limits on ${{\mathit Z}^{\,'}}$ induced neutrino non-standard interaction (NSI) parameter $\epsilon $ from LHC and IceCube data. See their Fig.2 for limits in ${{\mathit M}}_{{{\mathit Z}^{\,'}}}−\epsilon $ plane, where $\epsilon $ = ${{\mathit g}_{{q}}}{{\mathit g}}_{{{\mathit \nu}}}$ v${}^{2}$ $/$ (2 ${{\mathit M}}{}^{2}_{{{\mathit Z}^{\,'}}}$).
25 SIRUNYAN 2019AL search for a new resonance decaying to a top quark and a heavy vector-like top partner in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV. See their Fig. 8 for limits on ${{\mathit Z}^{\,'}}$ production cross section.
26 SIRUNYAN 2019AN search for a Dark Matter (DM) simplified model ${{\mathit Z}^{\,'}}$ decaying to ${{\mathit H}}$ DM DM in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV. See their Fig. 7 for limits on the signal strength modifiers.
27 SIRUNYAN 2019CB search in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for a new resonance decaying to ${{\mathit q}}{{\overline{\mathit q}}}$ . For a leptophobic ${{\mathit Z}^{\,'}}$ in the mass region $50 - 300$ GeV, the ${{\mathit Z}^{\,'}}$ coupling with quarks ${{\mathit g}_{{q}}^{\,'}}$ is constrained below 0.2. See their Figs. 4 and 5 for limits on ${{\mathit g}_{{q}}^{\,'}}$ in the mass range 50 $<$ ${{\mathit M}}_{{{\mathit Z}^{\,'}}}$ $<$ 450 GeV.
28 SIRUNYAN 2019CD search in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$=13 TeV for a leptophobic ${{\mathit Z}^{\,'}}$ produced in association of high $p_T$ ISR photon and decaying to ${{\mathit q}}{{\overline{\mathit q}}}$ . See their Fig. 2 for limits on the ${{\mathit Z}^{\,'}}$ coupling strength ${{\mathit g}_{{q}}^{\,'}}$ to ${{\mathit q}}{{\overline{\mathit q}}}$ in the mass range between 10 and 125 GeV.
29 SIRUNYAN 2019D search for a narrow neutral vector resonance decaying to ${{\mathit H}}{{\mathit \gamma}}$ . See their Fig. 3 for exclusion limit in $\mathit M_{{{\mathit Z}^{\,'}}}−\sigma \cdot{}\mathit B$ plane. Upper limits on the production of ${{\mathit H}}{{\mathit \gamma}}$ resonances are set as a function of the resonance mass in the range of $720 - 3250$ GeV.
30 AABOUD 2018AA search for a narrow neutral vector boson decaying to ${{\mathit H}}{{\mathit \gamma}}$ . See their Fig. 10 for the exclusion limit in M$_{{{\mathit Z}^{\,'}}}$ $−$ $\sigma $B plane.
31 AABOUD 2018CJ search for heavy-vector-triplet $Z'$ in $pp$ collisions at $\sqrt{s}=13$ TeV. The limit quoted above is for model with $g_V=3$ assuming $M_{Z'}=M_{W'}$. The limit becomes $M_{Z'}>5500$ GeV for model with $g_V=1$.
32 AABOUD 2018N search for a narrow resonance decaying to ${{\mathit q}}{{\overline{\mathit q}}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV using trigger level analysis to improve the low mass region sensitivity. See their Fig. 5 for limits in the mass-coupling plane in the ${{\mathit Z}^{\,'}}$ mass range $450 - 1800$ GeV.
33 AAIJ 2018AQ search for spin-0 and spin-1 resonances decaying to ${{\mathit \mu}^{+}}{{\mathit \mu}^{-}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 7 and 8 TeV in the mass region near 10 GeV. See their Figs. 4 and 5 for limits on $\sigma \cdot{}\mathit B$.
34 SIRUNYAN 2018DR searches for ${{\mathit \mu}^{+}}{{\mathit \mu}^{-}}$ resonances produced in association with ${{\mathit b}}$-jets in the ${{\mathit p}}{{\mathit p}}$ collision data with $\sqrt {s }$ = 8 TeV and 13 TeV. An excess of events near ${\mathit m}_{\mathrm { {{\mathit \mu}} {{\mathit \mu}} }}$ = 28 GeV is observed in the 8 TeV data. See their Fig. 3 for the measured fiducial signal cross sections at $\sqrt {s }$ = 8 TeV and the 95$\%$ CL upper limits at $\sqrt {s }$ = 13 TeV.
35 SIRUNYAN 2018G search for a new resonance decaying to dijets in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV in the mass range $50 - 300$ GeV. See their Fig.7 for limits in the mass-coupling plane.
36 SIRUNYAN 2018I search for a narrow resonance decaying to ${{\mathit b}}{{\overline{\mathit b}}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV using dedicated b-tagged dijet triggers to improve the sensitivity in the low mass region. See their Fig. 3 for limits on $\sigma \cdot{}{{\mathit B}}$ in the ${{\mathit Z}^{\,'}}$ mass range $325 - 1200$ GeV.
37 AABOUD 2017B search for resonances decaying to ${{\mathit H}}{{\mathit Z}}$ ( ${{\mathit H}}$ $\rightarrow$ ${{\mathit b}}{{\overline{\mathit b}}}$ , ${{\mathit c}}{{\overline{\mathit c}}}$ ; ${{\mathit Z}}$ $\rightarrow$ ${{\mathit \ell}^{+}}{{\mathit \ell}^{-}}$ , ${{\mathit \nu}}{{\overline{\mathit \nu}}}$ ) in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV. The quoted limit is for heavy-vector-triplet ${{\mathit Z}^{\,'}}$ with ${{\mathit g}_{{V}}}$ = 3. The limit becomes ${{\mathit M}}_{{{\mathit Z}^{\,'}}}>$ 1490 GeV for ${{\mathit g}_{{V}}}$ = 1. If we assume ${{\mathit M}}_{{{\mathit Z}^{\,'}}}$ = ${{\mathit M}}_{{{\mathit W}^{\,'}}}$, the limit increases ${{\mathit M}}_{{{\mathit Z}^{\,'}}}>$ 2310 GeV and ${{\mathit M}}_{{{\mathit Z}^{\,'}}}>$ 1730 GeV for ${{\mathit g}_{{V}}}$ = 3 and ${{\mathit g}_{{V}}}$ = 1, respectively. See their Fig.3 for limits on ${{\mathit \sigma}}\cdot{}{{\mathit B}}$.
38 KHACHATRYAN 2017AX search for lepto-phobic resonances decaying to four leptons in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV.
39 KHACHATRYAN 2017U search for resonances decaying to ${{\mathit H}}{{\mathit Z}}$ ( ${{\mathit H}}$ $\rightarrow$ ${{\mathit b}}{{\overline{\mathit b}}}$ ; ${{\mathit Z}}$ $\rightarrow$ ${{\mathit \ell}^{+}}{{\mathit \ell}^{-}}$ , ${{\mathit \nu}}{{\overline{\mathit \nu}}}$ ) in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV. The limit on the heavy-vector-triplet model is $\mathit M_{{{\mathit Z}^{\,'}}}$ = $\mathit M_{{{\mathit W}^{\,'}}}$ $>$ 2 TeV for ${{\mathit g}_{{V}}}$ = 3, in which constraints from the ${{\mathit W}^{\,'}}$ $\rightarrow$ ${{\mathit H}}{{\mathit W}}$ ( ${{\mathit H}}$ $\rightarrow$ ${{\mathit b}}{{\overline{\mathit b}}}$ ; ${{\mathit W}}$ $\rightarrow$ ${{\mathit \ell}}{{\mathit \nu}}$ ) are combined. See their Fig.3 and Fig.4 for limits on $\sigma \cdot{}\mathit B$.
40 SIRUNYAN 2017A search for resonances decaying to ${{\mathit W}}{{\mathit W}}$ with ${{\mathit W}}$ ${{\mathit W}}$ $\rightarrow$ ${{\mathit \ell}}{{\mathit \nu}}{{\mathit q}}{{\overline{\mathit q}}}$ , ${{\mathit q}}{{\overline{\mathit q}}}{{\mathit q}}{{\overline{\mathit q}}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV. The quoted limit is for heavy-vector-triplet ${{\mathit Z}^{\,'}}$ with ${{\mathit g}_{{V}}}$ = 3. The limit becomes $\mathit M_{{{\mathit Z}^{\,'}}}$ $>$ 1600 GeV for ${{\mathit g}_{{V}}}$ = 1. If we assume $\mathit M_{{{\mathit Z}^{\,'}}}$ = $\mathit M_{{{\mathit W}^{\,'}}}$, the limit increases $\mathit M_{{{\mathit Z}^{\,'}}}$ $>$ 2400 GeV and $\mathit M_{{{\mathit Z}^{\,'}}}$ $>$ 2300 GeV for ${{\mathit g}_{{V}}}$ = 3 and ${{\mathit g}_{{V}}}$ = 1, respectively. See their Fig.6 for limits on $\sigma \cdot{}\mathit B$.
41 SIRUNYAN 2017AP search for resonances decaying into a SM-like Higgs scalar ${{\mathit H}}$ and a light pseudo scalar ${{\mathit A}}$. ${{\mathit A}}$ is assumed to decay invisibly. See their Fig.9 for limits on ${{\mathit \sigma}}\cdot{}{{\mathit B}}$.
42 SIRUNYAN 2017T search for a new resonance decaying to dijets in $pp$ collisions at $\sqrt {s }$ = 13 TeV in the mass range $100 - 300$ GeV. See their Fig.3 for limits in the mass-coupling plane.
43 SIRUNYAN 2017V search for a new resonance decaying to a top quark and a heavy vector-like top partner ${{\mathit T}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV. See their table 5 for limits on the ${{\mathit Z}^{\,'}}$ production cross section for various values of $\mathit M_{{{\mathit Z}^{\,'}}}$ and $\mathit M_{T}$ in the range of $\mathit M_{{{\mathit Z}^{\,'}}}$ = $1500 - 2500$ GeV and $\mathit M_{T}$ = $700 - 1500$ GeV.
44 AABOUD 2016 search for a narrow resonance decaying into ${{\mathit b}}{{\overline{\mathit b}}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV. The limit quoted above is for a leptophobic ${{\mathit Z}^{\,'}}$ with SM-like couplings to quarks. See their Fig.6 for limits on $\sigma \cdot{}\mathit B$.
45 AAD 2016L search for ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\mathit a}}{{\mathit \gamma}}$ , ${{\mathit a}}$ $\rightarrow$ ${{\mathit \gamma}}{{\mathit \gamma}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV. See their Table 6 for limits on $\sigma \cdot{}\mathit B$.
46 AAD 2016S search for a new resonance decaying to dijets in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV. The limit quoted above is for a leptophobic ${{\mathit Z}^{\,'}}$ having coupling strength with quark ${{\mathit g}_{{q}}}$ = 0.3 and is taken from their Figure 3.
47 KHACHATRYAN 2016AP search for a resonance decaying to ${{\mathit H}}{{\mathit Z}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV. Both ${{\mathit H}}$ and ${{\mathit Z}}$ are assumed to decay to fat jets. The quoted limit is for heavy-vector-triplet ${{\mathit Z}^{\,'}}$ with ${{\mathit g}_{{V}}}$ = 3.
48 KHACHATRYAN 2016E search for a leptophobic top-color ${{\mathit Z}^{\,'}}$ decaying to ${{\mathit t}}{{\overline{\mathit t}}}$ using ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV. The quoted limit assumes that ${\Gamma}_{{\mathit Z}^{\,'}}/{\mathit m}_{{{\mathit Z}^{\,'}}}$ = 0.012. Also ${\mathit m}_{{{\mathit Z}^{\,'}}}$ $<$ 2.9 TeV is excluded for wider topcolor ${{\mathit Z}^{\,'}}$ with ${\Gamma}_{{\mathit Z}^{\,'}}/{\mathit m}_{{{\mathit Z}^{\,'}}}$ = 0.1.
49 AAD 2015AO search for narrow resonance decaying to ${{\mathit t}}{{\overline{\mathit t}}}$ using ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV. See Fig. 11 for limit on $\sigma \mathit B$.
50 AAD 2015AT search for monotop production plus large missing $\mathit E_{T}$ events in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV and give constraints on a ${{\mathit Z}^{\,'}}$ model having ${{\mathit Z}^{\,'}}{{\mathit u}}{{\overline{\mathit t}}}$ coupling. ${{\mathit Z}^{\,'}}$ is assumed to decay invisibly. See their Fig. 6 for limits on $\sigma \cdot{}\mathit B$.
51 AAD 2015CD search for decays of Higgs bosons to 4 ${{\mathit \ell}}$ states via ${{\mathit Z}^{\,'}}$ bosons, ${{\mathit H}}$ $\rightarrow$ ${{\mathit Z}}{{\mathit Z}^{\,'}}$ $\rightarrow$ 4 ${{\mathit \ell}}$ or ${{\mathit H}}$ $\rightarrow$ ${{\mathit Z}^{\,'}}{{\mathit Z}^{\,'}}$ $\rightarrow$ 4 ${{\mathit \ell}}$ . See Fig. 5 for the limit on the signal strength of the ${{\mathit H}}$ $\rightarrow$ ${{\mathit Z}}{{\mathit Z}^{\,'}}$ $\rightarrow$ 4 ${{\mathit \ell}}$ process and Fig. 16 for the limit on ${{\mathit H}}$ $\rightarrow$ ${{\mathit Z}^{\,'}}{{\mathit Z}^{\,'}}$ $\rightarrow$ 4 ${{\mathit \ell}}$ .
52 KHACHATRYAN 2015F search for monotop production plus large missing $\mathit E_{T}$ events in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV and give constraints on a ${{\mathit Z}^{\,'}}$ model having ${{\mathit Z}^{\,'}}{{\mathit u}}{{\overline{\mathit t}}}$ coupling. ${{\mathit Z}^{\,'}}$ is assumed to decay invisibly. See Fig. 3 for limits on $\sigma \mathit B$.
53 KHACHATRYAN 2015O search for narrow ${{\mathit Z}^{\,'}}$ resonance decaying to ${{\mathit Z}}{{\mathit H}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV. See their Fig. 6 for limit on $\sigma \mathit B$.
54 AAD 2014AT search for a narrow neutral vector boson decaying to ${{\mathit Z}}{{\mathit \gamma}}$ . See their Fig. 3b for the exclusion limit in ${\mathit m}_{{{\mathit Z}^{\,'}}}−\sigma \mathit B$ plane.
55 KHACHATRYAN 2014A search for new resonance in the ${{\mathit W}}{{\mathit W}}$ ( ${{\mathit \ell}}{{\mathit \nu}}{{\mathit q}}{{\overline{\mathit q}}}$ ) and the ${{\mathit Z}}{{\mathit Z}}$ ( ${{\mathit \ell}}{{\mathit \ell}}{{\mathit q}}{{\overline{\mathit q}}}$ ) channels using ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$=8 TeV. See their Fig.13 for the exclusion limit on the number of events in the mass-width plane.
56 MARTINEZ 2014 use various electroweak data to constrain the ${{\mathit Z}^{\,'}}$ boson in the 3-3-1 models.
57 AAD 2013AQ search for a leptophobic top-color ${{\mathit Z}^{\,'}}$ decaying to ${{\mathit t}}{{\overline{\mathit t}}}$ . The quoted limit assumes that ${\Gamma}_{{\mathit Z}^{\,'}}/{\mathit m}_{{{\mathit Z}^{\,'}}}$ = 0.012.
58 CHATRCHYAN 2013BM search for top-color ${{\mathit Z}^{\,'}}$ decaying to ${{\mathit t}}{{\overline{\mathit t}}}$ using ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$=8 TeV. The quoted limit is for ${\Gamma}_{{\mathit Z}^{\,'}}/{\mathit m}_{{{\mathit Z}^{\,'}}}$ = 0.012.
59 CHATRCHYAN 2013AP search for top-color leptophobic ${{\mathit Z}^{\,'}}$ decaying to ${{\mathit t}}{{\overline{\mathit t}}}$ using ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$=7 TeV. The quoted limit is for ${\Gamma}_{{\mathit Z}^{\,'}}/{\mathit m}_{{{\mathit Z}^{\,'}}}$ = 0.012.
60 AAD 2012BV search for narrow resonance decaying to ${{\mathit t}}{{\overline{\mathit t}}}$ using ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$=7 TeV. See their Fig. 7 for limit on $\sigma \cdot{}$B.
61 AAD 2012K search for narrow resonance decaying to ${{\mathit t}}{{\overline{\mathit t}}}$ using ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$=7 TeV. See their Fig. 5 for limit on $\sigma \cdot{}$B.
62 AALTONEN 2012AR search for chromophilic ${{\mathit Z}^{\,'}}$ in ${{\mathit p}}{{\overline{\mathit p}}}$ collisions at $\sqrt {s }$ = 1.96 TeV. See their Fig. 5 for limit on $\sigma \cdot{}$B.
63 AALTONEN 2012N search for ${{\mathit p}}$ ${{\overline{\mathit p}}}$ $\rightarrow$ ${{\mathit t}}{{\mathit Z}^{\,'}}$ , ${{\mathit Z}^{\,'}}$ $\rightarrow$ ${{\overline{\mathit t}}}{{\mathit u}}$ events in ${{\mathit p}}{{\overline{\mathit p}}}$ collisions. See their Fig. 3 for the limit on $\sigma \cdot{}$B.
64 ABAZOV 2012R search for top-color ${{\mathit Z}^{\,'}}$ boson decaying exclusively to ${{\mathit t}}{{\overline{\mathit t}}}$ . The quoted limit is for ${\Gamma}_{{\mathit Z}^{\,'}}/{\mathit m}_{{{\mathit Z}^{\,'}}}$= 0.012.
65 CHATRCHYAN 2012AI search for ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit t}}{{\mathit t}}$ events and give constraints on a ${{\mathit Z}^{\,'}}$ model having ${{\mathit Z}^{\,'}}{{\overline{\mathit u}}}{{\mathit t}}$ coupling. See their Fig. 4 for the limit in mass-coupling plane.
66 Search for resonance decaying to ${{\mathit t}}{{\overline{\mathit t}}}$ . See their Fig. 6 for limit on $\sigma \cdot{}$B.
67 Search for narrow resonance decaying to ${{\mathit t}}{{\overline{\mathit t}}}$ . See their Fig. 4 for limit on $\sigma \cdot{}$B.
69 CHATRCHYAN 2011O search for same-sign top production in ${{\mathit p}}{{\mathit p}}$ collisions induced by a hypothetical FCNC ${{\mathit Z}^{\,'}}$ at $\sqrt {s }$ = 7 TeV. See their Fig. 3 for limit in mass-coupling plane.
70 Search for narrow resonance decaying to ${{\mathit t}}{{\overline{\mathit t}}}$ . See their Fig.$~$3 for limit on $\sigma \cdot{}$B.
72 BARGER 2003B use the nucleosynthesis bound on the effective number of light neutrino $\delta \mathit N_{{{\mathit \nu}}}$. See their Figs.$~4 - 5$ for limits in general $\mathit E_{6}$ motivated models.
73 CHO 2000 use various electroweak data to constrain ${{\mathit Z}^{\,'}}$ models assuming ${\mathit m}_{{{\mathit H}}}$=100 GeV. See Fig.$~$2 for limits in general $\mathit E_{6}$-motivated models.
74 CHO 1998 study constraints on four-Fermi contact interactions obtained from low-energy electroweak experiments, assuming no ${{\mathit Z}}-{{\mathit Z}^{\,'}}$ mixing.
75 Search for ${{\mathit Z}^{\,'}}$ decaying to dijets at $\sqrt {\mathit s }=1.8$ TeV. For ${{\mathit Z}^{\,'}}$ with electromagnetic strength coupling, no bound is obtained.
AABOUD 2019D
PL B788 316 Search for light resonances decaying to boosted quark pairs and produced in association with a photon or a jet in proton-proton collisions at $\sqrt{s}=13$ TeV with the ATLAS detector
AABOUD 2019V
JHEP 1905 142 Constraints on mediator-based dark matter and scalar dark energy models using $\sqrt s = 13$ TeV $pp$ collision data collected by the ATLAS detector
AABOUD 2019AS
PR D99 092004 Search for heavy particles decaying into a top-quark pair in the fully hadronic final state in $pp$ collisions at $\sqrt{s} =$ 13 TeV with the ATLAS detector
AABOUD 2019AJ
PL B795 56 Search for low-mass resonances decaying into two jets and produced in association with a photon using $pp$ collisions at $\sqrt{s} = 13$ TeV with the ATLAS detector
AAD 2019L
PL B796 68 Search for high-mass dilepton resonances using 139 fb$^{-1}$ of $pp$ collision data collected at $\\sqrt{s}=$13 TeV with the ATLAS detector
AAD 2019D
JHEP 1909 091 Search for diboson resonances in hadronic final states in 139 fb$^{-1}$ of $pp$ collisions at $\sqrt{s} = 13$ TeV with the ATLAS detector
LONG 2019
NP B943 114629 Constraining heavy neutral gauge boson $Z'$ in the 3 - 3 - 1 models by weak charge data of Cesium and proton
PANDEY 2019
JHEP 1911 046 Strong constraints on non-standard neutrino interactions: LHC vs. IceCube
SIRUNYAN 2019AN
EPJ C79 280 Search for dark matter produced in association with a Higgs boson decaying to a pair of bottom quarks in proton?proton collisions at $\sqrt{s}=13\,\text {Te}\text {V} $
SIRUNYAN 2019AA
JHEP 1904 031 Search for resonant $ \mathrm{t}\overline{\mathrm{t}} $ production in proton-proton collisions at $ \sqrt{s}=13 $ TeV
SIRUNYAN 2019D
PRL 122 081804 Search for narrow H$\gamma$ resonances in proton-proton collisions at $\sqrt{s} =$ 13 TeV
SIRUNYAN 2019CD
PRL 123 231803 Search for low-mass quark-antiquark resonances produced in association with a photon at $\sqrt{s} = $ 13 TeV
SIRUNYAN 2019CP
PL B798 134952 Combination of CMS searches for heavy resonances decaying to pairs of bosons or leptons
SIRUNYAN 2019AL
EPJ C79 208 Search for a heavy resonance decaying to a top quark and a vector-like top quark in the lepton+jets final state in pp collisions at $\sqrt{s} =$ 13 TeV
SIRUNYAN 2019CB
PR D100 112007 Search for low mass vector resonances decaying into quark-antiquark pairs in proton-proton collisions at $\sqrt{s}=$ 13 TeV
SIRUNYAN 2019I
JHEP 1901 051 Search for heavy resonances decaying into two Higgs bosons or into a Higgs boson and a W or Z boson in proton-proton collisions at 13 TeV
AABOUD 2018F
PL B777 91 Search for Diboson Resonances with Boson-Tagged Jets in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 13 TeV with the ATLAS Detector
AABOUD 2018CJ
PR D98 052008 Combination of searches for heavy resonances decaying into bosonic and leptonic final states using 36??fb$^{-1}$ of proton-proton collision data at $\sqrt{s} = 13$ TeV with the ATLAS detector
AABOUD 2018BI
EPJ C78 565 Search for heavy particles decaying into top-quark pairs using lepton-plus-jets events in proton?proton collisions at $\sqrt{s} = 13$ $\text {TeV}$ with the ATLAS detector
AABOUD 2018AK
JHEP 1803 042 Search for $WW/WZ$ resonance production in $\ell \nu qq$ final states in $pp$ collisions at $\sqrt{s} =$ 13 TeV with the ATLAS detector
AABOUD 2018AI
JHEP 1803 174 Search for heavy resonances decaying into a $W$ or $Z$ boson and a Higgs boson in final states with leptons and $b$-jets in 36 fb$^{-1}$ of $\sqrt s = 13$ TeV $pp$ collisions with the ATLAS detector
AABOUD 2018AB
PR D98 032016 Search for resonances in the mass distribution of jet pairs with one or two jets identified as $b$-jets in proton-proton collisions at $\sqrt{s}=13$ TeV with the ATLAS detector
AABOUD 2018AA
PR D98 032015 Search for heavy resonances decaying to a photon and a hadronically decaying $Z/W/H$ boson in $pp$ collisions at $\sqrt{s}=13$ $\mathrm{TeV}$ with the ATLAS detector
AABOUD 2018B
EPJ C78 24 Search for Heavy Resonances Decaying into ${{\mathit W}}{{\mathit W}}$ in the ${{\mathit e}}{{\mathit \nu}}{{\mathit \mu}}{{\mathit \nu}}$ Final State in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 13 TeV with the ATLAS Detector
AABOUD 2018N
PRL 121 081801 Search for low-mass dijet resonances using trigger-level jets with the ATLAS detector in $pp$ collisions at $\sqrt{s}=13$ TeV
AAIJ 2018AQ
JHEP 1809 147 Search for a dimuon resonance in the $\Upsilon$ mass region
SIRUNYAN 2018ED
JHEP 1811 172 Search for heavy resonances decaying into a vector boson and a Higgs boson in final states with charged leptons, neutrinos and b quarks at $ \sqrt{s}=13 $ TeV
SIRUNYAN 2018DR
JHEP 1811 161 Search for resonances in the mass spectrum of muon pairs produced in association with b quark jets in proton-proton collisions at $\sqrt{s} =$ 8 and 13 TeV
SIRUNYAN 2018P
PR D97 072006 Search for massive resonances decaying into $WW$, $WZ$, $ZZ$, $qW$, and $qZ$ with dijet final states at $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$
PRL 120 201801 Search for narrow resonances in the b-tagged dijet mass spectrum in proton-proton collisions at $\sqrt{s} =$ 8 TeV
SIRUNYAN 2018G
JHEP 1801 097 Search for Low Mass Vector Resonances Decaying into Quark-Antiquark Pairs in Proton-Proton Collisions at $\sqrt {s }$ = 13 TeV
PL B765 32 Search for New Resonances Decaying to a ${{\mathit W}}$ or ${{\mathit Z}}$ Boson and a Higgs Boson in the ${{\mathit \ell}^{+}}{{\mathit \ell}^{-}}{\mathit {\mathit b}}{\mathit {\overline{\mathit b}}}$, ${{\mathit \ell}}{{\mathit \nu}}{\mathit {\mathit b}}{\mathit {\overline{\mathit b}}}$, and ${{\mathit \nu}}{{\overline{\mathit \nu}}}{\mathit {\mathit b}}{\mathit {\overline{\mathit b}}}$ Channels with ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 13 TeV with the ATLAS Detector
AABOUD 2017AO
PL B774 494 Search for Heavy Resonances Decaying to a ${{\mathit W}}$ or ${{\mathit Z}}$ Boson and a Higgs Boson in the ${\mathit {\mathit q}}{\mathit {\mathit b}}{\mathit {\overline{\mathit b}}}$ Final State in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 13 TeV with the ATLAS Detector
PR D96 052004 Search for New Phenomena in Dijet Events using 37 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ Collision Data Collected at $\sqrt {s }$ = 13 TeV with the ATLAS Detector
KHACHATRYAN 2017AX
PL B773 563 Search for Leptophobic ${{\mathit Z}^{\,'}}$ Bosons Decaying into Four-Lepton Final States in Proton–Proton Collisions at $\sqrt {s }$ = 8 TeV
KHACHATRYAN 2017U
PL B768 137 Search for Heavy Resonances Decaying into a Vector Boson and a Higgs Boson in Final States with Charged Leptons, Neutrinos, and ${\mathit {\mathit b}}$ Quarks
SIRUNYAN 2017T
PRL 119 111802 Search for Low Mass Vector Resonances Decaying to Quark-Antiquark Pairs in Proton-Proton Collisions at $\sqrt {s }$ = 13 TeV
SIRUNYAN 2017A
JHEP 1703 162 Search for Massive Resonances Decaying into ${{\mathit W}}{{\mathit W}}$, ${{\mathit W}}{{\mathit Z}}$ or ${{\mathit Z}}{{\mathit Z}}$ Bosons in Proton-Proton Collisions at $\sqrt {s }$ = 13 TeV
SIRUNYAN 2017Q
JHEP 1707 001 Search for ${\mathit {\mathit t}}{\mathit {\overline{\mathit t}}}$ Resonances in Highly Boosted Lepton+Jets and Fully Hadronic Final States in Proton-Proton Collisions at $\sqrt {s }$ = 13 TeV
SIRUNYAN 2017AK
PL B774 533 Combination of Searches for Heavy Resonances Decaying to ${{\mathit W}}{{\mathit W}}$, ${{\mathit W}}{{\mathit Z}}$, ${{\mathit Z}}{{\mathit Z}}$, ${{\mathit W}}{{\mathit H}}$, and ${{\mathit Z}}{{\mathit H}}$ Boson Pairs in Proton-Proton Collisions at $\sqrt {s }$ = 8 and 13 TeV
SIRUNYAN 2017AP
JHEP 1710 180 Search for Associated Production of Dark Matter with a Higgs Boson Decaying to ${\mathit {\mathit b}}{\mathit {\overline{\mathit b}}}$ or ${{\mathit \gamma}}{{\mathit \gamma}}$ at $\sqrt {s }$ = 13 TeV
SIRUNYAN 2017R
EPJ C77 636 Search for Heavy Resonances that Decay into a Vector Boson and a Higgs Boson in Hadronic Final States at $\sqrt {s }$ = 13 TeV
SIRUNYAN 2017V
JHEP 1709 053 Search for a Heavy Resonance Decaying to a Top Quark and a Vector-Like Top Quark at $\sqrt {s }$ = 13 TeV
AABOUD 2016
PL B759 229 Search for Resonances in the Mass Distribution of Jet Pairs with One or Two Jets Identified as ${\mathit {\mathit b}}$-Jets in Proton-proton Collisions at $\sqrt {s }$ =13 TeV with the ATLAS Detector
EPJ C76 210 Search for New Phenomena in Events with at Least Three Photons Collected in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 8 TeV with the ATLAS Detector
AAD 2016S
PL B754 302 Search for New Phenomena in Dijet Mass and Angular Distributions from ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 13 TeV with the ATLAS Detector
KHACHATRYAN 2016AP
JHEP 1602 145 Search for a Massive Resonance Decaying into a Higgs Boson and a ${{\mathit W}}$ or ${{\mathit Z}}$ Boson in Hadronic Final States in Proton-Proton Collisions at $\sqrt {s }$ = 8 TeV
KHACHATRYAN 2016E
PR D93 012001 Search for Resonant ${\mathit {\mathit t}}{\mathit {\overline{\mathit t}}}$ Production in Proton-Proton Collisions at $\sqrt {s }$ = 8 TeV
AAD 2015AT
EPJ C75 79 Search for Invisible Particles Produced in Association with Single-Top-Quarks in Proton-Proton Collisions at $\sqrt {s }$ = 8 TeV with the ATLAS Detector
AAD 2015CD
PR D92 092001 Search for New Light Gauge Bosons in Higgs Boson Decays to Four-Lepton Final States in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 8 TeV with the ATLAS Detector at the LHC
AAD 2015AO
JHEP 1508 148 A Search for ${\mathit {\mathit t}}{\mathit {\overline{\mathit t}}}$ Resonances using Lepton-plus-Jets Events in Proton-Proton Collisions at $\sqrt {s }$ = 8 TeV with the ATLAS Detector
KHACHATRYAN 2015F
PRL 114 101801 Search for Monotop Signatures in Proton-Proton Collisions at $\sqrt {s }$ = 8 TeV
KHACHATRYAN 2015O
PL B748 255 Search for Narrow High-Mass Resonances in Proton$−$Proton Collisions at$\sqrt {s }$ = 8 TeV Decaying to a ${{\mathit Z}}$ and a Higgs Boson
PL B738 428 Search for New Resonances in ${{\mathit W}}{{\mathit \gamma}}$ and ${{\mathit Z}}{{\mathit \gamma}}$ Final States in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 8 TeV with the ATLAS Detector
KHACHATRYAN 2014A
JHEP 1408 174 Search for Massive Resonances Decaying into Pairs of Boosted Bosons in Semi-Leptonic Final States at $\sqrt {s }$ = 8 TeV
MARTINEZ 2014
PR D90 015028 Constraints on 3-3-1 Models with Electroweak ${{\mathit Z}}$ Pole Observables and ${{\mathit Z}^{\,'}}$ Search at the LHC
AAD 2013AQ
PR D88 012004 A Search for ${\mathit {\mathit t}}{\mathit {\overline{\mathit t}}}$ Resonances in the Lepton Plus Jets Final State with ATLAS using 4.7 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 7 TeV
AAD 2013G
JHEP 1301 116 Search for Resonances Decaying into Top-Quark Pairs using Fully Hadronic Decays in ${{\mathit p}}{{\mathit p}}$ Collisions with ATLAS at $\sqrt {s }$ = 7 TeV
AALTONEN 2013A
PRL 110 121802 Search for Resonant Top-Antitop Production in the Lepton Plus Jets Decay Mode Using the Full CDF Data Set
CHATRCHYAN 2013AP
PR D87 072002 Search for ${{\mathit Z}^{\,'}}$ Resonances Decaying to ${\mathit {\mathit t}}{\mathit {\overline{\mathit t}}}$ in dilepton+jets Final States in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 7 TeV
CHATRCHYAN 2013BM
PRL 111 211804 Searches for New Physics using the ${\mathit {\mathit t}}{\mathit {\overline{\mathit t}}}$ Invariant Mass Distribution in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 8 TeV
AAD 2012BV
JHEP 1209 041 A Search for ${\mathit {\mathit t}}{\mathit {\overline{\mathit t}}}$ Resonances in Lepton+Jets Events with Highly Boosted Top Quarks Collected in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 7 TeV with the ATLAS Detector
AAD 2012K
EPJ C72 2083 A Search for ${\mathit {\mathit t}}{\mathit {\overline{\mathit t}}}$ Resonances with the ATLAS Detector in 2.05 ${\mathrm {fb}}{}^{-1}$ of proton-proton Collisions at $\sqrt {s }$ = 7 TeV
AALTONEN 2012AR
PR D86 112002 Search for a Heavy Vector Boson Decaying to Two Gluons in ${{\mathit p}}{{\overline{\mathit p}}}$ Collisions at $\sqrt {s }$ = 1.96 TeV
AALTONEN 2012N
PRL 108 211805 Search for a Heavy Particle Decaying to a Top Quark and a Light Quark in ${{\mathit p}}{{\overline{\mathit p}}}$ Collisions at $\sqrt {s }$ = 1.96 TeV
ABAZOV 2012R
PR D85 051101 Search for a Narrow ${\mathit {\mathit t}}{\mathit {\overline{\mathit t}}}$ Resonance in ${{\mathit p}}{{\overline{\mathit p}}}$ Collisions at $\sqrt {s }$ = 1.96 TeV
CHATRCHYAN 2012AQ
JHEP 1209 029 Search for Anomalous ${\mathit {\mathit t}}{\mathit {\overline{\mathit t}}}$ Production in the Highly-Boosted All-Hadronic Final State
CHATRCHYAN 2012BL
JHEP 1212 015 Search for Resonant ${\mathit {\mathit t}}{\mathit {\overline{\mathit t}}}$ Production in Lepton+Jets Events in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 7 TeV
CHATRCHYAN 2012AI
JHEP 1208 110 Search for New Physics in events with Same-Sign Dileptons and ${\mathit {\mathit b}}$-Tagged Jets in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 7 TeV
AALTONEN 2011AD
PR D84 072003 Search for Resonant Production of ${\mathit {\mathit t}}{\mathit {\overline{\mathit t}}}$ Decaying to Jets in ${{\mathit p}}{{\overline{\mathit p}}}$ Collisions at $\sqrt {s }$ = 1.96 TeV
AALTONEN 2011AE
PR D84 072004 Search for Resonant Production of ${\mathit {\mathit t}}{\mathit {\overline{\mathit t}}}$ pairs in 4.8 fb${}^{-1}$ of Integrated Luminosity of ${{\mathit p}}{{\overline{\mathit p}}}$ Collisions at $\sqrt {s }$ = 1.96 TeV
CHATRCHYAN 2011O
JHEP 1108 005 Search for Same-Sign Top-Quark Pair Production at $\sqrt {s }$ = 7 TeV and Limits on Flavour Changing Neutral Currents in the Top Sector
AALTONEN 2008Y
PRL 100 231801 Search for Resonant ${{\mathit t}}{{\overline{\mathit t}}}$ Production in ${{\mathit p}}{{\overline{\mathit p}}}$ Collisions at $\sqrt {s }$ = 1.96 TeV
AALTONEN 2008D
PR D77 051102 Limits on the Production of Narrow ${{\mathit t}}{{\overline{\mathit t}}}$ Resonances in ${{\mathit p}}{{\overline{\mathit p}}}$ Collisions at $\sqrt {s }$ = 1.96 TeV
ABAZOV 2008AA
PL B668 98 Search for ${{\mathit t}}{{\overline{\mathit t}}}$ Resonances in the Lepton Plus Jets Final State in ${{\mathit p}}{{\overline{\mathit p}}}$ Collisions at $\sqrt {s }$ = 1.96 TeV
ABAZOV 2004A
PRL 92 221801 Search for Narrow ${\mathit {\mathit t}}{\mathit {\overline{\mathit t}}}$ Resonances in ${{\mathit p}}{{\overline{\mathit p}}}$ Collisions at = 1.8 TeV
BARGER 2003B
PR D67 075009 Primordial Nucleosynthesis Constraints on ${{\mathit Z}^{\,'}}$ Properties
CHO 2000
MPL A15 311 Looking for ${{\mathit Z}^{\,'}}$ Bosons in Supersymmetric E(6) Models Through Electroweak Precision Data
EPJ C5 155 Constraints on Four Fermi Contact Interactions from Low-Energy Electroweak Experiments
ABE 1997G
PR D55 5263 Search for New Particles Decaying to Dijets at CDF | CommonCrawl |
Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system
EECT Home
Recovery of the sound speed and initial displacement for the wave equation by means of a single Dirichlet boundary measurement
June 2013, 2(2): 365-378. doi: 10.3934/eect.2013.2.365
Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces
Stefan Meyer 1, and Mathias Wilke 1,
Martin Luther University Halle-Wittenberg, NWF II - Institute of Mathematics, D - 06099 Halle (Saale), Germany, Germany
Received November 2012 Revised February 2013 Published March 2013
We investigate a quasilinear initial-boundary value problem for Kuznetsov's equation with non-homogeneous Dirichlet boundary conditions. This is a model in nonlinear acoustics which describes the propagation of sound in fluidic media with applications in medical ultrasound. We prove that there exists a unique global solution which depends continuously on the sufficiently small data and that the solution and its temporal derivatives converge at an exponential rate as time tends to infinity. Compared to the analysis of Kaltenbacher & Lasiecka, we require optimal regularity conditions on the data and give simplified proofs which are based on maximal $L_p$-regularity for parabolic equations and the implicit function theorem.
Keywords: quasilinear parabolic system, Kuznetsov's equation, global well-posedness, optimal regularity, exponential decay..
Mathematics Subject Classification: Primary: 35K59; Secondary: 35K51, 35Q35, 35B30, 35B35, 35B40, 35B45, 35B6.
Citation: Stefan Meyer, Mathias Wilke. Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces. Evolution Equations & Control Theory, 2013, 2 (2) : 365-378. doi: 10.3934/eect.2013.2.365
Robert A. Adams and John J. F. Fournier, "Sobolev Spaces," $2^{nd}$ edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003. Google Scholar
Herbert Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in "Function Spaces, Differential Operators and Nonlinear Analysis" (Friedrichroda, 1992), Teubner-Texte Math., 133, Teubner, Stuttgart, (1993), 9-126. Google Scholar
Herbert Amann, "Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory," Monographs in Mathematics, 89, Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6. Google Scholar
Sigurd B. Angenent, Nonlinear analytic semiflows, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 91-107. doi: 10.1017/S0308210500024598. Google Scholar
Klaus Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin, 1985. Google Scholar
Robert Denk, Jürgen Saal and Jörg Seiler, Inhomogeneous symbols, the Newton polygon, and maximal $L^p$-regularity, Russ. J. Math. Phys., 15 (2008), 171-191. doi: 10.1134/S1061920808020040. Google Scholar
Robert Denk, Matthias Hieber and Jan Prüss, $\mathcal R$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003). Google Scholar
Robert Denk, Matthias Hieber and Jan Prüss, Optimal $L^p$- $L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224. doi: 10.1007/s00209-007-0120-9. Google Scholar
Gabriella Di Blasio, Linear parabolic evolution equations in $L^p$-spaces, Ann. Mat. Pura Appl. (4), 138 (1984), 55-104. doi: 10.1007/BF01762539. Google Scholar
Barbara Kaltenbacher and Irena Lasiecka, An analysis of nonhomogeneous Kuznetsov's equation: Local and global well-posedness; exponential decay, Math. Nachr., 285 (2012), 295-321. doi: 10.1002/mana.201000007. Google Scholar
Manfred Kaltenbacher, "Numerical Simulation of Mechatronic Sensors and Actuators," Springer, 2007. Available from: http://dx.doi.org/10.1007/978-3-540-71360-9. Google Scholar
V. P. Kuznetsov, Equations of nonlinear acoustics, Sov. Phys. Acoust., 16 (1971), 467-470. Google Scholar
Yuri Latushkin, Jan Prüss and Roland Schnaubelt, Stable and unstable manifolds for quasilinear parabolic systems with fully nonlinear boundary conditions, J. Evol. Equ., 6 (2006), 537-576. doi: 10.1007/s00028-006-0272-9. Google Scholar
Alessandra Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Progress in Nonlinear Differential Equations and Their Applications, 16, Birkhäuser Verlag, Basel, 1995. Google Scholar
Stefan Meyer and Mathias Wilke, Optimal regularity and long-time behavior of solutions for the Westervelt equation, Appl. Math. Optim., 64 (2011), 257-271. doi: 10.1007/s00245-011-9138-9. Google Scholar
Martin Meyries and Roland Schnaubelt, Interpolation, embeddings and traces of anisotropic fractional Sobolev spaces with temporal weights, J. Funct. Anal., 262 (2012), 1200-1229. doi: 10.1016/j.jfa.2011.11.001. Google Scholar
Hans Triebel, "Theory of Function Spaces," Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1. Google Scholar
Hans Triebel, "Interpolation Theory, Function Spaces, Differential Operators," $2^{nd}$ edition, Johann Ambrosius Barth, Heidelberg, 1995. Google Scholar
Eberhard Zeidler, "Nonlinear Functional Analysis and its Applications. I. Fixed-Point Theorems," Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5. Google Scholar
Zhaohi Huo, Yueling Jia, Qiaoxin Li. Global well-posedness for the 3D Zakharov-Kuznetsov equation in energy space $H^1$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1797-1851. doi: 10.3934/dcdss.2016075
Francis Ribaud, Stéphane Vento. Local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 449-483. doi: 10.3934/dcds.2017019
Mohamad Darwich. Local and global well-posedness in the energy space for the dissipative Zakharov-Kuznetsov equation in 3D. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3715-3724. doi: 10.3934/dcdsb.2020087
Takamori Kato. Global well-posedness for the Kawahara equation with low regularity. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1321-1339. doi: 10.3934/cpaa.2013.12.1321
Barbara Kaltenbacher, Irena Lasiecka. Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions. Conference Publications, 2011, 2011 (Special) : 763-773. doi: 10.3934/proc.2011.2011.763
Xin Zhong. Global well-posedness and exponential decay for 3D nonhomogeneous magneto-micropolar fluid equations with vacuum. Communications on Pure & Applied Analysis, 2022, 21 (2) : 493-515. doi: 10.3934/cpaa.2021185
Lin Shen, Shu Wang, Yongxin Wang. The well-posedness and regularity of a rotating blades equation. Electronic Research Archive, 2020, 28 (2) : 691-719. doi: 10.3934/era.2020036
Abderrahmane Youkana, Salim A. Messaoudi. General and optimal decay for a quasilinear parabolic viscoelastic system. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021129
Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete & Continuous Dynamical Systems, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095
Baoyan Sun, Kung-Chien Wu. Global well-posedness and exponential stability for the fermion equation in weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021147
Takafumi Akahori. Low regularity global well-posedness for the nonlinear Schrödinger equation on closed manifolds. Communications on Pure & Applied Analysis, 2010, 9 (2) : 261-280. doi: 10.3934/cpaa.2010.9.261
Huafei Di, Yadong Shang, Xiaoxiao Zheng. Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 781-801. doi: 10.3934/dcdsb.2016.21.781
E. Compaan, N. Tzirakis. Low-regularity global well-posedness for the Klein-Gordon-Schrödinger system on $ \mathbb{R}^+ $. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 3867-3895. doi: 10.3934/dcds.2019156
Xujie Yang. Global well-posedness in a chemotaxis system with oxygen consumption. Communications on Pure & Applied Analysis, 2022, 21 (2) : 471-492. doi: 10.3934/cpaa.2021184
Tadahiro Oh, Yuzhao Wang. On global well-posedness of the modified KdV equation in modulation spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2971-2992. doi: 10.3934/dcds.2020393
Zhaoyang Yin. Well-posedness, blowup, and global existence for an integrable shallow water equation. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 393-411. doi: 10.3934/dcds.2004.11.393
Seung-Yeal Ha, Jinyeong Park, Xiongtao Zhang. A global well-posedness and asymptotic dynamics of the kinetic Winfree equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1317-1344. doi: 10.3934/dcdsb.2019229
Hideo Takaoka. Global well-posedness for the Kadomtsev-Petviashvili II equation. Discrete & Continuous Dynamical Systems, 2000, 6 (2) : 483-499. doi: 10.3934/dcds.2000.6.483
Jian-Wen Sun, Seonghak Kim. Exponential decay for quasilinear parabolic equations in any dimension. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021280
Kei Matsuura, Mitsuharu Otani. Exponential attractors for a quasilinear parabolic equation. Conference Publications, 2007, 2007 (Special) : 713-720. doi: 10.3934/proc.2007.2007.713
Stefan Meyer Mathias Wilke | CommonCrawl |
G-module
In mathematics, given a group G, a G-module is an abelian group M on which G acts compatibly with the abelian group structure on M. This widely applicable notion generalizes that of a representation of G. Group (co)homology provides an important set of tools for studying general G-modules.
The term G-module is also used for the more general notion of an R-module on which G acts linearly (i.e. as a group of R-module automorphisms).
Definition and basics
Let $G$ be a group. A left $G$-module consists of[1] an abelian group $M$ together with a left group action $\rho :G\times M\to M$ such that
g·(a1 + a2) = g·a1 + g·a2
where g·a denotes ρ(g,a). A right G-module is defined similarly. Given a left G-module M, it can be turned into a right G-module by defining a·g = g−1·a.
A function f : M → N is called a morphism of G-modules (or a G-linear map, or a G-homomorphism) if f is both a group homomorphism and G-equivariant.
The collection of left (respectively right) G-modules and their morphisms form an abelian category G-Mod (resp. Mod-G). The category G-Mod (resp. Mod-G) can be identified with the category of left (resp. right) ZG-modules, i.e. with the modules over the group ring Z[G].
A submodule of a G-module M is a subgroup A ⊆ M that is stable under the action of G, i.e. g·a ∈ A for all g ∈ G and a ∈ A. Given a submodule A of M, the quotient module M/A is the quotient group with action g·(m + A) = g·m + A.
Examples
• Given a group G, the abelian group Z is a G-module with the trivial action g·a = a.
• Let M be the set of binary quadratic forms f(x, y) = ax2 + 2bxy + cy2 with a, b, c integers, and let G = SL(2, Z) (the 2×2 special linear group over Z). Define
$(g\cdot f)(x,y)=f((x,y)g^{t})=f\left((x,y)\cdot {\begin{bmatrix}\alpha &\gamma \\\beta &\delta \end{bmatrix}}\right)=f(\alpha x+\beta y,\gamma x+\delta y),$
where
$g={\begin{bmatrix}\alpha &\beta \\\gamma &\delta \end{bmatrix}}$
and (x, y)g is matrix multiplication. Then M is a G-module studied by Gauss.[2] Indeed, we have
$g(h(f(x,y)))=gf((x,y)h^{t})=f((x,y)h^{t}g^{t})=f((x,y)(gh)^{t})=(gh)f(x,y).$
• If V is a representation of G over a field K, then V is a G-module (it is an abelian group under addition).
Topological groups
If G is a topological group and M is an abelian topological group, then a topological G-module is a G-module where the action map G×M → M is continuous (where the product topology is taken on G×M).[3]
In other words, a topological G-module is an abelian topological group M together with a continuous map G×M → M satisfying the usual relations g(a + a′) = ga + ga′, (gg′)a = g(g′a), and 1a = a.
Notes
1. Curtis, Charles W.; Reiner, Irving (1962), Representation Theory of Finite Groups and Associative Algebras, John Wiley & Sons (Reedition 2006 by AMS Bookstore), ISBN 978-0-470-18975-7.
2. Kim, Myung-Hwan (1999), Integral Quadratic Forms and Lattices: Proceedings of the International Conference on Integral Quadratic Forms and Lattices, June 15–19, 1998, Seoul National University, Korea, American Mathematical Soc.
3. D. Wigner (1973). "Algebraic cohomology of topological groups". Trans. Amer. Math. Soc. 178: 83–93. doi:10.1090/s0002-9947-1973-0338132-7.
References
• Chapter 6 of Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259.
| Wikipedia |
# Linear programming: formulating optimization problems
Linear programming is a fundamental optimization technique used in machine learning and other fields. It is based on the idea of finding the best solution to a problem by minimizing or maximizing a linear function subject to a set of linear constraints.
To formulate a linear programming problem, we need to define the objective function, which represents the value we want to optimize, and the constraints, which represent the limitations imposed on the solution.
The objective function is usually written as:
$$\min \text{ or } \max \quad c^Tx$$
where $c$ is a vector of coefficients, and $x$ is the vector of variables.
The constraints are written as:
$$Ax \le b$$
where $A$ is a matrix of coefficients, $x$ is the vector of variables, and $b$ is a vector of constants.
## Exercise
Write the objective function and constraints for a linear programming problem that represents the following scenario:
A company wants to maximize its profit by producing two products, A and B. The company has a limited amount of raw material, and each product requires a different amount of raw material. The company also has a limited amount of time to produce each product.
Objective function: $\max \quad \text{profit}$
Constraints:
- Raw material for product A: $x_1 \le 100$
- Raw material for product B: $x_2 \le 150$
- Time for product A: $x_3 \le 8$
- Time for product B: $x_4 \le 12$
### Solution
Objective function: $\max \quad \text{profit} = c^T(x_1, x_2, x_3, x_4)$
Constraints:
- $A = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{bmatrix}$
- $b = \begin{bmatrix} 100 \\ 150 \\ 8 \\ 12 \end{bmatrix}$
# Linear programming: simplex method and dual solutions
The simplex method is an algorithm used to solve linear programming problems. It is based on the idea of iteratively improving a feasible solution until it becomes optimal.
The simplex method starts with an initial feasible solution and iteratively improves it by pivoting around the vertices of the feasible region. The algorithm terminates when the solution cannot be further improved.
The dual solution of a linear programming problem is another way to represent the optimal solution. It is obtained by solving a dual problem, which is a different formulation of the original problem.
## Exercise
Solve the linear programming problem from the previous section using the simplex method.
### Solution
To solve the linear programming problem using the simplex method, we can follow these steps:
1. Formulate the dual problem:
Objective function: $\min \quad \text{cost} = b^T(y_1, y_2, y_3, y_4)$
Constraints:
- Profit for product A: $y_1 \ge 0$
- Profit for product B: $y_2 \ge 0$
- Raw material for product A: $-y_1 \le 0$
- Raw material for product B: $-y_2 \le 0$
- Time for product A: $-y_3 \le 0$
- Time for product B: $-y_4 \le 0$
2. Solve the dual problem using the simplex method.
3. Compute the optimal solution to the original problem by finding the corresponding values of $x_1, x_2, x_3, x_4$.
# Linear programming: applications in machine learning
Linear programming has numerous applications in machine learning, including:
- Feature selection: selecting the most important features in a dataset to reduce the dimensionality and improve the performance of a machine learning model.
- Regularization: adding a regularization term to the objective function to prevent overfitting in regression models.
- Linear support vector machines: using linear programming to solve the optimization problem for training a linear support vector machine.
## Exercise
Apply linear programming to select the most important features in a dataset.
### Solution
1. Formulate the linear programming problem:
Objective function: $\max \quad \text{sum of feature values} = c^T(x_1, x_2, x_3, \dots, x_n)$
Constraints:
- Feature values: $Ax \le b$
- Feature importance: $x \ge 0$
2. Solve the linear programming problem using the simplex method.
3. Select the features with non-zero values in the optimal solution.
# Integer programming: formulating optimization problems
Integer programming is an extension of linear programming that allows the variables to be integers instead of real numbers. It is used to solve optimization problems with discrete variables.
To formulate an integer programming problem, we need to define the objective function and the constraints, as in linear programming. The additional constraint is that the variables must be integers.
The objective function is usually written as:
$$\min \text{ or } \max \quad c^Tx$$
where $c$ is a vector of coefficients, and $x$ is the vector of integer variables.
The constraints are written as:
$$Ax \le b$$
where $A$ is a matrix of coefficients, $x$ is the vector of integer variables, and $b$ is a vector of constants.
## Exercise
Write the objective function and constraints for an integer programming problem that represents the following scenario:
A company wants to maximize its profit by producing two products, A and B. The company has a limited amount of raw material, and each product requires a different amount of raw material. The company also has a limited amount of time to produce each product. The number of units of each product that can be produced must be an integer.
Objective function: $\max \quad \text{profit}$
Constraints:
- Raw material for product A: $x_1 \le 100$
- Raw material for product B: $x_2 \le 150$
- Time for product A: $x_3 \le 8$
- Time for product B: $x_4 \le 12$
### Solution
Objective function: $\max \quad \text{profit} = c^T(x_1, x_2, x_3, x_4)$
Constraints:
- $A = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{bmatrix}$
- $b = \begin{bmatrix} 100 \\ 150 \\ 8 \\ 12 \end{bmatrix}$
- $x \in \mathbb{Z}$
# Integer programming: branch and bound method
The branch and bound method is an algorithm used to solve integer programming problems. It is a divide-and-conquer approach that explores the solution space by generating a tree of partial solutions.
The algorithm starts with an initial solution, which is usually the trivial solution (all variables set to 0). Then, it selects a variable to branch on, and generates two subproblems: one with the lower bound of the selected variable and one with the upper bound.
The algorithm continues to branch on the variable with the most promising subproblem until a feasible solution is found or the solution space is exhausted.
## Exercise
Instructions:
Explain how the branch and bound method can be used to solve the integer programming problem from the previous exercise.
### Solution
The branch and bound method can be used to solve the integer programming problem by following these steps:
1. Start with the trivial solution (all variables set to 0).
2. Select a variable to branch on, for example, $x_1$.
3. Generate two subproblems: one with the lower bound of $x_1$ (0) and one with the upper bound of $x_1$ (100).
4. Solve both subproblems using linear programming methods.
5. If both subproblems are infeasible, the problem is unbounded. If one subproblem is feasible, branch on the next variable and repeat steps 2-4.
6. If a feasible solution is found, return it. If the solution space is exhausted, the problem is infeasible.
# Integer programming: applications in machine learning
Integer programming has several applications in machine learning, such as feature selection, model selection, and combinatorial optimization problems.
Feature selection is the process of selecting a subset of relevant features from a dataset to improve the performance of a machine learning model. Integer programming can be used to solve the feature selection problem by formulating it as an optimization problem and using the branch and bound method to find the optimal subset.
Model selection is the process of choosing the best model from a set of candidate models. Integer programming can be used to solve the model selection problem by formulating it as an optimization problem and using the branch and bound method to find the optimal model.
Combinatorial optimization problems are problems that involve selecting the best combination of items from a set to maximize or minimize a certain objective function. Integer programming can be used to solve combinatorial optimization problems by formulating the problem as an optimization problem and using the branch and bound method to find the optimal combination.
## Exercise
Instructions:
Explain how integer programming can be used to solve a feature selection problem in machine learning.
### Solution
Integer programming can be used to solve a feature selection problem in machine learning by following these steps:
1. Formulate the feature selection problem as an optimization problem.
2. Define the objective function as the improvement in the performance of the machine learning model after selecting the subset of features.
3. Define the constraints as the conditions that must be satisfied by the selected features, such as a minimum number of features or a maximum correlation between features.
4. Use the branch and bound method to find the optimal subset of features.
5. Select the optimal subset of features for the machine learning model.
# Decision trees: basics and applications
A decision tree is a flowchart-like structure used to model decisions and their possible consequences. It is a popular machine learning algorithm used for classification and regression tasks.
A decision tree consists of nodes and branches. The root node represents the initial decision, and each internal node represents a decision based on a feature. The leaf nodes represent the final decisions or the predicted outcomes.
Decision trees have several advantages, such as their interpretability, ease of implementation, and ability to handle both numerical and categorical data. However, they can also suffer from overfitting and lack of generalization to new data.
## Exercise
Instructions:
Explain the difference between classification and regression in the context of decision trees.
### Solution
Classification and regression are two types of supervised learning tasks that can be solved using decision trees. The main difference between them is the type of prediction they make.
In classification, the goal is to predict a categorical target variable, such as whether an email is spam or not spam. Decision trees for classification use the mode (most frequent class) of the training data in each leaf node to make predictions.
In regression, the goal is to predict a continuous target variable, such as the price of a house. Decision trees for regression use the mean (average) of the training data in each leaf node to make predictions.
# Random forests: ensemble learning method
Random forests are an ensemble learning method that combines multiple decision trees to improve the performance of the model. They are particularly effective in reducing overfitting and improving the model's generalization ability.
The random forest algorithm works by creating multiple decision trees and then combining their predictions using a voting mechanism. This ensemble of trees provides a more robust and accurate model than a single decision tree.
Random forests can be used for both classification and regression tasks. They have several advantages, such as their interpretability, ease of implementation, and ability to handle both numerical and categorical data.
## Exercise
Instructions:
Explain how the random forest algorithm works.
### Solution
The random forest algorithm works as follows:
1. Create a training dataset by sampling with replacement from the original dataset. This is done to reduce overfitting and improve the model's generalization ability.
2. Create multiple decision trees by randomly selecting a subset of features for each tree. This is done to reduce overfitting and improve the model's generalization ability.
3. For each tree, use the training dataset to learn the decision rules that minimize the error.
4. When making predictions, each tree in the forest votes for the class that maximizes the sum of its votes.
5. Combine the votes from all the trees to make the final prediction.
# Neural networks: basic structure and function
A neural network is a computational model inspired by the structure and function of the human brain. It is a collection of interconnected nodes, or neurons, organized into layers.
The basic structure of a neural network consists of an input layer, one or more hidden layers, and an output layer. The neurons in the input layer receive input from the external environment, the neurons in the hidden layers process the input, and the neurons in the output layer produce the network's output.
The function of a neural network is to learn a mapping from the input space to the output space. This is done by adjusting the weights and biases of the network based on the error between the predicted output and the actual output.
## Exercise
Instructions:
Explain how a neural network can be used to approximate a nonlinear function.
### Solution
A neural network can be used to approximate a nonlinear function by following these steps:
1. Create a neural network with a sufficient number of hidden layers and neurons.
2. Initialize the weights and biases of the network randomly.
3. Use a training algorithm, such as gradient descent, to adjust the weights and biases of the network based on the error between the predicted output and the actual output.
4. Use the trained network to approximate the nonlinear function.
# Neural networks: types of neural networks
Neural networks come in various types, each designed for a specific task or problem. Some common types of neural networks include:
- Artificial neural networks: These are the most general type of neural networks, used for a wide range of tasks, such as classification, regression, and optimization.
- Convolutional neural networks (CNNs): These are specifically designed for image recognition and classification tasks. They have a hierarchical structure, with layers that detect specific patterns, such as edges and textures.
- Recurrent neural networks (RNNs): These are designed for sequence prediction tasks, such as natural language processing and time series forecasting. They have a loop-like structure that allows them to maintain a "memory" of previous inputs.
- Long short-term memory (LSTM) networks: These are a type of RNN that can learn long-term dependencies. They are particularly effective for sequence prediction tasks that require a long-term memory.
- Autoencoders: These are neural networks that learn to encode and decode input data. They are used for dimensionality reduction, noise reduction, and anomaly detection.
## Exercise
Instructions:
Explain the difference between a feedforward neural network and a recurrent neural network.
### Solution
The main difference between a feedforward neural network and a recurrent neural network is the way they process input data.
In a feedforward neural network, the input data is processed in a forward direction, from the input layer to the output layer. The weights and biases are updated based on the error between the predicted output and the actual output.
In a recurrent neural network, the input data is processed in a loop-like structure, with the output of each neuron being fed back into the network as input for the next neuron. This allows the network to maintain a "memory" of previous inputs and process sequences of data.
# Genetic algorithms: basic concepts and applications
Genetic algorithms (GAs) are a class of evolutionary algorithms inspired by the process of natural selection. They are used to find optimal solutions to optimization and search problems.
The basic concepts of genetic algorithms include:
- Population: A population is a set of candidate solutions to the problem. Each solution is represented as a chromosome, which is a string of genes.
- Fitness function: The fitness function evaluates the quality of a solution. It is used to select the fittest individuals in the population.
- Selection: The selection process selects a subset of individuals from the population based on their fitness. This subset is used to create the next generation of the population.
- Crossover: The crossover process combines the genes of two individuals to create a new individual. This is done to introduce new variations in the population.
- Mutation: The mutation process introduces small random changes to the genes of an individual. This is done to introduce diversity in the population.
## Exercise
Instructions:
Explain how genetic algorithms can be used to solve an optimization problem.
### Solution
Genetic algorithms can be used to solve an optimization problem by following these steps:
1. Define a population of candidate solutions to the problem. Each solution is represented as a chromosome, which is a string of genes.
2. Define a fitness function that evaluates the quality of a solution.
3. Implement the selection, crossover, and mutation processes.
4. Initialize the population with random solutions.
5. Evaluate the fitness of each individual in the population.
6. Select a subset of individuals based on their fitness.
7. Create a new generation of the population by combining the selected individuals using the crossover and mutation processes.
8. Repeat steps 5-7 until a solution with a high fitness is found or a stopping criterion is met.
# Table Of Contents
1. Linear programming: formulating optimization problems
2. Linear programming: simplex method and dual solutions
3. Linear programming: applications in machine learning
4. Integer programming: formulating optimization problems
5. Integer programming: branch and bound method
6. Integer programming: applications in machine learning
7. Decision trees: basics and applications
8. Random forests: ensemble learning method
9. Neural networks: basic structure and function
10. Neural networks: types of neural networks
11. Genetic algorithms: basic concepts and applications
Course | Textbooks |
-1 0 1 2 3 4 5 6 7 8 9 →
List of numbers — Integers
← 0 10 20 30 40 50 60 70 80 90 →
Ordinal
(eighth)
Numeral system
1, 2, 4,
Greek numeral
Roman numeral
VIII, viii
Greek prefix
octa-/oct-
Latin prefix
octo-/oct-
Duodecimal
η (or Η)
Arabic, Kurdish, Persian, Sindhi, Urdu
፰
Chinese numeral
八,捌
Devanāgarī
Ը ը
8 (eight) is the oul' natural number followin' 7 and precedin' 9.
1 In mathematics
1.1 List of basic calculations
3 Evolution of the bleedin' Arabic digit
4 In science
4.1 Physics
4.2 Astronomy
4.3 Chemistry
4.4 Geology
4.5 Biology
4.6 Psychology
5 In technology
5.1 In measurement
6 In culture
6.1 Currency
6.3 In religion, folk belief and divination
6.3.1 Hinduism
6.3.2 Buddhism
6.3.3 Judaism
6.3.4 Christianity
6.3.5 Islam
6.3.6 Taoism
6.3.8 As a feckin' lucky number
6.3.9 In astrology
6.4 In music and dance
6.5 In film and television
6.6 In sports and other games
6.7 In foods
6.8 In literature
6.9 In shlang
8 is:
a composite number, its proper divisors bein' 1, 2, and 4, Lord bless us and save us. It is twice 4 or four times 2.
a power of two, bein' 23 (two cubed), and is the first number of the bleedin' form p3, p bein' an integer greater than 1.
the first number which is neither prime nor semiprime.
the base of the oul' octal number system,[1] which is mostly used with computers. In octal, one digit represents three bits. Whisht now and eist liom. In modern computers, a byte is a bleedin' groupin' of eight bits, also called an octet.
a Fibonacci number, bein' 3 plus 5. Sure this is it. The next Fibonacci number is 13, be the hokey! 8 is the bleedin' only positive Fibonacci number, aside from 1, that is a feckin' perfect cube.[2]
the only nonzero perfect power that is one less than another perfect power, by Mihăilescu's Theorem.
the order of the smallest non-abelian group all of whose subgroups are normal.
the dimension of the octonions and is the highest possible dimension of a normed division algebra.
the first number to be the oul' aliquot sum of two numbers other than itself; the oul' discrete biprime 10, and the feckin' square number 49.
A number is divisible by 8 if its last three digits, when written in decimal, are also divisible by 8, or its last three digits are 0 when written in binary.
There are a holy total of eight convex deltahedra.[3]
A polygon with eight sides is an octagon.[4] Figurate numbers representin' octagons (includin' eight) are called octagonal numbers.
A polyhedron with eight faces is an octahedron.[5] A cuboctahedron has as faces six equal squares and eight equal regular triangles.[6]
A cube has eight vertices.[7]
Sphenic numbers always have exactly eight divisors.[8]
The number 8 is involved with a feckin' number of interestin' mathematical phenomena related to the oul' notion of Bott periodicity. Whisht now and listen to this wan. For example, if O(∞) is the bleedin' direct limit of the feckin' inclusions of real orthogonal groups
O ( 1 ) ↪ O ( 2 ) ↪ … ↪ O ( k ) ↪ … {\displaystyle O(1)\hookrightarrow O(2)\hookrightarrow \ldots \hookrightarrow O(k)\hookrightarrow \ldots } ,
π k + 8 ( O ( ∞ ) ) ≅ π k ( O ( ∞ ) ) {\displaystyle \pi _{k+8}(O(\infty ))\cong \pi _{k}(O(\infty ))} .
Clifford algebras also display a feckin' periodicity of 8.[9] For example, the oul' algebra Cl(p + 8,q) is isomorphic to the bleedin' algebra of 16 by 16 matrices with entries in Cl(p,q). We also see a period of 8 in the bleedin' K-theory of spheres and in the oul' representation theory of the rotation groups, the feckin' latter givin' rise to the 8 by 8 spinorial chessboard. All of these properties are closely related to the feckin' properties of the bleedin' octonions.
The spin group Spin(8) is the feckin' unique such group that exhibits the oul' phenomenon of triality.
The lowest-dimensional even unimodular lattice is the bleedin' 8-dimensional E8 lattice. Even positive definite unimodular lattices exist only in dimensions divisible by 8.
A figure 8 is the common name of an oul' geometric shape, often used in the feckin' context of sports, such as skatin'.[10] Figure-eight turns of an oul' rope or cable around a bleedin' cleat, pin, or bitt are used to belay somethin'.[11]
List of basic calculations
8 × x 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120
8 ÷ x 8 4 2.6 2 1.6 1.3 1.142857 1 0.8 0.8
0.72 0.6 0.615384 0.571428 0.53
x ÷ 8 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1 1.125 1.25
1.375 1.5 1.625 1.75 1.875
8x 8 64 512 4096 32768 262144 2097152 16777216 134217728 1073741824
8589934592 68719476736 549755813888
x8 1 256 6561 65536 390625 1679616 5764801 16777216 43046721 100000000
214358881 429981696 815730721
English eight, from Old English eahta, æhta, Proto-Germanic *ahto is a direct continuation of Proto-Indo-European *oḱtṓ(w)-, and as such cognate with Greek ὀκτώ and Latin octo-, both of which stems are reflected by the feckin' English prefix oct(o)-, as in the bleedin' ordinal adjective octaval or octavary, the feckin' distributive adjective is octonary. The adjective octuple (Latin octu-plus) may also be used as a bleedin' noun, meanin' "a set of eight items"; the feckin' diminutive octuplet is mostly used to refer to eight siblings delivered in one birth.
The Semitic numeral is based on a feckin' root *θmn-, whence Akkadian smn-, Arabic ṯmn-, Hebrew šmn- etc. The Chinese numeral, written 八 (Mandarin: bā; Cantonese: baat), is from Old Chinese *priāt-, ultimately from Sino-Tibetan b-r-gyat or b-g-ryat which also yielded Tibetan brgyat.
It has been argued that, as the feckin' cardinal number 7 is the feckin' highest number of items that can universally be cognitively processed as a single set, the feckin' etymology of the numeral eight might be the feckin' first to be considered composite, either as "twice four" or as "two short of ten", or similar. Would ye swally this in a minute now? The Turkic words for "eight" are from a bleedin' Proto-Turkic stem *sekiz, which has been suggested as originatin' as a negation of eki "two", as in "without two fingers" (i.e., "two short of ten; two fingers are not bein' held up");[12] this same principle is found in Finnic *kakte-ksa, which conveys a feckin' meanin' of "two before (ten)". The Proto-Indo-European reconstruction *oḱtṓ(w)- itself has been argued as representin' an old dual, which would correspond to an original meanin' of "twice four". Jesus, Mary and holy Saint Joseph. Proponents of this "quaternary hypothesis" adduce the feckin' numeral 9, which might be built on the oul' stem new-, meanin' "new" (indicatin' the feckin' beginnin' of a bleedin' "new set of numerals" after havin' counted to eight).[13]
Evolution of the Arabic digit
Evolution of the numeral 8 from the bleedin' Indians to the feckin' Europeans
The modern digit 8, like all modern Arabic numerals other than zero, originates with the bleedin' Brahmi numerals. The Brahmi digit for eight by the oul' 1st century was written in one stroke as a curve └┐ lookin' like an uppercase H with the oul' bottom half of the bleedin' left line and the feckin' upper half of the right line removed. However, the feckin' digit for eight used in India in the oul' early centuries of the bleedin' Common Era developed considerable graphic variation, and in some cases took the oul' shape of a feckin' single wedge, which was adopted into the Perso-Arabic tradition as ٨ (and also gave rise to the later Devanagari form ८); the bleedin' alternative curved glyph also existed as an oul' variant in Perso-Arabic tradition, where it came to look similar to our digit 5.[year needed]
The digits as used in Al-Andalus by the bleedin' 10th century were a distinctive western variant of the oul' glyphs used in the oul' Arabic-speakin' world, known as ghubār numerals (ghubār translatin' to "sand table"). Bejaysus here's a quare one right here now. In these digits, the line of the oul' 5-like glyph used in Indian manuscripts for eight came to be formed in ghubār as a closed loop, which was the oul' 8-shape that became adopted into European use in the oul' 10th century.[14]
Just as in most modern typefaces, in typefaces with text figures the bleedin' character for the digit 8 usually has an ascender, as, for example, in .
The infinity symbol ∞, described as a "sideways figure eight", is unrelated to the bleedin' digit 8 in origin; it is first used (in the bleedin' mathematical meanin' "infinity") in the 17th century, and it may be derived from the oul' Roman numeral for "one thousand" CIƆ, or alternatively from the oul' final Greek letter, ω.
In nuclear physics, the bleedin' second magic number.[15]
In particle physics, the oul' eightfold way is used to classify sub-atomic particles.[16]
In statistical mechanics, the eight-vertex model has 8 possible configurations of arrows at each vertex.[17]
Messier object M8, an oul' magnitude 5.0 nebula in the oul' constellation of Sagittarius.[18]
The New General Catalogue object NGC 8, a double star in the oul' constellation Pegasus.
Since the oul' demotion of Pluto to a dwarf planet on 24 August 2006, in our Solar System, eight of the bleedin' bodies orbitin' the feckin' Sun are considered to be planets.
The atomic number of oxygen.[19]
The most stable allotrope of a sulfur molecule is made of eight sulfur atoms arranged in a rhombic form.[20]
The maximum number of electrons that can occupy an oul' valence shell.
The red pigment lycopene consists of eight isoprene units.[21]
A disphenoid crystal is bounded by eight scalene triangles arranged in pairs. A ditetragonal prism in the tetragonal crystal system has eight similar faces whose alternate interfacial angles only are equal.
All spiders, and more generally all arachnids, have eight legs.[22] Orb-weaver spiders of the feckin' cosmopolitan family Areneidae have eight similar eyes.[23]
The octopus and its cephalopod relatives in genus Argonauta have eight arms (tentacles).
Compound coelenterates of the subclass or order Alcyonaria have polyps with eight-branched tentacles and eight septa.[24]
Sea anemones of genus Edwardsia have eight mesenteries.[25]
Animals of phylum Ctenophora swim by means of eight meridional bands of transverse ciliated plates, each plate representin' a feckin' row of large modified cilia.[26]
The eight-spotted forester (genus Alypia, family Zygaenidae) is a bleedin' diurnal moth havin' black wings with brilliant white spots.
The ascus in fungi of the bleedin' class Ascomycetes, followin' nuclear fusion, bears within it typically eight ascospores.[27]
Herbs of genus Coreopsis (tickseed) have showy flower heads with involucral bracts in two distinct series of eight each.
In human adult dentition there are eight teeth in each quadrant.[28] The eighth tooth is the oul' so-called wisdom tooth.
There are eight cervical nerves on each side in man and most mammals.[29]
There are eight Jungian cognitive functions, accordin' to the oul' MBTI models by John Beebe and Linda Berens.[30]
Timothy Leary identified an oul' hierarchy of eight levels of consciousness.
NATO signal flag for 8
A byte is eight bits.[31]
Many (mostly historic) computer architectures are eight-bit, among them the oul' Nintendo Entertainment System.
Standard-8 and Super-8 are 8 mm film formats.[32]
Video8, Hi8 and Digital8 are related 8 mm video formats.[33]
On most phones, the 8 key is associated with the feckin' letters T, U, and V, but on the feckin' BlackBerry it is the oul' key for B, N, and X.
An eight may refer to an eight-cylinder engine or automobile.[34] A V8 engine is an internal combustion engine with eight cylinders configured in two banks (rows) of four formin' a holy "V" when seen from the end.
A figure-eight knot (so named for its configuration) is a holy kind of stopper knot.[35]
The number eight written in parentheses is the oul' code for the feckin' musical note in Windows Live Messenger.
In a feckin' seven-segment display, when an 8 is illuminated, all the display bulbs are on.
In measurement
The SI prefix for 10008 is yotta (Y), and for its reciprocal, yocto (y).
In liquid measurement (United States customary units), there are eight fluid ounces in a holy cup, eight pints in a gallon and eight tablespoonfuls in a holy gill.[36]
There are eight furlongs in a mile.[37]
The clove, an old English unit of weight, was equal to eight pounds when measurin' cheese.[38]
An eight may be an article of clothin' of the eighth size.
Force eight is the oul' first wind strength attributed to a gale on the oul' Beaufort scale when announced on a holy Shippin' Forecast.[39]
Sailors and civilians alike from the bleedin' 1500s onward referred to evenly divided parts of the feckin' Spanish dollar as "pieces of eight", or "bits".
This section does not cite any sources. Please help improve this section by addin' citations to reliable sources, would ye believe it? Unsourced material may be challenged and removed. (September 2021) (Learn how and when to remove this template message)
Various types of buildings are usually eight-sided (octagonal), such as single-roomed gazebos and multi-roomed pagodas (descended from stupas; see religion section below).
Eight caulicoles rise out of the bleedin' leafage in an oul' Corinthian capital, endin' in leaves that support the volutes.
In religion, folk belief and divination
Also known as Ashtha, Aṣṭa, or Ashta in Sanskrit, it is the number of wealth and abundance.
The goddess of wealth and prosperity, Lakshmi, has eight forms known as Ashta Lakshmi and worshipped as:
"Maha-lakshmi, Dhana-lakshmi, Dhanya-lakshmi, Gaja-lakshmi,
Santana-lakshmi, Veera-lakshmi, Vijaya-lakshmi and Vidhya-lakshmi"[40]
There are eight nidhi, or seats of wealth, accordin' to Hinduism.
There are eight guardians of the directions known as Astha-dikpalas.[41]
There are eight Hindu monasteries established by the bleedin' saint Madhvacharya in Udupi, India popularly known as the oul' Ashta Mathas of Udupi.[42]
In Buddhism, the feckin' 8-spoked Dharmacakra represents the Noble Eightfold Path
The Dharmacakra, a Buddhist symbol, has eight spokes.[43] The Buddha's principal teachin'—the Four Noble Truths—ramifies as the oul' Noble Eightfold Path and the feckin' Buddha emphasizes the bleedin' importance of the eight attainments or jhanas.
In Mahayana Buddhism, the oul' branches of the oul' Eightfold Path are embodied by the bleedin' Eight Great Bodhisattvas: (Manjusri, Vajrapani, Avalokiteśvara, Maitreya, Ksitigarbha, Nivaranavishkambhi, Akasagarbha, and Samantabhadra).[44] These are later (controversially) associated with the Eight Consciousnesses accordin' to the oul' Yogacara school of thought: consciousness in the bleedin' five senses, thought-consciousness, self-consciousness, and unconsciousness-"consciousness" or "store-house consciousness" (alaya-vijñana). Stop the lights! The "irreversible" state of enlightenment, at which point a holy Bodhisattva goes on "autopilot", is the feckin' Eight Ground or bhūmi. Holy blatherin' Joseph, listen to this. In general, "eight" seems to be an auspicious number for Buddhists, e.g., the bleedin' "eight auspicious symbols" (the jewel-encrusted parasol; the feckin' goldfish (always shown as a feckin' pair, e.g., the feckin' glyph of Pisces); the bleedin' self-replenishin' amphora; the oul' white kamala lotus-flower; the bleedin' white conch; the eternal (Celtic-style, infinitely loopin') knot; the oul' banner of imperial victory; the bleedin' eight-spoked wheel that guides the feckin' ship of state, or that symbolizes the Buddha's teachin'), that's fierce now what? Similarly, Buddha's birthday falls on the oul' 8th day of the bleedin' 4th month of the Chinese calendar.
The religious rite of brit milah (commonly known as circumcision) is held on a bleedin' baby boy's eighth day of life.[45]
Hanukkah is an eight-day Jewish holiday that starts on the bleedin' 25th day of Kislev.[46]
Shemini Atzeret (Hebrew: "Eighth Day of Assembly") is a one-day Jewish holiday immediately followin' the oul' seven-day holiday of Sukkot.[47]
The spiritual Eighth Day, because the number 7 refers to the feckin' days of the week (which repeat themselves).
The number of Beatitudes.[48]
1 Peter 3:20 states that there were eight people on Noah's Ark.[49]
The Antichrist is the eighth kin' in the bleedin' Book of Revelation.[50]
In Islam, eight is the bleedin' number of angels carryin' the throne of Allah in heaven.[51]
The number of gates of heaven.
Ba Gua[52]
Ba Xian[53]
In Wicca, there are eight Sabbats, festivals, seasons, or spokes in the bleedin' Wheel of the feckin' Year.[54]
In Ancient Egyptian mythology, the feckin' Ogdoad represents the oul' eight primordial deities of creation.[55]
In Scientology there are eight dynamics of existence.[56]
There is also the Ogdoad in Gnosticism.[57]
As a lucky number
The number eight is considered to be a holy lucky number in Chinese and other Asian cultures.[58] Eight (八; accountin' 捌; pinyin bā) is considered a lucky number in Chinese culture because it sounds like the oul' word meanin' to generate wealth (發(T) 发(S); Pinyin: fā). Jesus, Mary and Joseph. Property with the number 8 may be valued greatly by Chinese. For example, a holy Hong Kong number plate with the bleedin' number 8 was sold for $640,000.[59] The openin' ceremony of the Summer Olympics in Beijin' started at 8 seconds and 8 minutes past 8 pm (local time) on 8 August 2008.[60]
In Pythagorean numerology (a pseudoscience) the bleedin' number 8 represents victory, prosperity and overcomin'.
Eight (八, hachi, ya) is also considered a holy lucky number in Japan, but the reason is different from that in Chinese culture.[61] Eight gives an idea of growin' prosperous, because the oul' letter (八) broadens gradually.
The Japanese thought of eight (や, ya) as a holy number in the feckin' ancient times. The reason is less well-understood, but it is thought that it is related to the bleedin' fact they used eight to express large numbers vaguely such as manyfold (やえはたえ, Yae Hatae) (literally, eightfold and twentyfold), many clouds (やくも, Yakumo) (literally, eight clouds), millions and millions of Gods (やおよろずのかみ, Yaoyorozu no Kami) (literally, eight millions of Gods), etc. Bejaysus this is a quare tale altogether. It is also guessed that the feckin' ancient Japanese gave importance to pairs, so some researchers guess twice as four (よ, yo), which is also guessed to be an oul' holy number in those times because it indicates the bleedin' world (north, south, east, and west) might be considered a feckin' very holy number.
In numerology, 8 is the feckin' number of buildin', and in some theories, also the number of destruction.
In astrology
In astrology, Scorpio is the feckin' 8th astrological sign of the Zodiac.[62]
In the bleedin' Middle Ages, 8 was the bleedin' number of "unmovin'" stars in the bleedin' sky, and symbolized the oul' perfection of incomin' planetary energy.
In music and dance
A note played for one-eighth the feckin' duration of a whole note is called an eighth note, or quaver.[63]
An octave, the oul' interval between two musical notes with the same letter name (where one has double the bleedin' frequency of the other), is so called because there are eight notes between the bleedin' two on a standard major or minor diatonic scale, includin' the notes themselves and without chromatic deviation.[64] The ecclesiastical modes are ascendin' diatonic musical scales of eight notes or tones comprisin' an octave.
There are eight notes in the bleedin' octatonic scale.
There are eight musicians in a bleedin' double quartet or an octet.[65] Both terms may also refer to a feckin' musical composition for eight voices or instruments.[66]
Caledonians is a square dance for eight, resemblin' the quadrille.
Albums with the oul' number eight in their title include 8 by the bleedin' Swedish band Arvingarna, 8 by the feckin' American rock band Incubus,[67] The Meanin' of 8 by Minnesota indie rock band Cloud Cult and 8ight by Anglo-American singer-songwriter Beatie Wolfe.[68]
Dream Theater's eighth album Octavarium contains many different references to the feckin' number 8, includin' the bleedin' number of songs and various aspects of the music and cover artwork.
"Eight maids a-milkin'" is the gift on the bleedin' eighth day of Christmas in the oul' carol "The Twelve Days of Christmas".[69]
The 8-track cartridge is an oul' musical recordin' format.
"#8" is the stage name of Slipknot vocalist Corey Taylor.
"Too Many Eights" is an oul' song by Athens, Georgia's Supercluster.[70]
Eight Seconds, a feckin' Canadian musical group popular in the feckin' 1980s with their most notable song "Kiss You (When It's Dangerous)".[71]
"Eight Days a Week" is a #1 single for the bleedin' music group The Beatles.[72]
Figure 8 is the oul' fifth studio album by singer-songwriter Elliott Smith, released in the feckin' year 2000,[73] an album released by Julia Darlin' in 1999,[74] and an album released by Outasight in 2011.[75]
Min' Hao from the k-pop group Seventeen goes by the name "The8".[76]
"8 (circle)" is the eighth song on the album 22, A Million by the feckin' American band Bon Iver.[77]
"8" is the bleedin' eighth song on the oul' album When We All Fall Asleep, Where Do We Go? by Billie Eilish.[78]
Russian NSBM band M8l8th have 8's in their name instead of the feckin' o's (see 88)
In film and television
8 Guys is an oul' 2003 short film written and directed by Dane Cook.
8 Man (or Eightman): 1963 Japanese manga and anime superhero.
8 Mile is a holy 2002 film directed by Curtis Hanson.[79]
8mm is an oul' 1999 film directed by Joel Schumacher.[80]
8 Women (Original French title: 8 femmes) is a 2001 film directed by François Ozon.[81]
Eight Below is a feckin' 2006 film directed by Frank Marshall.[82]
Eight Legged Freaks is a bleedin' 2002 film directed by Ellory Elkayem.[83]
Eight Men Out is a feckin' 1988 film directed by John Sayles.[84]
Jennifer Eight, also known as Jennifer 8, is an oul' 1992 film written and directed by Bruce Robinson.[85]
Eight Is Enough is an American television comedy-drama series.
In Stargate SG-1 and Stargate Atlantis, dialin' an 8-chevron address will open a bleedin' wormhole to another galaxy.
The Hateful Eight is a feckin' 2015 American western mystery film written and directed by Quentin Tarantino.[86]
Kate Plus 8 is an American reality television show.[87]
In sports and other games
An 8-ball in pool
Eight-ball pool is played with a cue ball and 15 numbered balls, the feckin' black ball numbered 8 bein' the feckin' middle and most important one, as the bleedin' winner is the oul' player or side that legally pockets it after first pocketin' its numerical group of 7 object balls (for other meanings see Eight ball (disambiguation)).
In chess, each side has eight pawns and the oul' board is made of 64 squares arranged in an eight by eight lattice. The eight queens puzzle is a challenge to arrange eight queens on the feckin' board so that none can capture any of the oul' others.
In the feckin' game of eights or Crazy Eights, each successive player must play a card either of the bleedin' same suit or of the bleedin' same rank as that played by the feckin' precedin' player, or may play an eight and call for any suit. G'wan now. The object is to get rid of all one's cards first.
In association football, the feckin' number 8 has historically been the oul' number of the oul' Central Midfielder.
In Australian rules football, the oul' top eight teams at the feckin' end of the bleedin' Australian Football League regular season qualify for the finals series (i.e. C'mere til I tell ya. playoffs).
In baseball:
The center fielder is designated as number 8 for scorekeepin' purposes.
The College World Series, the oul' final phase of the bleedin' NCAA Division I tournament, features eight teams.
In rugby union, the only position without a bleedin' proper name is the feckin' Number 8, a feckin' forward position.
In rugby league:
Most competitions (though not the bleedin' Super League, which uses static squad numberin') use a bleedin' position-based player numberin' system in which one of the oul' two startin' props wears the oul' number 8.
The Australia-based National Rugby League has its own 8-team finals series, similar but not identical in structure to that of the bleedin' Australian Football League.
In rowin', an "eight" refers to a holy sweep-oar racin' boat with a crew of eight rowers plus a coxswain.[88]
In the 2008 Games of the XXIX Olympiad held in Beijin', the oul' official openin' was on 08/08/08 at 8:08:08 p.m. CST.
In The Stanley Parable Demonstration, there is an eight button that, when pressed, says the bleedin' word eight.
In Mario Kart:
The Crazy Eight item in Mario Kart 8 allows the bleedin' player to use eight items at once.
The number of characters in the feckin' startin' grid in most games except for Mario Kart Wii and Mario Kart 8.
In foods
Nestlé sells a bleedin' brand of chocolates filled with peppermint-flavoured cream called After Eight, referrin' to the bleedin' time 8 p.m.[89]
There are eight vegetables in V8 juice.
In literature
Eights may refer to octosyllabic, usually iambic, lines of verse.
The drott-kvaett, an Old Icelandic verse, consisted of a bleedin' stanza of eight regular lines.[90]
In Terry Pratchett's Discworld series, eight is a magical number[91] and is considered taboo. In fairness now. Eight is not safe to be said by wizards on the bleedin' Discworld and is the oul' number of Bel-Shamharoth. C'mere til I tell yiz. Also, there are eight days in a feckin' Disc week and eight colours in a holy Disc spectrum, the eighth one bein' octarine.
Lewis Carroll's poem The Huntin' of the bleedin' Snark has 8 "fits" (cantos), which is noted in the feckin' full name "The Huntin' of the oul' Snark – An Agony, in Eight Fits."[92]
Eight apparitions appear to Macbeth in Act 4 scene 1 of Shakespeare's Macbeth as representations of the feckin' eight descendants of Banquo.
In shlang
An "eighth" is a feckin' common measurement of marijuana, meanin' an eighth of an ounce. Stop the lights! It is also a common unit of sale for psilocybin mushrooms, grand so. Also, an eighth of an ounce of cocaine is commonly referred to as an "8-ball."[93]
The numeral "8" is sometimes used in informal writin' and Internet shlang to represent the syllable "ate", as in writin' "H8" for "hate", or "congratul8ions" for "congratulations". I hope yiz are all ears now. Avril Lavigne's song "Sk8er Boi" uses this convention in the feckin' title.
"Section 8" is common U.S. shlang for "crazy", based on the feckin' U.S. Whisht now and eist liom. military's Section 8 discharge for mentally unfit personnel.
The Housin' Choice Voucher Program, operated by the United States Department of Housin' and Urban Development, is commonly referred to as the feckin' Section 8 program, as this was the original section of the bleedin' Act which instituted the feckin' program.[94]
In Colombia and Venezuela, "volverse un ocho" (meanin' to tie oneself in a figure 8) refers to gettin' in trouble or contradictin' oneself.
In China, "8" is used in chat speak as an oul' term for partin'. Sufferin' Jaysus listen to this. This is due to the closeness in pronunciation of "8" (bā) and the English word "bye".
The Magical Number Seven, Plus or Minus Two
List of highways numbered 8
^ 2020.
^ Weisstein, Eric W. Here's a quare one for ye. "Octagon". Jasus. mathworld.wolfram.com. Retrieved 7 August 2020.
^ Weisstein, Eric W. G'wan now. "Octahedron". mathworld.wolfram.com, begorrah. Retrieved 7 August 2020.
^ Weisstein, Eric W. "Cuboctahedron", would ye swally that? mathworld.wolfram.com. Retrieved 7 August 2020.
^ Weisstein, Eric W. "Cube", would ye swally that? mathworld.wolfram.com. Here's a quare one for ye. Retrieved 7 August 2020.
^ Weisstein, Eric W. "Sphenic Number", for the craic. mathworld.wolfram.com. Sufferin' Jaysus listen to this. Retrieved 7 August 2020. C'mere til I tell yiz. ...then every sphenic number n=pqr has precisely eight positive divisors
^ Lounesto, Pertti (3 May 2001). Clifford Algebras and Spinors. Cambridge University Press. In fairness now. p. 216. Would ye swally this in a minute now?ISBN 978-0-521-00551-7, would ye believe it? ...Clifford algebras, contains or continues with two kinds of periodicities of 8...
^ Inc, Boy Scouts of America (1931). Here's a quare one. Boys' Life. Boy Scouts of America, Inc, that's fierce now what? p. 20. lunge forward upon this skate in an oul' left outside forward circle, in just the oul' reverse of your right outside forward circle, until you complete an oul' figure 8.
^ Day, Cyrus Lawrence (1986), you know yerself. The Art of Knottin' & Splicin', the shitehawk. Naval Institute Press. p. 231. Right so. ISBN 978-0-87021-062-4. To make a feckin' line temporarily fast by windin' it , figure - eight fashion , round a bleedin' cleat , an oul' belayin' pin , or a holy pair of bitts.
^ Etymological Dictionary of Turkic Languages: Common Turkic and Interturkic stems startin' with letters «L», «M», «N», «P», «S», Vostochnaja Literatura RAS, 2003, 241f. Whisht now. (altaica.ru Archived 31 October 2007 at the oul' Wayback Machine)
^ the hypothesis is discussed critically (and rejected as "without sufficient support") by Werner Winter, 'Some thought about Indo-European numerals' in: Jadranka Gvozdanović (ed.), Indo-European Numerals, Walter de Gruyter, 1992, 14f.
^ Georges Ifrah, The Universal History of Numbers: From Prehistory to the oul' Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 395, Fig. Sure this is it. 24.68.
^ Ilangovan, K. Arra' would ye listen to this shite? (10 June 2019). Nuclear Physics. MJP Publisher. C'mere til I tell ya now. p. 30.
^ Gell-Mann, M, begorrah. (15 March 1961). Whisht now. THE EIGHTFOLD WAY: A THEORY OF STRONG INTERACTION SYMMETRY (Technical report), the cute hoor. OSTI 4008239.
^ Baxter, R. I hope yiz are all ears now. J. C'mere til I tell yiz. (5 April 1971), be the hokey! "Eight-Vertex Model in Lattice Statistics". In fairness now. Physical Review Letters, you know yourself like. 26 (14): 832–833. Bibcode:1971PhRvL..26..832B, Lord bless us and save us. doi:10.1103/PhysRevLett.26.832.
^ "Messier Object 8". www.messier.seds.org. Retrieved 7 August 2020.
^ Thomas, Mary Ann (15 August 2004). Oxygen, begorrah. The Rosen Publishin' Group, Inc. Story? p. 12. ISBN 978-1-4042-0159-0, Lord bless us and save us. Knowin' that oxygen has an atomic number of 8,
^ Choppin, Gregory R.; Johnsen, Russell H, that's fierce now what? (1972). Introductory chemistry, the hoor. Addison-Wesley Pub. Whisht now and eist liom. Co. Arra' would ye listen to this. p. 366. Here's a quare one for ye. ISBN 9780201010220. under normal conditions the oul' most stable allotropic form (Fig. 23-8a). Soft oul' day. Sulfur molecules within the oul' crystal consist of puckered rings of eight sulfur atoms linked by single...
^ Puri, Basant; Hall, Anne (16 December 1998). Arra' would ye listen to this shite? Phytochemical Dictionary: A Handbook of Bioactive Compounds from Plants, Second Edition. Sufferin' Jaysus listen to this. CRC Press, grand so. p. 810. ISBN 978-0-203-48375-6. C'mere til I tell yiz. The chemical structure of lycopene consists of an oul' long chain of eight isoprene units joined head to tail
^ Parker, Barbara Keevil (28 December 2006), to be sure. Ticks. Lerner Publications. Be the hokey here's a quare wan. p. 7, Lord bless us and save us. ISBN 978-0-8225-6464-5. Arachnids have eight legs
^ Jackman, J. Whisht now and listen to this wan. A. Jaykers! (1997). Here's another quare one. A Field Guide to Spiders & Scorpions of Texas. Arra' would ye listen to this. Gulf Publishin' Company. p. 70, the shitehawk. ISBN 978-0-87719-264-0, enda story. Araneids have eight eyes
^ Fisher, James; Huxley, Julian (1961). The Doubleday Pictorial Library of Nature: Earth, Plants, Animals, Lord bless us and save us. Doubleday, the hoor. p. 311. Polyps with eight branched tentacles and eight septa
^ Bourne, Gilbert Charles (1911). "Anthozoa" , be the hokey! In Chisholm, Hugh (ed.), you know yerself. Encyclopædia Britannica. 02 (11th ed.). Stop the lights! Cambridge University Press. pp. 97–105, see page 100. Be the holy feck, this is a quare wan. Zoantharia.....It is not known whether all the eight mesenteries of Edwardsia are developed simultaneously or not, but in the feckin' youngest form which has been studied all the bleedin' eight mesenteries were present
^ The Century Dictionary and Cyclopedia: A work of Universal Reference in all Departments of Knowledge with a holy New Atlas of the bleedin' World. Arra' would ye listen to this shite? 1906. Here's another quare one for ye. p. 1384. ...are radially symmetrical, and swim by means of eight meridional ciliated bands, ...
^ Parrish, Fred K. (1975). Arra' would ye listen to this. Keys to Water Quality Indicative Organisms of the oul' Southeastern United States, so it is. Environmental Protection Agency, Office of Research and Development, Environmental Monitorin' and Support Laboratory, Biological Methods Branch, Aquatics Biology Section. C'mere til I tell ya. p. 11. Jesus Mother of Chrisht almighty. ... C'mere til I tell ya now. the feckin' ascospores, are borne in sac like structures termed asci. Jesus, Mary and holy Saint Joseph. The ascus usually contains eight as cospores,...
^ Dofka, Charline M. (1996). Competency Skills for the oul' Dental Assistant, would ye swally that? Cengage Learnin'. p. 83, for the craic. ISBN 978-0-8273-6685-5. G'wan now and listen to this wan. ...In each quadrant of the feckin' permanent set of teeth (dentition), there are eight teeth
^ Quain, Jones (1909). C'mere til I tell ya. Quain's Elements of Anatomy. Listen up now to this fierce wan. Longmans, Green, & Company. Arra' would ye listen to this shite? p. 52. These eight pairs are usually reckoned as eight cervical nerves ...
^ Beebe, John (17 June 2016). Energies and Patterns in Psychological Type: The reservoir of consciousness. Routledge, the shitehawk. p. 124. Be the hokey here's a quare wan. ISBN 978-1-317-41366-0. Linda Berens used the term 'cognitive processes' (1999) to refer to the bleedin' eight types of consciousness that Jung discovered.
^ "Definition of byte | Dictionary.com", you know yerself. www.dictionary.com. Retrieved 8 August 2020.
^ Kindem, Gorham; PhD, Robert B. I hope yiz are all ears now. Musburger (21 August 2012). Introduction to Media Production: The Path to Digital Media Production. CRC Press, what? p. 320. ISBN 978-1-136-05322-1. Soft oul' day. There used to be two 8 mm formats: standard 8 mm and Super-8 mm.
^ The Library of Congress Veterans History Project: Field Kit : Conductin' and Preservin' Interviews. Veterans History Project, American Folklife Center, Library of Congress. Chrisht Almighty. 2008, like. p. 15. Betacam SX 8mm Hi8, Digital8, Video8 DVD-Video";
^ "Definition of eight | Dictionary.com". Sure this is it. www.dictionary.com. Sufferin' Jaysus listen to this. Retrieved 8 August 2020.
^ Griffiths, Garth (1971). Boatin' in Canada: Practical Pilotin' and Seamanship. Right so. University of Toronto Press, like. p. 32, the cute hoor. ISBN 978-0-8020-1817-5, what? First is a bleedin' stopper knot , the bleedin' figure of eight , ...
^ The Milwaukee Cook Book. Press of Houtkamp Printin', begorrah. 1907.
^ "Definition of furlong | Dictionary.com". Story? www.dictionary.com. Whisht now and listen to this wan. Retrieved 8 August 2020.
^ "Definition of clove | Dictionary.com", be the hokey! www.dictionary.com. Retrieved 8 August 2020.
^ Fairhall, David; Peyton, Mike (17 May 2013). Would ye believe this shite?Pass Your Yachtmaster. Whisht now. A&C Black. ISBN 978-1-4081-5627-8. Gale warnings will be given if mean wind speeds of force 8 (34–40 knots)
^ Hatcher, Brian A. (5 October 2015). Hinduism in the bleedin' Modern World. Jasus. Routledge. ISBN 978-1-135-04630-9. a group manifestation of eight forms
^ Jeyaraj, Daniel (23 September 2004). Bejaysus. Genealogy of the South Indian Deities: An English Translation of Bartholomäus Ziegenbalg's Original German Manuscript with a feckin' Textual Analysis and Glossary. Routledge. p. 168. ISBN 978-1-134-28703-1. Here's a quare one. He is one of the bleedin' eight guardians of the oul' world
^ Ramachandran, Nirmala (2000). C'mere til I tell ya now. Hindu Heritage, game ball! Stamford Lake Publication. p. 72. ISBN 978-955-8733-09-7, like. The temple has eight monasteries, founded by Madhvacharya ,
^ Issitt, Micah; Main, Carlyn (16 September 2014), like. Hidden Religion: The Greatest Mysteries and Symbols of the feckin' World's Religious Beliefs: The Greatest Mysteries and Symbols of the World's Religious Beliefs. Sure this is it. ABC-CLIO. p. 186, enda story. ISBN 978-1-61069-478-0, the cute hoor. The dharmachakra is typically depicted with eight spokes,
^ Hay, Jeff (6 March 2009). C'mere til I tell ya now. World Religions. I hope yiz are all ears now. Greenhaven Publishin' LLC. p. 61. ISBN 978-0-7377-4627-3. The focus of ordinary believers' religious life is on followin' a holy relevant version of the feckin' Eightfold Path ...
^ Rosten, Leo (14 April 2010). Here's another quare one for ye. The New Joys of Yiddish: Completely Updated. Here's another quare one. Potter/Ten Speed/Harmony/Rodale. p. 48, would ye swally that? ISBN 978-0-307-56604-1. Brit Milah is observed on a feckin' boy's eighth day of life
^ Ross, Kathy (1 August 2012). Whisht now. Crafts for Hanukkah. Me head is hurtin' with all this raidin'. Millbrook Press. Sufferin' Jaysus. p. 7. Jaykers! ISBN 978-0-7613-6836-6. Bejaysus here's a quare one right here now. Hanukkah is an eight-day Jewish holiday
^ Axelrod, Cantor Matt (24 December 2013), to be sure. Your Guide to the bleedin' Jewish Holidays: From Shofar to Seder. Rowman & Littlefield. Jesus, Mary and Joseph. p. 58. ISBN 978-0-7657-0990-5. Jesus, Mary and Joseph. Shemini Atzeret—literally, "the eighth day of assembly"
^ "CATHOLIC ENCYCLOPEDIA: The Eight Beatitudes". Jesus Mother of Chrisht almighty. www.newadvent.org. Would ye believe this shite?Retrieved 9 August 2020.
^ Akintola, Olufolahan Olatoye (2011). Here's another quare one. Nations of the bleedin' World…How They Evolved!: Families and Nations That Came Out of Ham. Hilldew View International Limited. p. 8. Bejaysus this is a quare tale altogether. ISBN 978-0-9569702-2-0, bejaysus. These eight souls in Noah's ark were the oul' survivors...
^ "Life Application New Testament Commentary" Archived 11 May 2013 at the feckin' Wayback Machine, Bruce B. Whisht now and listen to this wan. Barton, that's fierce now what? Tyndale House Publishers, Inc., 2001. Sure this is it. ISBN 0-8423-7066-8, ISBN 978-0-8423-7066-0, the cute hoor. p. Whisht now and eist liom. 1257
^ Mahmutćehajić, Rusmir (2011). Maintainin' the Sacred Center: The Bosnian City of Stolac. World Wisdom, Inc. Here's a quare one for ye. p. 201. ISBN 978-1-935493-91-4. Right so. ... Here's another quare one for ye. at the oul' last, eight Angels will carry the bleedin' Throne...
^ Little, Stephen; Eichman, Shawn; Shipper, Kristofer; Ebrey, Patricia Buckley (1 January 2000). Taoism and the oul' Arts of China. Jasus. University of California Press. Holy blatherin' Joseph, listen to this. p. 139. G'wan now and listen to this wan. ISBN 978-0-520-22785-9. Sufferin' Jaysus listen to this. Evidence for the oul' early use of the bleedin' Eight Trigrams in an oul' religious Taoist...
^ Ho, Peter Kwok Man; Kwok, Man-Ho; O'Brien, Joanne (1990). The Eight Immortals of Taoism: Legends and Fables of Popular Taoism. Meridian. Right so. p. 7, to be sure. ISBN 978-0-452-01070-3. ...famous Eight Immortals of China...
^ Zimmermann, Denise; Gleason, Katherine; Liguana, Miria (2006), game ball! The Complete Idiot's Guide to Wicca and Witchcraft. Arra' would ye listen to this shite? Penguin. Arra' would ye listen to this shite? p. 172. ISBN 978-1-59257-533-6. There are eight Sabbats
^ Remler, Pat (2010). Here's a quare one. Egyptian Mythology, A to Z. Be the hokey here's a quare wan. Infobase Publishin'. Here's another quare one. p. 79. ISBN 978-1-4381-3180-1, that's fierce now what? ...of the bleedin' gods of the Ogdoad, or the oul' eight deities of the oul' Egyptian creation...
^ Neusner, Jacob (1 January 2003). World Religions in America: An Introduction. Here's a quare one. Westminster John Knox Press. p. 222. Right so. ISBN 978-0-664-22475-2. Here's a quare one for ye. represent what Scientology refers to as the oul' eight "dynamics" of existence.
^ David, Fideler (1993). Jesus Christ, Sun of God: Ancient Cosmology and Early Christian Symbolism. Would ye swally this in a minute now?Quest Books. Bejaysus. p. 128. G'wan now. ISBN 978-0-8356-0696-7.
^ Ang, Swee Hoon (1997). Sufferin' Jaysus listen to this. "Chinese consumers' perception of alpha-numeric brand names". Be the holy feck, this is a quare wan. Journal of Consumer Marketin'. C'mere til I tell ya. 14 (3): 220–233. doi:10.1108/07363769710166800. Archived from the oul' original on 5 December 2011.
^ Steven C. Listen up now to this fierce wan. Bourassa; Vincent S, be the hokey! Peng (1999). Here's a quare one. "Hedonic Prices and House Numbers: The Influence of Feng Shui" (PDF). Jesus, Mary and holy Saint Joseph. International Real Estate Review. 2 (1): 79–93. Whisht now and listen to this wan. Archived from the original (PDF) on 13 April 2015. Retrieved 11 May 2011.
^ "Patriot games: China makes its point with greatest show" Archived 5 March 2017 at the oul' Wayback Machine by Richard Williams, The Guardian, published 9 August 2008
^ Jefkins, Frank (6 December 2012). Modern Marketin' Communications. Springer Science & Business Media. p. 36, be the hokey! ISBN 978-94-011-6868-7, bejaysus. ...eight bein' a lucky number in Japanese.
^ "Definition of SCORPIO". Jesus Mother of Chrisht almighty. www.merriam-webster.com. Jesus Mother of Chrisht almighty. Retrieved 10 August 2020.
^ "Definition of eighth note | Dictionary.com". www.dictionary.com, for the craic. Retrieved 9 August 2020.
^ "Definition of OCTAVE". www.merriam-webster.com. Retrieved 9 August 2020. Jesus, Mary and holy Saint Joseph. a tone or note that is eight steps above or below another note or tone
^ "Definition of octet | Dictionary.com". Chrisht Almighty. www.dictionary.com, bejaysus. Retrieved 9 August 2020. a company of eight singers or musicians.
^ "Definition of octet | Dictionary.com". www.dictionary.com. Retrieved 9 August 2020. In fairness now. a musical composition for eight voices or instruments.
^ "Incubus Premiere New Song "Glitterbomb", Detail New Album "8"". Would ye believe this shite?Theprp.com. Holy blatherin' Joseph, listen to this. 17 March 2017. Here's another quare one for ye. Retrieved 9 August 2020.
^ Beatie Wolfe-8ight, retrieved 9 August 2020
^ Tribble, Mimi (2004). 300 Ways to Make the feckin' Best Christmas Ever!: Decorations, Carols, Crafts & Recipes for Every Kind of Christmas Tradition, grand so. Sterlin' Publishin' Company, Inc, bedad. ISBN 978-1-4027-1685-0.
^ Too Many Eights - Supercluster | Song Info | AllMusic, retrieved 10 August 2020
^ "Eight Seconds | Biography & History", what? AllMusic. Retrieved 10 August 2020.
^ Eight Days a feckin' Week - The Beatles | Song Info | AllMusic, retrieved 10 August 2020
^ Figure 8 - Elliott Smith | Songs, Reviews, Credits | AllMusic, retrieved 10 August 2020
^ Figure 8 - Julia Darlin' | Songs, Reviews, Credits | AllMusic, retrieved 10 August 2020
^ Figure 8 - Outasight | Songs, Reviews, Credits | AllMusic, retrieved 10 August 2020
^ "The8 | Credits". AllMusic. C'mere til I tell yiz. Retrieved 10 August 2020.
^ 22, A Million - Bon Iver | Songs, Reviews, Credits | AllMusic, retrieved 10 August 2020
^ When We All Fall Asleep, Where Do We Go? - Billie Eilish | Songs, Reviews, Credits | AllMusic, retrieved 10 August 2020
^ 8 Mile (2002) - Curtis Hanson | Synopsis, Characteristics, Moods, Themes and Related | AllMovie, retrieved 10 August 2020
^ 8MM (1999) - Joel Schumacher | Synopsis, Characteristics, Moods, Themes and Related | AllMovie, retrieved 10 August 2020
^ 8 Women (2001) - François Ozon | Synopsis, Characteristics, Moods, Themes and Related | AllMovie, retrieved 10 August 2020
^ Eight Below (2006) - Bruce Hendricks, Frank Marshall | Synopsis, Characteristics, Moods, Themes and Related | AllMovie, retrieved 10 August 2020
^ Eight Legged Freaks (2002) - Ellory Elkayem | Synopsis, Characteristics, Moods, Themes and Related | AllMovie, retrieved 10 August 2020
^ Eight Men Out (1988) - John Sayles | Synopsis, Characteristics, Moods, Themes and Related | AllMovie, retrieved 10 August 2020
^ Jennifer Eight (1992) - Bruce Robinson | Synopsis, Characteristics, Moods, Themes and Related | AllMovie, retrieved 10 August 2020
^ The Hateful Eight (2015) - Quentin Tarantino | Synopsis, Characteristics, Moods, Themes and Related | AllMovie, retrieved 10 August 2020
^ Jon & Kate Plus 8 (2007) - | Synopsis, Characteristics, Moods, Themes and Related | AllMovie, retrieved 10 August 2020
^ "Definition of EIGHT". www.merriam-webster.com, so it is. Retrieved 10 August 2020.
^ "Buy After Eight® Online | Nestlé Family ME". www.nestle-family.com, the shitehawk. Retrieved 10 August 2020.
^ "Definition of DROTT-KVAETT", grand so. www.merriam-webster.com, enda story. Retrieved 10 August 2020.
^ Collins, Robert; Latham, Robert (1988). Sufferin' Jaysus listen to this. Science Fiction & Fantasy Book Review Annual. Meckler. p. 289. Jesus, Mary and Joseph. ISBN 9780887362491.
^ "The Huntin' of the Snark". G'wan now. www.gutenberg.org. Sure this is it. Retrieved 10 August 2020.
^ "Cocaine - Frequently Asked Questions", the hoor. thegooddrugsguide.com. Archived from the bleedin' original on 6 October 2008. C'mere til I tell yiz. Retrieved 29 September 2008.
^ "CT Housin' Choice Voucher Program". G'wan now and listen to this wan. www.cthcvp.org. In fairness now. Retrieved 11 August 2020. Welcome to the oul' Housin' Choice Voucher Program (also known as Section 8)
The Octonions, John C. Here's a quare one for ye. Baez
Retrieved from "https://en.wikipedia.org/w/index.php?title=8&oldid=1060741715"
8 (number)
Mickopedia articles incorporatin' a citation from the feckin' 1911 Encyclopaedia Britannica with Wikisource reference
Mickopedia semi-protected pages
Articles containin' Chinese-language text
Articles needin' the year an event occurred from October 2014
Articles containin' Japanese-language text | CommonCrawl |
\begin{document}
\title
{ Regularity results on a class of doubly nonlocal problems }
\author{ Jacques Giacomoni$^{\,1}$ \footnote{e-mail: {\tt [email protected]}}, \ Divya Goel$^{\,2}$\footnote{e-mail: {\tt [email protected]}}, \
and \ K. Sreenadh$^{\,2}$\footnote{
e-mail: {\tt [email protected]}} \\
\\ $^1\,${\small Universit\'e de Pau et des Pays de l'Adour, LMAP (UMR E2S-UPPA CNRS 5142) }\\ {\small Bat. IPRA, Avenue de l'Universit\'e F-64013 Pau, France}\\
$^2\,${\small Department of Mathematics, Indian Institute of Technology Delhi,}\\
{\small Hauz Khaz, New Delhi-110016, India } }
\date{}
\maketitle
\begin{abstract}
\noi The purpose of this article is twofold. First, an issue of regularity of weak solution to the problem $(P)$ (See below) is addressed. Secondly, we investigate the question of $H^s$ versus $C^0$- weighted minimizers of the functional associated to problem $(P)$ and then give applications to existence and multiplicity results.
\noi \textbf{Key words}: Choquard equation, fractional Laplacian, regularity, uniform bound, singular nonlinearity, local minimizers.
\noi \textit{2010 Mathematics Subject Classification: 35B45, 35R09, 35B65, 35B33 }
\end{abstract} \section{Introduction} In this article we will study the following problem: \begin{equation*} (P)\; \left\{\begin{array}{rllll} (-\De)^s u
&=g(x,u)+ \left(\ds \int_{\Om}\frac{F(u)(y)}{|x-y|^{\mu}}dy\right) f(u) \; \text{in}\; \Om,\\ u&=0 \; \text{ in } \mathbb{R}^N \setminus \Om \end{array} \right. \end{equation*} where $\Om$ is a smooth bounded domain in $\mathbb R^n, N\ge 2$, $s \in (0,1)$, $\mu <N$, $g:\Om\times \mathbb R\to \mathbb R$ Carath\'edory function, $f:\mathbb R \rightarrow \mathbb R$ is a continuous function and $F$ is the primitive of $f$. Here the operator $(-\De)^s$ is the fractional Laplacian defined up to a positive multiplicative constant as \begin{align*}
(-\De)^su(x)=\text{P.V. } \int_{ \mathbb{R}^N} \frac{u(x)-u(y)}{|x-y|^{N+2s}} dy \end{align*} where P.V. denotes the Cauchy principal value.
The existence and regularity of weak solutions have been a fascinating topic for the researchers for a long time. The work on Choquard equations was started with the quantum theory of a polaron model given by S. Pekar \cite{pekar}. In 1976, in the modeling of a one component plasma, P. Choquard \cite{ehleib} used the following equation with $\mu=1,\; p=2$ and $N=3$: \begin{equation}\label{ch26}
-\De u +u = \left(\frac{1}{|x|^{\mu}}* F(u)\right) f(u) \text{ in } \mathbb{R}^3 \end{equation}
where $f(u)=|u|^{p-2}u$ and $F^\prime =f$. In \cite{moroz2}, Moroz and Schaftingen established the existence of a ground state solution and the regularity of weak solutions of the problem \eqref{ch26} in higher dimensions $N \geq 3 , \mu \in (0,N)$ and with more general functions $F\in C^1(\mathbb{R},\mathbb{R})$ satisfying certain growth conditions.
For more results on the existence of solutions we refer to \cite{moroz4,moroz5} and the references therein. In \cite{yang}, Yang and Gao studied the Brezis-Nirenberg type result for the following equation \begin{equation*}
-\Delta u = \la u+ \left(\int_{\Om}\frac{|u(y)|^{2^*_{\mu}}}{|x-y|^{\mu}}dy\right)|u|^{2^*_{\mu}-2}u \text{ in } \Om, \quad u=0 \text{ on } \pa \Om, \end{equation*} \noi where $\Om\subset \mb R^N, N\ge 3$ is a bounded domain having smooth boundary $\pa \Om$, $\la>0$, $0<\mu<N$ and $ 2^*_\mu = \frac{2N-\mu}{N-2}$. Later, many researchers studied the Choquard equation for the existence and multiplicity of solutions, for instance see \cite{alves,yang1, ts} and references therein. \\
On the other hand, in recent years, the subject of nonlocal elliptic equations involving fractional Laplacian
has gained more popularity because of many applications such as continuum mechanics, game theory and phase transition phenomena.
For an extensive survey on fractional Laplacian and its applications, one may refer to \cite{valdi, stinga} and references therein.
The nonlocal equations with Hartree-type nonlinearities were used to model the dynamics of pseudo-relativistic boson stars. In fractional quantum mechanics, fractional Schr\"odinger equations play an important role, for instance see \cite{frank,zhang,ts}. For the existence and multiplicity results on fractional Laplacian, readers can refer to \cite{servadei} and references therein. For the doubly nonlocal problem, precisely, the nonlocal elliptic equation involving fractional Laplacian and Choquard type nonlinearity, there are articles which discuss the existence and multiplicity of solutions, we cite \cite{ambrosio,avenia,pucci,zhang} and references therein, with no attempt to provide a complete list.
Regularity results about problem involving fractional diffusion are also attracting a large number of researchers.
Consider the following nonlocal problem
\begin{equation}\label{ch24}
(-\Delta)^s u =g\text{ in } \Om, \quad
u=h \text{ in } \mathbb{R}^N\setminus \Om.
\end{equation} The interior regularity of solutions to \eqref{ch24} is primarily determined by Caffarelli and Silvestre. In \cite{caffe1}, authors developed the $C^{1+\al}$ interior regularity for viscosity solutions to
nonlocal equations with bounded measurable coefficients. For the convex equation, authors proved $C^{2s+\al}$ regularity in \cite{caffe2} while in \cite{caffe3}, authors established a perturbative theory for non translation invariant equations. In \cite{silves}, Silvestre studied regularity of weak solutions to free boundary problem.
For the boundary regularity, Ros-Oton and Serra \cite{RS} studied the regularity of weak solutions to \eqref{ch24} with $h=0$ and $g \in L^\infty(\Om)$. By using a suitable
upper barrier and the interior regularity results for the fractional Laplacian they prove that $u\in C^s(\mathbb R^N)$ and $\|u\|_{C^s} \le c \|g\|_{L^\infty(\Om)}$ for some constant $c$. Moreover, authors established a fractional analog of the Krylov boundary Harnack method to further prove $u \in C_d^{0,\al}(\overline{\Om})$ for some $\al \in (0,1)$.
In \cite{RS1}, authors proved the high integrability of the weak solution by using the regularity of Riesz potential established in \cite{stein}.
In \cite{adi}, authors discussed the existence and regularity of weak solution to the following problem
\begin{equation*}
(-\De)^s u
=u^{-q}+ f(u) \; \text{in}\;
\Om,\;
u=0 \; \text{ in } \mathbb{R}^N \setminus \Om
\end{equation*}
where $q>0$ and the function $f$ is of subcritical growth. When $f$ has critical growth then the question of existence and regularity have been answered in \cite{gts1}.
Despite the ample amount of research on doubly nonlocal problems, there is very little done in respect of regularity of weak solutions to these problems. For instance, in \cite{avenia}, authors proved the regularity of a ground state solution of doubly nonlocal equation with subcritical growth in the sense of Hardy-Littlewood-Sobolev inequality, by generalizing the idea of \cite{moroz4} in fractional framework. In \cite{su}, authors establish the $L^\infty(\mathbb R)$ bound of the nonnegative ground state solution of doubly local problem with critical growth in the sense of Hardy-Littlewood-Sobolev inequality under the assumption that $\mu < \min\{ N, 4s\}$.
In \cite{yangjmaa}, Gao and Yang studied the Dirichlet problem involving Choquard nonlinearity with Laplacian operator.
Here authors aim to prove the regularity for weak solutions. The boot-strap techniques as it is developed in \cite{yangjmaa} work for the subcritical growth
and seems to fail in handling the critical non linearity in the sense of Hardy-Littlewood-Sobolev inequality. For the critical case, Moroz and Schaftingen \cite{moroz2}, studied problem \eqref{ch26} and prove the $W^{2,p}_{\text{loc}}(\mathbb R^N), \; p>1$, regularity of the weak solution for problems in the whole space without a perturbation term $g(x,u)$. The techniques given in \cite{moroz2} cannot be straightforward carried to problem $(P)$ in a general setting. The regularity of positive solution to the following singular problem
\begin{equation}\label{ch34}
-\De u
= u^{q-1} + \left(\ds \int_{\Om}\frac{F(u)(y)}{|x-y|^{\mu}}dy\right) f(u) \; \text{in}\;
\Om,
u=0 \; \text{ in } \mathbb{R}^N \setminus \Om, ~ 0<q<1
\end{equation}
was also an open problem.
Motivated by the above discussion and the stated issues, the first part of the present article is intended to address the question of $L^\infty(\Om) $ bound for weak solutions of the problem $(P)$ covering large classes of $f$ and $g$. Since once $L^\infty(\Om)$ is there then one can use the result given by Ros-Oton \cite{RS,silves} coupled with Hardy-Littlewood-Sobolev inequality, to prove the desired regularity results.
To prove the $L^\infty(\Om)$ bound, we develop an unified approach handling both subcritical and critical case of the perturbation $g$. In this article we also provide an answer to the regularity of weak solutions to doubly nonlocal equation involving singular nonlinearity, particularly problem \eqref{ch34}. The existence and multiplicity of solutions to problem \eqref{ch34}, is specially address in \cite{jds}. The novelty of the obtained results here is that they hold true for all $\mu<N$, contrasting to previous regularity results in literature.
The techniques and tools which are used here to prove the $L^\infty(\Om)$ estimate are contemporary and new. Precisely, we extend further the classical Brezis-Kato techniques \cite{kato} to improve the integrability of weak solutions to (P). In addition, we mention that to the best of our knowledge, there is no article which establish the proof of $L^\infty(\Om)$ bound to problem involving singular nonlinearity. The results in this article can be used similarly to Laplacian operator (that is, $s=1$) and are also new to the literature.
The second part of this article is destined to prove the $H^s$ versus $C^0$- weighted minimizers. That is, we show that the local minima with respect to $C_d^0(\overline{\Om})$ topology will also be a local minima with respect to $X_0$ topology.
In variational problems this result illustrate a significant role as it helps to prove that the solutions to constraint minimization of the energy functional emerge as solutions to unconstraint local minimization of the energy functional.
This procedure of constraint minimizations has ample amount of applications such as to prove the existence and multiplicity of solutions to elliptic problems, for instance see Theorem \ref{thmch4}.
In case of local framework this result was first done by Brezis and Nirenberg \cite{niren}.
Here authors prove that local minima in $C^1 $ will remain so in $H^1$ topology despite of the fact that latter one is weaker than the former one. In fractional framework, this result is proved by Iannizzotto, Mosconi and Squassina \cite{squassina}.
But in case of nonlocal nonlinearity, in particular, Choquard equation, a particular case to our result had been answered by \cite{yangjmaa} for the Laplacian operator. For the general nonlinearity, this issue is recently posed as an open problem in \cite{ts}.
In this article, we also provide a full answer to this open problem. Since there is significant amount of difference in handling doubly nonlocal problem, so we cannot stick around the tools given in \cite{niren,squassina} to establish the result.
\begin{Remark}
We would like to remark that the results of our article can be adapted to the following fractional Schr\"odinger problem
\begin{equation*}
(-\De)^s u + Vu
=g(x,u)+ \left(\ds \int_{\Om}\frac{F(u)(y)}{|x-y|^{\mu}}dy\right) f(u) \; \text{in}\;
\Om, u=0 \; \text{ in } \mathbb{R}^N \setminus \Om,
\end{equation*}
where $ V\in L^2(\Om)$ and $ (-\De)^s + V$ should be coercive in the energy space $X_0$. \end{Remark}
\section{Functional framework and main results} This section of the article is intended to provide the fractional Sobolev space setting. For the complete and rigid details, one can refer \cite{nezzaH,servadei}. Further in this section we state the main results of current article with a short sketch of proof. \\ For $0<s<1$, the fractional Sobolev space is defined as \begin{align*}
H^{s}(\mathbb{R}^N)= \left\lbrace u \in L^2(\mathbb{R}^N): \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}~dxdy <+ \infty \right\rbrace \end{align*} \noi endowed with the norm \begin{align*}
\|u\|_{H^{s}(\mathbb{R}^N)}:= \|u\|_{L^2(\mathbb{R}^N)}+ [u]_{H^s(\mathbb R^N)} = \|u\|_{L^2(\mathbb{R}^N)}+\left(\int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}~dxdy \right)^{\frac{1}{2}}. \end{align*} \noi Consider the space \begin{align*} X_0:= \{u \in H^{s}(\mathbb{R}^N): u=0 \text{ a.e in } \mathbb{R}^N \setminus \Om \} \end{align*}
equipped with the norm \begin{align*}
\ld u,v \rd = \int_{Q} \frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+2s}}~dxdy \end{align*} where $ Q= \mathbb{R}^N \setminus (\Om^c\times \Om^c)$. From the embedding results (\cite{servadei}), the space $X_0$ is continuously embedded into $L^r(\mathbb{R}^N)$ with $ r\in [1,2^*_s]$ where $2^*_s= \frac{2N}{N-2s}$. The best constant $S_s$ is defined \begin{align}\label{ch27}
S_s= \inf_{u \in X_0\setminus \{ 0\}} \frac{\int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}~dxdy }{\left( \int_{ \Om} |u|^{2^*_s}~dx\right)^{2/2^*_s}}. \end{align} Let $d: \overline{\Om} \ra \mathbb{R}_+ $ by $d(x):= \text{dist}(x,\mathbb{R}^N\setminus \Om),\; x \in \overline{\Om}$. The best constant $S_H$ is defined as \begin{align}\label{ch35}
S_H= \inf_{u \in X_0\setminus \{ 0\}} \frac{\int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}~dxdy }{ \int_{ \Om}\frac{ |u|^{2}}{d^{2s}}~dx}. \end{align} Now we define the weighted H\"older-type spaces \begin{align*} & C^0_d(\overline{\Om}) := \bigg \{ u \in C^0(\overline{\Om}): u/d^s \text{ admits a continuous extension to } \overline{\Om} \bigg\},\\ & C^{0,\al}_d(\overline{\Om}) := \bigg \{ u \in C^0(\overline{\Om}): u/d^s \text{ admits a } \al \text{ -H\"older continuous extension to } \overline{\Om} \bigg\} \end{align*} endowed with the norms \begin{align*}
\|u\|_{0,d}:= \|u/d^s\|_{\infty}, \quad \|u\|_{\al,d}:= \|u\|_{0,d}+ \sup_{x, y \in \overline{\Om}, x\not = y } \frac{|u(x)/(x)d^s- u(y)/d(y)^s|}{|x-y|^{\al}} \end{align*} respectively. We assume that $f$ satisfies the following growth conditions throughout the current article.\\ $(\mc F) \quad F\in C^1(\mathbb{R},\mathbb{R})$, $ F^\prime= f$ and there exists $C>0$ such that for all $t\in \mathbb R$, \begin{align*}
|tf(t)| \leq C(|t|^{\frac{2N-\mu}{N}}+ |t|^{\frac{2N-\mu}{N-2s}}). \end{align*} \begin{Definition} A function $u\in X_0$ with $u\equiv 0$ in $\mathbb R^N \backslash \Omega$ is said to be a solution to (P) if
\begin{equation*}
\int_{Q} \frac{(u(x)-u(y))(\phi(x)-\phi(y))}{|x-y|^2}~dxdy= \la\int_{\Om} g(x,u )\phi ~dx + \iint_{\Om\times \Om} \frac{F(u)f(u)}{|x-y|^\mu} \phi ~dxdy
\end{equation*}
for all $\phi \in X_0$.
\end{Definition} Let $G(x,u) = \int_{0}^{u} g(x, \tau )~d\tau$ then functional associated with problem $(P)$ is defined as \begin{align*}
J(u) = \frac{\|u\|^2}{2}-\int_{\Om}G(x,u)~dx -\frac{1}{2} \iint_{\Om\times \Om} \frac{F(u)F(u)}{|x-y|^\mu} ~dxdy, \text{ for all } u \in X_0. \end{align*}
With this functional framework, we state the main results of the article. First we state the result about the regularity of weak solution to problem $(P)$.
\begin{Theorem}\label{thmch1}
Let $ g: \overline{\Om} \times \mathbb{R} \ra \mathbb{R}$ be a Carath\'eodory function satisfying
\begin{align*}
g(x, u) = O(|u|^{2^*_s-1}), & \text{ if } |u|\ra \infty
\end{align*}
uniformly for all $ x \in \overline{\Om}$. Then any solution $u \in X_0$ of $(P)$ belongs to $L^\infty(\mathbb{R}^N) \cap C^s(\mathbb{R}^N)$. Furthermore, there exists positive constant $C$ depending on $N,\mu, s, |\Om|$ such that\\ $|u|_\infty\leq C (1+|u|_{2^*_s})^{\frac{2}{(2^*-1)(2^*-2)}}\left( 1+ \left( (1+|u|_{2^*_s}) \left(|u|_{2^*_s}^{2^*_s} + R^{2^*_s} |u|_{2^*_s-1}^{2^*_s-1} \right) \right)^{\frac{2^*_s}{2}}
\right)^{\frac{2}{2^*_s(2^*_s-1)}}$
and $R>0$ large enough such that $ \left( \int_{|u|>R} |u|^{2^*_s} ~dx\right)^{\frac{2^*_s-2}{2^*_s}} \leq \frac{1}{2 C (1+|u|_{2^*_s})}.$
\end{Theorem} Next we consider the regularity for singular problems. \begin{Theorem}\label{thmch2}
Let $ q \in (0,1)$ and $ g(x,u)=u^{q-1}$. Then any positive solution $u \in X_0$ of $(P)$ belongs to $L^\infty(\mathbb{R}^N) \cap C^s(\mathbb{R}^N)$. Moreover, there exists $C>0 $ depending on $N,\mu, s $ and $|\Om|$ and a positive constant $C_1$ s.t. \begin{align*}
|u|_{\infty} \leq 1+ C_1\mc S_1^{\frac{2}{(2^*-1)(2^*-2)}}\left( 1+ \left( \mc S_1\left(|(u-1)^+|_{2^*_s}^{2^*_s} + R^{2^*_s} |(u-1)^+|_{2^*_s-1}^{2^*_s-1} \right) \right)^{\frac{2^*_s}{2}} \right)^{\frac{2}{2^*_s(2^*_s-1)}} \end{align*}
with $\mc S_1= \max\{ 1, C(N,\mu ,|\Om|) |u|_{2^*_s} \}$, $R>0$ such that $\left( \int_{|u|>R} |(u-1)^+|^{2^*_s} ~dx\right)^{\frac{2^*_s-2}{2^*_s}} \leq \frac{1}{2(2^*_s+1)\mc S_1}.$ \end{Theorem}
\begin{Remark} Replacing $u^{q-1}$ by $g(x,u)$ with $g:\Om\times \mathbb R^+\backslash\{0\}\to \mathbb R^+$ satisfying $g(x,t)t^{1-q}$ uniformly bounded as $t\to 0^+$ and $t\to g(x,t)$ nonincreasing for a.e $x\in \Om$, then Theorem \ref{thmch2} holds. \end{Remark} To achieve the intended goal in the above results, we first prove the non local version of Brezis-Kato estimates (See Lemma \ref{lemch3} and \ref{lemch4}) in a similar manner as in \cite{kato, moroz2}. Subsequently we construct a sequence of coercive, bilinear maps. This sequence allows us to further construct a sequence of function $u_n$ will converge weakly to $u$ (weak solution to $(P)$). Then we inherit some classical technique of Brezis-Kato \cite{kato, moroz2}. We prove that $u_n \in L^p(\Om) $ with $2^*_s<p< p_0$ for some $p_0$. Consequently, $u \in L^p(\Om) $ with $2^*_s<p< p_0$. Using these estimates, we establish \begin{align*}
\int_{\Om}\frac{F(u(y))}{|x-y|^{\mu}}dy \in L^\infty(\Om). \end{align*}
Then by Moser iterations proved established in Lemma \ref{lemch5}, we prove that $u \in L^\infty(\Om) $. For the $C^{0,\al}(\overline{\Om})$ regularity we can conclude by using Ros-Oton and Serra \cite{RS} mentioned above. We mention here that the construction of the bilinear forms for the Theorem \ref{thmch2} is most sensitive part and require more technicality. We remark that if we use Moser iterations without employing the method we present above then we can achieve $ L^\infty(\Om) $ bound of weak solutions to $(P)$ under the additional assumption $\mu < \min \{N, 4s \}$ and $f= |u|^{\frac{N-\mu+2}{N-2s}}$, see for instance \cite{jds}. To incorporate the case $\mu \geq \min \{N, 4s \}$, we develop the above stated unified course of steps.
The second main aim of this paper is to give an application of $L^\infty(\Om)$ estimate. In that direction we have the following. \begin{Theorem}\label{thmch3}
Let $ g: \overline{\Om} \times \mathbb{R} \ra \mathbb{R}$ be a Carath\'eodory function satisfying
\begin{align*}
g(x, u) = O(|u|^{2^*_s-1}), & \text{ if } |u|\ra \infty
\end{align*}
uniformly for all $ x \in \overline{\Om}$. Let $v_0 \in X_0$. Then the following assertions holds are equivalent:
\begin{itemize}
\item[(i)] there exists $\varepsilon>0 $ such that $ J(v_0+v) \geq J(v_0) $ for all $ c\in X_0,\; \|v\|\leq \varepsilon$.
\item[(ii)] there exists $\rho>0$ such that $J(v_0+ v) \geq J(v_0)$ for all $ v \in X_0 \cap C^0_d(\overline{\Om})$, $\|v\|_{0,\de}\leq \rho$.
\end{itemize} \end{Theorem} To prove the above result we have modified the techniques which have been developed by \cite{niren,squassina}.
As an application of the $H^s$ versus $C^0$- weighted minimizers, in section 6, we proved the existence of weak solution to Choquard equation, which is also a local minimizer in $X_0$ topology (See Theorem \ref{thmch4}). To prove the desired result, instead by trapping the nonlinearity between sub and supersolution, we generalize Perron's method for the doubly nonlocal problem \cite[Theorem 2.4]{struwe2}. An advantage to proceed by this alternative method is that we don't need strong assumptions on sub and supersolution except the fact, they belong to $X_0$.
For simplicity of illustration, we set some notations. We denote $\|u\|_{L^p(\Om)} $ by $|u|_p$ and $\|u\|_{X_0}$ by $\|u\|$. $B^X_\rho(u),\bar{B}^X_\rho(u)\;(B^d_\rho(u),\bar{B}^d_\rho(u))$ denote the open and closed ball, centered at $u$ with radius $\rho$, respectively in $X_0\;(C^0_d(\overline{\Om}))$. The positive constant $C$ values change case by case.
Rest of the paper organized as follows: In section 3, we give some preliminary results. In section 4, we give some technical lemmas which will help us to prove the main theorems of the paper. In section 5, we prove the Theorem \ref{thmch1} and \ref{thmch2}. In section 6, we give the proof of Theorem \ref{thmch3} and provide an application to Theorem \ref{thmch3}.
\section{Preliminary results} In this section we contribute some preliminary results, though rather straightforward, do not appear explicitly in former literature, and are worthy to archive them here. \\ The Hardy-Littlewood-Sobolev Inequality, foundational in study of Choquard equation is stated here. \begin{Proposition} \cite{leib}
Let $t,r>1$ and $0<\mu <N$ with $1/t+\mu/N+1/r=2$, $f\in L^t(\mathbb{R}^N)$ and $h\in L^r(\mathbb{R}^N)$. There exists a sharp constant $C(t,r,\mu,N)$ independent of $f,h$, such that
\begin{equation*}\label{co9}
\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{f(x)h(y)}{|x-y|^{\mu}}~dydx \leq C(t,r,\mu,N) |f|_{t}|h|_{r}.
\end{equation*}
\end{Proposition}
\begin{Lemma}\label{lemch3}
If $V\in L^\infty(\Om)+ L^{N/2s}(\Om)$ then for every $\varepsilon>0$ there exists $C_\varepsilon$ such that for every $u \in X_0$, we have
\begin{align*}
\int_{ \Om} V|u|^2~dx \leq \varepsilon^2 \|u\|^2+ C_\varepsilon \int_{ \Om}|u|^2~dx.
\end{align*} \end{Lemma} \begin{proof} Let $V=V_1+V_2$ where $V_1 \in L^\infty(\Om)$ and $V_2 \in L^{N/2s}(\Om)$. For each $k>0$ we have \begin{align*}
\int_{ \Om} V|u|^2~dx & \leq \|V_1\|_{L^\infty(\Om)}\int_{ \Om}|u|^2~dx + k \int_{|V_2|\leq k} |u|^2~dx + \int_{|V_2|> k} |V_2||u|^2~dx\\
&\leq \|V_1\|_{L^\infty(\Om)}\int_{ \Om}|u|^2~dx + k \int_{|V_2|\leq k} |u|^2~dx + S^{-1}_s \left(\int_{|V_2|> k} |V_2|^{N/2s}~dx\right)^{2s/N}\|u\|^2 \end{align*} where $S_s$ is the best constant of the embedding $X_0$ into $L^{\frac{2N}{N-2s}}.$ For a given $\varepsilon>0$, choose $k>0$ such that \begin{align*}
S^{-1}_s \left(\int_{|V_2|> k} |V_2|^{N/2s}~dx\right)^{2s/N} < \varepsilon^2. \end{align*} It implies that \begin{align*}
\int_{ \Om} V|u|^2~dx & \leq \varepsilon^2\|u\|^2+ C_\varepsilon \int_{ \Om}|u|^2~dx. \end{align*}
{$\square$}\goodbreak
\end{proof}
\begin{Lemma}\label{lemch4} \cite[Lemma 3.3]{moroz2}
Let $p,q,r,t \in [1,\infty)$ and $\la \in [0,2]$ such that
\begin{align*}
1+\frac{N-\mu}{N}-\frac{1}{p} -\frac{1}{t} = \frac{\la}{q}+ \frac{2-\la}{r}.
\end{align*}
If $\theta \in (0,2)$ satisfies
\begin{align*} & \min\{ q,r\} \left(\frac{N-\mu}{N}-\frac{1}{p} \right)< \theta < \max\{ q,r\} \left(1-\frac{1}{p} \right)\\ & \min\{ q,r\} \left(\frac{N-\mu}{N}-\frac{1}{t} \right)< 2-\theta < \max\{ q,r\} \left(1-\frac{1}{t} \right)
\end{align*}
then for $H \in L^p(\mathbb{R}^N), K \in L^t(\mathbb{R}^N)$ and $ u \in L^q(\mathbb{R}^N) \cap L^r(\mathbb{R}^N)$,
\begin{align*}
\int_{ \mathbb{R}^N}(|x|^{-\mu}* (H|u|^{\theta}))K|u|^{2-\theta}~ dx \leq C \|H\|_{L^p(\mathbb R^N)} \|K\|_{L^t(\mathbb R^N)}
\left( \int_{ \mathbb{R}^N} |u|^q\right)^{\la/q} \left( \int_{ \mathbb{R}^N} |u|^r\right)^{\frac{(2-\la)}{r}} .
\end{align*} \end{Lemma} \begin{Lemma}\label{lemch1}
Let $N\geq 2s ,\; 0<\mu<N$ and $\theta \in (0,2)$. If $H,\; K \in L^{\frac{2N}{N-\mu+2s}}(\mathbb{R}^N) + L^{\frac{2N}{N-\mu}}(\mathbb{R}^N) $ and $ 1-\frac{\mu}{N} < \theta <1+ \frac{\mu}{N}$ then for every $\varepsilon>0$ there exists $C_{\varepsilon,\theta} \in \mathbb{R}$ such that for every $u \in H^s(\mathbb{R}^N)$,
\begin{align*}
\int_{ \mathbb{R}^N}(|x|^{-\mu}* (H|u|^{\theta}))K|u|^{2-\theta}~ dx \leq \varepsilon^2 \left(\int_{ \mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}~dxdy\right)^2 + C_{\varepsilon,\theta} \int_{ \mathbb{R}^N}|u|^2~ dx.
\end{align*} \end{Lemma} \begin{proof}
We follow the proof of \cite[Lemma 3.2]{moroz2} in the nonlocal framework.
Let $H= H_1+H_2$ and $K= K_1+K_2$ with $H_1,K_1 \in L^{\frac{2N}{N-\mu}}(\mathbb{R}^N)$ and $H_2, K_2 \in L^{\frac{2N}{N-\mu+2s}}(\mathbb{R}^N)$. Now using Lemma \ref{lemch4} iteratively with appropriate values of $p,q,r,t, \theta$ and $\la$ (See \cite[Lemma 3.2]{moroz2}), we have \begin{align*}
\int_{ \mathbb{R}^N}(|x|^{-\mu}* (H|u|^{\theta}))K|u|^{2-\theta}~ dx & \leq C \left( |H_2|_{\frac{2N}{N-\mu+2s}} + |K_2|_{\frac{2N}{N-\mu+2s}} \right)^2 \left(\int_{ \mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}~dxdy\right)^2\\& \quad + C \left( |H_1|_{\frac{2N}{N-\mu}} + |K_1|_{\frac{2N}{N-\mu}} \right)^2 \int_{ \mathbb{R}^N}|u|^2~ dx. \end{align*} For given $\varepsilon>0$, choose $H_2, K_2$ such that \begin{align*}
|H_2|_{\frac{2N}{N-\mu+2s}} , |K_2|_{\frac{2N}{N-\mu+2s}} < \frac{\varepsilon}{2\sqrt{C}}. \end{align*} Therefore, the result holds.
{$\square$}\goodbreak
\end{proof}
\begin{Lemma}\label{lemch2}
For $a,b \in \mathbb R, r\geq 2 , k\geq 0 $, we have
\begin{align*}
\frac{4(r-1)}{r^2} \left( |a_k|^{r/2} -|b_k|^{r/2}\right)^2 \leq (a-b)(a_k|a_k|^{r-2}-b_k|b_k|^{r-2})
\end{align*}
where \begin{align*}
a_{k} = \max\{-k , \min\{ a,k\} \}=\left\{
\begin{array}{ll}
-k , & \text{ if } a\leq -k, \\
a, & \text{ if } -k< a<k,\\
k , & \text{ if } a\geq k. \\
\end{array}
\right.
\end{align*}
\end{Lemma} \begin{proof}
From \cite[Lemma 3.1]{squassina}, we have
\begin{align}\label{ch1}
\frac{4(r-1)}{r^2} \left( a|a_k|^{\frac{r}{2}-1} -b|b_k|^{\frac{r}{2}-1} \right)^2 \leq (a-b)(a_k|a_k|^{r-2}-b_k|b_k|^{r-2}).
\end{align}
By symmetry of the inequality, it is enough to show that result hold for $a\leq b$. For this, let $a= a_k $ and $b=b_k$ in \eqref{ch1}, we have
\begin{align*}
\frac{4(r-1)}{r^2} \left( a_k|a_k|^{\frac{r}{2}-1} -b_k|b_k|^{\frac{r}{2}-1} \right)^2 \leq (a_k-b_k)(a_k|a_k|^{r-2}-b_k|b_k|^{r-2}).
\end{align*}
\begin{enumerate}
\item[Case 1:] $0\leq b<a$\\
Clearly $0\leq b_k< a_k$ and $a_k-b_k \leq a-b$. This implies \[(a_k-b_k)(a_k|a_k|^{r-2}-b_k|b_k|^{r-2}) \leq (a-b)(a_k|a_k|^{r-2}-b_k|b_k|^{r-2}).\]
\item[Case 2:] $b\leq 0 \leq a$\\
Again notice that $b_k\leq 0\leq a_k , a_k-b_k \leq a-b $ and $ a_kb_k \leq |a_kb_k|$ we have \[(a_k-b_k)(a_k|a_k|^{r-2}-b_k|b_k|^{r-2}) \leq (a-b)(a_k|a_k|^{r-2}-b_k|b_k|^{r-2})\] and \[\left( |a_k|^{r/2} -|b_k|^{r/2}\right)^2 \leq
\left( a_k|a_k|^{\frac{r}{2}-1} -b_k|b_k|^{\frac{r}{2}-1} \right)^2.\] Hence the proof.
{$\square$}\goodbreak
\end{enumerate} \end{proof} \section{Technical results} This section is devoted to the study of weak solutions to the following problem \begin{equation*} (P_1)\; \left\{\begin{array}{rllll} (-\De)^s u
&=g(x,u)+ \left(\ds \int_{\Om}\frac{H(y)u(y)}{|x-y|^{\mu}}dy\right) K(x) \; \text{in}\; \Om,\\ u&=0 \; \text{ in } \mathbb{R}^N \setminus \Om, \end{array} \right. \end{equation*} where $H, K \in L^{\frac{2N}{N-\mu+2s}}(\Om) + L^{\frac{2N}{N-\mu}}(\Om) $. Here we use the results, established in last section to improve the integrability regularity of weak solutions to the above mentioned problem.
\begin{Proposition}\label{Propch1}
Let $H, K \in L^{\frac{2N}{N-\mu+2s}}(\Om) + L^{\frac{2N}{N-\mu}}(\Om) $. Let
$g: \overline{\Om} \times \mathbb{R} \ra \mathbb{R}$ be a continuous function satisfying
\begin{align*}
g(x, u) = O(|u|^{2^*_s-1}), & \text{ if } |u|\ra \infty
\end{align*}
uniformly for all $ x \in \overline{\Om}$. Then any solution $u \in X_0$ of the problem $(P_1)$ belongs to $L^r(\Om)$ where $r \in [2, \frac{2N^2}{(N-\mu)(N-2s)})$. \end{Proposition} \begin{proof} For $\theta=1 $ in Lemma \ref{lemch1}, there exists $\al >0$ such that for every $ \phi \in X_0$, \begin{align}\label{ch2}
\int_{ \Om} \int_{\Om}\frac{|H(y)\phi(y)K(x)\phi(x)|}{|x-y|^{\mu}}~dxdy \leq \frac12 \left(\int_{ Q}\frac{|\phi(x)-\phi(y)|^2}{|x-y|^{N+2s}}~dxdy\right)^2 + \frac{\al}{2} \int_{ \Om}|\phi|^2~ dx. \end{align}
If $3\leq 2^*_s\leq 2^*_s$ then $|u|^{2^*_s-2} \in L^{N/2s}(\Om)$. If $2<2^*_s<3$ then choose $p>1$ such that $1\leq \frac{p(2^*_s-2)N}{2s}\leq 2^*_s$ then using H\"older's inequality gives us
\begin{align*}
\left( \int_{\Om} |u|^{\frac{(2^*_s-2)N}{2s}}~dx\right)^{2s/N} \leq C \left( \int_{ \Om} |u|^{\frac{p(2^*_s-2)N}{2s}}~dx\right)^{2s/Np}<\infty.
\end{align*}
Choose $L_1>0$ such that $ \left( \int_{|u|>L_1} |u|^{\frac{(2^*_s-2)N}{2s}}~dx\right)^{2s/N} \leq \frac{S_s}{2}$ where $S_s$ is the best Sobolev constant defined in \eqref{ch27}.
Since $g(x,u)= O(|u|^{2^*_s-1})$ for $u$ large enough, there exist $L/2> L_1>0$ such that $ g(x,u)\leq |u|^{2^*_s-1}$ uniformly for $x \in \overline{\Om}$ and $|u|>L/2$. Define $\eta \in C_c^\infty[0,\infty)$ such that $0\leq \eta\leq 1$ and \begin{align*} \eta(u) = \left\{ \begin{array}{ll}
1 , & \text{ if } |u|< L/2, \\
0, & \text{ if } |u|>L. \end{array} \right. \end{align*} Define $V:= (1-\eta) \frac{g(x,u)}{u}$ and $T := \eta g(x,u)+ \al u $. By the choice of $\eta$, we obtain \begin{align}\label{ch28}
|V|_{N/2s}< S_s/2 \text{ and } T \in X_0^\prime. \end{align} Observe that $u$ is the unique solution to the following problem \begin{equation*} (-\De)^s u + \al u
=Vu+ \left(\ds \int_{\Om}\frac{H(y)u(y)}{|x-y|^{\mu}}dy\right) K +T \; \text{in}\; \Om,u=0 \; \text{ in } \mathbb{R}^N \setminus \Om. \end{equation*}
Choose sequence $ \{ H_n \}_{n\in {I\!\!N}}$ and $\{ K_n \}_{n\in {I\!\!N}}$ in $L^{\frac{2N}{N-\mu}}(\Om)$ such that $|H_n |\leq |H|,\; |K_n|\leq |K| $ and $H_n \ra H $, $K_n \ra K $ a.e in $\Om$.
For each $ n\in {I\!\!N}$, $V_n$ denotes the truncated potential defined as $V_n= V$ if $|V|\leq n$ and $V_n = n $ if $|V|>n$. Now we introduce the bilinear form \begin{align*}
B_n(v, w )= & \int_{ Q} \frac{(v(x)-v(y))(w(x)-w(y))}{|x-y|^{N+2s}}~dxdy +\al \int_{ \Om} vw~dx \\& \quad - \int_{ \Om} \int_{\Om}\frac{H_n(y)v(y)K_n(x)w(x)}{|x-y|^{\mu}}~dxdy- \int_{ \Om}V_n vw~dx. \end{align*}
In view of H\"older's inequality, Sobolev embedding, \eqref{ch28} and \eqref{ch2}, one can easily conclude that $ B_n$ is continuous coercive bilinear form. Hence by Lax-Miligram Lemma (See \cite[Corollary 5.8]{brezis}) there exists a unique $u_n \in X_0$ such that for all $w \in X_0$ we have \begin{align}\label{ch3} B_n(u_n, w)= \int_{ \Om}T w~dx . \end{align}
Subsequently, $u_n$ is a unique solution to the problem \begin{equation}\label{ch12} (-\De)^s u_n + \al u_n
= \left(\ds \int_{\Om}\frac{H_n(y)u_n(y)}{|x-y|^{\mu}}dy\right) K_n +V_nu_n +T \; \text{in}\; \Om, u_n=0 \; \text{ in } \mathbb{R}^N \setminus \Om. \end{equation}
Furthermore, using \eqref{ch3} we can easily prove that $u_n $ is a bounded sequence in $X_0$. It implies that up to a subsequence, $u_n \rp u$ weakly in $X_0$. Let $u_{n, \tau}= \max\{-\tau , \min\{ u_n,\tau \} \} $ for $ \tau>0$ and $x \in \Om$. Testing Problem \eqref{ch12} with $ \phi = |u_{n, \tau}|^{r-2} u_{n, \tau} \in X_0$ ($2\leq r< \frac{2N}{N-\mu}$), with the help of Lemma \ref{lemch2}, we get \begin{equation}\label{ch6} \begin{aligned}
& \frac{4(r-1)}{r^2} \| |u_{n, \tau}|^{r/2}\|^2 + \al \int_{ \Om} || u_{n, \tau}|^{r/2}|~dx \\
& \leq \int_{Q} \frac{(u_n(x)-u_n(y))(\phi(x)-\phi(y))}{|x-y|^{N+2s}}~dxdy + \al \int_{ \Om} u_n \phi~dx \\
& = \int_{\Om} \int_{ \Om} \frac{H_n(y)u_n(y)K_n(x) |u_{n, \tau}|^{r-2} u_{n, \tau} }{|x-y|^{\mu}}dy + \int_{ \Om} V_n u_n |u_{n, \tau}|^{r-2} u_{n, \tau}~dx + \int_{\Om}T |u_{n, \tau}|^{r-2} u_{n, \tau}~dx. \end{aligned} \end{equation} Using Lemma \ref{lemch1} with $\varepsilon^2 =\frac{(r-1)}{r^2}$, we obtain \begin{equation}\label{ch7} \begin{aligned}
\int_{\Om} \int_{ \Om} \frac{H_n(y)u_n(y)K_n(x) |u_{n, \tau}|^{r-2} u_{n, \tau} }{|x-y|^{\mu}}dxdy & \leq \int_{\Om} \int_{ \Om} \frac{|H_n(y)u_{n,\tau}(y)||K_n(x)| |u_{n, \tau}(x)|^{r-1} }{|x-y|^{\mu}}dxdy\\
& + \int_{E_{n,\tau}} \int_{ \Om} \frac{|H_n(y)u_{n}(y)||K_n(x)| |u_{n}(x)|^{r-1}}{|x-y|^{\mu}}dxdy \\
&\leq \frac{(r-1)}{r^2} \||u_{n \tau}|^{r/2}\|^2 + C_r \int_{ \Om}|u_{n, \tau}|^{r}~dx \\
& + \int_{E_{n,\tau}} \int_{ \Om} \frac{|H_n(y)u_{n}(y)||K_n(x)||u_{n}(x)|^{r-1}}{|x-y|^{\mu}}dxdy \end{aligned} \end{equation}
where $E_{n,\tau}= \{ x \in \mathbb{R}^N : |u_n(x)|\geq \tau \}$. By Hardy-Littlewood-Sobolev inequality and H\"older's inequality, we have \begin{align}\label{ch5}
\int_{E_{n,\tau}} \int_{ \Om} \frac{|H_n(y)u_{n}(y)||K_n(x)||u_{n}(x)|^{r-1}}{|x-y|^{\mu}}dxdy\leq C \left( \int_{ \mathbb{R}^N} \bigg||K_n||u_n|^{r-1}\bigg|^j~ d \xi\right)^{\frac{1}{j}}\left( \int_{E_{n,\tau}} |H_nu_n|^l~ d \xi\right)^{\frac{1}{l}} \end{align}
where $j$ and $l$ satisfy the relation $\frac{1}{j}= 1+ \frac{N-\mu}{2N}-\frac{1}{r}$ and $\frac{1}{l}= \frac{N-\mu}{2N}+\frac{1}{r}$. Using the fact that $H_n, K_n \in L^{\frac{2N}{N-\mu}}(\Om)$ and again the H\"older's inequality, $u_n \in L^r(\mathbb{R}^N)$ implies that $|K_n||u_n|^{r-1} \in L^j(\mathbb{R}^N)$ and $ |H_nu_n| \in L^l(\mathbb{R}^N)$. Therefore, as $\tau\ra \infty$, \eqref{ch5} gives \begin{align}\label{ch8}
\lim_{\tau \ra \infty}\int_{E_{n,\tau}} \int_{ \Om} \frac{|H_n(y)u_{n}(y)||K_n(x)||u_{n}(x)|^{r-1}}{|x-y|^{\mu}}dxdy =0. \end{align} Using the Sobolev inequality, \eqref{ch6}, \eqref{ch7} and \eqref{ch8}, we have
\begin{equation}\label{ch13}
\begin{aligned}
\frac{3(r-1)S_s}{r^2}& \left( \int_{ \Om} |u_{n,\tau}|^{\frac{rN}{(N-2s)}}~dx\right)^{\frac{N-2s}{N}}\\
& \leq C_r \int_{ \Om}|u_{n}|^{r} + \int_{E_{n,\tau}} \int_{ \Om} \frac{|H_n(y)u_{n}(y)||K_n(x)||u_{n}(x)|^{r-1}}{|x-y|^{\mu}}dxdy \\
& \quad + \int_{ \Om} V_n u_n |u_{n, \tau}|^{r-2} u_{n, \tau}~dx + \int_{\Om}g |u_{n, \tau}|^{r-2} u_{n, \tau}~dx.
\end{aligned}
\end{equation} Employing the fact that $g$ is a Carath\'eodory function,
\begin{equation}\label{ch14}
\begin{aligned}
\int_{\Om}T |u_{n, \tau}|^{r-2} u_{n, \tau}~dx & \leq \int_{|u|\leq L }g(x,u) |u_{n}|^{r-1} ~dx + \al \int_{\Om} u |u_{n, \tau}|^{r-1} ~dx \\
& \leq C(L_1) \left( \int_{\Om} |u|^{r} ~dx + \int_{\Om} |u_{n}|^{r} ~dx\right).
\end{aligned}
\end{equation}
By Lemma \ref{lemch3} for $\varepsilon^2 = \frac{r-1}{r^2}$, we have
\begin{equation}\label{ch15}
\begin{aligned}
\int_{ \Om} V_n u_n |u_{n, \tau}|^{r-2} u_{n, \tau}~dx & \leq 2 \int_{E_{n,\tau}} V_n |u_{n}|^{r} ~dx + \int_{ \Om} V_n |u_{n, \tau}|^{r} ~dx \\
& \leq \frac{(r-1)}{r^2} \||u_{n \tau}|^{r/2}\|^2 + C_r \int_{ \Om}|u_{n, \tau}|^{r}~dx + 2 \int_{E_{n,\tau}} V_n |u_{n}|^{r} ~dx.
\end{aligned}
\end{equation}
Using Dominated Convergence theorem, one can easily shows that $\ds \lim_{\tau \ra \infty} \int_{E_{n,\tau}} V_n |u_{n}|^{r} ~dx=0$. Now taking into account \eqref{ch13}, \eqref{ch14}, \eqref{ch15} and letting $\tau \ra \infty$, we have
\begin{equation*}
\begin{aligned}
\left( \int_{ \Om} |u_{n}|^{\frac{rN}{(N-2s)}}~dx\right)^{\frac{N-2s}{N}}
& \leq C_r \left( \int_{\Om}|u_{n}|^r~dx + \int_{\Om}|u|^r~dx \right).
\end{aligned}
\end{equation*}
Therefore,
\begin{align*}
\limsup_{n \ra \infty} \left( \int_{ \Om} |u_{n}|^{\frac{rN}{(N-2s)}}~dx\right)^{\frac{N-2s}{N}} \leq C_r \limsup_{n \ra \infty} \left( \int_{\Om}|u_{n}|^r~dx + \int_{\Om}|u|^r~dx \right).
\end{align*}
Hence, by iterating a finite number of times, we infer that $ u \in L^q(\Om)$ for all $ q\in \left[2, \frac{2N^2}{(N-\mu)(N-2s)}\right) $. Moreover, there exists a positive constant $C(q,N,\mu, |\Om|)$ such that $|u|_q\leq C(q,N,\mu, |\Om|) |u|_{2^*_s}$.
{$\square$}\goodbreak
\end{proof} \begin{Definition}
For $\phi \in C^0(\overline{\Om})$ with $\phi >0$ in $\Om$, the set $C_\phi(\Om)$ is defined as
\begin{align*}
C_\phi(\Om)= \{ u \in C^0(\overline{\Om})\; :\; \text{there exists } c \geq 0 \text{ such that } |u(x)|\leq c\phi(x), \text{ for all } x \in \Om \},
\end{align*}
endowed with the natural norm $\bigg\|\ds \frac{u}{\phi}\bigg\| _{L^{\infty}(\Om)}$. \end{Definition} \begin{Definition}
The positive cone of $C_\phi(\Om)$ is the open convex subset of $C_\phi(\Om)$ defined as
\begin{align*}
C_\phi^+(\Om)= \left\{ u \in C_\phi(\Om)\; :\; \inf_{x \in \Om} \frac{u(x)}{\phi(x)}>0 \right\}.
\end{align*} \end{Definition}
\begin{Proposition}\label{propch1}
\cite[Theorem 1.2]{adi} Let $\phi_1 \in C^s(\mathbb{R}^N) \cap C^+_{d^s}(\Om)$ be the normalized eigenvalue of $(-\De)^s$ in $X_0$. If $q \in (0,1)$ then there exists a unique positive $ \underline{u} \in X_0 \cap C^+_{\phi_1}(\Om) \cap C_0(\overline{\Om}) $ classical solution to the following problem
\begin{equation}\label{ch25}
(-\De)^s u
=u^{q-1},\; u>0 \; \text{in}\;
\Om,\;
u=0 \; \text{ in } \mathbb{R}^N \setminus \Om.
\end{equation} \end{Proposition} \begin{Proposition}\label{Propch3}
Let $ q \in (0,1), g(x,u)=u^{q-1}$ and $0 \leq H, K \in L^{\frac{2N}{N-\mu+2s}}(\Om) + L^{\frac{2N}{N-\mu}}(\Om) $. Let $ u \in X_0$ be a positive weak solution of problem $(P_1)$. Then $u \in L^p(\Om)$ where $p \in [2, \frac{2N^2}{(N-\mu)(N-2s)})$. \end{Proposition}
\begin{proof}
Since $0 \leq H, K,$ we see that $\underline{u}\in X_0$ is a subsolution to problem $(P_1)$. \\
\textbf{Claim:} $\underline{u}\leq u $ a.e in $\Om$. \\ Assuming by contradiction, assume that the Claim is not true. Since for any $ u \in X_0$ we have \begin{align*}
\|u^+\|^2 \leq \int_{ Q} \frac{(u(x)-u(y))(u^+(x)-u^+(y))}{|x-y|^{N+2s}}~dxdy. \end{align*}
Testing $(-\De)^s\underline{u}- (-\De)^s u\leq \underline{u}^{q-1}- u^{q-1} $ with $(\underline{u}-u)^+$, we obtain
\begin{align*}
0\leq \|(\underline{u}-u)^+\|^2 & \leq \int_{ Q} \frac{((\underline{u}-u)^+(x)-(\underline{u}-u)^+(y))((\underline{u}-u)(x)-(\underline{u}-u)(y))}{|x-y|^{N+2s}}~dxdy\\
& \leq \int_{ \Om} ( \underline{u}^{q-1}- u^{q-1})(\underline{u}-u)^+~dx \leq 0.
\end{align*}
It implies $|\{ x \in \Om \; : \; \underline{u } \geq u \text{ a.e in } \Om \}| = 0$. It provides the expected contradiction. Hence $\underline{u}\leq u $ a.e in $\Om$.\\
Observe that using Proposition \ref{propch1}, for all $\ba >0$, we have
\begin{align*}
\chi_{\{u< \ba \} }u^{q-1}\leq \chi_{\{u< \ba \} }\frac{u}{\underline{u}^2} u^{q} < \chi_{\{u< \ba \} }\frac{u}{C_1^2\phi_1^2} \ba^{q} \leq \chi_{\{u< \ba \} }\frac{u}{C_1^2C_2^2d^{2s}} \ba^{q}.
\end{align*}
where $C_1$ and $C_2$ are appropriate positive constants. Hence we can choose $\de:= \de(\ba)>0$ such that $\chi_{\{u< \ba \} }u^{q-1}= \de(\ba) \chi_{\{u< \ba \} }\frac{u}{d^{2s}}$. Now choose $\ba >0$ such that $\ga_1:= \frac12 - S_{H} \de(\ba)>0$ and $\ga_2:= \frac{3(r-1)}{r^2} - S_{H}\de(\ba)>0$ for $2\leq r< \frac{2N}{N-\mu}$ and with $S_H$ defined on \eqref{ch35}. The choice of $\ba, \de(\ba)$ and Lax-Milgram Lemma, imply that $u$ is the unique solution of the following problem:
\begin{equation*}
(-\De)^s u+ \al u -\de(\ba) \chi_{\{u< \ba \} }\frac{u}{d^{2s}}
= \left(\ds \int_{\Om}\frac{H(y)u(y)}{|x-y|^{\mu}}dy\right) K +\chi_{\{u\geq \ba \} }u^{q-1} + \al u \; \text{in}\;
\Om,
u=0 \; \text{ in } \mathbb{R}^N \setminus \Om
\end{equation*}
where $\al>0$ is chosen as in Proposition \ref{Propch1}. Now we will follow the same arguments as in Proposition \ref{Propch1} to achieve the result. Notice that $T= \chi_{\{u\geq \ba \} }u^{q-1} + \al u \in X_0^\prime$. For each $n \in {I\!\!N}$, we define the bilinear form
\begin{align*}
B_n(v, w )= & \int_{ Q} \frac{(v(x)-v(y))(w(x)-w(y))}{|x-y|^{N+2s}}~dxdy +\al \int_{ \Om} vw~dx \\& \quad - \int_{ \Om} \int_{\Om}\frac{H_n(y)v(y)K_n(x)w(x)}{|x-y|^{\mu}}~dxdy- \int_{ \Om}\de(\ba) \chi_{\{u< \ba \} }\frac{vw}{d^{2s}} ~dx.
\end{align*}
Using as the arguments as in Proposition \ref{Propch1}, there exist unique $u_n \in X_0$ such that for all $w \in X_0$ we have
\begin{align*}\label{ch16}
B_n(u_n, w)= \int_{ \Om}T w~dx.
\end{align*}
Moreover, $u_n$ is a unique solution to the problem
\begin{equation*}\label{ch17}
(-\De)^s u_n + \al u_n
= \left(\ds \int_{\Om}\frac{H_n(y)u_n(y)}{|x-y|^{\mu}}dy\right) K_n +\de(\ba) \chi_{\{u< \ba \} }\frac{u_n}{d^{2s}} +T \; \text{in}\;
\Om, u_n=0 \; \text{ in } \mathbb{R}^N \setminus \Om.
\end{equation*}
Clearly, $u_n \rp u$ weakly in $X_0$. Let $u_{n, \tau}= \max\{-\tau , \min\{ u_n,\tau \} \} $ for $ \tau>0$ and $x \in \Om$. Choose $ \phi = |u_{n, \tau}|^{r-2} u_{n, \tau} \in X_0$ ($2\leq r< \frac{2N}{N-\mu}$) as the test function in \eqref{ch12}. Using the same arguments as in Proposition \ref{Propch1}, we have
\begin{equation}\label{ch18}
\begin{aligned}
\frac{3(r-1)S_s}{r^2}& \left( \int_{ \Om} |u_{n,\tau}|^{\frac{rN}{(N-2s)}}~dx\right)^{\frac{N-2s}{N}}\\
& \leq C_r \int_{ \Om}|u_{n}|^{r} + \int_{E_{n,\tau}} \int_{ \Om} \frac{|H_n(y)u_{n}(y)||K_n(x)||u_{n}(x)|^{r-1}}{|x-y|^{\mu}}dxdy \\
& \quad + \int_{ \Om}\de(\ba) \chi_{\{u< \ba \} }\frac{u_n}{d^{2s}}|u_{n, \tau}|^{r-2} u_{n, \tau}~dx + \int_{\Om}g |u_{n, \tau}|^{r-2} u_{n, \tau}~dx.
\end{aligned}
\end{equation}
Consider
\begin{equation}\label{ch19}
\begin{aligned}
\int_{\Om}T |u_{n, \tau}|^{r-2} u_{n, \tau}~dx & = \int_{ \Om} \chi_{\{u\geq \ba \} }(u^{q-1} + \al u) |u_{n, \tau}|^{r-2} u_{n, \tau}~dx\\
& \leq C(N,\mu , r, |\Om|) \left( \int_{\Om} |u|^{r} ~dx + \int_{\Om} |u_{n}|^{r} ~dx\right).
\end{aligned}
\end{equation}
With the help of Hardy inequality, we have
\begin{equation}\label{ch20}
\begin{aligned}
\int_{ \Om} \de(\ba) \chi_{\{u< \ba \} }\frac{u_n}{d^{2s}}|u_{n, \tau}|^{r-2} u_{n, \tau}~dx & \leq 2 \int_{E_{n,\tau}} \de(\ba) \frac{|u_{n}|^{r}}{d^{2s}} ~dx + \int_{ \Om} \frac{|u_{n, \tau}|^{r}}{d^{2s}}~dx \\
& \leq S_{H}\de(\ba) \||u_{n \tau}|^{r/2}\|^2 + 2 \int_{E_{n,\tau}} \de(\ba) \frac{|u_{n}|^{r}}{d^{2s}} ~dx .
\end{aligned}
\end{equation}
Using Dominated Convergence theorem, it follows that $\ds \lim_{\tau \ra \infty} \int_{E_{n,\tau}} \frac{ \de(\ba)|u_{n}|^{r}}{d^{2s}} ~dx=0$.
Now taking into account \eqref{ch18}, \eqref{ch19}, \eqref{ch20}, definition of $\ga_2$ and letting $\tau \ra \infty$, we have
\begin{equation*}
\begin{aligned}
\left( \int_{ \Om} |u_{n}|^{\frac{rN}{(N-2s)}}~dx\right)^{\frac{N-2s}{N}}
& \leq C(N,\mu , r, |\Om|) \left( \int_{\Om}|u_{n}|^r~dx + \int_{\Om}|u|^r~dx \right).
\end{aligned}
\end{equation*}
Therefore,
\begin{align*}
\limsup_{n \ra \infty} \left( \int_{ \Om} |u_{n}|^{\frac{rN}{(N-2s)}}~dx\right)^{\frac{N-2s}{N}} \leq C(N,\mu , r, |\Om|) \limsup_{n \ra \infty} \left( \int_{\Om}|u_{n}|^r~dx + \int_{\Om}|u|^r~dx \right).
\end{align*}
Hence, $ u \in L^r(\Om)$ for all $ r\in \left[2, \frac{2N^2}{(N-\mu)(N-2s)}\right) $. As earlier we remark that there exists a positive constant $ C(N,\mu , q, |\Om|) $ such that $|u|_q\leq C(N,\mu , q, |\Om|) |u|_{2^*_s}$.
{$\square$}\goodbreak
\end{proof}
\begin{Remark} We highlight here that the next lemma investigates the $L^\infty(\Om)$ bound for the fractional Laplacian with critical Sobolev exponent.
In \cite{servadei} authors already proved this type of result for a positive solution. Here we used the ideas form \cite{ squassina,servadei} to extend the result of \cite{servadei} to any weak solution. \end{Remark} \begin{Lemma}\label{lemch5}
Let $u$ be any weak solution to the following problem
\begin{align}\label{ch36}
(-\De)^su = k(x,u) \text{ in } \Om,\; u=0 \text{ in } \mathbb{R}^N \setminus \Om
\end{align}
where $|k(x,u)|\leq C(1+|u|^{2^*_s-1})$ and $C>0$. Then $u \in L^\infty(\Om)$. \end{Lemma} \begin{proof}
Let $u \in X_0$ be any weak solution to \eqref{ch36}. Let $u_\tau= \max\{-\tau , \min\{ u,\tau \} \} $ for $ \tau>0$. Let $ \phi = u|u_\tau|^{r-2} \in X_0$ ($ r\geq 2$) be a test function to problem \eqref{ch36}, then by inequality \eqref{ch1}, we deduce that
\begin{equation}\label{ch37}
\begin{aligned}
|u|u_\tau|^{\frac{r}{2}-1}|^2_{2^*_s}& \leq C \|u|u_\tau|^{\frac{r}{2}-1}\|^2 \leq \frac{Cr^2}{r-1} \int_{Q} \frac{(u(x)-u(y))(\phi(x)-\phi(y))}{|x-y|^{N+2s}}~dxdy\\
&\leq Cr \int_{ \Om} |k(x,u)||u ||u_\tau|^{r-2}~dx\\
& \leq Cr \int_{ \Om} |u ||u_\tau|^{r-2}+|u|^{2^*_s}|u_\tau|^{r-2} ~dx.
\end{aligned} \end{equation}
\textbf{Claim:} Let $r_1= 2^*_s+1$. Then $ u \in L^{\frac{2^*_s r_1}{2}}(\Om)$.\\ For this, consider
\begin{equation}\label{ch38}
\begin{aligned}
\int_{ \Om} |u|^{2^*_s}|u_\tau|^{r_1-2} ~dx& = \int_{ |u|\leq R} |u|^{2^*_s}|u_\tau|^{r_1-2} ~dx+ \int_{|u|>R} |u|^{2^*_s}|u_\tau|^{r-2} ~dx\\
& \int_{ |u|\leq R} R^{2^*_s}|u_\tau|^{r_1-2} ~dx+\left(\int_{\Om}( u^2|u_\tau|^{r-2})^{\frac{2^*_s}{2}} ~dx\right)^{\frac{2}{2^*_s}} \left( \int_{|u|>R} |u|^{2^*_s} ~dx\right)^{\frac{2^*_s-2}{2^*_s}}.
\end{aligned}
\end{equation}
Choose $R>0$ large enough such that
\begin{align}\label{ch39}
\left( \int_{|u|>R} |u|^{2^*_s} ~dx\right)^{\frac{2^*_s-2}{2^*_s}} \leq \frac{1}{2Cr_1}.
\end{align}
Taking into account \eqref{ch37}, \eqref{ch38} jointly with \eqref{ch39}, we obtain
\begin{align*}
|u|u_\tau|^{\frac{r_1}{2}-1}|^2_{2^*_s}& \leq Cr_1\left(\int_{ \Om} |u |^{2^*_s}~dx + \int_{ \Om} R^{2^*_s}|u|^{2^*_s-1} ~dx\right).
\end{align*}
Appealing Fatou's Lemma as $\tau \ra \infty$, we obtain
\begin{align}\label{ch41}
||u|^{\frac{r_1}{2}}|^2_{2^*_s}& \leq Cr_1\left(\int_{ \Om} |u |^{2^*_s}~dx + \int_{ \Om} R^{2^*_s}|u|^{2^*_s-1} ~dx\right)<\infty.
\end{align}
This establishes the Claim. Now let $\tau \ra \infty$ in \eqref{ch37}, we deduce that
\begin{equation*}
\begin{aligned}
||u|^{\frac{r}{2}}|^2_{2^*_s}& \leq Cr \int_{ \Om} |u |^{r-1}+|u|^{r+2^*_s-2} ~dx \leq 2Cr(1+|\Om|) \left(1+ \int_{ \Om} |u|^{r+2^*_s-2} \right).
\end{aligned}
\end{equation*}
It implies that
\begin{equation}\label{ch40}
\begin{aligned}
\left(1+ \int_{ \Om}|u|^{\frac{2^*_s r}{2}}\right)^{\frac{2}{2^*_s(r-2)}}& \leq \mc C_r^{\frac{1}{(r-2)}} \left(1+ \int_{ \Om} |u|^{r+2^*_s-2} \right)^{\frac{1}{(r-2)}}
\end{aligned}
\end{equation}
where $\mc C_r= 4Cr(1+|\Om|)$. For $j\geq 1$, we define $r_{j+1}$ iteratively as $ r_{j+1} +2^*_s-2= \frac{2^*_s r_j}{2}$. It implies
\begin{align*}
\left(r_{j+1}-2 \right)= \left(\frac{2^*_s}{2}\right)^j \left(r_1-2 \right).
\end{align*}
From \eqref{ch40}, we get
\begin{align*}
\left(1+ \int_{ \Om}|u|^{\frac{2^*_s r_{j+1}}{2}}\right)^{\frac{2}{2^*_s(r_{j+1}-2)}}& \leq \mc C_{j+1}^{\frac{1}{(r_{j+1}-2)}} \left(1+ \int_{ \Om} |u|^{\frac{2^*_s r_j}{2}} \right)^{\frac{2}{2^*_s(r_j-2)}}
\end{align*}
where $\mc C_{j+1}:= 4Cr_{j+1}(1+|\Om|)$. Denote $ D_j= \left(1+ \int_{ \Om} |u|^{\frac{2^*_s r_j}{2}} \right)^{\frac{2}{2^*_s(r_j-2)}}$, for $j\geq 1$.
By limiting arguments, one can easily prove that, for $j>1$, \begin{align*} D_{j+1}\leq \prod_{k=2}^{j+1} \mc C_{k}^{\frac{1}{(r_{k}-2)}} D_1 \leq \mc C_0 D_1. \end{align*}
It implies that $|u|_\infty\leq \mc C_0 D_1$ where $D_1$ is explicitly given in \eqref{ch41}.
{$\square$}\goodbreak
\end{proof}
\section{Proof of Theorem \ref{thmch1} and \ref{thmch2}} In this section we will conclude the proofs of Theorem \ref{thmch1} and Theorem \ref{thmch2}. Before this we recall the following result, which can be consulted in \cite{RS}. \begin{Proposition}\label{Propch5}
Let $\Om$ be a bounded Lipschitz domain satisfying the exterior ball condition, $g \in L^\infty(\Om)$ and $u$ be a solution of \eqref{ch24}. Then $u \in C^s(\mathbb{R}^N)$ and
\begin{align*}
\|u\|_{C^s(\mathbb{R}^N)}\leq C\|g\|_{L^\infty(\Om)}
\end{align*}
where $C$ is a constant depending on $\Om$ and $s$. \end{Proposition} \textbf{Proof of Theorem \ref{thmch1} :} Let $ u \in X_0$ be a positive weak solution to problem $(P)$ and $H= F(u)/u$ and $K=f$. Then From Proposition \ref{Propch1}, we get $ u \in L^r(\Om)$ for all $ r\in \left[2, \frac{2N^2}{(N-\mu)(N-2s)}\right) $. It implies $F(u) \in L^r(\Om)$ for all $ r\in \left[\frac{2N}{2N-\mu}, \frac{2N^2}{(N-\mu)(2N-\mu)}\right)$. Observe that $
\frac{2N}{2N-\mu}< \frac{N}{N-\mu}< \frac{2N^2}{(N-\mu)(2N-\mu)}$ and there exists a constant $C(N,\mu, |\Om|)>0$ such that $|F(u)|_{\frac{N}{N-\mu}} \leq C(N,\mu, |\Om|) |u|_{2^*_s}$. Therefore, we infer that $\int_{\Om} \frac{F(u)}{|x-y|^\mu} ~dy \in L^{\infty}(\Om)$ and \begin{align*}
\bigg| \int_{\Om} \frac{F(u)}{|x-y|^\mu} ~dy\bigg|_{\infty} \leq C(N,\mu, |\Om|) |u|_{2^*_s}. \end{align*} Using the assumptions on $f$ and $g$, we obtain \begin{align*}
(-\De)^s u& = g(x,u)+ \left(\ds \int_{\Om}\frac{F(u)(y)}{|x-y|^{\mu}}dy\right) f \\
& \leq C(N,\mu, |\Om|) (1+ |u|_{2^*_s}) (1+ |u|^{2^*_s-1})= \mc S_0 (1+ |u|^{2^*_s-1})(\text{say}). \end{align*}
From Lemma \ref{lemch5}, we have $ u \in L^\infty(\Om)$. Furthermore, there exists a function $ C_0>0 $ independent of $N,\mu, s$ and $|\Om|$ such that \begin{align*}
|u|_{\infty} \leq C_0 \mc S_0^{\frac{2}{(2^*-1)(2^*-2)}}D_1 \end{align*} \begin{align*}
\text{ with }\quad D_1\leq\left( 1+ \left( (2^*_s+1)\mc S_0 \left(\int_{ \Om} |u |^{2^*_s}~dx + \int_{ \Om} R^{2^*_s}|u|^{2^*_s-1} ~dx\right) \right)^{\frac{2^*_s}{2}} \right)^{\frac{2}{2^*_s(2^*_s-1)}} \end{align*} and $R>0$ chosen large enough such that
\begin{align}\label{ch42}
\left( \int_{|u|>R} |u|^{2^*_s} ~dx\right)^{\frac{2^*_s-2}{2^*_s}} \leq \frac{1}{2(2^*_s+1)\mc S_0}. \end{align} Now using Proposition \ref{Propch5}, we obtain that $u \in C^s(\mathbb{R}^N)$.
{$\square$}\goodbreak
\textbf{Proof of Theorem \ref{thmch2} :} From Proposition \ref{Propch3}, and the assumption on $f$, we have \begin{align*}
\int_{\Om} \frac{F(u)}{|x-y|^\mu} ~dy \in L^{\infty}(\Om). \end{align*}
Furthermore, there exists a constant $C(N,\mu , |\Om|) >0$ such that $|F(u)|_{\frac{N}{N-\mu}} \leq C(N,\mu , |\Om|) |u|_{2^*_s}$. Therefore, we infer that \begin{align*}
(-\De)^s u& \leq u^{q-1}+ C(N,\mu , |\Om|) |u|_{2^*_s} |f| \leq u^{q-1}+ C(N,\mu , |\Om|) |u|_{2^*_s} (1+ |u|^{2^*_s-1}). \end{align*} Let $ \psi \in \mathbb{R} \ra [0,1]$ be a $C^\infty(\mathbb{R})$ convex increasing function such that $\psi^\prime(t)\leq 1$ for all $t \in [0,1]$ and $\psi^\prime(t)=1$ when $t\geq 1$. Define $\psi_\varepsilon(t)= \varepsilon \psi(\frac{t}{\varepsilon})$ then using the fact that $\psi_\varepsilon$ is smooth, we obtain $\psi_\varepsilon \ra (t-1)^+$ uniformly as $\varepsilon \ra 0$.
It implies \begin{align*} (-\De)^s \psi_\varepsilon(u) \leq \psi_\varepsilon^\prime(u)(-\De)^s u& \leq \chi_{\{ u >1\}}(-\De)^s u\\
& \leq \chi_{\{ u >1\}} (u^{q-1}+ C(N,\mu , |\Om|) |u|_{2^*_s} (1+ |u|^{2^*_s-1}))\\
& \leq \max\{ 1, C(N,\mu ,|\Om|) |u|_{2^*_s} \}(1+ ((u-1)^+)^{2^*_s-1})\\ &= \mc S_1 (1+ ((u-1)^+)^{2^*_s-1}) \text{ (say)}. \end{align*} Hence, as $\varepsilon \ra 0$, we deduce that \begin{align*} (-\De)^s (u-1)^+ \leq \mc S_1 (1+ ((u-1)^+)^{2^*_s-1}). \end{align*} Employing Lemma \ref{lemch5}, we deduce that $(u-1)^+ \in L^\infty(\Om)$, that is, $ u \in L^\infty(\Om)$. Furthermore, since $u$ is a positive solution, there exists $C_1>0 $ such that
independent of $N,\mu, s $ and $|\Om|$ such that \begin{align*}
|u|_{\infty} \leq 1+ C_1 \mc S_1^{\frac{2}{(2^*-1)(2^*-2)}}D_1 \end{align*} \begin{align*}
\text{ with }\quad D_1\leq\left( 1+ \left( (2^*_s+1)\mc S_1 \left(\int_{ \Om} |(u-1)^+ |^{2^*_s}~dx + \int_{ \Om} R^{2^*_s}|(u-1)^+|^{2^*_s-1} ~dx\right) \right)^{\frac{2^*_s}{2}} \right)^{\frac{2}{2^*_s(2^*_s-1)}} \end{align*} and $R>0$ chosen large enough such that \begin{align*}
\left( \int_{|u|>R} |(u-1)^+|^{2^*_s} ~dx\right)^{\frac{2^*_s-2}{2^*_s}} \leq \frac{1}{2(2^*_s+1)\mc S_1}. \end{align*}
Let $\overline{u}$ be the unique solution (See \cite[Theorem 1.2, Remark 1.5]{adi}) to the following problem \begin{align*} (-\De)^s \overline{u} = \overline{u}^{-q}+ c, u>0 \text{ in } \Om, u =0 \text{ in } \mathbb{R}^N\setminus \Om \end{align*}
where $c= C_1|F(u) f(u)|_\infty $ with $C_1= \bigg|\ds \int_{\Om}\frac{dy}{|x-y|^{\mu}}\bigg|_\infty$. Then following similar lines as in the proof of Claim in Proposition \ref{Propch3}, we have $\underline{u}\leq u \leq \overline{u}$ a.e in $\Om$ where $\underline{u}$ is the unique solution to \eqref{ch25}. Therefore, $ u \in X_0 \cap L^\infty(\Om)\cap C^+_{\phi_1}(\Om) $. Now from \cite[Theorem 1.3]{gts1}, we have the desired result.
{$\square$}\goodbreak
\section{Applications} The purpose of this section is to derive applications from the uniform estimates given in Theorems \ref{thmch1} and \ref{thmch2}. Precisely, here, we prove the theorem \ref{thmch3} which deals with $H^s$ versus $C^0_d(\overline{\Om})$ weighted minimizers. Furthermore, we provide an application of this result, concerning the existence and multiplicity of solutions.
\textbf{Proof of Theorem \ref{thmch3}: (i) implies (ii).} Assume by contradiction that there exists a sequence $v_n \ra v_0$ in $C^0_d(\overline{\Om})$ and $J(v_n)< J(v_0)$. It follows that \begin{align*}
\int_{\Om}G(x,v_n)~dx \ra \int_{\Om}G(x,v_0)~dx \text{ and } \iint_{\Om\times \Om} \frac{F(v_n)F(v_n)}{|x-y|^\mu} ~dxdy \ra \iint_{\Om\times \Om} \frac{F(v_0)F(v_0)}{|x-y|^\mu} ~dxdy. \end{align*}
Taking into account above statements, we infer that $ \ds\limsup_{n \ra \infty} \|v_n\|^2 \leq \|v_0\|^2$. Hence upto a subsequence $v_n$ converges to $v_0$ weakly in $X_0$. By Fatou's Lemma and above conclusion one obtains $\| v_n\|\ra \|v\| $. This settles the proof. \\ \textbf{Proof of Theorem \ref{thmch3}: (ii) implies (i).} To show the result, we will first consider the case $v_0=0 $. It implies that \begin{align}\label{ch29} \inf_{v \in X_0 \cap \bar{B}_\rho ^d(0)} J(v)= J(v_0)=0. \end{align} Assume that (i) doesn't hold. Then we can choose $\varepsilon_n \in (0,\infty),\; \varepsilon_n \ra 0$ such that there exist $z_n \in \bar{B}_{\varepsilon_n} ^X(0)$ with
$J(z_n)<0$. For each $m\in {I\!\!N}$, define the functions $g_m,\;G_m: \overline{\Om} \times \mathbb{R} \ra \mathbb{R}$ and $f_m,\; F_m: \mathbb{R} \ra \mathbb{R}^+$ as \begin{align*} & g_{m}(x,t) = \max\{g(x,-m) , \min\{ g(x,t),g(x,m)\} \}, \quad G_m(x,t):= \int_0^tg_m(x,\tau) ~d \tau\\ & \text{ and } f_m(t):= \max\{f(-m) , \min\{ f(t),f(m)\} \},\quad F_m(t):= \int_0^tf_m(\tau) ~d \tau. \end{align*} Subsequently, we define the truncated functional $J_m$ as \begin{align*}
J_m(v)= \frac{\|v\|^2}{2} - \int_{ \Om} G_m(x,v)~dx - \frac12 \iint_{ \Om\times \Om} \frac{F_m(v)F_m(v)}{|x-y|^\mu} ~dxdy \text{ for all } v \in X_0. \end{align*} Notice that $J_m \in C^1(X_0)$ and by appealing Dominated convergence theorem, we infer that $J_m(v) \ra J(v)$ as $m \ra \infty$ and for all $v \in X_0$. Thus, for every $n \in {I\!\!N}$ we pick $ m_n \in {I\!\!N}$ such that $J_{m_n}(z_n)<0$.
Observe that $|G_m(x,v)|\leq (|g(x,-m)|+|g(x,m)|)|v|$ and $F_m(v)|\leq (f(-m)+f(m))|v|$. That is, $G_m$ and $F_m$ has subcritical growth in the sense of Sobolev inequality and Hardy-Littlewood-Sobolev inequality respectively. Therefore, $J_m$ is weakly lower semicontinuous functional. Since $ \bar{B}_{\varepsilon_n} ^X(0)$ is a closed convex set, it implies that there exists $w_n \in \bar{B}_{\varepsilon_n} ^X(0)$ such that \begin{align*} J_{m_n}(w_n)= \inf_{v \in \bar{B}_{\varepsilon_n} ^X(0)} J_{m_n}(v) \leq J_{m_n}(w_n). \end{align*} With the help of Lagrange multiplier's rule, one can easily prove that there exists $\la_n \in (0,1]$ such that $w_n$ is a weak solution of \begin{equation*} \left\{\begin{array}{rllll} (-\De)^s u
&=\la_n \left(g_{k_n}(x,u)+ \left(\ds \int_{\Om}\frac{F_{k_n}(u)(y)}{|x-y|^{\mu}}dy\right) f_{k_n}(u) \right) \; \text{in}\; \Om,\\ u&=0 \; \text{ in } \mathbb{R}^N \setminus \Om. \end{array} \right. \end{equation*}
Since $\|w_n\|\in \bar{B}_{\varepsilon_n} ^X(0),\; \|w_n\|\ra 0 $ as $|\varepsilon_n| \ra 0 $. It implies $ |w_n|_{2^*_s} \ra 0$ and hence for $n$ large enough we can choose $R=0$ in \eqref{ch42}. Subsequently there exists $K>0$ such that $|w_n|_\infty\leq K$ for all $n$. By appealing \cite[Theorem 1.2]{RS}, we obtain that for all $n$, $ w_n \in C_d^0(\overline{\Om})$ and $\|w_n\|_{C^{0,\al}_d(\overline{\Om})}\leq K_1$ for some suitable $K_1>0$. Since $ C^{0,\al}_d(\overline{\Om})$ is compactly embedded into $C_d^0(\overline{\Om}) $, $w_n $ is strongly convergent in $C_d^0(\overline{\Om})$. Consequently, taking in account the fact that $w_n \ra 0 $ a.e in $\Om$, we get $ w_n \ra 0 $ in $C_d^0(\overline{\Om})$. We conclude that for $n$ large enough, $\|w_n\|_{C^0_d(\overline{\Om})}\leq \rho$ and $|w_n|_\infty <1$. From this we infer that \begin{align*} J(w_m)= J_{m_n}(w_m)<0 \end{align*} and we obtain the desired contradiction to the assumption \eqref{ch29}. Now we will consider the case $v\not = 0$. By given assumption (ii), it follows that $ J^\prime(v_0)(v)=0$ for all $v \in C_c^\infty(\Om)$ and applying the standard density arguments we infer that \begin{align}\label{ch30} J^\prime(v_0)(v)=0 \text{ for all } v \in X_0. \end{align} In view of Theorem \ref{thmch1}, we have $ u \in L^\infty(\Om) \cap C_d^0(\overline{\Om})$. For all $v \in X_0$, let \begin{align*}
\widehat{F}(x,v) := & \left(\ds \int_{\Om}\frac{F(v_0+v)(y)}{|x-y|^{\mu}}dy\right) F(v_0+v)(x)- \left(\ds \int_{\Om}\frac{F(v_0)(y)}{|x-y|^{\mu}}dy\right)\left( F(v_0)+2f(v_0)v\right)(x)\\
\text{and }\quad & \widehat{G}(x,v) := G(x,(v_0+v)(x)) - G(x,v_0(x))- g(x,v_0(x))v(x). \end{align*} Set \begin{align*}
\widehat{J}(v)= \frac{\|v\|^2}{2}- \int_{\Om} \widehat{G}(x,v) ~dx -\frac12 \int_{\Om} \widehat{F}(x,v) ~dx \text{ for all } v \in X_0. \end{align*} Note that $ \widehat{J}\in C^1(X_0)$. Employing \eqref{ch30} and the definition of $\widehat{F} $ and $\widehat{G}$, we have \begin{align*}
\widehat{J}(v)& = \frac{\|v_0+v\|^2}{2}- \frac{\|v_0\|^2}{2} - \int_{ \Om} G(x,(v_0+v)(x)) - G(x,v_0(x))~dx\\
& \quad -\frac{1}{2} \iint_{\Om\times \Om}\frac{F(v_0+v)F(v_0+v)}{|x-y|^{\mu}}~dxdy +\frac{1}{2} \iint_{\Om \times \Om}\frac{F(v_0)F(v_0)}{|x-y|^{\mu}}~dxdy\\ & = \widehat{J}(v_0+v)- \widehat{J}(v_0). \end{align*} We may deduce that $\tilde{J}(0)=0$. Therefore given assumptions can be expressed as \begin{align*} \inf_{v \in X_0 \cap \bar{B}_\rho ^d(0)} \widehat{J}(v)=0. \end{align*} Now by using above case we get the desired result and hence the proof of (ii) implies (i).
{$\square$}\goodbreak
\begin{Theorem}\label{thmch4}
Let $ G: \overline{\Om} \times \mathbb{R} \ra \mathbb{R}$ be a Carath\'eodory function satisfying
\begin{align*}
g(x, u) = O(|u|^{2^*_s-1}), & \text{ if } |u|\ra \infty
\end{align*}
uniformly for all $ x \in \overline{\Om}$. Let $f$ satisfies $(\mc F)$. Let $f(\cdot)$ and $G(x,\cdot)$ be non decreasing functions for all $x \in \Om$. Suppose $\underline{w}, \overline{w} \in X_0$ are a weak subsolution and a weak supersolution, respectively to $(P)$, which are not solutions. Then, there exists a solution $ w_0\in X_0$ to $(P)$ such that $\underline{w} \leq w_0 \leq \overline{w}$ a.e in $\Om$ and $w_0$ is a local minimizer of $J$ on $X_0$. \end{Theorem} \begin{proof}
Consider a closed convex set $W$ of $X_0$ as \begin{align*} W: = \{ w \in X_0\; :\; \underline{w} \leq w_0 \leq \overline{w} \text{ a.e in } \Om \}. \end{align*} Using the definition of $W$, one can easily prove that \begin{align*}
J(w)\geq \frac{\|w\|^2}{2}- c_1-c_2 \end{align*} for appropriate positive constants $c_1$ and $c_2$. This implies $J$ is coercive on $W$. $J$ is weakly lower semi continuous on $W$. Indeed, let $\{ v_n\} \subset W$ such that $v_n \rp v$ weakly in $X_0$ as $n\ra \infty$. For each $n$, \begin{align*} & \int_{ \Om} G(x,v_n)~dx \leq \int_{ \Om} G(x,v)~dx< +\infty,\\
& \iint_{\Om \times \Om}\frac{F(v_n)F(v_n)}{|x-y|^{\mu}}~dxdy\leq \iint_{\Om \times \Om}\frac{F(\overline{w})F(\overline{w})}{|x-y|^{\mu}}~dxdy< +\infty. \end{align*} Now we may invoke Dominated convergence theorem and the weak lower semicontinuity of norms to get that $J$ is weakly lower semi continuous on $W$. Hence, there exists $ w_0 \in X_0$ such that \begin{align}\label{ch31} \inf_{w \in W} J(w) = J(w_0). \end{align} \textbf{Claim:} $w_0$ is a weak solution to $(P)$. \\ Let $\phi \in C_c^\infty(\Om)$ and $\varepsilon>0$. Define \begin{align*} u_\varepsilon = \min\{ \overline{w}, \max\{ \underline{w}, w_0+ \varepsilon\phi\} \}= v_0+\varepsilon\phi - \phi^{\varepsilon}+\phi_{\varepsilon} \end{align*} where $\phi^{\varepsilon}= \max\{ 0, w_0+ \varepsilon\phi -\overline{w} \}$ and $\phi_{\varepsilon}= \max\{ 0, \underline{w}-w_0- \varepsilon\phi\}$. Observe that $\phi_{\varepsilon},\phi^{\varepsilon} \in X_0 \cap L^\infty(\Om)$. In view of the fact that $w_0+ t(u_\varepsilon-w_0) \in W$ for all $ t \in (0,1)$ and \eqref{ch31}, we obtain \begin{align*}
\int_{ \mathbb{R}^N}(-\De)^s w_0 (u_\varepsilon-w_0)~dx - \int_{\Om} g(x,w_0)(u_\varepsilon-w_0)~dx- \iint_{\Om \times \Om}\frac{F(w_0)f(w_0) (u_\varepsilon-w_0)}{|x-y|^{\mu}}~dxdy \geq0. \end{align*} Set \begin{align*} & A^\varepsilon= \int_{ \mathbb{R}^N}(-\De)^s (w_0-\overline{w}) \phi^{\varepsilon} ~dx + \int_{ \mathbb{R}^N}(-\De)^s \overline{w}\phi^{\varepsilon} ~dx- \int_{\Om} g(x,w_0)\phi^{\varepsilon}~dx\\
&\qquad - \iint_{\Om \times \Om}\frac{F(w_0)f(w_0) \phi^{\varepsilon}}{|x-y|^{\mu}}~dxdy, \\ & A_\varepsilon= \int_{ \mathbb{R}^N}(-\De)^s (w_0-\underline{w}) \phi_{\varepsilon} ~dx + \int_{ \mathbb{R}^N}(-\De)^s \underline{w}\phi_{\varepsilon} ~dx- \int_{\Om} g(x,w_0)\phi_{\varepsilon}~dx\\
&\qquad - \iint_{\Om \times \Om}\frac{F(w_0)f(w_0) \phi_{\varepsilon}}{|x-y|^{\mu}}~dxdy . \end{align*} Then by simple computations, we get \begin{align}\label{ch32}
\int_{ \mathbb{R}^N}(-\De)^s w_0 \phi~dx - \int_{\Om} g(x,w_0)\phi~dx- \iint_{\Om \times \Om}\frac{F(w_0)f(w_0) \phi}{|x-y|^{\mu}}~dxdy \geq \frac{1}{\varepsilon} \left( A^\varepsilon- A_\varepsilon \right). \end{align} Using the assertions as in \cite[Propostion 3.2]{gts1} with $\overline{w}$ in spite of $u_{\la^\prime}$, we have \begin{align*} \frac{1}{\varepsilon}\int_{ \mathbb{R}^N}(-\De)^s (w_0-\overline{w}) \phi^{\varepsilon} ~dx\geq o(1) \text{ as } \varepsilon \ra 0^+. \end{align*} To this end, employing the fact that $\overline{w} $, we deduce that \begin{align*}
& \frac{1}{\varepsilon} \int_{ \mathbb{R}^N}(-\De)^s \overline{w}\phi^{\varepsilon} ~dx- \frac{1}{\varepsilon}\int_{\Om} g(x,w_0)\phi^{\varepsilon}~dx- \frac{1}{\varepsilon}\iint_{\Om \times \Om}\frac{F(w_0)f(w_0) \phi^{\varepsilon}}{|x-y|^{\mu}}~dxdy\\
&\geq \frac{1}{\varepsilon} \int_{\Om}(g(x,\overline{w})- g(x,w_0))\phi^{\varepsilon}~dx+ \frac{1}{\varepsilon} \iint_{\Om \times \Om}\frac{(F(\overline{w})f(\overline{w})-F(w_0)f(w_0)) \phi^{\varepsilon}}{|x-y|^{\mu}}~dxdy\\ & \geq \frac{1}{\varepsilon} \int_{\Om}(g(x,\overline{w})- g(x,w_0))\phi^{\varepsilon}~dx = o(1) \text{ as } \varepsilon \ra 0^+. \end{align*} Hence we infer that $ \frac{1}{\varepsilon} A^{\varepsilon}\geq o(1)\text{ as } \varepsilon \ra 0^+$. On the similar lines, one can prove that $ \frac{1}{\varepsilon} A_{\varepsilon}\leq o(1)\text{ as } \varepsilon \ra 0^+$. From \eqref{ch32}, for all $\phi \in C_c^\infty(\Om)$, it follows that \begin{align}
\int_{ \mathbb{R}^N}(-\De)^s w_0 \phi~dx - \int_{\Om} g(x,w_0)\phi~dx- \iint_{\Om \times \Om}\frac{F(w_0)f(w_0) \phi}{|x-y|^{\mu}}~dxdy \geq 0 \text{ as } \varepsilon \ra 0^+. \end{align} As $\phi \in C_c^\infty(\Om)$ was arbitrarily chosen, it implies that $w_0$ is weak solution to $(P)$. From this, we follows that there exists a solution $ w_0\in X_0$ to $(P)$ such that $\underline{w} \leq w_0 \leq \overline{w}$ a.e in $\Om$. To prove that $w_0$ is a local minimizer in $X_0$, we proceed as follows. Using Theorem \ref{thmch1} and \cite[Theorem 1.2]{RS}, we deduce $ w_0 \in C^{0,\al}_d(\overline{\Om})$. Now consider \begin{align*}
(-\De)^s(w_0-\underline{w})& \geq( g(x,w_0)- g(x,\underline{w})) + \left( \int_{\Om }\frac{F(w_0)}{|x-y|^{\mu}}~dy\right) f(w_0)- \left( \int_{\Om }\frac{F(\underline{w})}{|x-y|^{\mu}}~dy\right) f(\underline{w})\\ & \geq 0 . \end{align*}
Using the fact that $\underline{w}$ is not solution to $(P)$, we have $w_0 \not = \underline{w}$ and by definition, $w_0-\underline{w}\geq 0$ in $\mathbb{R}^N \setminus \Om$. From \cite[Lemma 2.7]{squassina}, we infer that $w_0 -\underline{w}>Cd^s$ for some $C>0$. On a similar note $\overline{w}-w_0 > Cd^s$ for some $C>0$. For each $w \in \bar{B}^d_{C/2}(w_0)$, we have \begin{align*} \frac{w_0 -\underline{w}}{d^s} = \frac{w_0 -w}{d^s}+ \frac{w -\underline{w}}{d^s} \geq \frac{C}{2}. \end{align*} From above, it can read that $w_0 -\underline{w}>0 $ in $\Om$. Likewise, $\overline{w}-w_0>0$ in $\Om$. Therefore, $w_0$ emerge as a local minimizer of $J$ on $X_0\cap \bar{B}^d_{C/2}(w_0)$ and this completes the proof.
{$\square$}\goodbreak
\end{proof} \begin{Remark} Consider the following problem \begin{equation}\label{ch33} \left\{\begin{array}{rllll} (-\De)^s u
&=\la \left( |u|^{q-2}u + \left(\ds \int_{\Om}\frac{F(u)(y)}{|x-y|^{\mu}}dy\right) f(u)\right) ,\; u>0 \; \text{in}\; \Om,\\ u&=0 \; \text{ in } \mathbb{R}^N \setminus \Om, \end{array} \right. \end{equation} where $\la>0,\; 1<q<2$ and $f$ is a non decreasing function and satisfies $(\mc F)$. Let $\underline{v}$ denotes the solution to \begin{equation*} (-\De)^s u
=\la |u|^{q-2}u , \; u>0 \; \text{in}\; \Om, u=0 \; \text{ in } \mathbb{R}^N \setminus \Om, \end{equation*} and let $\overline{v}$ is a solution to \begin{equation*} (-\De)^s u =1 , \; u>0 \; \text{in}\; \Om, u=0 \; \text{ in } \mathbb{R}^N \setminus \Om. \end{equation*} Then for all $\la>0,\; \underline{v}$ is a subsolution to \eqref{ch33}. And for $\la$ small enough, $\overline{v}$ is a supersolution to \eqref{ch33}. Now using Theorem \ref{thmch4}, there exists a solution to \eqref{ch33}, which is a local minimizer in $X_0$. The moutain pass lemma provides then the existence of a second solution. \end{Remark}
\end{document} | arXiv |
An existence criterion for the $\mathcal{PT}$-symmetric phase transition
Discontinuity waves as tipping points: Applications to biological & sociological systems
September 2014, 19(7): 1935-1953. doi: 10.3934/dcdsb.2014.19.1935
Singular limit of an integrodifferential system related to the entropy balance
Elena Bonetti 1, , Pierluigi Colli 1, and Gianni Gilardi 1,
Dipartimento di Matematica "F. Casorati", Università di Pavia, Via Ferrata 1, 27100 Pavia
Received January 2013 Revised May 2013 Published August 2014
A thermodynamic model describing phase transitions with thermal memory, in terms of an entropy equation and a momentum balance for the microforces, is adressed. Convergence results and error estimates are proved for the related integrodifferential system of PDE as the sequence of memory kernels converges to a multiple of a Dirac delta, in a suitable sense.
Keywords: thermal memory, phase field model, asymptotics on the memory term., nonlinear partial differential equations, Entropy equation.
Mathematics Subject Classification: Primary: 35K55, 35B40; Secondary: 35Q79, 80A2.
Citation: Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935
G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of Materials with Memory. Theory and Applications,, Springer, (2012). doi: 10.1007/978-1-4614-1692-0. Google Scholar
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces,, Noordhoff International Publishing, (1976). Google Scholar
E. Bonetti, P. Colli and M. Frémond, A phase field model with thermal memory governed by the entropy balance,, Math. Models Methods Appl. Sci., 13 (2003), 1565. doi: 10.1142/S0218202503003033. Google Scholar
E. Bonetti, P. Colli, M. Fabrizio and G. Gilardi, Modelling and long-time behaviour for phase transitions with entropy balance and thermal memory conductivity,, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1001. doi: 10.3934/dcdsb.2006.6.1001. Google Scholar
E. Bonetti, P. Colli, M. Fabrizio and G. Gilardi, Global solution to a singular integrodifferential system related to the entropy balance,, Nonlinear Anal., 66 (2007), 1949. doi: 10.1016/j.na.2006.02.035. Google Scholar
E. Bonetti, P. Colli, M. Fabrizio and G. Gilardi, Existence and boundedness of solutions for a singular phase field system,, J. Differential Equations, 246 (2009), 3260. doi: 10.1016/j.jde.2009.02.007. Google Scholar
E. Bonetti, M. Frémond and E. Rocca, A new dual approach for a class of phase transitions with memory: Existence and long-time behaviour of solutions,, J. Math. Pures Appl., 88 (2007), 455. doi: 10.1016/j.matpur.2007.09.005. Google Scholar
H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,, North-Holland Math. Stud., (1973). Google Scholar
G. Canevari and P. Colli, Solvability and asymptotic analysis of a generalization of the Caginalp phase field system,, Commun. Pure Appl. Anal., 11 (2012), 1959. doi: 10.3934/cpaa.2012.11.1959. Google Scholar
G. Canevari and P. Colli, Convergence properties for a generalization of the Caginalp phase field system,, Asymptot. Anal., 82 (2013), 139. doi: 10.3233/ASY-2012-1142. Google Scholar
M. Frémond, Non-smooth Thermomechanics,, Springer-Verlag, (2002). doi: 10.1007/978-3-662-04800-9. Google Scholar
G. Gilardi and E. Rocca, Convergence of phase field to phase relaxation governed by the entropy balance with memory,, Math. Methods Appl. Sci., 29 (2006), 2149. doi: 10.1002/mma.765. Google Scholar
A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermo-mechanics,, Proc. Roy. Soc. Lond. A, 432 (1991), 171. doi: 10.1098/rspa.1991.0012. Google Scholar
G. Gripenberg, S-O. Londen and O. Staffans, Volterra Integral and Functional Equations,, Encyclopedia Math. Appl., (1990). doi: 10.1017/CBO9780511662805. Google Scholar
M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds,, Arch. Rational Mech. Anal., 31 (1968), 113. doi: 10.1007/BF00281373. Google Scholar
P. Podio-Guidugli, A virtual power format for thermomechanics,, Contin. Mech. Thermodyn., 20 (2009), 479. doi: 10.1007/s00161-009-0093-5. Google Scholar
J. Simon, Compact sets in the space $L^p(0,T; B)$,, Ann. Mat. Pura Appl. (4), 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar
Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089
Elena Bonetti, Pierluigi Colli, Mauro Fabrizio, Gianni Gilardi. Modelling and long-time behaviour for phase transitions with entropy balance and thermal memory conductivity. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1001-1026. doi: 10.3934/dcdsb.2006.6.1001
Federico Mario Vegni. Dissipativity of a conserved phase-field system with memory. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 949-968. doi: 10.3934/dcds.2003.9.949
Michela Eleuteri, Luca Lussardi, Ulisse Stefanelli. Thermal control of the Souza-Auricchio model for shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 369-386. doi: 10.3934/dcdss.2013.6.369
Kung-Ching Chang. In memory of professor Rouhuai Wang (1924-2001): A pioneering Chinese researcher in partial differential equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 571-575. doi: 10.3934/dcds.2016.36.571
Andrei Halanay, Luciano Pandolfi. Lack of controllability of thermal systems with memory. Evolution Equations & Control Theory, 2014, 3 (3) : 485-497. doi: 10.3934/eect.2014.3.485
Sergiu Aizicovici, Hana Petzeltová. Convergence to equilibria of solutions to a conserved Phase-Field system with memory. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 1-16. doi: 10.3934/dcdss.2009.2.1
S. Gatti, M. Grasselli, V. Pata, M. Squassina. Robust exponential attractors for a family of nonconserved phase-field systems with memory. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 1019-1029. doi: 10.3934/dcds.2005.12.1019
Umberto Biccari, Mahamadi Warma. Null-controllability properties of a fractional wave equation with a memory term. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020011
Yongqin Liu. Asymptotic behavior of solutions to a nonlinear plate equation with memory. Communications on Pure & Applied Analysis, 2017, 16 (2) : 533-556. doi: 10.3934/cpaa.2017027
Monica Conti, Elsa M. Marchini, V. Pata. Global attractors for nonlinear viscoelastic equations with memory. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1893-1913. doi: 10.3934/cpaa.2016021
Michela Eleuteri, Luca Lussardi. Thermal control of a rate-independent model for permanent inelastic effects in shape memory materials. Evolution Equations & Control Theory, 2014, 3 (3) : 411-427. doi: 10.3934/eect.2014.3.411
S. Gatti, Elena Sartori. Well-posedness results for phase field systems with memory effects in the order parameter dynamics. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 705-726. doi: 10.3934/dcds.2003.9.705
Pierluigi Colli, Gianni Gilardi, Philippe Laurençot, Amy Novick-Cohen. Uniqueness and long-time behavior for the conserved phase-field system with memory. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 375-390. doi: 10.3934/dcds.1999.5.375
Monica Conti, Stefania Gatti, Alain Miranville. A singular cahn-hilliard-oono phase-field system with hereditary memory. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3033-3054. doi: 10.3934/dcds.2018132
Gianluca Mola. Global attractors for a three-dimensional conserved phase-field system with memory. Communications on Pure & Applied Analysis, 2008, 7 (2) : 317-353. doi: 10.3934/cpaa.2008.7.317
Eugenio Sinestrari. Wave equation with memory. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 881-896. doi: 10.3934/dcds.1999.5.881
Kai Liu. On regularity of stochastic convolutions of functional linear differential equations with memory. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1279-1298. doi: 10.3934/dcdsb.2019220
Yves Achdou, Mathieu Laurière. On the system of partial differential equations arising in mean field type control. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3879-3900. doi: 10.3934/dcds.2015.35.3879
Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703
Elena Bonetti Pierluigi Colli Gianni Gilardi | CommonCrawl |
\begin{document}
\title{Euler hydrodynamics for attractive particle systems in random environment} \author{C. Bahadoran$^{a,e}$, H. Guiol$^{b,e}$, K. Ravishankar$^{c,e,f}$, E. Saada$^{d,e}$} \date{} \maketitle
$$ \begin{array}{l} ^a\,\mbox{\small Laboratoire de Math\'ematiques, Universit\'e Clermont 2, 63177 Aubi\`ere, France} \\ \quad \mbox{\small e-mail: [email protected]}\\ ^b\, \mbox{\small UJF-Grenoble 1 / CNRS / Grenoble INP / TIMC-IMAG UMR 5525, Grenoble, F-38041, France } \\ \quad \mbox{\small e-mail: [email protected]}\\ ^c\, \mbox{\small Dep. of Mathematics, SUNY, College at New Paltz, NY, 12561, USA} \\ \quad \mbox{\small e-mail:
[email protected]}\\ ^d\, \mbox{\small CNRS, UMR 8145, MAP5, Universit\'e Paris Descartes,
Sorbonne Paris Cit\'e, France}\\ \quad \mbox{\small e-mail: [email protected]}\\ \\
^e\, \mbox{\small Supported by grants ANR-07-BLAN-0230, ANR-2010-BLAN-0108, PICS 5470 }\\ ^f\, \mbox{\small Supported by NSF grant DMS 0104278}\\
\end{array} $$
\begin{abstract} \noindent We prove quenched hydrodynamic limit under hyperbolic time scaling for bounded attractive particle systems on ${\mathbb Z}$ in random ergodic environment. Our result is a strong law of large numbers, that we illustrate with various examples. \end{abstract}
\textbf{Keywords:} Hydrodynamic limit, attractive particle system, scalar conservation law, entropy solution, random environment, quenched disorder, generalized misanthropes and $k$-step models. \\ \\
\textbf{AMS 2000 Subject Classification: }Primary 60K35; Secondary 82C22.\\ \\
\section{Introduction}
Hydrodynamic limit
describes the time evolution (usually governed by a limiting PDE,
called the hydrodynamic equation) of empirical density fields in interacting particle systems (\emph{IPS}). For usual models, such as the simple exclusion process, the limiting PDE is a nonlinear diffusion equation or hyperbolic conservation law (see \cite{kl} and references therein). In this context, a random environment leads to homogeneization-like effects, where an effective diffusion matrix or flux function is expected to capture the effect of inhomogeneity. Hydrodynamic limit in random environment has been widely addressed and robust methods have been developed in the diffusive case (\cite{fa,fam,fri,gj,jar,kou,nag,qua}).\\ \\
In the hyperbolic setting, due to non-existence of strong solutions and non-uniqueness of weak solutions, the key issue is to establish convergence to the so-called entropy solution (see e.g. \cite{serre}) of the Cauchy problem. The first such result without restrictive assumptions is due to \cite{fraydoun} for spatially homogeneous attractive systems with product invariant measures.
In random environment, the few available results depend on particular features of the investigated models. In \cite{bfl}, the authors consider the asymmetric zero-range process with site disorder on ${\mathbb Z}^d$, extending a model introduced in \cite{ev}. They prove a quenched hydrodynamic limit given by a hyperbolic conservation law with an effective homogeneized flux function. To this end, they use in particular the existence of explicit product invariant measures for the disordered zero-range process below some critical value of the disorder parameter. In \cite{ks}, extension to the supercritical case is carried out in the totally asymmetric case with constant jump rate.
In \cite{timo}, under a strong mixing assumption, the author establishes a quenched hydrodynamic limit for the totally asymmetric nearest-neighbor $K$-exclusion process on ${\mathbb Z}$ with site disorder, for which explicit invariant measures are not known. The last two results rely on
a microscopic version of the Lax-Hopf formula. However, the simple exclusion process beyond the totally asymmetric nearest-neighbor case, or more complex models with state-dependent jump rates, remain outside the scope of the above approaches. \\ \\
In this paper, we prove quenched hydrodynamics for attractive particle systems in random environment on ${\mathbb Z}$ with a bounded number of particles per site. Our method is quite robust with respect to the model and disorder. We only require the environment to be ergodic. Besides, we are not restricted to site or bond disorder. However, for simplicity we treat in detail the misanthropes' process with site disorder, and explain in the last section how our method applies to various other models.
An essential difficulty for the disordered system is the simultaneous loss of translation invariance {\em and} lack of knowledge of explicit invariant measures. Note that even if the system without disorder has explicit invariant measures, the disordered system in general does not, with the above exception of the zero-range process. In particular, one does not have an effective characterization theorem for invariant measures of the quenched process.
Our strategy is to prove hydrodynamic limit for a joint disorder-particle process which, unlike the quenched process, is translation invariant. The idea is that hydrodynamic limit for the joint process should imply quenched hydrodynamic limit.
This is false for limits in the usual (weak) sense, but becomes true if a \textit{strong} hydrodynamic limit is proved for the joint process. We are able to do it
by characterizing the extremal invariant and translation invariant measures of the joint process, and by adapting the tools developed in \cite{bgrs3}.\\ \\
The paper is organized as follows. In Section \ref{sec_results}, we define the model and state our main result. Section \ref{sec_disorder_particle} is devoted to the study of the joint disorder-particle process and characterization of its invariant measures. The hydrodynamic limit is proved in Section \ref{proof_hydro}. Finally, in Section \ref{sec:general} we consider
models other than the misanthropes' process: We detail generalizations of misanthropes and $k$-step exclusion processes, as well as a traffic model.
\section{Notation and results}\label{sec_results}
Throughout this paper ${\mathbb N}=\{1,2,...\}$ will denote the set of natural numbers, and ${\mathbb Z}^+=\{0,1,2,...\}$ the set of non-negative integers. The integer part $\lfloor x\rfloor\in{\mathbb Z}$ of $x\in{\mathbb R}$ is uniquely defined by $\lfloor x\rfloor \leq x<\lfloor x\rfloor +1$.
We consider particle configurations on ${\mathbb Z}$ with at most $K$ particles per site, $K\in{\mathbb N}$. Thus the state space of the process is ${\mathbf X}=\{0,1,\cdots,K\}^{{\mathbb Z}}$, which we endow with the product topology, that makes ${\mathbf X}$ a compact metrisable space, with the product (partial) order.\\ \\
The set $\mathbf A$ of environments
is a compact metric space endowed with its Borel $\sigma$-field.
A function $f$ defined on ${\mathbf A}\times{\mathbf X}$
(resp. $g$ on ${\mathbf A}\times{\mathbf X}^2$)
is called {\em local} if there is a finite subset $\Lambda$ of ${\mathbb Z}$ such that $f(\alpha,\eta)$ depends only on $\alpha$ and $(\eta(x),x\in\Lambda)$
(resp. $g(\alpha,\eta,\xi)$ depends only on $\alpha$ and $(\eta(x),\xi(x),x\in\Lambda)$).
We denote by $\tau_x$ either the spatial translation operator on the real line for $x\in{\mathbb R}$, defined by $\tau_x y=x+y$, or its restriction to $x\in{\mathbb Z}$. By extension, if $f$ is a function defined on ${\mathbb Z}$ (resp. ${\mathbb R}$), we set $\tau_x f=f\circ\tau_x$ for $x\in{\mathbb Z}$ (resp. ${\mathbb R}$). In the sequel this will be applied to particle configurations $\eta\in\mathbf X$, disorder configurations $\alpha\in\mathbf{A}$, or joint disorder-particle configurations $(\alpha,\eta)\in\mathbf{A}\times\mathbf{X}$. In the latter case, unless mentioned explicitely, $\tau_x$
applies simultaneously to both components.\\ \\
If $\tau_x$ acts on some set and $\mu$ is a measure on this set, $\tau_x\mu=\mu\circ\tau_{x}^{-1}$.
We let ${\mathcal M}^+({\mathbb R})$ denote the set of nonnegative measures on ${\mathbb R}$ equipped with the metrizable topology of vague convergence, defined by convergence on continuous test functions with compact support. The set of probability measures on $\mathbf{X}$ is denoted by ${\mathcal P}(\mathbf{X})$. If $\eta$ is an ${\mathbf X}$-valued random variable and $\nu\in{\mathcal P}(\mathbf{X})$, we write $\eta\sim\nu$ to specify that $\eta$ has distribution $\nu$. Similarly, for $\alpha\in\mathbf{A}, Q\in{\mathcal P}(\mathbf{A})$, $\alpha\sim Q$ means that $\alpha$ has distribution $Q$. \\ \\
A sequence $(\nu_n,n\in{\mathbb N})$ of probability measures on ${\mathbf X}$ converges weakly to some $\nu\in{\mathcal P}(\mathbf{X})$, if and only if $\lim_{n\to\infty}\int f\,d\nu_n=\int f\,d\nu$ for every continuous function $f$ on ${\mathbf X}$. The topology of weak convergence is metrizable and makes ${\mathcal P}({\mathbf X})$ compact.
A partial stochastic order is defined on ${\mathcal P}(\mathbf{X})$; namely, for $\mu_1,\mu_2\in{\mathcal P}(\mathbf{X})$, we write $\mu_1\leq\mu_2$ if the following equivalent conditions hold (see {\em e.g.} \cite{lig1, strassen}): \textit{(i)} For every non-decreasing nonnegative function $f$ on $\mathbf X$, $\int f\,d\mu_1\leq\int f\,d\mu_2$. \textit{(ii)} There exists a coupling measure $\overline{\mu}$ on $ \mathbf {X}\times\mathbf {X}$ with marginals $\mu_1$ and $\mu_2$, such that $\overline{\mu}\{(\eta,\xi):\,\eta\leq\xi\}=1$. \\ \\
In the following model, we fix a constant $c>0$ and define ${\mathbf A}=[c,1/c]^{\mathbb Z}$ to be the set of environments (or disorders) $\alpha=(\alpha(x):\,x\in{\mathbb Z})$ such that
\begin{equation}\label{alpha_bounds}\forall x\in{\mathbb Z},\quad c\leq\alpha(x)\leq c^{-1}\end{equation}
For each realization $\alpha\in{\mathbf A}$ of the disorder, the {\em quenched process} $(\eta_t)_{t\geq 0}$ is a Feller process on $\mathbf X$ with generator given by, for any local function $f$ on ${\mathbf X}$,
\begin{equation} \label{generator} L_\alpha f(\eta)=\sum_{x,y\in{{\mathbb Z}}}\alpha(x) p(y-x)b(\eta(x),\eta(y)) \left[ f\left(\eta^{x,y} \right)-f(\eta) \right] \end{equation}
where $\eta^{x,y}$ denotes the new state after a particle has jumped from $x$ to $y$ (that is $\eta^{x,y}(x)=\eta(x)-1,\,\eta^{x,y}(y)=\eta(y)+1,\, \eta^{x,y}(z)=\eta(z)$ otherwise), the particles' jump kernel $p$ is a probability distribution on ${\mathbb Z}$, and $b\ :\ {\mathbb Z}^+\times{\mathbb Z}^+\to {\mathbb R}^+$ is the jump rate.
We assume that $p$ and $b$ satisfy: \\ \\
{\em (A1)} Irreducibility: For every $z\in{\mathbb Z}$, $\sum_{n\in{\mathbb N}}[p^{*n}(z)+p^{*n}(-z)]>0$, where $*n$ denotes $n$-th convolution power;\\ {\em (A2)} finite mean: $\sum_{z\in{\mathbb Z}}\abs{z}p(z)<+\infty$;\\ {\em (A3)} $K$-exclusion rule: $b(0,.)=0,\,b(.,K)=0$;\\ {\em (A4)} non-degeneracy: $b(1,K-1)>0$; \\ {\em (A5)} attractiveness: $b$ is nondecreasing (nonincreasing) in its first (second) argument. \\ \\
For the graphical construction of the system given by \eqref{generator} (see \cite{bgrs3} and references therein), let us introduce a general framework that applies to a larger class of models (see Section \ref{sec:general} below).
Given a measurable space $(\mathcal V,{\mathcal F}_\mathcal V,m)$,
for $m$ a finite nonnegative measure, we consider the probability space $(\Omega,{\mathcal F},{\rm I\hspace{-0.8mm}P})$ of locally finite point measures $\omega(dt,dx,dv)$ on ${\mathbb R}^+\times{\mathbb Z}\times\mathcal V$, where $\mathcal F$ is generated by the mappings $\omega\mapsto\omega(S)$ for Borel sets $S$ of ${\mathbb R}^+\times{\mathbb Z}\times\mathcal V$, and ${\rm I\hspace{-0.8mm}P}$ makes $\omega$ a Poisson process with intensity
\[ M(dt,dx,dv)= \lambda_{{\mathbb R}^+}(dt)\lambda_{\mathbb Z}(dx)m(dv) \]
denoting by $\lambda$ either the Lebesgue or
the counting measure. We write ${\rm I\hspace{-0.8mm}E}$ for expectation
with respect to ${\rm I\hspace{-0.8mm}P}$. For the particular model \eqref{generator} we take
\begin{equation}\label{special_choice} \mathcal V:={\mathbb Z}\times[0,1],\quad v=(z,u)\in\mathcal V,\quad
m(dv)=c^{-1}||b||_\infty p(dz)\lambda_{[0,1]}(du)
\end{equation}
Thanks to assumption {\em (A2)}, for ${\rm I\hspace{-0.8mm}P}$-a.e. $\omega$, there exists a unique mapping
\begin{equation} \label{unique_mapping} (\alpha,\eta_0,t)\in{{\mathbf A}}\times{\mathbf X}\times{\mathbb R}^+\mapsto\eta_t=\eta_t(\alpha,\eta_0,\omega)\in{\mathbf X} \end{equation}
satisfying: \textit{(a)} $t\mapsto\eta_t(\alpha,\eta_0,\omega)$ is right-continuous; \textit{(b)} $\eta_0(\alpha,\eta_0,\omega)=\eta_0$; \textit{(c)} the particle configuration is updated at points $(t,x,v)\in\omega$ (and only at such points; by $(t,x,v)\in\omega$ we mean $\omega\{(t,x,v)\}=1$) according to the rule
\begin{equation}\label{update_rule}
\eta_t(\alpha,\eta_0,\omega)={\mathcal T}^{\alpha,x,v}\eta_{t^-}(\alpha,\eta_0,\omega)
\end{equation}
where, for $v=(z,u)\in\mathcal V$, ${\mathcal T}^{\alpha,x,v}$ is defined by
\begin{equation}\label{update_misanthrope}
{\mathcal T}^{\alpha,x,v}\eta=\left\{
\begin{array}{lll} \eta^{x,x+z} & \mbox{if} & u<\displaystyle\alpha(x){\frac{b(\eta(x),\eta(x+z))}
{c^{-1}||b||_\infty}}\\
\eta & & \mbox{otherwise}
\end{array}
\right.
\end{equation}
Notice the shift commutation property
\begin{equation}\label{shift_t}
{\mathcal T}^{\tau_x\alpha,y,v}\tau_x=\tau_x{\mathcal T}^{\alpha,y+x,v}
\end{equation}
where $\tau_x$ on the right-hand side acts only on $\eta$.
By assumption {\em (A5)},
\begin{equation}\label{attractive_0}{\mathcal T}^{\alpha,x,v}:{\mathbf X}\to{\mathbf X}\mbox{ is nondecreasing}\end{equation}
Hence,
\begin{equation} \label{attractive_1} (\alpha,\eta_0,t)\mapsto\eta_t(\alpha,\eta_0,\omega) \mbox{ is nondecreasing w.r.t. } \eta_0\end{equation}
For every $\alpha\in{\mathbf A}$, under ${\rm I\hspace{-0.8mm}P}$, $(\eta_t(\alpha,\eta_0,\omega))_{t\geq 0}$ is a Feller process with generator
\begin{equation}\label{gengen} L_\alpha f(\eta)=\sum_{x\in{\mathbb Z}}\int_{\mathcal V}\left[
f\left({\mathcal T}^{\alpha,x,v}\eta\right)-f(\eta)
\right]m(dv) \end{equation}
With \eqref{update_misanthrope}, \eqref{gengen} reduces to \eqref{generator}. Thus for any $t\in{\mathbb R}^+$ and continuous function $f$ on ${\mathbf X}$, ${\rm I\hspace{-0.8mm}E}[f(\eta_t(\alpha,\eta_0,\omega))]=S_{\alpha}(t)f(\eta_0)$, where $S_\alpha$ denotes the semigroup generated by $L_\alpha$.
{}From \eqref{attractive_1}, for $\mu_1,\mu_2\in{\mathcal P}(\mathbf{X})$,
\begin{equation} \label{attractive_2} \mu_1\leq\mu_2\Rightarrow\forall t\in{\mathbb R}^+,\,\mu_1 S_\alpha(t)\leq\mu_2 S_\alpha(t) \end{equation}
Property \eqref{attractive_2} is usually called {\em attractiveness}. Condition \eqref{attractive_0} implies the stronger {\em complete monotonicity} property (\cite{fm, dplm}), that is, existence of a monotone Markov coupling for an arbitrary number of processes with generator \eqref{generator}, see \eqref{coupling_t} below; we also say that the process is \textit{strongly attractive}. \\ \\
Let $N\in{\mathbb N}$ be the scaling parameter for the hydrodynamic limit, that is, the inverse of the macroscopic distance between two consecutive sites. The empirical measure of a configuration $\eta$ viewed on scale $N$ is given by
\[ \pi^N(\eta)(dx)=N^{-1}\sum_{y\in{\mathbb Z}}\eta(y)\delta_{y/N}(dx) \in{\mathcal M}^+({\mathbb R}) \]
where, for $x\in{\mathbb R}$, $\delta_x$ denotes the Dirac measure at $x$. \\ Our main result is
\begin{theorem}\label{th:hydro} Assume $p(.)$ has finite third moment.
Let $Q$ be an ergodic probability distribution on $\mathbf A$. Then there exists a Lipschitz-continuous function $G^Q$ on $[0,K]$ defined below (depending only on $p(.)$, $b(.,.)$ and $Q$) such that the following holds.
Let $(\eta^N_0,\,N\in{\mathbb N})$ be a sequence of ${\mathbf X}$-valued random variables on a probability space $(\Omega_0,\mathcal F_0,{\rm I\hspace{-0.8mm}P}_0)$ such that
\begin{equation}\label{initial_profile_vague} \lim_{N\to\infty}\pi^N(\eta^N_0)(dx)= u_0(.)dx\quad{\rm I\hspace{-0.8mm}P}_0\mbox{-a.s.}\end{equation}
for some measurable $[0,K]$-valued profile $u_0(.)$.
Then for $Q$-a.e. $\alpha\in{\mathbf A}$, the ${\rm I\hspace{-0.8mm}P}_0\otimes{\rm I\hspace{-0.8mm}P}$-a.s. convergence
\[ \lim_{N\to\infty}\pi^N(\eta_{Nt}(\alpha,\eta^N_0(\omega_0),\omega))(dx)=u(.,t)dx
\]
holds uniformly on all bounded time intervals, where $(x,t)\mapsto u(x,t)$ denotes the unique entropy solution with initial condition $u_0$ to the conservation law
\begin{equation} \label{hydrodynamics} \partial_t u+\partial_x[G^Q(u)]=0 \end{equation}
\end{theorem}
We refer the reader (for instance) to \cite{serre} for the definition of entropy solutions.
To define the {\sl macroscopic flux} $G^Q$, let
the {\sl microscopic flux} through site 0 be
\begin{eqnarray}\label{def_f} j(\alpha,\eta)&=&j^+(\alpha,\eta)-j^-(\alpha,\eta)\\\nonumber j^+(\alpha,\eta) &=& \sum_{y,z\in{\mathbb Z}:\,y\leq 0<y+z}\alpha(y) p(z)b(\eta(y),\eta(y+z))\\\nonumber
j^-(\alpha,\eta) & = & \sum_{y,z\in{\mathbb Z}:\,y+z\leq 0<y}\alpha(y) p(z)b(\eta(y),\eta(y+z)) \end{eqnarray}
We will show in Corollary \ref{corollary_invariant} below that there exists a closed subset $\mathcal R^Q$ of $[0,K]$, a subset $\mathbf{\widetilde{A}}^Q$ of ${\mathbf A}$ with $Q$-probability $1$ (both depending also on $p(.)$ and $b(.,.)$), and a family of probability measures $(\nu^{Q,\rho}_\alpha:\,\alpha\in\mathbf{\widetilde{A}}^Q,\rho\in\mathcal R^Q)$ on ${\mathbf X}$, such that, for every $\rho\in\mathcal R^Q$:\\
\label{properties_b}
\noindent {\em (B1)} For every $\alpha\in\mathbf{\widetilde{A}}^Q$, $\nu^{Q,\rho}_\alpha$ is an invariant measure for $L_\alpha$.\\
{\em (B2)} For every $\alpha\in\mathbf{\widetilde{A}}^Q$, $\nu^{Q,\rho}_\alpha$-a.s.,
$$\lim_{l\to\infty}(2l+1)^{-1}\sum_{x\in{\mathbb Z}:\,|x|\leq l}\eta(x)=\rho$$
{\em (B3)} The quantity
\begin{equation} \label{flux} G^Q_\alpha(\rho):=\int j(\alpha,\eta)\nu^{Q,\rho}_\alpha(d\eta) \end{equation}
does not depend on $\alpha\in\mathbf{\widetilde{A}}^Q$. Hence we define $G^Q(\rho)$ as \eqref{flux} for $\rho\in\mathcal R^Q$ and extend it by linear interpolation on the complement of $\mathcal R^Q$, which is a finite or countably infinite union of disjoint open intervals. \\ \\
The function $G^Q$ is Lipschitz continuous (see Remark \ref{remark_lipschitz} below), which is the minimum regularity required for the classical theory of entropy solutions. We cannot say more about $G^Q$ in general, because the measures $\nu^{Q,\rho}_\alpha$ are not explicit. This is true even in the spatially homogeneous case $\alpha(x)\equiv 1$, unless $b$ satisfies additional algebraic relations introduced in \cite{coc}. In the absence of disorder, for the exclusion process and a few simple models satisfying these relations (see for instance \cite[Section 4]{bgrs1}), we have an explicit flux function. Nevertheless, invariant measures are no longer computable when introducing disorder, so that the effect of the latter on the flux function is difficult to evaluate.
However, in the special case $b(n,m)={\bf 1}_{\{n>0\}}{\bf 1}_{\{m<K\}},\,p(1)=1$, $G^Q$ is shown to be concave in \cite{timo}, as a consequence of the variational approach used to derive hydrodynamic limit. But this approach does not apply to the models we consider in the present paper.
\section{The disorder-particle process} \label{sec_disorder_particle}
In this section we study invariant measures for the markovian {\em joint process} $(\alpha_t,\eta_t)_{t\ge 0}$ on ${\mathbf A}\times {\mathbf X}$ with generator given by, for any local function $f$ on ${\mathbf A}\times{\mathbf X}$,
\begin{equation} \label{generator_joint-general} L f(\alpha,\eta)=\sum_{x\in{{\mathbb Z}}}\int_{\mathcal V}\left[
f\left(\alpha,{\mathcal T}^{\alpha,x,v}\eta\right)-f(\alpha,\eta)
\right]m(dv) \end{equation}
that is, for the particular model \eqref{update_misanthrope},
\begin{equation} \label{generator_joint} L f(\alpha,\eta)=\sum_{x,y\in{{\mathbb Z}}}\alpha(x) p(y-x)b(\eta(x),\eta(y)) \left[ f\left(\alpha,\eta^{x,y} \right)-f(\alpha,\eta) \right] \end{equation}
We denote by $(S(t),t\in{\mathbb R}^+)$ the semigroup generated by $L$. Given $\alpha_0=\alpha$, this dynamics simply means that $\alpha_t=\alpha$ for all $t\geq 0$, while $(\eta_t)_{t\ge 0}$ is a Markov process with generator $L_\alpha$ given by \eqref{generator}. Note that $L$ is \textit{translation invariant}, that is
\begin{equation}\label{eq:L-transl-inv}\tau_x L=L\tau_x\end{equation}
where $\tau_x$ acts jointly on $(\alpha,\eta)$. This is equivalent to a \textit{shift commutation property} for the quenched dynamics:
\begin{equation}\label{commutation}
L_\alpha\tau_x=\tau_x L_{\tau_x\alpha}
\end{equation}
where, since $L_\alpha$ is a Markov generator on ${\mathbf X}$, the first $\tau_x$ on the r.h.s. acts only on $\eta$.
We need to introduce a conditional stochastic order. For the sequel, we define the set $\overline{\mathcal O} = \overline{\mathcal O}_+\cup\overline{\mathcal O}_-$, where
\begin{eqnarray} \overline{\mathcal O}_+ & = & \{(\alpha,\eta,\xi)\in{\mathbf A}\times{\mathbf X}^2: \eta\leq\xi \}\cr \overline{\mathcal O}_- & = & \{(\alpha,\eta,\xi)\in{\mathbf A}\times{\mathbf X}^2: \xi\leq\eta \} \label{def_ordered_set} \end{eqnarray}
\begin{lemma} \label{lemma_conditional}
For two probability measures $\mu^1=\mu^1(d\alpha,d\eta)$, $\mu^2=\mu^2(d\alpha,d\eta)$ on ${\mathbf A}\times{\mathbf X}$, the following properties (denoted by $\mu^1\ll\mu^2$) are equivalent:
(i) For every bounded measurable local function $f$ on ${\mathbf A}\times{\mathbf X}$, such that $f(\alpha,.)$ is nondecreasing for all $\alpha\in{\mathbf A}$, we have $\int f\,d\mu^1\leq\int f\,d\mu^2$.
(ii) The measures $\mu^1$ and $\mu^2$ have a common $\alpha$-marginal $Q$, and
$\mu^1(d\eta|\alpha)\leq\mu^2(d\eta|\alpha)$ for $Q$-a.e. $\alpha\in{\mathbf A}$.
(iii) There exists a coupling measure $\overline{\mu}(d\alpha,d\eta,d\xi)$ supported on $\overline{\mathcal O}_+$ under which $(\alpha,\eta)\sim\mu^1$ and $(\alpha,\xi)\sim\mu^2$.
\end{lemma}
\begin{proof}{lemma}{lemma_conditional}
\textit{(ii)}$\Rightarrow$\textit{(i)} follows from conditioning.
For \textit{(i)}$\Rightarrow$\textit{(ii)}, consider $f(\alpha,\eta)=g(\alpha)h(\eta)$, where $g$ is a nonnegative measurable function on $\mathbf A$ and $h$ a nondecreasing local function on ${\mathbf X}$. Specializing to $h\equiv 1$, using both $f$ and $-f$ in \textit{(i)}, we obtain
$$\displaystyle\int g(\alpha)\mu^1(d\alpha,d\eta)= \int g(\alpha)\mu^2(d\alpha,d\eta)$$
Thus $\mu^1$ and $\mu^2$ have a common $\alpha$-marginal $Q$. Now with a general $h$, by conditioning, we have
$$
\int g(\alpha)\left(\int h(\eta)\mu^1(d\eta|\alpha)\right)Q(d\alpha)\leq\int g(\alpha)\left(\int h(\eta)\mu^2(d\eta|\alpha)\right)Q(d\alpha) $$
Thus, for any nondecreasing local function $h$ on ${\mathbf X}$,
$$\int h(\eta)\mu^1(d\eta|\alpha)\leq\int h(\eta)\mu^2(d\eta|\alpha)$$
holds $Q(d\alpha)$-a.e. Since the set of nondecreasing local functions on ${\mathbf X}$ has a countable dense subset (w.r.t. uniform convergence), we can exchange ``for any $h$'' and ``$Q$-a.e.'' In other words,
$\mu^1(d\eta|\alpha)\leq\mu^2(d\eta|\alpha)$ for $Q$-a.e. $\alpha\in\mathbf A$.\\ \\
For \textit{(ii)}$\Rightarrow$\textit{(iii)}, by Strassen's theorem
(\cite{strassen}), for $Q$-a.e. $\alpha\in{\mathbf A}$, there exists a coupling measure $\overline{\mu}_\alpha(d\eta,d\xi)$ on ${\mathbf X}^2$ under which $\eta\sim\mu^1(.|\alpha)$,
$\xi\sim\mu^2(.|\alpha)$, and $\eta\leq\xi$ a.s. Then $\overline{\mu}(d\alpha,d\eta,d\xi):=\int_{\mathbf A}[\delta_\beta(d\alpha)\overline{\mu}_\alpha(d\eta,d\xi)]Q(d\beta)$ yields the desired coupling.
\textit{(iii)}$\Rightarrow$\textit{(i)} is straightforward.
\end{proof}
\mbox{}\\ \\
We now state the main result of this section. Let $\mathcal I_L$, $\mathcal S$ and ${\mathcal S}^{\mathbf A}$ denote the sets of probability measures that are respectively invariant for $L$, shift-invariant
on $\mathbf{A}\times\mathbf{X}$ and shift-invariant on $\mathbf A$.
\begin{proposition}\label{invariant}
For every $Q\in{\mathcal S}^{\mathbf A}_e$, there exists a closed subset $\mathcal R^Q$ of $[0,K]$ containing $0$ and $K$, such that
\begin{eqnarray*}
\left(\mathcal I_{L}\cap\mathcal S\right)_e & = & \left\{\nu^{Q,\rho},\,Q\in{\mathcal S}^{\mathbf A}_e,\,\rho\in{\mathcal R}^Q\right\}
\end{eqnarray*}
where index $e$ denotes the set of extremal elements, and $(\nu^{Q,\rho}:\,\rho\in\mathcal R^Q)$ is a family of shift-invariant measures on ${\mathbf A}\times{\mathbf X}$, weakly continuous with respect to $\rho$, such that
\begin{eqnarray}\label{densite-rho} \int\eta(0)\nu^{Q,\rho}(d\alpha,d\eta)&=&\rho\\ \label{Rezakhanlou}
\lim_{l\to\infty}(2l+1)^{-1}\sum_{x\in{\mathbb Z}:|x|\le l}\eta(x)&=&\rho, \quad \nu^{Q,\rho}-\mbox{a.s.} \\ \label{ordered_measures}\rho\leq\rho'&\Rightarrow&\nu^{Q,\rho}\ll\nu^{Q,\rho'}\end{eqnarray}
\end{proposition}
For $\rho=0\in {\mathcal R^Q}$ (resp. $\rho=K\in {\mathcal R^Q}$) we get the invariant distribution $\delta_0^{\otimes{\mathbb Z}}$ (resp. $\delta_K^{\otimes{\mathbb Z}}$), the deterministic distribution of the configuration with no particles (resp. with maximum number of particles $K$ everywhere).
\begin{remark}
The set ${\mathcal R^Q}$ and measures $\nu^{Q,\rho}$ also depend on $p(.)$ and $b(.,.)$, but we did not reflect this in the notation because only $Q$ varies in Proposition \ref{invariant}.
\end{remark}
\begin{corollary} \label{corollary_invariant}
(i) The family of probability measures
$\nu^{Q,\rho}_\alpha(.):=\nu^{Q,\rho}(.|\alpha)$ on ${\mathbf X}$ satisfies properties (B1)--(B3) on page \pageref{properties_b};
(ii) for $\rho\in\mathcal R^Q$, $G^Q(\rho)=\int j(\alpha,\eta)\nu^{Q,\rho}(d\alpha,d\eta)$.
\end{corollary}
\begin{remark} \label{remark_flux}
By \textit{(ii)} of Corollary \ref{invariant}, and shift-invariance of $\nu^{Q,\rho}(d\alpha,d\eta)$,
\begin{equation} G^Q(\rho) = \int j(\alpha,\eta)\nu^{Q,\rho}(d\alpha,d\eta) =\int \widetilde{\jmath}(\alpha,\eta)\nu^{Q,\rho}(d\alpha,d\eta) \end{equation}
for every $\rho\in\mathcal R^Q$, where
\begin{equation}\label{other_flux} \widetilde{\jmath}(\alpha,\eta):=\alpha(0)\sum_{z\in{\mathbb Z}}zp(z)b(\eta(0),\eta(z)) \end{equation}
Thus one can alternatively take $\widetilde{\jmath}(\alpha,\eta)$ as a microscopic flux function
(we refer to \cite[p. 1347]{bgrs2} for an analogous remark in the non-disordered setting).
\end{remark}
\begin{proof}{corollary}{corollary_invariant}
Properties {\em (B1)} and {\em (B2)} follow from Proposition \ref{invariant} by conditioning (here and after, we proceed as in the proof of Lemma \ref{lemma_conditional}). By translation invariance of $\nu^{Q,\rho}$ and conditioning we have, for $Q$-a.e. $\alpha\in\mathbf A$,
\begin{equation}\label{commutation_inv} \tau_x\nu^{Q,\rho}_\alpha=\nu^{Q,\rho}_{\tau_x\alpha} \end{equation}
where $\tau_x$ on the l.h.s. acts on ${\mathbf X}$. For property {\em (B3)} the result will follow from ergodicity of $Q$ once we show that, for every $\rho\in\mathcal R^Q$, $G^Q_\alpha(\rho)=G^Q_{\tau_1\alpha}(\rho)$ holds $Q$-a.s. To this end we note that, as a result of \eqref{commutation},
$$L_\alpha[\eta(1)]=j(\alpha,\eta)-j(\tau_1\alpha,\tau_1\eta)$$
Taking expectation w.r.t. invariant measure $\nu_\alpha^{Q,\rho}$, and using \eqref{commutation_inv}, we obtain
\begin{eqnarray*} G^Q_\alpha(\rho)=\int j(\alpha,\eta)\nu^{Q,\rho}_\alpha(d\eta) & = & \int j(\tau_1\alpha,\tau_1\eta)\nu^{Q,\rho}_\alpha(d\eta)\\
& = & \int j(\tau_1\alpha,\eta)\nu^{Q,\rho}_{\tau_1\alpha}(d\eta)=G^Q_{\tau_1\alpha}(\rho)
\end{eqnarray*}
\end{proof}
\mbox{}\\ \\
To prove Proposition \ref{invariant}, we need some definitions and lemmas.
For every $\alpha\in{\mathbf A}$, we denote by $\overline{L}_\alpha$ the coupled generator on ${\mathbf X}^2$ given by
\begin{equation}\label{coupling_t}
\overline{L}_\alpha f(\eta,\xi):=\sum_{x\in{\mathbb Z}}\int_{\mathcal V}\left[
f\left({\mathcal T}^{\alpha,x,v}\eta,{\mathcal T}^{\alpha,x,v}\xi\right)-f(\eta,\xi)
\right]m(dv)
\end{equation}
for any local function $f$ on ${\mathbf X}^2$. For the particular model \eqref{update_misanthrope}, this is equivalent to the ``basic coupling'' of $L_\alpha$
defined in \cite{coc}, namely $\overline{L}_\alpha=\sum_{x,y\in{\mathbb Z}:\,x\neq y}\overline{L}^{x,y}_\alpha$, with $\overline{L}^{x,y}_\alpha f(\eta,\xi)$ given by
\begin{eqnarray}\label{xy-gen-coupl}
& & \alpha(x)p(y-x)[b(\eta(x),\eta(y))\wedge b(\xi(x),\xi(y))]\left[f(\eta^{x,y},\xi^{x,y})-f(\eta,\xi)\right]\cr
& + & \alpha(x)p(y-x)[b(\eta(x),\eta(y))-b(\xi(x),\xi(y))]^+\left[f(\eta^{x,y},\xi)-f(\eta,\xi)\right]\cr
& + & \alpha(x)p(y-x)[b(\xi(x),\xi(y))- b(\eta(x),\eta(y))]^+\left[f(\eta,\xi^{x,y})-f(\eta,\xi)\right]
\end{eqnarray}
If $(\eta_t,\xi_t)$ is a Markov process with generator $\overline{L}_\alpha$, and $\eta_0\leq\xi_0$, then $\eta_t\leq\xi_t$ a.s. for every $t>0$. We indicate this by saying
that $\overline{L}_\alpha$ is a \textit{monotone coupling} of $L_\alpha$.
We denote by $\overline{L}$ the coupled generator for the joint process $(\alpha_t,\eta_t,\xi_t)_{t\ge 0}$ on ${\mathbf A}\times{\mathbf X}^2$ defined by
\begin{equation}\label{joint_coupling}\overline{L}f(\alpha,\eta,\xi)=(\overline{L}_\alpha f(\alpha,.))(\eta,\xi)\end{equation}
for any local function $f$ on ${\mathbf A}\times{\mathbf X}^2$. Given $\alpha_0=\alpha$, this means that $\alpha_t=\alpha$ for all $t\geq 0$, while $(\eta_t,\xi_t)_{t\ge 0}$ is a Markov process with generator $\overline{L}_\alpha$. Let $\overline{S}(t)$ denote the semigroup generated by $\overline{L}$.
We denote by $\overline{\mathcal S}$ the set of probability measures on ${\mathbf A}\times{\mathbf X}^2$ that are invariant by space shift $\tau_x(\alpha,\eta,\xi)=(\tau_x\alpha,\tau_x\eta,\tau_x\xi)$.
In the following, if $\overline{\nu}(d\alpha,d\eta,d\xi)$ is a probability measure on ${\mathbf A}\times{\mathbf X}^2$, $\overline{\nu}_1$, $\overline{\nu}_2$ and $\overline{\nu}_3$ (resp. $\overline{\nu}_{12}$ and $\overline{\nu}_{13}$) denote marginal distributions of $\alpha$, $\eta$ and $\xi$ (resp. $(\alpha,\eta)$ and $(\alpha,\xi)$) under $\overline{\nu}$.
\begin{lemma} \label{coupling_extremal}
Let $\mu',\mu''\in(\mathcal I_{L}\cap\mathcal S)_e$ with a common $\alpha$-marginal $Q$. Then there exists $\overline{\nu}\in\left(\mathcal I_{\overline{L}}\cap\overline{\mathcal S}\right)_e$ such that $\overline{\nu}_{12}=\mu'$ and $\overline{\nu}_{13}=\mu''$.
\end{lemma}
\begin{proof}{lemma}{coupling_extremal}
Let $\overline{\mathcal M}(\mu',\mu'')$ denote the set of probability measures $\overline{\nu}\in\mathcal I_{\overline{L}}\cap\overline{\mathcal S}$ with $\overline{\nu}_{12}=\mu'$ and $\overline{\nu}_{13}=\mu''$. We show that $\overline{\mathcal M}(\mu',\mu'')$ is a nonempty set.
Set
$\overline{\nu}^0(d\alpha,d\eta,d\xi):=Q(d\alpha)[\mu'(d\eta|\alpha)\otimes\mu''(d\xi|\alpha)]$.
Then $\overline{\nu}^0_{12}=\mu'$, $\overline{\nu}^0_{13}=\mu''$ and $\overline{\nu}^0\in\overline{\mathcal S}$. Let
$$ \overline{\nu}^t:=\frac{1}{t}\int_0^t\overline{\nu}^0\overline{S}(s) ds $$
The set $\{\overline{\nu}^t,t>0\}$ is relatively compact because $\overline{\nu}^t_1=Q$ is independent of $t$ and, for $i\in\{2,3\}$, $\overline{\nu}^t_i\leq\delta_K^{\otimes{\mathbb Z}}$. Let $\overline{\nu}^\infty$ be any subsequential weak limit of $\overline{\nu}^t$ as $t\to\infty$. Then $\overline{\nu}^\infty$ retains the above properties of $\overline{\nu}^0$, and
$\overline{\nu}^\infty\in \mathcal I_{\overline{L}}$, thus $\overline{\nu}^\infty\in\overline{\mathcal M}(\mu',\mu'')$. Let $\overline{\nu}$ be an extremal element of the compact convex set $\overline{\mathcal M}(\mu',\mu'')$. We now prove that $\overline{\nu}\in\left(\mathcal I_{\overline{L}}\cap\overline{\mathcal S}\right)_e$.
Assume there exist $\lambda\in (0,1)$ and probability measures $\overline{\nu}^l$, $\overline{\nu}^r$ on ${\bf A}\times{\bf X}^2$, such that
\begin{equation}\label{decomp_2}
\overline{\nu}=\lambda \overline{\nu}^l+(1-\lambda)\overline{\nu}^r
\end{equation}
with $\overline{\nu}^i\in\mathcal I_{\overline{L}}\cap\overline{\mathcal S}$ for $i\in\{l,r\}$.
Since $\overline{\nu}\in\overline{\mathcal M}(\mu',\mu'')$, the projections of \eqref{decomp_2} on $(\alpha,\eta)$ and $(\alpha,\xi)$ yield
\begin{eqnarray}
\mu' & = & \lambda \overline{\nu}^l_{12}+(1-\lambda)\overline{\nu}^r_{12}\label{decomp_21}\\
\mu'' & = & \lambda \overline{\nu}^l_{13}+(1-\lambda)\overline{\nu}^r_{13}\label{decomp_22}
\end{eqnarray}
For $i\in\{l,r\}$, $\overline{\nu}^i\in\mathcal I_{\overline{L}}\cap\overline{\mathcal S}$ implies $\overline{\nu}^i_{1j}\in\mathcal I_{L}\cap\mathcal S$ for $j\in\{2,3\}$. Since $\mu',\mu''$ belong to $(\mathcal I_{L}\cap\mathcal S)_e$, $\overline{\nu}^i_{12}=\mu'$, $\overline{\nu}^i_{13}=\mu''$ by \eqref{decomp_21}--\eqref{decomp_22}, that is, $\overline{\nu}^i\in\overline{\mathcal M}(\mu',\mu'')$. Since $\overline{\nu}$ is extremal in $\overline{\mathcal M}(\mu',\mu'')$, \eqref{decomp_2} yields $\overline{\nu}^l=\overline{\nu}^r=\overline{\nu}$.
\end{proof}
\begin{lemma}\label{stable}
Let $\nu$ be a stationary distribution for some Markov transition semigroup, and $(X_t)_{t\ge 0}$ be a Markov process associated to this semigroup with initial distribution $\nu$. Assume $A$ is a subset of $E$ such that, for every $t>0$, ${\bf 1}_A(X_t)\geq {\bf 1}_A(X_0)$ almost surely. Then
$\nu_A(dx)=\nu(dx|x\in A)$ and $\nu_{A^c}(dx)=\nu(dx|x\in A^c)$ are stationary for the considered semigroup.
\end{lemma}
\begin{proof}{lemma}{stable}
Since $(X_t)_{t\ge 0}$ is stationary, we have ${\rm I\hspace{-0.8mm}E} [{\bf 1}_A(X_t)]={\rm I\hspace{-0.8mm}E} [{\bf 1}_A(X_0)]$, thus ${\bf 1}_A(X_0)={\bf 1}_A(X_t)$ and ${\bf 1}_{A^c}(X_t)={\bf 1}_{A^c}(X_0)$ almost surely. Stationarity of $\nu_A$ amounts to $\displaystyle {\rm I\hspace{-0.8mm}E}[f(X_t)|X_0\in A]={\rm I\hspace{-0.8mm}E}[f(X_0)|X_0\in A]$ for every bounded $f$. We conclude with
$$ \begin{array}{lllll}
{\rm I\hspace{-0.8mm}E}[f(X_t)|X_0\in A] & = & \displaystyle\frac{{\rm I\hspace{-0.8mm}E}[f(X_t){\bf 1}_A(X_0)]}{{\rm I\hspace{-0.8mm}E}[{\bf 1}_A(X_0)]} & = & \displaystyle\frac{{\rm I\hspace{-0.8mm}E}[f(X_t){\bf 1}_A(X_t)]}{{\rm I\hspace{-0.8mm}E}[{\bf 1}_A(X_0)]}\\
& = & \displaystyle\frac{{\rm I\hspace{-0.8mm}E}[f(X_0){\bf 1}_A(X_0)]}{{\rm I\hspace{-0.8mm}E}[{\bf 1}_A(X_0)]}
& = & \displaystyle{\rm I\hspace{-0.8mm}E}[f(X_0)|X_0\in A]
\end{array} $$
\end{proof}
\begin{lemma} \label{order_extremal}
Let $\overline{\nu}\in\left(\mathcal I_{\overline{L}}\cap\overline{\mathcal S}\right)_e$. Then $\overline{\nu}\left(\overline{\mathcal O}_+\right)$ and $\overline{\nu}\left(\overline{\mathcal O}_-\right)$ belong to $\{0,1\}$.
\end{lemma}
\begin{proof}{lemma}{order_extremal}
Let $A=\{(\alpha,\eta,\xi)\in{\mathbf A}\times{\mathbf X}^2:\,\eta\leq\xi\}$ and assume $\lambda:=\overline{\nu}(A)\in(0,1)$. Since the coupling defined by $\overline{L}$ is monotone, we have ${\bf 1}_A(\alpha_t,\eta_t,\xi_t)\geq {\bf 1}_A(\alpha_0,\eta_0,\xi_0)$. By Lemma \ref{stable},
$$\overline{\nu}_A:=\overline{\nu}(d\alpha,d\eta,d\xi|(\alpha,\eta,\xi)\in A)\in\mathcal I_{\overline{L}}$$
From
$ \overline{\nu}=\lambda\overline{\nu}_A+(1-\lambda)\overline{\nu}_{A^c}, $
we deduce $\overline{\nu}_{A^c}\in\mathcal I_{\overline{L}}$. Since $A$ is shift invariant in ${\mathbf A}\times{\mathbf X}^2$, $\overline{\nu}_A$ and $\overline{\nu}_{A^c}$ lie in $\overline{\mathcal{S}}$. By extremality of $\overline{\nu}$, we must have $\overline{\nu}_A=\overline{\nu}_{A^c}$ which is impossible since these measures are supported on disjoint sets.
\end{proof}
\mbox{}\\ \\
Attractiveness assumption \textit{(A5)} ensures that an initially ordered pair of coupled configurations remains ordered at later times. Assumptions \textit{(A1), (A4)} induce a stronger property: pairs of opposite discrepancies between two coupled configurations eventually get killed, so that the two configurations become ordered.
\begin{proposition}\label{prop_irred}
Every $\overline{\nu}\in{\mathcal I}_{\overline{L}}\cap\overline{\mathcal S}$ is supported on $\overline{\mathcal O}$.
\end{proposition}
\begin{proof}{proposition}{prop_irred}
We follow the scheme used in \cite{lig,andjel,coc,guiol,gs} for the non-disordered case,
and only sketch the arguments needed for the disordered setting. \\ \\
{\em Step 1.} For $x\in{\mathbb Z}$, let $f_x(\eta,\xi)=(\eta(x)-\xi(x))^+$. By translation invariance of $\overline{\nu}$, the shift commutation property \eqref{commutation} and \eqref{coupling_t}, \eqref{xy-gen-coupl},
\begin{eqnarray*} 0&=&\int\overline{L}_\alpha f_0 (\alpha,\eta,\xi)\overline{\nu}(d\alpha,d\eta,d\xi) \\ & = & \sum_{v\in{\mathbb Z}} \int\overline{L}_\alpha^{0,v} [f_0+f_v] (\alpha,\eta,\xi)\overline{\nu}(d\alpha,d\eta,d\xi)
\end{eqnarray*}
On the other hand (see \cite{coc,gs})
\begin{eqnarray*}
\overline{L}_\alpha^{0,v}(f_0+f_v)& \leq & -p(v)\alpha(0)|b(\eta(0),\eta(v))-b(\xi(0),\xi(v))|\\
& & \times\left(
{\bf 1}_{\{\eta(0)>\xi(0),\,\eta(v)<\xi(v)\}}+{\bf 1}_{\{\eta(0)<\xi(0),\,\eta(v)>\xi(v)\}}
\right)
\end{eqnarray*}
Using Assumptions \textit{(A4)--(A5)}, \eqref{alpha_bounds} and translation invariance of $\overline{\nu}$, we obtain
\begin{equation}\label{opposite_discrepancies} \begin{array}{ll}
& \overline{\nu}\left(
(\alpha,\eta,\xi):\,\eta(x)>\xi(x),\,\eta(y)<\xi(y)
\right)\\
+ & \overline{\nu}\left(
(\alpha,\eta,\xi):\,\eta(x)<\xi(x),\,\eta(y)>\xi(y)
\right)=0 \end{array} \end{equation}
for $x\neq y$ with $p(y-x)+p(x-y)>0$. Whenever one of the events in \eqref{opposite_discrepancies} holds, we say there is \textit{a pair of opposite discrepancies} at $(x,y)$. \\ \\
{\em Step 2.} One proves by induction that, for all $n\in{\mathbb N}$, \eqref{opposite_discrepancies} holds if $x\neq y$ with $p^{*n}(y-x)+p^{*n}(x-y)>0$. The induction step is based on the following idea. Assume $(\eta,\xi)$ has
a pair of opposite discrepancies at $(x,y)$. Then one can find a finite path of coupled transitions (with rates uniformly bounded below thanks to \textit{(A4)--(A5)} and \eqref{alpha_bounds}), leading to a coupled state with a pair of opposite discrepancies, either at $(x,z)$ for some $z$ with $p^{*(n-1)}(z-x)+p^{*(n-1)}(x-z)>0$, or at $(z,y)$ with $p^{*(n-1)}(y-z)+p^{*(n-1)}(z-y)>0$. This part of the argument is insensitive to the presence of disorder so long as $\alpha(x)$ is uniformly bounded below.\\ \\
{\em Conclusion.} By irreducibility assumption \textit{(A1)}, \eqref{opposite_discrepancies} holds for all $(x,y)\in{\mathbb Z}^2$ with $x\neq y$. This implies $\overline{\nu}(\overline{\mathcal O})=1$.
\end{proof}
\mbox{}\\ \\
We are now in a position to prove Proposition \ref{invariant}.\\
\begin{proof}{proposition}{invariant}
We define
$$\mathcal R^Q:=\left\{\int\eta(0)\nu(d\alpha,d\eta):\, \nu\in\left(\mathcal I_{L}\cap\mathcal S\right)_e,\,\nu\mbox{ has }\alpha\mbox{-marginal }Q\right\}$$
Let $\nu^i\in\left(\mathcal I_{L}\cap\mathcal S\right)_e$ with $\alpha$-marginal $Q$ and $\rho^i:=\int\eta(0)\nu^i(d\alpha,d\eta)\in{\mathcal R}^Q$ for $i\in\{1,2\}$. Assume $\rho^1\leq\rho^2$. Using Lemma \ref{lemma_conditional},\textit{(iii)}, Lemmas \ref{coupling_extremal} and \ref{order_extremal}, and Proposition \ref{prop_irred}, we obtain $\nu^1\ll\nu^2$, that is \eqref{ordered_measures}.
Existence \eqref{Rezakhanlou} of an asymptotic particle density can be obtained by a proof analogous to \cite[Lemma 14]{mrs}, where the space-time ergodic theorem is applied to the joint disorder-particle process. Then, closedness of ${\mathcal R}^Q$ is established as in \cite[Proposition 3.1]{bgrs2}.
We end up proving the weak continuity statement given the rest of the proposition. Let $\rho,\rho'\in{\mathcal R}^Q$ with $\rho\leq\rho'$. By \eqref{ordered_measures} and Lemma \ref{lemma_conditional}, there exists a coupling $\overline{\nu}^{Q,\rho,\rho'}(d\alpha,d\eta,d\xi)$ of $\nu^{Q,\rho}(d\alpha,d\eta)$ and $\nu^{Q,\rho'}(d\alpha,d\xi)$ supported on $\overline{\mathcal O}_+$. Thus, for $x\in{\mathbb Z}$
\begin{equation}\label{standard_coupling}\int|\eta(x)-\xi(x)|\overline{\nu}^{Q,\rho,\rho'}(d\alpha,d\eta,d\xi)=|\rho-\rho'|\end{equation}
from which weak continuity follows by a coupling argument.
\end{proof}
\begin{remark}\label{remark_lipschitz}
Since
\begin{equation}\label{compare_fluxes} G^Q(\rho)-G^Q(\rho')=\int [\widetilde{\jmath}(\alpha,\eta)-\widetilde{\jmath}(\alpha,\xi)]\overline{\nu}^{Q,\rho,\rho'}(d\alpha,d\eta,d\xi) \end{equation}
a Lipschitz constant $V$ for $G^Q$ follows from \eqref{other_flux}, \eqref{standard_coupling}:
\begin{equation}\label{maxspeed}V=2c^{-1}||b||_\infty\sum_{z\in{\mathbb Z}}|z|p(z)\end{equation}
\end{remark}
\section{ Proof of hydrodynamics} \label{proof_hydro}
In this section, we prove the hydrodynamic limit following the strategy introduced in \cite{bgrs1,bgrs2} and significantly strengthened in \cite{bgrs3}. That is, we reduce general Cauchy data to step initial conditions (the so-called Riemann problem) and use a constructive approach (as in \cite{av}).
Some technical details similar to \cite{bgrs3} will be omitted. We shall rather focus on how to deal with the disorder, which is the substantive part of this paper.
The measure $Q$ being fixed once and for all by Theorem \ref{th:hydro}, we simply write $\nu^\rho$, $\mathcal R$, $G$.
\subsection{Riemann problem}\label{riemann}
Let $\lambda,\rho\in[0,K]$ with $\lambda<\rho$ (for $\lambda>\rho$ replace infimum with supremum below), and
\begin{equation}\label{eq:rie} R_{\lambda,\rho}(x,0)=\lambda \mathbf 1_{\{x<0\}}+\rho \mathbf 1_{\{x\geq 0\}} \end{equation}
The entropy solution to the conservation law \eqref{hydrodynamics} with initial condition \eqref{eq:rie}, denoted by $R_{\lambda,\rho}(x,t)$, is given (\cite[Proposition 4.1]{bgrs2}) by a variational formula, and satisfies
\begin{eqnarray}
\int_v^w R_{\lambda,\rho}(x,t)dx & = & t[\mathcal G_{v/t}(\lambda,\rho)-\mathcal G_{w/t}(\lambda,\rho)],\mbox{ with}\cr \mathcal G_v(\lambda,\rho) & := & \inf\left\{
G(r)-vr:\,r\in[\lambda,\rho]\cap\mathcal R
\right\}\label{limiting_current}
\end{eqnarray}
for all $v,w\in{\mathbb R}$.
Microscopic states with profile \eqref{eq:rie} will be constructed using the following lemma, established in Subsection \ref{technical} below.
\begin{lemma}\label{big_coupling} There exist random variables $\alpha$ and $(\eta^\rho:\,\rho\in\mathcal R)$ on a probability space $(\Omega_{\mathbf A},\mathcal F_{\mathbf A},{\rm I\hspace{-0.8mm}P}_{\mathbf A})$ such that
\begin{eqnarray}\label{marginals}(\alpha,\eta^\rho)\sim\nu^\rho,&& \alpha\sim Q\\ \label{strassen} {\rm I\hspace{-0.8mm}P}_{\mathbf A}-a.s.,&& \rho\mapsto\eta^\rho\mbox{ is nondecreasing}\end{eqnarray}
\end{lemma}
Let $\overline{\nu}^{\lambda,\rho}$ denote the distribution of $(\alpha,\eta^\lambda,\eta^\rho)$, and $\overline{\nu}^{\lambda,\rho}_\alpha$ the conditional distribution of $(\alpha,\eta^\lambda,\eta^\rho)$ given $\alpha$.
For $(x_0,t_0)\in{\mathbb Z}\times{\mathbb R}^+$, the {\sl space-time shift} $\theta_{x_0,t_0}$ is defined for any $\omega\in\Omega$, for any $(t,x,z,u)\in{\mathbb R}^+\times{\mathbb Z}\times{\mathbb Z}\times[0,1]$, by
\[ (t,x,z,u)\in\theta_{x_0,t_0}\omega\mbox{ if and only if }(t_0+t,x_0+x,z,u)\in\omega\]
By its definition and property \eqref{shift_t}, the mapping introduced in \eqref{unique_mapping} satisfies, for all
$s,t\geq 0$, $x\in{\mathbb Z}$ and $(\alpha,\eta,\omega)\in{{\mathbf A}}\times{\mathbf X}\times{\Omega}$:
\begin{eqnarray*} \label{mapping_markov} \eta_s(\alpha,\eta_t(\alpha,\eta,\omega),\theta_{0,t}\omega)&=&\eta_{t+s}(\alpha,\eta,\omega) \\ \label{mapping_shift} \tau_x\eta_t(\alpha,\eta,\omega)&=&\eta_t(\tau_x\alpha,\tau_x\eta,\theta_{x,0}\omega) \end{eqnarray*}
We now introduce an extended shift $\theta'$ on $\Omega'={\mathbf A}\times{\mathbf X}^2\times\Omega$. If
$\omega'=(\alpha,\eta,\xi,\omega)$
denotes a generic element of $\Omega'$, we set
\begin{equation}\label{extended_shift}\theta'_{x,t}\omega' = (\tau_x\alpha,\tau_x\eta_t(\alpha,\eta,\omega),\tau_x\eta_t(\alpha,\xi,\omega),\theta_{x,t}\omega) \end{equation}
It is important to note that this shift incorporates disorder.
Let $T:{\mathbf X}^2\to{\mathbf X}$ be given by
\begin{equation}\label{transfo_T} T(\eta,\xi)(x)=\eta(x){\bf 1}_{\{x< 0\}}+\xi(x){\bf 1}_{\{x\geq 0\}} \end{equation}
The main result of this subsection is
\begin{proposition} \label{corollary_2_2} Set, for $t\ge 0$,
\begin{equation} \label{def_empirical_shift} \beta^N_t(\omega')(dx):=\pi^N(\eta_t(\alpha,T(\eta,\xi),\omega))(dx) \end{equation}
For all $t>0$, $s_0\geq 0$ and $x_0\in{\mathbb R}$, we have that, for $Q$-a.e. $\alpha\in{\mathbf A}$,
\[ \lim_{N\to\infty}\beta^N_{Nt}(\theta'_{\lfloor Nx_0\rfloor ,Ns_0}\omega')(dx)=R_{\lambda,\rho}(.,t)dx, \quad\overline{\nu}_\alpha^{\lambda,\rho}\otimes{\rm I\hspace{-0.8mm}P}\mbox{-a.s.} \]
\end{proposition}
Proposition \ref{corollary_2_2} will follow from a law of large numbers for currents.
Let $x_.=(x_t,\,t\geq 0)$ be a ${\mathbb Z}$-valued {\em cadlag} random path, with $\abs{x_t-x_{t^-}}\leq 1$, independent of the Poisson measure $\omega$.
We define the particle current seen by an observer travelling along this path by
\begin{equation} \label{current_3} \varphi^{x_.}_t(\alpha,\eta_0,\omega) =\varphi^{x_.,+}_t(\alpha,\eta_0,\omega) -\varphi^{x_.,-}_t(\alpha,\eta_0,\omega)+\widetilde{\varphi}^{x_.}_t(\alpha,\eta_0,\omega) \end{equation}
where
$\varphi^{x_.,\pm}_t(\alpha,\eta_0,\omega)$ count the number of rightward/leftward crossings of $x_.$ due to particle jumps, and
$\widetilde{\varphi}^{x_.}_t(\alpha,\eta_0,\omega)$ is the current due to the self-motion of the observer.
We shall write $\varphi^v_t$ in the particular case $x_t=\lfloor vt\rfloor$.
Set $\phi^{v}_t(\omega'):=\varphi^{v}_t(\alpha,T(\eta,\xi),\omega)$. Note that for $(v,w)\in{\mathbb R}^2$,
$\beta^N_{Nt}(\omega')([v,w])=t(Nt)^{-1}(\phi^{v/t}_{Nt}(\omega')-\phi^{w/t}_{Nt}(\omega'))$.
By \eqref{limiting_current}, Proposition \ref{corollary_2_2} is reduced to
\begin{proposition} \label{proposition_2_2}
For all $t>0$, $a\in{\mathbb R}^+,b\in{\mathbb R}$ and $v\in{\mathbb R}$,
\begin{equation}\lim_{N\to\infty}(Nt)^{-1}\phi^{v}_{Nt}(\theta'_{\lfloor b N\rfloor ,a N}\omega') = \mathcal G_v(\lambda,\rho)\qquad\overline{\nu}^{\lambda,\rho}\otimes{\rm I\hspace{-0.8mm}P}-a.s. \label{as}\end{equation}
\end{proposition}
To prove Proposition \ref{proposition_2_2}, we introduce a probability space $\Omega^+$, whose generic element is denoted by $\omega^+$, on which is defined a Poisson process $(N_t(\omega^+))_{t\ge 0}$ with intensity $\abs{v}$ ($v\in{\mathbb R}$). Denote by ${{\rm I\hspace{-0.8mm}P}}^+$ the associated probability. Set
\begin{eqnarray}\label{def_poisson} x_s^N(\omega^+)&:=&({\rm sgn}(v))\left[N_{a N+s}(\omega^+)-N_{aN}(\omega^+)\right]\\ \label{mapping_tilde}\widetilde{\eta}^N_s(\alpha,\eta_0,\omega,\omega^+) &:=&\tau_{x_s^N(\omega^+)}\eta_s(\alpha,\eta_0,\omega)\\
\label{mapping_tilde_alpha}
\widetilde{\alpha}^N_s(\alpha,\omega^+) & := & \tau_{x^N_s(\omega^+)}\alpha
\end{eqnarray}
Thus $(\widetilde{\alpha}_s^N,\widetilde{\eta}_s^N)_{s\ge 0}$ is a Feller process with generator
\[
L^v=L+S^v,\quad S^v f(\alpha,\zeta)=\abs{v} [f(\tau_{{\rm sgn}(v)}\alpha,\tau_{{\rm sgn}(v)}\zeta)-f(\alpha,\zeta)] \]
for $f$ local and $\alpha\in\mathbf A$, $\zeta\in{\mathbf X}$. Since any translation invariant measure on ${\mathbf A}\times{\mathbf X}$ is stationary for the pure shift generator $S^v$, we have
${\mathcal I}_L\cap{\mathcal S}={\mathcal I}_{L^v}\cap{\mathcal S}$.
Define the time and space-time empirical measures (where $\varepsilon>0$) by
\begin{eqnarray} \label{def_empirical} m_{tN}(\omega',\omega^+)&:=&(Nt)^{-1}\int_0^{tN} \delta_{(\widetilde{\alpha}^N_s(\alpha,\omega^+),\widetilde{\eta}^N_s(\alpha,T(\eta,\xi),\omega,\omega^+))}ds\\ \label{def_empirical_2} m_{tN,\varepsilon}(\omega',\omega^+)&:=&\abs{{\mathbb Z}\cap[-\varepsilon N,\varepsilon N]}^{-1}\sum_{x\in{\mathbb Z}:\,\abs{x}\leq \varepsilon N}\tau_xm_{tN}(\omega',\omega^+) \end{eqnarray}
Notice that there is a disorder component we cannot omit in the empirical measure, although ultimately we are only interested in the behavior of the $\eta$-component. Let ${\mathcal M}_{\lambda,\rho}$ denote the compact set of probability measures $\mu(d\alpha,d\eta)\in \mathcal I_{L}\cap\mathcal S$ such that $\mu$ has $\alpha$-marginal $Q$, and $\nu^\lambda\ll\mu\ll\nu^\rho$. By Proposition \ref{invariant},
\begin{equation}\label{setofmeasures}
\mathcal M_{\lambda,\rho}=\left\{\nu(d\alpha,d\eta) =\int\nu^r(d\alpha,d\eta)\gamma(dr):\,\gamma\in\mathcal P([\lambda,\rho]\cap\mathcal R)\right\}
\end{equation}
The key ingredients for Proposition \ref{proposition_2_2} are the following lemmas, proved in Subsection \ref{technical} below.
\begin{lemma}\label{current_comparison}
The function $\phi^v_t(\alpha,\eta,\xi,\omega)$ is increasing in $\eta$, decreasing in $\xi$.
\end{lemma}
\begin{lemma}\label{lemma_empirical}
With $\overline{\nu}^{\lambda,\rho}\otimes{{\rm I\hspace{-0.8mm}P}}\otimes{{\rm I\hspace{-0.8mm}P}}^+$-probability one, every subsequential limit as $N\to\infty$ of $m_{tN,\varepsilon }(\theta'_{\lfloor b N\rfloor ,a N}\omega',\omega^+)$ lies in $\mathcal M_{\lambda,\rho}$.
\end{lemma}
\begin{proof}{Proposition}{proposition_2_2}
We will show that
\begin{eqnarray} \liminf_{N\to\infty}\,(Nt)^{-1}\phi^{v}_{tN}\circ\theta'_{\lfloor b N\rfloor ,a N}(\omega') & \geq & \mathcal G_v(\lambda,\rho),\quad \overline{\nu}^{\lambda,\rho}\otimes{\rm I\hspace{-0.8mm}P}\mbox{-a.s.}\label{liminf_better}\\
\label{limsup} \limsup_{N\to\infty}\,(Nt)^{-1}\phi^{v}_{tN}\circ\theta'_{\lfloor b N\rfloor ,a N}(\omega') & \leq & \mathcal G_v(\lambda,\rho),\quad \overline{\nu}^{\lambda,\rho}\otimes{\rm I\hspace{-0.8mm}P}\mbox{-a.s.} \end{eqnarray}
{\em Step one: proof of \eqref{liminf_better}.}\par\noindent
Setting
$\varpi_{a N}=\varpi_{a N}(\omega'):=T\left( \tau_{\lfloor b N\rfloor }\eta_{a N}(\alpha,\eta,\omega), \tau_{\lfloor b N\rfloor }\eta_{a N}(\alpha,\xi,\omega)
\right)$,
we have
\begin{equation} \label{sothat} (Nt)^{-1}\phi^v_{tN}\circ\theta'_{\lfloor b N\rfloor ,a N}(\omega')=(Nt)^{-1}\varphi^{v}_{tN}(\tau_{\lfloor b N\rfloor }\alpha,\varpi_{a N},\theta_{\lfloor b N\rfloor ,a N}\omega) \end{equation}
Let, for every $(\alpha,\zeta,\omega,\omega^+)\in{\mathbf A}\times{\mathbf X}\times\Omega\times\Omega^+$ and $x^N_.(\omega^+)$ given by \eqref{def_poisson},
\begin{equation}\label{psi-t-v} \psi_{tN}^{v,\varepsilon}(\alpha,\zeta,\omega,\omega^+):=\abs{{\mathbb Z}\cap[-\varepsilon N,\varepsilon N]}^{-1}\sum_{y\in{\mathbb Z}:\,\abs{y}\leq\varepsilon N}\varphi^{x^N_.(\omega^+)+y}_{tN}(\alpha,\zeta,\omega) \end{equation}
Note that $\lim_{N\to\infty}(Nt)^{-1}x_{tN}^N(\omega^+)= v$, ${\rm I\hspace{-0.8mm}P}^+$-a.s., and that for two paths $y_.,z_.$ (see \eqref{current_3}),
$$ \abs{\varphi^{y_.}_{tN}(\alpha,\eta_0,\omega)-\varphi^{z_.}_{tN}(\alpha,\eta_0,\omega)} \leq K\left(\abs{y_{tN}-z_{tN}}+ \abs{y_0-z_0}\right) $$
Hence the proof of \eqref{liminf_better} reduces to that of the same inequality where we replace $(Nt)^{-1}\phi^{v}_{tN}\circ\theta'_{\lfloor b N\rfloor ,a N}(\omega')$ by
$ (Nt)^{-1}\psi^{v,\varepsilon}_{tN}(\tau_{\lfloor b N\rfloor }\alpha,\varpi_{a N},\theta_{\lfloor b N\rfloor ,a N}\omega,\omega^+)$
and $\overline{\nu}^{\lambda,\rho}\otimes{\rm I\hspace{-0.8mm}P}$ by $\overline{\nu}^{\lambda,\rho}\otimes{\rm I\hspace{-0.8mm}P}\otimes{\rm I\hspace{-0.8mm}P}^+$.
By definitions \eqref{def_f}, \eqref{current_3} of flux and current, for any $\alpha\in\mathbf{A}$, $\zeta\in\mathbf{X}$,
\begin{eqnarray*} &&M^{x,v}_{tN}(\alpha,\zeta,\omega,\omega^+):= \varphi^{x^N_.(\omega^+)+x}_{tN}(\alpha,\zeta,\omega)-\nonumber\\ && \int_0^{tN} \tau_x\left\{ j(\widetilde{\alpha}^N_{s}(\alpha,\omega^+),\widetilde{\eta}^N_{s}(\alpha,\zeta,\omega,\omega^+)) -v (\widetilde{\eta}^N_{s}(\alpha,\zeta,\omega,\omega^+))({\bf 1}_{\{v>0\}})\right\}ds\label{martingale} \end{eqnarray*}
is a mean $0$ martingale under ${\rm I\hspace{-0.8mm}P}\otimes{\rm I\hspace{-0.8mm}P}^+$. Let
\begin{eqnarray}\nonumber R_{tN}^{\varepsilon,v}&:=&\label{as_2} \left(Nt\abs{{\mathbb Z}\cap[-\varepsilon N,\varepsilon N]}\right)^{-1}\sum_{x\in{\mathbb Z}:\,\abs{x}\leq\varepsilon N}M^{x,v}_{tN}( \tau_{\lfloor b N\rfloor }\alpha,\varpi_{a N},\theta_{\lfloor b N\rfloor ,a N}\omega,\omega^+)\\\nonumber &=&(Nt)^{-1}\psi^{v,\varepsilon}_{tN}(\tau_{\lfloor b N\rfloor }\alpha,\varpi_{a N},\theta_{\lfloor b N\rfloor ,a N}\omega,\omega^+)\\ &&-\int [j(\alpha,\eta)-v\eta({\bf 1}_{\{v>0\}})]m_{tN,\varepsilon}(\theta'_{\lfloor b N\rfloor ,a N}\omega',\omega^+)(d\alpha,d\eta) \label{replace_oncemore} \end{eqnarray}
where the last equality comes from \eqref{def_empirical_2}, \eqref{psi-t-v}.
The exponential martingale associated with $M^{x,v}_{tN}$ yields a Poissonian bound, uniform in $(\alpha,\zeta)$, for the exponential moment of $M_{tN}^{x,v}$ with respect to ${\rm I\hspace{-0.8mm}P}\otimes{\rm I\hspace{-0.8mm}P}^+$. Since $\varpi_{a N}$ is independent of $(\theta_{\lfloor b N\rfloor ,a N}\omega,\omega^+)$ under $\overline{\nu}^{\lambda,\rho}\otimes{\rm I\hspace{-0.8mm}P}\otimes{\rm I\hspace{-0.8mm}P}^+$, the bound is also valid under this measure, and Borel-Cantelli's lemma implies $\lim_{N\to\infty}R_{tN}^{\varepsilon,v}=0$.
{}From \eqref{replace_oncemore}, Lemma \ref{lemma_empirical} and Corollary \ref{corollary_invariant}, \textit{(ii)} imply \eqref{liminf_better}, as well as
\begin{equation} \limsup_{N\to\infty}\,(Nt)^{-1}\phi^{v}_{tN}\circ\theta'_{\lfloor b N\rfloor ,a N}(\omega') \leq \sup_{r\in[\lambda,\rho]\cap{\mathcal R}} [G(r)-v r],\quad \overline{\nu}^{\lambda,\rho}\otimes{\rm I\hspace{-0.8mm}P}\mbox{-a.s.}\label{liminf_better_2}\end{equation}
{\em Step two: proof of \eqref{limsup}.}
Let $r\in[\lambda,\rho]\cap{\mathcal R}$. We define $\overline{\nu}^{\lambda,r,\rho}$ as the distribution of $(\alpha,\eta^\lambda,\eta^r,\eta^\rho)$.
By \eqref{liminf_better} and \eqref{liminf_better_2},
\[ \lim_{N\to\infty}\,(Nt)^{-1}\phi^{v}_{tN}\circ\theta'_{\lfloor b N\rfloor ,a N}(\alpha,\eta^r,\eta^r,\omega) = G(r)-vr\]
By Lemma \ref{current_comparison},
\[ \phi^v_{tN}\circ\theta'_{\lfloor bN\rfloor ,aN}(\omega')\leq\phi^v_{tN}\circ\theta'_{\lfloor bN\rfloor ,aN}(\alpha,\eta^r,\eta^r,\omega)\]
The result follows by continuity of $G$ and minimizing over $r$.
\end{proof}
\subsection{Cauchy problem} \label{Cauchy}
For two measures $\mu,\nu\in{\mathcal M}^+({\mathbb R})$ with compact support, we define
\begin{equation}\label{def_delta}\Delta(\mu,\nu):=\sup_{x\in{\mathbb R}}\abs{ \nu((-\infty,x])-\mu((-\infty,x])}\end{equation}
which satisfies: \emph{(P1)}
For a sequence $(\mu_n)_{n\ge 0}$ of measures with uniformly bounded support,
$\mu_n\to \mu$ vaguely is equivalent to $\lim_{n\to\infty}\Delta(\mu_n,\mu)=0$;
\emph{(P2)} the macroscopic stability property (\cite{bm, mrs}) states that $\Delta$ is, with high probability, an ``almost'' nonincreasing function of two coupled particle systems; \emph{(P3)} correspondingly, there is $\Delta$-stability for \eqref{hydrodynamics}, that is, $\Delta$ is a nonincreasing function along two entropy solutions (\cite[Proposition 4.1, \textit{iii), b)}]{bgrs3}).
\begin{proposition}\label{hydro_finite} Assume $(\eta^N_0)$ is a sequence of configurations such that:
(i) there exists $C>0$ such that for all $N\in{\mathbb N}$, $\eta^N_0$ is supported on ${\mathbb Z}\cap[-CN,CN]$;
(ii) $\pi^N(\eta^N_0)\to u_0(.)dx$ as $N\to\infty$, where $u_0$
has compact support, is a.e. ${\mathcal R}$-valued and has finite space variation.
Let $u(.,t)$ denote the unique entropy solution to \eqref{hydrodynamics} with Cauchy datum $u_0(.)$.
Then, $Q\otimes{\rm I\hspace{-0.8mm}P}$-a.s. as $N\to\infty$,
\[ \Delta^N(t):=\Delta(\pi^N(\eta^N_{Nt}(\alpha,\eta^N_0,\omega)),u(.,t)dx) \]
converges uniformly to $0$ on $[0,T]$ for every $T>0$.
\end{proposition}
Theorem \ref{th:hydro} follows for general initial data $u_0$ by coupling and approximation arguments (see \cite[Section 4.2.2]{bgrs3}). \\ \\
\begin{proof}{proposition}{hydro_finite}
By initial assumption \eqref{initial_profile_vague}, $\lim_{N\to\infty}\Delta^N(0)=0$.
Let $\varepsilon>0$, and $\varepsilon'=\varepsilon/(2V)$, for $V$ given by \eqref{maxspeed}. Set $t_k=k\varepsilon'$ for $k\le \kappa:=\lfloor T/\varepsilon'\rfloor $, $t_{\kappa+1}=T$. Since the number of steps is proportional to $\varepsilon$, if we want to bound the total error, the main step is to prove
\begin{equation}\label{wearegoing_1} \limsup_{N\to\infty}\sup_{k=0,\ldots,{\mathcal K}-1}\left[\Delta^N(t_{k+1})-\Delta^N(t_k)\right]\leq 3\delta\varepsilon,\quad Q\otimes{\rm I\hspace{-0.8mm}P}\mbox{-a.s.} \end{equation}
where $\delta:=\delta(\varepsilon)$ goes to 0 as $\varepsilon$ goes to 0; the gaps between discrete times are filled by an
estimate for the time modulus of continuity of $\Delta^N(t)$ (see \cite[Lemma 4.5]{bgrs3}). \\ \\
{\em Proof of \eqref{wearegoing_1}}. Since $u(.,t_k)$ has locally finite variation, by \cite[Lemma 4.2]{bgrs3}, for all $\varepsilon>0$
we can find
functions
\begin{equation} \label{decomp_approx} v_k=\sum_{l=0}^{l_k}r_{k,l}{\bf 1}_{[x_{k,l},x_{k,l+1})} \end{equation}
with
$ -\infty=x_{k,0}<x_{k,1}<\ldots<x_{k,l_k}<x_{k,l_k+1}=+\infty$, $r_{k,l}\in{\mathcal R}$, $r_{k,0}=r_{k,l_k}=0$,
such that $x_{k,l}-x_{k,l-1}\geq \varepsilon$, and
\begin{equation}\label{uniform_approx}
\Delta(u(.,t_k)dx,v_kdx) \leq \delta\varepsilon
\end{equation}
For $t_k\leq t< t_{k+1}$, we denote by $v_k(.,t)$ the entropy solution to \eqref{hydrodynamics} at time $t$ with Cauchy datum $v_k(.)$.
The configuration $\xi^{N,k}$ defined on $(\Omega_{\mathbf A}\otimes\Omega,\mathcal F_{\mathbf A}\otimes\mathcal F,{\rm I\hspace{-0.8mm}P}_{\mathbf A}\otimes{\rm I\hspace{-0.8mm}P})$ (see Lemma \ref{big_coupling}) by
\[ \xi^{N,k}(\omega_{\mathbf A},\omega)(x):=\eta_{Nt_k}(\alpha(\omega_{\mathbf A}),\eta^{r_{k,l}}(\omega_{\mathbf A}), \omega)(x),\, \mbox{ if }\, \lfloor Nx_{k,l}\rfloor \leq x<\lfloor Nx_{k,l+1}\rfloor \]
is a microscopic version of $v_k(.)$, since by Proposition \ref{corollary_2_2} with $\lambda=\rho=r^{k,l}$,
\begin{equation} \label{profile_xi} \lim_{N\to\infty}\pi^N(\xi^{N,k}(\omega_{\mathbf A},\omega))(dx) =v_k(.)dx,\quad{\rm I\hspace{-0.8mm}P}_{\mathbf A}\otimes{\rm I\hspace{-0.8mm}P}\mbox{-a.s.} \end{equation}
We denote by
$\xi^{N,k}_t(\omega_{\mathbf A},\omega) = \eta_{t}(\alpha(\omega_{\mathbf A}),\xi^{N,k}(\omega_{\mathbf A},\omega), \theta_{0,Nt_k}\omega)$ evolution starting from $\xi^{N,k}$.
By triangle inequality,
\begin{eqnarray}
\Delta^N(t_{k+1})-\Delta^N(t_k) & \leq & \Delta\left[ \pi^N( \eta^N_{Nt_{k+1}} ),\pi^N( \xi^{N,k}_{N\varepsilon'} ) \right]-\Delta^N(t_k)\label{decomp_delta_1}\\
& + & \Delta\left[ \pi^N( \xi^{N,k}_{N\varepsilon'} ),v_k(.,\varepsilon')dx \right]\label{decomp_delta_2}\\ & + & \Delta(v_k(.,\varepsilon')dx,u(.,t_{k+1})dx)\label{decomp_delta_3}
\end{eqnarray}
To conclude, we rely on Properties \emph{(P1)--(P3)} of $\Delta$: Since $\varepsilon'=\varepsilon/(2V)$, finite propagation property for \eqref{hydrodynamics} and for the particle system (see \cite[Proposition 4.1, \emph{iii), a)} and Lemma 4.3]{bgrs3}) and Proposition \ref{corollary_2_2} imply
$$ \lim_{N\to\infty}\pi^N(
\xi^{N,k}_{N\varepsilon'}(\omega_{\mathbf A},\omega)
)=v_k(.,\varepsilon')dx,\qquad {\rm I\hspace{-0.8mm}P}_A\otimes{\rm I\hspace{-0.8mm}P}\mbox{-a.s.} $$
Hence, the term \eqref{decomp_delta_2} converges a.s. to $0$ as $N\to\infty$. By $\Delta$-stability for \eqref{hydrodynamics}, the term \eqref{decomp_delta_3} is bounded by $\Delta(v_k(.)dx,u(.,t_k)dx)\leq\delta\varepsilon$.
We now consider the term \eqref{decomp_delta_1}.
By macroscopic stability (\cite[Theorem 2, Equation (4) and Remark 1]{mrs}), outside probability $e^{-CN\delta\varepsilon}$, \begin{equation}\label{macrostablast}\Delta\left[ \pi^N( \eta^N_{Nt_{k+1}}),\pi^N( \xi^{N,k}_{N\varepsilon'} ) \right]\leq
\Delta\left[ \pi^N( \eta^N_{Nt_{k}} ),\pi^N( \xi^{N,k} ) \right]+\delta\varepsilon
\end{equation}
Thus the event \eqref{macrostablast} holds a.s. for $N$ large enough.
By triangle inequality,
\begin{eqnarray*} &&\Delta\left[ \pi^N( \eta^N_{Nt_{k}} ),\pi^N( \xi^{N,k} ) \right] - \Delta^N(t_k) \\ &\leq&\Delta\left(u(.,t_k)dx,v_k(.)dx\right)+\Delta\left[v_k(.)dx,\pi^N(\xi^{N,k})\right]
\end{eqnarray*}
for which \eqref{uniform_approx}, \eqref{profile_xi} yield as $N\to\infty$ an upper bound $2\delta\varepsilon$, hence $3\delta\varepsilon$ for the term \eqref{decomp_delta_1}.
\end{proof}
\subsection{Proofs of lemmas} \label{technical}
\begin{proof}{lemma}{big_coupling}
Let $\mathcal R_d$ be a countable dense subset of $\mathcal R$ that contains all the isolated points of $\mathcal R$. We denote by $\mathcal R_d^+$, resp. $\mathcal R_d^-$, the set of $\rho\in[0,K]$ that lie in the closure of $[0,\rho)\cap\mathcal R_d$, resp. $(\rho,K]\cap\mathcal R_d$.
Because $\mathcal R$ is closed, we have ${\mathcal R}={\mathcal R_d}\cup{\mathcal R_d^+}\cup{\mathcal R_d^-}$.
By \eqref{ordered_measures} there exists a subset ${\mathbf A}'$ of ${\mathbf A}$ with $Q$-probability $1$, such that $\nu^\rho_\alpha\leq\nu^{\rho'}_\alpha$ for all $\alpha\in{\mathbf A}'$ and $\rho,\rho'\in{\mathcal R}_d$.
By \cite[Theorem 6]{kk}, for every $\alpha\in{\mathbf A}'$, there exists a family of random variables $(\eta^\rho_\alpha:\,\rho\in{\mathcal R}_d)$ on a probability space $(\Omega_\alpha,\mathcal F_\alpha,{\rm I\hspace{-0.8mm}P}_\alpha)$, such that \eqref{marginals}--\eqref{strassen} hold for $\rho\in{\mathcal R}_d$.
Let $\Omega_{\mathbf A}=\{(\alpha,\omega_\alpha):\,\alpha\in{\mathbf A'},\,\omega_\alpha\in\Omega_\alpha\}$, $\mathcal F_{\mathbf A}$ be the $\sigma$-field generated by mappings $(\alpha,\omega_\alpha)\mapsto\eta^\rho(\alpha,\omega_\alpha):=\eta_\alpha^\rho(\omega_\alpha)$ for $\rho\in{\mathcal R}_d$, and ${\rm I\hspace{-0.8mm}P}_{\mathbf A}(d\alpha,d\omega_\alpha)=Q(d\alpha)\otimes{\rm I\hspace{-0.8mm}P}_\alpha(d\omega_\alpha)$.
Now consider $\rho\in{\mathcal R}\setminus{\mathcal R}_d$. Since $\eta^r_\alpha$ is a nondecreasing function of $r$, for every $\alpha\in{\mathbf A}'$ and $\omega_\alpha\in\Omega_\alpha$, $\eta^{\rho+}(\alpha,\omega_\alpha):=\lim_{r\to\rho,r<\rho,r\in{\mathcal R}_d}\eta^r_\alpha(\omega_\alpha)$ exists if $\rho\in{\mathcal R}_d^+$, and $\eta^{\rho-}(\alpha,\omega_\alpha):=\lim_{r\to\rho,r>\rho,r\in{\mathcal R}_d}\eta^r_\alpha(\omega_\alpha)$ exists if $\rho\in{\mathcal R}_d^-$.
We set $\eta^\rho(\alpha,\omega_\alpha)=\eta^{\rho+}(\alpha,\omega_\alpha)$ if $\rho\in{\mathcal R}_d^+$, $\eta^\rho(\alpha,\omega_\alpha)=\eta^{\rho-}(\alpha,\omega_\alpha)$ otherwise. Suppose for instance $\rho\in\mathcal R_d^+$. Since $\eta^{\rho+}$ is a ${\rm I\hspace{-0.8mm}P}_{\mathbf A}$-a.s. limit of $\eta^r$ as $r\to\rho$, $r<\rho$, $r\in\mathcal R_d$, it is a limit in distribution. Weak continuity of $\nu^\rho$ then implies \eqref{marginals}. Property \eqref{strassen} on $\mathcal R$ follows from the property on $\mathcal R_d$ and definitions of $\eta^{\rho\pm}$.
\end{proof}
\mbox{}\\ \\
To prove Lemma \ref{lemma_empirical}, we need the following uniform upper bound (proved in \cite[Lemma 3.4]{bgrs3}).
\begin{lemma}\label{deviation_empirical}
Let ${\bf P}_\nu^v$ denote the law of a Markov process $(\widetilde\alpha_.,\widetilde\xi_.)$ with generator $L^v$ and initial distribution $\nu$. For $\varepsilon>0$, let
\begin{equation} \label{def_empirical-gen} \pi_{t,\varepsilon}:=\abs{{\mathbb Z}\cap[-\varepsilon t,\varepsilon t]}^{-1}\sum_{x\in{\mathbb Z}\cap[-\varepsilon t,\varepsilon t]}t^{-1}\int_0^t\delta_{(\tau_x\widetilde\alpha_s,\tau_x\widetilde\xi_s)}ds
\end{equation}
Then, there exists a functional ${\mathcal D}_v$ which is nonnegative, l.s.c., and satisfies ${\mathcal D}_v^{-1}(0)={\mathcal I}_{L^v}$, such that, for every closed subset $F$ of ${\mathcal P}({\bf A}\times\bf X)$,
\begin{equation} \label{ld} \limsup_{t\to\infty}t^{-1}\log\sup_{\nu\in\mathcal P({\bf A}\times{\mathbf X})}{\bf P}_\nu^v\left(\pi_{t,\varepsilon}(\widetilde\xi_.)\in F\right)\leq -\inf_{\mu\in F}{\mathcal D}_v(\mu) \end{equation}
\end{lemma}
\begin{proof}{Lemma}{lemma_empirical}
We give a brief sketch of the arguments (details are similar to \cite[Lemma 3.3]{bgrs3}). Spatial averaging in \eqref{def_empirical-gen} implies that any subsequential limit $\mu$ lies in $\mathcal S$. Lemma \ref{deviation_empirical} and Borel-Cantelli's Lemma imply that $\mu$ lies in $\mathcal I_{L^v}$ (uniformity in \eqref{ld} is important because $\theta$-shifts make the initial distribution of the process unknown). Finally, the inequality $\nu^\lambda\ll\mu\ll\nu^\rho$ is obtained by coupling the initial distribution with $\eta^\lambda$ and $\eta^\rho$, using attractiveness and space-time ergodicity for the equilibrium processes.
\end{proof}
\mbox{}\\ \\
\begin{proof}{lemma}{current_comparison}
Assume for instance $\eta\leq\eta'$. Let $\gamma:=T(\eta,\xi)$ and $\gamma':=T(\eta',\xi)$, $\gamma_t=\eta_t(\alpha,\gamma,\omega)$ and $\gamma'_t=\eta_t(\alpha,\gamma',\omega)$. By \eqref{attractive_0}, $\gamma_t\leq\gamma'_t$ for all $t\geq 0$. By definition of the current, $\phi^v_t(\alpha,\eta',\xi,\omega)-\phi^v_t(\alpha,\eta,\xi,\omega) =\sum_{x>vt}[\gamma'_t(x)-\gamma_t(x)]\geq 0$.
\end{proof}
\section{Other models}\label{sec:general}
For the proof of Theorem \ref{th:hydro} we have not used the particular form of $L_\alpha$ in \eqref{generator}, but the following properties.\\ \\
1) The set of environments is a probability space $({\mathbf A},{\mathcal F}_{\mathbf A},Q)$, where $\mathbf A$ is a compact metric space and ${\mathcal F}_{\mathbf A}$ its Borel $\sigma$-field. On $\mathbf A$ we have a group of space shifts $(\tau_x:\,x\in{\mathbb Z})$, with respect to which $Q$ is ergodic. For each $\alpha\in{\mathbf A}$, $L_\alpha$ is the generator of a Feller process on ${\mathbf X}$
that satisfies \eqref{commutation}. The latter should be viewed as the assumption on ``how the disorder enters the dynamics''. It is equivalent to $L$ satisfying \eqref{eq:L-transl-inv}, that is being a translation-invariant generator on ${\mathbf A}\times{\mathbf X}$.\\ \\
2) For $L_\alpha$ we can define a graphical construction \eqref{update_rule} on a space-time Poisson space $(\Omega,\mathcal F,{\rm I\hspace{-0.8mm}P})$ such that $L_\alpha$ coincides with \eqref{gengen}, for some mapping ${\mathcal T}^{\alpha,z,v}$ satisfying the shift commutation and strong attractiveness properties \eqref{shift_t} and \eqref{attractive_0}. The existence of this graphical construction for the infinite-volume system follows from assumption \textit{(A2)}, which controls the rate of faraway jumps.
This assumption is also responsible for the finite propagation property of discrepancies in the particle system, and its macroscopic counterpart, the Lipschitz continuity of the flux function
(see \eqref{flux}, Remarks \ref{remark_flux} and \ref{remark_lipschitz}). \\ \\
3) Irreducibility and non-degeneracy assumptions \textit{(A1), (A4)} (combined with attractiveness assumption \textit{(A5)}) imply Proposition \ref{prop_irred}. \\ \\
In the sequel we consider other models satisfying 1) and 2), for which appropriate assumptions replacing \textit{(A1)--(A5)} imply existence of a graphical construction, and Proposition \ref{prop_irred} as in 3). In these examples, the transition defined by $\mathcal T^{\alpha,z,v}$ in \eqref{update_rule} is a particle jump, that is of the form
$\mathcal T^{\alpha,z,v}\eta=\eta^{x(\alpha,z,v),y(\alpha,z,v)}$. It follows that \eqref{gengen} yields (in replacement of \eqref{generator})
\begin{equation}\label{generic_form}
L_\alpha f(\eta)=\sum_{x,y\in{{\mathbb Z}}}c_\alpha(x,y,\eta)\left[ f\left(\eta^{x,y} \right)-f(\eta) \right] \end{equation}
where
\begin{equation}\label{general_rate} c_\alpha(x,y,\eta)=\sum_{z\in{\mathbb Z}}m\left(
\left\{
v\in\mathcal V:\,\mathcal T^{\alpha,z,v}\eta=\eta^{x,y}
\right\}
\right)
\end{equation}
and the shift-commutation property \eqref{shift_t} implies
\begin{equation}\label{shift_c}
c_\alpha(x,y,\eta)=c_{\tau_x\alpha}(0,y-x,\tau_x\eta)
\end{equation}
which, for \eqref{generic_form}, is equivalent to \eqref{commutation}.
Microscopic fluxes \eqref{def_f} and \eqref{other_flux} more generally write
\begin{eqnarray}\nonumber j^+(\alpha,\eta) &=& \sum_{y,z\in{\mathbb Z}:\,y\leq 0<y+z} c_\alpha(\eta(y),\eta(y+z))\\\nonumber
\nonumber j^-(\alpha,\eta) & = & \sum_{y,z\in{\mathbb Z}:\,y+z\leq 0<y}c_\alpha(\eta(y),\eta(y+z)) \\
\widetilde{\jmath}(\alpha,\eta) & = & \sum_{z\in{\mathbb Z}}zc_\alpha(0,z,\eta)\label{other_flux_general}
\end{eqnarray}
\subsection{Generalized misanthropes' process}
Let $c\in(0,1)$, and $p(.)$ (resp. $P(.)$), be a probability distribution on ${\mathbb Z}$ satisfying assumption \textit{(A1)} (resp. \textit{(A2)}). Define $\mathbf A$ to be the set of functions
$B:{\mathbb Z}^2\times\{0,\ldots,K\}^2\to{\mathbb R}^+$ such that for all $(x,z)\in{\mathbb Z}^2$, $B(x,z,.,.)$ satisfies assumptions \textit{(A3)--(A5)} and
\begin{eqnarray}\label{genmis_1}
B(x,z,1,K-1) & \geq & cp(z)\\ \label{genmis_2} \quad B(x,z,K,0) & \leq & c^{-1}P(z)
\end{eqnarray}
The shift operator $\tau_y$ on $\mathbf A$ is defined by
$ (\tau_y B)(x,z,n,m)=B(x+y,z,n,m) $.
We generalize \eqref{generator} by setting
\begin{equation}\label{generator_genmis}
L_\alpha f(\eta)=\sum_{x,y\in{{\mathbb Z}}}B(x,y-x,\eta(x),\eta(y)) \left[ f\left(\eta^{x,y} \right)-f(\eta) \right]
\end{equation}
where we assume that the distribution $Q$ of $B(.,.,.,.)$ is ergodic with respect to the above spatial shift (we kept the notation $L_\alpha$ to be consistent with the rest of the paper, but we should have written $L_B$). Assumption \eqref{genmis_1} replaces \textit{(A1)} and implies Proposition \ref{prop_irred}. Assumption \eqref{genmis_2} replaces \textit{(A2)} and implies existence of the infinite volume dynamics given by the following graphical construction.
For $v=(z,u)$, set $m(dv)=c^{-1}P(dz)\lambda_{[0,1]}(du)$ in \eqref{special_choice}, and replace \eqref{update_misanthrope} with
\begin{equation}\label{update_genmis}
{\mathcal T}^{\alpha,x,v}\eta=\left\{
\begin{array}{lll} \eta^{x,x+z} & \mbox{if} & \displaystyle{ u<\frac{B(x,z,\eta(x),\eta(x+z))} {c^{-1}P(z)}}\\
\eta & & \mbox{otherwise}
\end{array}
\right.
\end{equation}
Here the microscopic flux \eqref{other_flux_general} writes
$$ \widetilde{\jmath}(\alpha,\eta)=\sum_{z\in{\mathbb Z}}zB(0,z,\eta(0),\eta(z)) $$
and the Lipschitz constant $V=2c^{-1}\sum_{z\in{\mathbb Z}}|z|P(z)$ for $G^Q$ follows as in \eqref{maxspeed} from \eqref{standard_coupling}--\eqref{compare_fluxes}.
The basic model \eqref{generator} is recovered with $B(x,z,n,m)=\alpha(x)p(z)b(n,m)$, for $p(.)$ a probability distribution on ${\mathbb Z}$ satisfying \textit{(A1)--(A2)}, $\alpha(.)$ an ergodic $(c,1/c)$-valued random field, and $b(.,.)$ a function satisfying \textit{(A3)--(A5)}. In this case \eqref{genmis_1}--\eqref{genmis_2} hold
with $P(.)=p(.)$. Here are two other examples.\\ \\
{\em Example 1.1.} This is the bond-disorder version of \eqref{generator}: we have $B(x,z,n,m)=\alpha(x,x+z)b(n,m)$, where $\alpha=(\alpha(x,y):\,x,y\in{\mathbb Z})$ is a positive random field on ${\mathbb Z}^2$, bounded away from $0$, ergodic with respect to the space shift $\tau_z\alpha=\alpha(.+z,.+z)$. Sufficient assumptions replacing \textit{(A1)} and \textit{(A2)} are
\begin{equation}\label{assumptions_bond}c\,p(y-x)\leq \alpha(x,y)\leq c^{-1}P(y-x)\end{equation}
for some constant $c>0$, and probability distributions $p(.)$ and $P(.)$ on ${\mathbb Z}$, respectively satisfying \textit{(A1)} and \textit{(A2)}.\\ \\
{\em Example 1.2.} This is a model that switches between two rate functions according to the environment: we have $B(x,z,n,m)=p(z)[(1-\alpha(x))b_0(n,m)+\alpha(x)b_1(n,m)]$, where $(\alpha(x),\,x\in{\mathbb Z})$ is an ergodic $\{0,1\}$-valued field, $p(.)$ satisfies assumption \textit{(A1}), and $b_0$, $b_1$ assumptions \textit{(A3)--(A5)}.
\subsection{Generalized $k$-step $K$-exclusion process}\label{subsec:k-step}
We first recall the definition of the $k$-step exclusion process, introduced in \cite{guiol}. Let $K=1$, $k\in{\mathbb N}$, and $p(.)$ be a jump kernel on ${\mathbb Z}$ satisfying assumptions \textit{(A1)--(A2)}. A particle at $x$ performs a random walk with kernel $p(.)$ and jumps to the first vacant site it finds along this walk, unless it returns to $x$ or does not find an empty site within $k$ steps, in which case it stays at $x$. \\ \\
To generalize this, let $K\geq 1$, $k\geq 1$, $c\in(0,1)$, and $\mathcal D$ denote the set of functions $\beta=(\beta^1,\ldots,\beta^k)$ from ${\mathbb Z}^k$ to $(0,1]^k$ such that
\begin{eqnarray}\label{eq:(o)} \beta^1(.)&\in&[c,1]\\ \label{eq:(i)} \beta^i(.)&\geq&\beta^{i+1}(.),\,\forall i\in\{1,\ldots,k-1\} \end{eqnarray}
In the sequel, an element of ${\mathbb Z}^k$ is denoted by $\underline{z}=(z_1,\ldots,z_k)$. Let $q$ be a probability distribution on ${\mathbb Z}^k$, and $\beta\in\mathcal D$. We define the $(q,\beta)$-$k$ step $K$-exclusion process as follows. A particle at $x$ (if some) picks a $q$-distributed random vector $\underline{Z}=(Z_1,\ldots,Z_k)$, and jumps to the first site $x+Z_i$ ($i\in\{1,\ldots,k\})$ with strictly less than $K$ particles along the path $(x+Z_1,\ldots,x+Z_k)$, if such a site exists, with rate $\beta^i(\underline{Z})$. Otherwise, it stays at $x$.
The $k$-step exclusion process corresponds to the particular case where $K=1$, $q$ is the distribution (hereafter denoted by $q^k_{RW}(p))$ of the first $k$ steps of a random walk with kernel $p(.)$ absorbed at $0$, and $\beta^i(\underline{z})=1$.
Outside the fact that $K$ can take values $\geq 1$, our model extends $k$-step exclusion in different directions: \\ \textit{(1)} The random path followed by the particle need not be a Markov process.\\
\textit{(2)} The distribution $q$ is not necessarily supported on paths absorbed at 0.\\
\textit{(3)} Different rates can be assigned to jumps according to the number of steps, and the collection of these rates may depend on the path realization. \\ \\
Next, disorder is introduced: the environment is a field $\alpha=((q_x,\beta_x):\,x\in{\mathbb Z})\in{\mathbf A}:=(\mathcal P({\mathbb Z}^k)\times\mathcal D)^{\mathbb Z}$. For a given realization of the environment,
the distribution of the path $\underline{Z}$ picked by a particle at $x$ is $q_x$, and the rate at which it jumps to $x+Z_i$ is $\beta^i_x(\underline{Z})$.
The corresponding generator is given by \eqref{generic_form} with $c_\alpha=\sum_{i=1}^k c_\alpha^i$, where
(with the convention that an empty product is equal to $1$)
$$c_\alpha^i(x,y,\eta )={\bf 1}_{\{\eta(x)>0\}}{\bf 1}_{\{\eta(y)<K\}} \int\left[
\beta^i_x(\underline{z}) {\bf 1}_{\{x+z_i=y\}} \prod_{j=1}^{i-1}{\bf 1}_{\{\eta(x+z_j)=K\}}
\right]\,dq_x(\underline{z}) $$
The distribution $Q$ of the environment on $\mathbf A$ is assumed ergodic with respect to the space shift $\tau_y$, where $\tau_y\alpha=((q_{x+y},\beta_{x+y}):\,x\in{\mathbb Z})$. \\ \\ For the existence of the process and graphical construction below, and for Proposition \ref{prop_irred}, sufficient assumptions to replace \textit{(A1)--(A2)} are
\begin{eqnarray}\label{irreducibility_kstep} \inf_{x\in{\mathbb Z}} q^1_x(.) & \geq & c p(.)\\\label{summability_kstep}
\sup_{i=1,...,k}\sup_{x\in{\mathbb Z}}q^i_x(.) & \leq & c^{-1}P(.)
\end{eqnarray}
for some constant $c>0$, where $q_x^i$ denotes the $i$-th marginal of $q_x$, and $p(.)$, resp. $P(.)$, are probability distributions satisfying \textit{(A1)}, resp. \textit{(A2)}.
To write the microscopic flux and define a graphical construction, we introduce the following notation: for $(x,\underline{z},\eta)\in{\mathbb Z}\times{\mathbb Z}^k\times{\mathbf X}$, $\beta\in\mathcal D$ and $u\in[0,1]$,
\begin{eqnarray*}\label{nbsteps} N(x,\underline{z},\eta) &=& \inf\left\{i\in\{1,\ldots,k\}:\,\eta\left(x+z_i\right)<K\right\} \mbox{ with} \inf\emptyset=+\infty \\ \label{finloc} Y(x,\underline{z},\eta) &=& \left\{
\begin{array}{lll}
x+z_{N(x,\underline{z},\eta)} & \mbox{if} & N(x,\underline{z},\eta)<+\infty\\
x & \mbox{if} & N(x,\underline{z},\eta)=+\infty
\end{array}
\right. \\
{{\mathcal T}_0}^{x,\underline{z},\beta,u}\eta&=&\left\{
\begin{array}{lll}
\eta^{x,Y(x,\underline{z},\eta)} & \mbox{if} & \eta(x)>0\mbox{ and }u< \beta^{N(x,\underline{z},\eta)}(\underline{z}) \\
\eta & & \mbox{otherwise}
\end{array}
\right.
\end{eqnarray*}
(where the definition of $\beta^{+\infty}(\underline{z})$ has no importance).
With these notations, we have
\begin{eqnarray}
c_\alpha(x,y,\eta) & = & {\bf 1}_{\{\eta(x)>0\}}{\rm I\hspace{-0.8mm}E}_{q_0}\left[
\beta^{N(x,\underline{Z},\eta)}_0 {\bf 1}_{\{Y(x,\underline{Z},\eta)=y\}}
\right]\label{rate_genkstep} \\
\label{flux_genkstep} \widetilde{\jmath}(\alpha,\eta) & = & {\bf 1}_{\{\eta(0)>0\}}{\rm I\hspace{-0.8mm}E}_{q_0}\left[
\beta^{N(0,\underline{Z},\eta)}_0 Y(0,\underline{Z},\eta)
\right] \end{eqnarray}
where expectation is with respect to $\underline{Z}$. Since
$$\left| \beta^{N(0,\underline{Z},\eta)}_0 Y(0,\underline{Z},\eta)- \beta^{N(0,\underline{Z},\xi)}_0 Y(0,\underline{Z},\xi)
\right|\leq 2\sum_{i=1}^k |Z_i|\sum_{i=1}^k|\eta(Z_i)-\xi(Z_i)| $$
\eqref{standard_coupling}--\eqref{compare_fluxes} yield for $G^Q$ the Lipschitz constant
$V=2k^2c^{-1}\sum_{z\in{\mathbb Z}}|z|P(z)$.\\ \\
Let $\mathcal V=[0,1]\times[0,1]$, $m=\lambda_{[0,1]}\otimes\lambda_{[0,1]}$. For each probability distribution $q$ on ${\mathbb Z}^k$, there exists a mapping $F_q:[0,1]\to{\mathbb Z}^k$ such that $F_q(V_1)$ has distribution $q$ if $V_1$ is uniformly distributed on $[0,1]$. Then the transformation $\mathcal T$ in \eqref{update_rule} is defined by (with $v=(v_1,v_2)$ and $\alpha=((q_x,\beta_x):\,x\in{\mathbb Z}))$
\begin{equation}\label{update_kstep}
{\mathcal T}^{\alpha,x,v}\eta={\mathcal T}_0^{x,F_{q_x}(v_1),\beta_x(F_{q_x}(v_1)),v_2}\eta \end{equation}
Strong attractiveness of our process will follow from
\begin{lemma}\label{attractive_kstep}
For every $(x,\underline{z},u)\in{\mathbb Z}\times{\mathbb Z}^k\times[0,1]$, ${\mathcal T}_0^{x,\underline{z},\beta,u}$ is an increasing mapping from ${\mathbf X}$ to ${\mathbf X}$.
\end{lemma}
\begin{proof}{lemma}{attractive_kstep}
Let $(\eta,\xi)\in{\mathbf X}^2$ with $\eta\leq\xi$. To prove that ${\mathcal T}_0^{x,\underline{z},\beta,u}\eta\leq {\mathcal T}_0^{x,\underline{z},\beta,u}\xi$, since $\eta$ and $\xi$ can only possibly change at sites $x$, $y:=Y(x,\underline{z},\eta)$ and $y':=Y(x,\underline{z},\xi)$, it is sufficient to verify the inequality at these sites. \\ \\ If $\xi(x)=0$, then by \eqref{update_kstep}, $\eta$ and $\xi$ are both unchanged by ${\mathcal T}_0^{x,\underline{z},\beta,u}$. If $\eta(x)=0<\xi(x)$, then ${\mathcal T}_0^{x,\underline{z},\beta,u}\xi(y') \geq\xi(y')\geq\eta(y')={\mathcal T}_0^{x,\underline{z},\beta,u}\eta(y')$. \\ \\
Now assume $\eta(x)>0$. Then $\eta\leq\xi$ implies $N(x,\underline{z},\eta)\leq N(x,\underline{z},\xi)$. If $N(x,\underline{z},\eta)=+\infty$, $\eta$ and $\xi$ are unchanged. If $N(x,\underline{z},\eta)<N(x,\underline{z},\xi)=+\infty$, then ${\mathcal T}_0^{x,\underline{z},\beta,u}\eta=\eta^{x,y}$ and $\xi(y)=K$. Thus, ${\mathcal T}_0^{x,\underline{z},\beta,u}\eta(x)=\eta(x)-1\leq\xi(x)={\mathcal T}_0^{x,\underline{z},\beta,u}\xi(x)$ and ${\mathcal T}_0^{x,\underline{z},\beta,u}\xi(y)=\xi(y)=K\geq{\mathcal T}_0^{x,\underline{z},\beta,u}\eta(y)$.
If $N(x,\underline{z},\eta)=N(x,\underline{z},\xi)<+\infty$, then $\beta^{N(x,\underline{z},\eta)}=\beta^{N(x,\underline{z},\xi)}=:\beta$. If $u\geq\beta$ both $\eta$ and $\xi$ are unchanged. Otherwise ${\mathcal T}_0^{x,\underline{z},\beta,u}\eta=\eta^{x,y}$ and ${\mathcal T}_0^{x,\underline{z},\beta,u}\xi=\xi^{x,y}$, whence the conclusion. Finally, assume $N(x,\underline{z},\eta)<N(x,\underline{z},\xi)<+\infty$, hence $\beta:=\beta^{N(x,\underline{z},\eta)}\geq\beta^{N(x,\underline{z},\xi)}=:\beta'$ by \eqref{eq:(i)} and $\eta(y)<\xi(y)=K$. If $u\geq\beta$, $\eta$ and $\xi$ are unchanged.
If $u<\beta'$, then ${\mathcal T}_0^{x,\underline{z},\beta,u}\eta(y)=\eta(y)+1\leq\xi(y)={\mathcal T}_0^{x,\underline{z},\beta,u}\xi(y)=K$ and ${\mathcal T}_0^{x,\underline{z},\beta,u}\xi(y')=\xi(y')+1\geq {\mathcal T}_0^{x,\underline{z},\beta,u}\eta(y')$. If $\beta'\leq u<\beta$, then ${\mathcal T}_0^{x,\underline{z},\beta,u}\eta(x)=\eta(x)-1\leq {\mathcal T}_0^{x,\underline{z},\beta,u}\xi(x)$ and ${\mathcal T}_0^{x,\underline{z},\beta,u}\eta(y)=\eta(y)+1\leq {\mathcal T}_0^{x,\underline{z},\beta,u}\xi(y)=\xi(y)=K$.
\end{proof}
\mbox{}\\ \\
We now describe a few examples.\\ \\
{\em Example 2.1.} Let $K=1$, $(\alpha_x:\,x\in{\mathbb Z})$ be an ergodic $[c,1/c]$-valued random field, and $r(.)$ be a probability measure on ${\mathbb Z}$ satisfying \textit{(A1)--(A2)}. A disordered version of the $k$-step exclusion process with jump kernel $r$ is obtained by multiplying the rate of any jump starting from $x$ by $\alpha_x$. This means that the random field $(q_x,\beta_x)_{x\in{\mathbb Z}}$ is defined by $q_x=q^k_{RW}(r)$, and $\beta_x(\underline{z})=(\alpha_x,\ldots,\alpha_x)$ for every $\underline{z}\in{\mathbb Z}^k$.\\ \\
{\em Example 2.2.} Let $(\gamma_x,\iota_x)_{x\in{\mathbb Z}}$ be an ergodic $[c,1]^{2k}$-valued random field, where $\gamma_x=(\gamma_x^n,\,1\le n\le k)$ and $\iota_x=(\iota_x^n,\,1\le n\le k)$. The random field $(q_x,\beta_x)_{x\in{\mathbb Z}}$ is defined by
\begin{eqnarray*} q_x&=&\frac{1}{2}\delta_{(1,2,\ldots,k)}+\frac{1}{2}\delta_{(-1,-2,\ldots,-k)}\\ \beta^i_x(1,2,\ldots,k)=2\gamma^i_x,&&\beta^i_x(-1,-2,\ldots,-k)=2\iota^i_x \end{eqnarray*}
Hence the rates are disordered but not the distribution of the random path followed by particles: the stationary random field $(q_x)_{x\in{\mathbb Z}}$ is deterministic and uniform. Here, the jump rate and microscopic flux \eqref{rate_genkstep}--\eqref{flux_genkstep} have a fairly explicit form:
\begin{eqnarray}
\label{fairly_explicit_rate_1} c_\alpha(x,y,\eta) & = & \gamma^{y-x}_x {\bf 1}_{\{\eta(x)>0\}}{\bf 1}_{\{\eta(y)<K\}}\prod_{z=x+1}^{y-1}{\bf 1}_{\{\eta(z)=K\}}\mbox{ if }y>x\\
\label{fairly_explicit_rate_2} c_\alpha(x,y,\eta) & = & \iota^{x-y}_x {\bf 1}_{\{\eta(x)>0\}}{\bf 1}_{\{\eta(y)<K\}}\prod_{z=y+1}^{x-1}{\bf 1}_{\{\eta(z)=K\}}\mbox{ if }y<x\\
\nonumber \widetilde{\jmath}(\alpha,\eta) & = & \eta(0)\sum_{n=1}^k n\gamma^n_0(1-\eta(n))\prod_{j=1}^{n-1}\eta(j)\\
& - & \eta(0)\sum_{n=1}^k n\iota^n_0(1-\eta(-n))\prod_{j=1}^{n-1}\eta(-j)\label{fairly_explicit}
\end{eqnarray}
{\em Example 2.3.} Set $q_x=q^k_{RW}(r_x)$, for $(r_x)_{x\in{\mathbb Z}}$ an ergodic random field with values in the probability measures on ${\mathbb Z}$ satisfying \textit{(A1)--(A2)}. The simplest case is nearest-neighbor jumps, that is, $r_x=p_x\delta_1+(1-p_x)\delta_{-1}$, where, for some $c\in(0,1)$, $(p_x)_{x\in{\mathbb Z}}$ is an ergodic $[c,1/c]$-random field.
Due to the nearest-neighbor assumption, a particle starting from $x$ can only jump to $y>x$ (resp. $y<x$) if $y$ is not full and all sites between $x$ and $y$ (resp. $y$ and $x$) are full. Hence, the jump rate \eqref{rate_genkstep} is identical (see example below) to the one obtained by taking in \eqref{fairly_explicit_rate_1}--\eqref{fairly_explicit_rate_2}
$$\gamma^n_x=\sum_{l=0}^{\lfloor(k-n)/2\rfloor}p_x^{n+l}(1-p_x)^l C_n(n+l,l), \quad\iota^n_x=\sum_{l=0}^{\lfloor(k-n)/2\rfloor}(1-p_x)^{n+l}p_x^l C_n(n+l,l)
$$
for $n\in\{1,\ldots,k\}$, where $C_n(i,j)$, for $i,j\in{\mathbb Z}^+$ and $i+j>0$, is the number of paths $(z_0=0,\ldots,z_{i+j})$ such that $0<z_m<n$
for $m=1,\ldots,i+j-1$, $|z_{m+1}-z_{m}|=1$ for $m=1,\ldots,i+j$, and ${\rm Card}\{m\in\{1,\ldots,i+j\}:\,z_m-z_{m-1}=1\}=i$.
With this choice of $\gamma^n_x$ and $\iota^n_x$, the microscopic flux is given by \eqref{fairly_explicit}.
For instance if $k=5$, we obtain, for $n\in\{1,\ldots,k\}$:
\begin{eqnarray}
c_\alpha(x,x+n,\eta) & = & p_x^n{\bf 1}_{\{\eta(x)>0\}}{\bf 1}_{\{\eta(x+n)<K\}}\prod_{j=1}^{n-1}{\bf 1}_{\{\eta(x+j)=K\}}\quad\mbox{ if }n\neq 3\\
\nonumber
c_\alpha(x,x+3,\eta) & = & p_x^3[1+p_x(1-p_x)]\times\\ \label{step3_right}&&\qquad\eta(x)\eta(x+1)\eta(x+2)(1-\eta(x+3))\\
\nonumber
c_\alpha(x,x-n,\eta) & = & (1-p_x)^n{\bf 1}_{\{\eta(x)>0\}}{\bf 1}_{\{\eta(x-n)<K\}}\prod_{j=1}^{n-1}{\bf 1}_{\{\eta(x-j)=K\}}\quad\mbox{ if }n\neq 3\\
\nonumber
c_\alpha(x,x-3,\eta) & = & (1-p_x)^3[1+p_x(1-p_x)]\times\\ \nonumber&&\qquad\eta(x)\eta(x-1)\eta(x-2)(1-\eta(x-3))
\end{eqnarray}
Indeed, for $n>0$ and $n\neq 3$, the only path from $x$ to $x+n$ that reaches $x+n$ in at most $k$ steps before returning to $0$ is $x,x+1,\ldots,x+n$. For $n=3$, the additional path $x,x+1,x+2,x+1,x+2,x+3$ yields the factor $p_x(1-p_x)$ in \eqref{step3_right}. For $n<0$, we change $p_x$ to $1-p_x$.\\ \\
Note that in this process a given particle does not follow a random walk in random environment (RWRE) before it finds a non full site,
but a homogeneous random walk depending (randomly) on its initial location. For instance, in a $3$-step process, a particle initially at $x\in{\mathbb Z}$ will follow the path $x,x+1,x+2,x+1$ with probability $p_x^2(1-p_{x})$.\\ \\
{\em Example 2.4.} The same random field $(p_x)_{x\in{\mathbb Z}}$ gives a different model if, at each transition, the selected particle follows a RWRE $(X_n)_{n\ge 0}$ with transition probabilities
\begin{equation}\label{rwre}{\rm I\hspace{-0.8mm}P}(X_{n+1}=x+1|X_n=x)=p_x,\quad {\rm I\hspace{-0.8mm}P}(X_{n+1}=x-1|X_n=x)=1-p_x\end{equation}
That is, we let $q_x$ be the distribution of $(X_1^x-x,\ldots,X_k^x-x)$, for $(X_n^x,\,1\le n\le k)$ a length $k$ Markov chain starting at $x$ with transition probabilities \eqref{rwre}. There, unlike in Example 2.3 above, a particle initially at $x\in{\mathbb Z}$ follows the path $x,x+1,x+2,x+1$ with probability $p_x p_{x+1}(1-p_{x+2})$. The generator of this process is also identical to that of example 2.2, with $\gamma^n_x$ and $\iota^n_x$ of the form
$$ \gamma^n_x=\gamma^n(p_y:\,x\leq y< x+n), \quad\iota^n_x=\iota^n(p_y:x-n<y\leq x) $$
for some polynomial functions $\gamma^n,\iota^n:[0,1]^n\to[0,+\infty)$, where $n\in\{1,\ldots,k\}$.
\subsection{ $K$-exclusion process with speed change and traffic flow model}
Let $\mathcal K:=\{-k,\ldots,k\}\setminus\{0\}$, and $\alpha=((\upsilon(x),\beta^1_x):x\in{\mathbb Z})$ be an ergodic $[0,+\infty)^{2k}\times(0,+\infty)$-valued field, where $\upsilon(x)=(\upsilon_z(x):\, z\in{\mathcal K})$.
We define the following dynamics. Set
\begin{eqnarray*} \Theta(x,\eta) & := & \{y\in{\mathbb Z}:\,y-x\in\mathcal K,\,\eta(y)<K\}\\
Z(\alpha,x,\eta) & := & \sum_{z\in\Theta(x,\eta)}\upsilon_{z-x}(x)
\end{eqnarray*}
In configuration $\eta$, if $Z(\alpha,x,\eta)>0$, a particle at $x$ picks a site $y$ at random in $\Theta(x,\eta)$ with probability
$Z(\alpha,x,\eta)^{-1}\upsilon_{y-x}(x)$, and jumps to this site at rate $\beta^1_x$. If $Z(\alpha,x,\eta)=0$, nothing happens. For instance, if $\upsilon_z(x)\equiv 1$, the particle uniformly chooses a site with strictly less than $K$ particles. The corresponding generator is given by \eqref{generic_form}, with
\[
c_\alpha(x,y,\eta)={\bf 1}_{\{\eta(x)>0\}}{\bf 1}_{\{Z(\alpha,x,\eta)>0\}}{\bf 1}_{\Theta(x,\eta)}(y) Z(\alpha,x,\eta)^{-1}\upsilon_{y-x}(x) \]
Hence, the microscopic flux \eqref{other_flux} writes
$$ \widetilde{\jmath}(\alpha,\eta)=\beta^1_0{\bf 1}_{\{\eta(0)>0\}}Z(\alpha,0,\eta)^{-1}\sum_{z\in\mathcal K}z\upsilon_z(0){\bf 1}_{\{\eta(z)<K\}} $$
This process can be compared with a bond-disordered $K$- exclusion process in which a particle at $x$ jumps to $y$ with rate $\alpha(x,y)=\upsilon_{y-x}(x)$. The difference is that in the latter, the particle could pick a location occupied by $K$ particles, in which case the jump is suppressed. In the former, the particle first eliminates sites occupied by $K$ particles and picks a site occupied by strictly less than $K$ particles whenever there is at least one.
This results in a speed change $K$-exclusion process, that is the jump rate from $x$ to $y$ has the form $c_{x,y}(\eta){\bf 1}_{\{\eta(x)>0\}}{\bf 1}_{\{\eta(y)<K\}}$.
To illustrate this, consider a nearest-neighbor example: we take $K=1$, $k=1$, $\upsilon_1(x)=p(x)\in[0,1]$, $\upsilon_{-1}(x)=1-p(x)$. If sites
$x-1$ and $x+1$ are free, in both processes the particle at $x$ moves with rate $\beta^1_x$ to a site picked in $\{x-1,x+1\}$ with probabilities $p(x)$ and $1-p(x)$. Now assume $x+1$ is free and $x-1$ occupied. If $p(x)=0$, nothing happens in either process. If $p(x)>0$, at rate $\beta^1_x$, the particle at $x$ moves to $x+1$ in the speed change process, while in the bond-disordered process it moves to $x+1$ with probability $p(x)$ and attempts in vain to jump to $x-1$ with probability $1-p(x)$.\\ \\
Assume $K=1$, and consider the totally asymmetric case, where $\upsilon_z(x)=0$ for $z<0$. Recalling that the totally asymmetric exclusion process is a classical simplified model of single-lane traffic flow (without overtaking) where particles represent cars, the above model can be viewed as a traffic-flow model with maximum overtaking distance $k$.
This is true also for Example 2.2 in Subsection \ref{subsec:k-step}, in the totally asymmetric setting $\iota_x^i=0,1\le i\le k$. However in the latter model, an overtaking car has only one choice for its new position. \\ \\
Though it is not clear from this formulation, we can rephrase this dynamics as a
$2k$-step model, which is thus strongly attractive by Lemma \ref{attractive_kstep}. To this end we take a random field of the form $\beta_x=(\beta^1_x,\ldots,\beta^1_x)$, and define $q_x:=q(\upsilon(x))$, where $q(\upsilon_z:\,z\in{\mathcal K})$ is the distribution of a random self-avoiding path $(Z_1,\ldots,Z_{2k})$ in $\mathcal K$ such that
\begin{eqnarray}\label{self_avoiding} \mathbf{P}(Z_1=y)&=&\frac{\upsilon_y}{\sum_{z\in\mathcal K}
\upsilon_z}\\\label{self_avoiding2} {\mathbf P}\left(Z_i=y|Z_1,\ldots,Z_{i-1}\right)&=&\frac{\upsilon_{y}} {\sum_{z\in\mathcal K\backslash\{Z_1,\ldots,Z_{i-1}\}} \upsilon_{z}}\quad\mbox{ for } 2\le i\le 2k
\end{eqnarray}
For this model, assumption \eqref{summability_kstep} is always satisfied, while \eqref{irreducibility_kstep} reduces to the existence of a constant $c>0$ and a probability distribution $p(.)$ on ${\mathbb Z}$ satisfying assumption \textit{(A1)}, such that
$$ \inf_{x\in{\mathbb Z}}\upsilon_.(x)\geq c\,p(.)$$
The link between the two models comes from
\begin{lemma}
\label{small_computation} Assume $(Z_1,\ldots,Z_{2k})\sim q(\upsilon_z:\,z\in{\mathcal K})$. Let $\Theta$ be a nonempty subset of $\{z\in{\mathcal K}:\upsilon_z\not=0\}$, $\tau:=\inf\{i\in\{1,\ldots,2k\}:\,Z_i\in\Theta\}$, and $Y=Z_\tau$. Then
$$ {\mathbf P}\left(Y=y\right)={\bf 1}_{\Theta}(y)\frac{\upsilon_{y}}{
\sum_{y'\in\Theta}\upsilon_{y'}} $$
\end{lemma}
\begin{proof}{lemma}{small_computation}
For all $t\geq 2$, let $\Theta_{t-1}$ be the set of self-avoiding paths $(z_1,\ldots,z_{t-1})$ of size $t-1$ on ${\mathcal K}\setminus\Theta$. For $y\in\Theta$, by \eqref{self_avoiding}--\eqref{self_avoiding2},
\begin{eqnarray*}
{\mathbf P}(Y=y) & = & \sum_{t=1}^{2k}{\mathbf P}(Z_t=y,\tau=t)\\
&=& \mathbf{P}(Z_1=y)+\\
\sum_{t=2}^{2k}\sum_{(z_1,\ldots,z_{t-1})\in \Theta_{t-1}} &&{\mathbf P}(Z_1=z_1,\ldots,Z_{t-1}=z_{t-1})
\frac{\upsilon_y}{
\sum_{z\in{\mathcal K}\backslash\{z_1,\ldots,z_{t-1}\}}\upsilon_z }\\
& = & C\upsilon_y
\end{eqnarray*}
where $C$ is independent of $y\in\Theta$, whence the result.
\end{proof}
\mbox{}\\ \\
\noindent {\bf Acknowledgments:} K.R. was supported by NSF grant DMS 0104278.
We thank BCM at TIMC - IMAG, Universit\'{e}s de Rouen, Paris Descartes and Clermont 2, and SUNY College at New Paltz, for hospitality.
\end{document} | arXiv |
Quantum supremacy using a programmable superconducting processor
Many-body Hilbert space scarring on a superconducting processor
Pengfei Zhang, Hang Dong, … Ying-Cheng Lai
Error mitigation extends the computational reach of a noisy quantum processor
Abhinav Kandala, Kristan Temme, … Jay M. Gambetta
Extra levels give extra functionality
Zhang Jiang
A look at the full stack
Gaia Donati
A programmable two-qubit solid-state quantum processor under ambient conditions
Yang Wu, Ya Wang, … Jiangfeng Du
Saving superconducting quantum processors from decay and correlated errors generated by gamma and cosmic rays
John M. Martinis
Leaps of quantum phase
Leonid Glazman
Realizing quantum convolutional neural networks on a superconducting quantum processor to recognize quantum phases
Johannes Herrmann, Sergi Masot Llima, … Christopher Eichler
Coherent simulation with thousands of qubits
David Bernal Neira
Frank Arute1,
Kunal Arya1,
Ryan Babbush1,
Dave Bacon1,
Joseph C. Bardin1,2,
Rami Barends1,
Rupak Biswas3,
Sergio Boixo1,
Fernando G. S. L. Brandao1,4,
David A. Buell1,
Brian Burkett1,
Yu Chen1,
Zijun Chen1,
Ben Chiaro5,
Roberto Collins1,
William Courtney1,
Andrew Dunsworth1,
Edward Farhi1,
Brooks Foxen1,5,
Austin Fowler1,
Craig Gidney1,
Marissa Giustina1,
Rob Graff1,
Keith Guerin1,
Steve Habegger1,
Matthew P. Harrigan1,
Michael J. Hartmann1,6,
Alan Ho1,
Markus Hoffmann1,
Trent Huang1,
Travis S. Humble7,
Sergei V. Isakov1,
Evan Jeffrey1,
Zhang Jiang1,
Dvir Kafri1,
Kostyantyn Kechedzhi1,
Julian Kelly1,
Paul V. Klimov1,
Sergey Knysh1,
Alexander Korotkov1,8,
Fedor Kostritsa1,
David Landhuis1,
Mike Lindmark1,
Erik Lucero1,
Dmitry Lyakh9,
Salvatore Mandrà3,10,
Jarrod R. McClean1,
Matthew McEwen5,
Anthony Megrant1,
Xiao Mi1,
Kristel Michielsen11,12,
Masoud Mohseni1,
Josh Mutus1,
Ofer Naaman1,
Matthew Neeley1,
Charles Neill1,
Murphy Yuezhen Niu1,
Eric Ostby1,
Andre Petukhov1,
John C. Platt1,
Chris Quintana1,
Eleanor G. Rieffel3,
Pedram Roushan1,
Nicholas C. Rubin1,
Daniel Sank1,
Kevin J. Satzinger1,
Vadim Smelyanskiy1,
Kevin J. Sung1,13,
Matthew D. Trevithick1,
Amit Vainsencher1,
Benjamin Villalonga1,14,
Theodore White1,
Z. Jamie Yao1,
Ping Yeh1,
Adam Zalcman1,
Hartmut Neven1 &
John M. Martinis1,5
Nature volume 574, pages 505–510 (2019)Cite this article
956k Accesses
6192 Altmetric
The promise of quantum computers is that certain computational tasks might be executed exponentially faster on a quantum processor than on a classical processor1. A fundamental challenge is to build a high-fidelity processor capable of running quantum algorithms in an exponentially large computational space. Here we report the use of a processor with programmable superconducting qubits2,3,4,5,6,7 to create quantum states on 53 qubits, corresponding to a computational state-space of dimension 253 (about 1016). Measurements from repeated experiments sample the resulting probability distribution, which we verify using classical simulations. Our Sycamore processor takes about 200 seconds to sample one instance of a quantum circuit a million times—our benchmarks currently indicate that the equivalent task for a state-of-the-art classical supercomputer would take approximately 10,000 years. This dramatic increase in speed compared to all known classical algorithms is an experimental realization of quantum supremacy8,9,10,11,12,13,14 for this specific computational task, heralding a much-anticipated computing paradigm.
You have full access to this article via your institution.
In the early 1980s, Richard Feynman proposed that a quantum computer would be an effective tool with which to solve problems in physics and chemistry, given that it is exponentially costly to simulate large quantum systems with classical computers1. Realizing Feynman's vision poses substantial experimental and theoretical challenges. First, can a quantum system be engineered to perform a computation in a large enough computational (Hilbert) space and with a low enough error rate to provide a quantum speedup? Second, can we formulate a problem that is hard for a classical computer but easy for a quantum computer? By computing such a benchmark task on our superconducting qubit processor, we tackle both questions. Our experiment achieves quantum supremacy, a milestone on the path to full-scale quantum computing8,9,10,11,12,13,14.
In reaching this milestone, we show that quantum speedup is achievable in a real-world system and is not precluded by any hidden physical laws. Quantum supremacy also heralds the era of noisy intermediate-scale quantum (NISQ) technologies15. The benchmark task we demonstrate has an immediate application in generating certifiable random numbers (S. Aaronson, manuscript in preparation); other initial uses for this new computational capability may include optimization16,17, machine learning18,19,20,21, materials science and chemistry22,23,24. However, realizing the full promise of quantum computing (using Shor's algorithm for factoring, for example) still requires technical leaps to engineer fault-tolerant logical qubits25,26,27,28,29.
To achieve quantum supremacy, we made a number of technical advances which also pave the way towards error correction. We developed fast, high-fidelity gates that can be executed simultaneously across a two-dimensional qubit array. We calibrated and benchmarked the processor at both the component and system level using a powerful new tool: cross-entropy benchmarking11. Finally, we used component-level fidelities to accurately predict the performance of the whole system, further showing that quantum information behaves as expected when scaling to large systems.
A suitable computational task
To demonstrate quantum supremacy, we compare our quantum processor against state-of-the-art classical computers in the task of sampling the output of a pseudo-random quantum circuit11,13,14. Random circuits are a suitable choice for benchmarking because they do not possess structure and therefore allow for limited guarantees of computational hardness10,11,12. We design the circuits to entangle a set of quantum bits (qubits) by repeated application of single-qubit and two-qubit logical operations. Sampling the quantum circuit's output produces a set of bitstrings, for example {0000101, 1011100, …}. Owing to quantum interference, the probability distribution of the bitstrings resembles a speckled intensity pattern produced by light interference in laser scatter, such that some bitstrings are much more likely to occur than others. Classically computing this probability distribution becomes exponentially more difficult as the number of qubits (width) and number of gate cycles (depth) grow.
We verify that the quantum processor is working properly using a method called cross-entropy benchmarking11,12,14, which compares how often each bitstring is observed experimentally with its corresponding ideal probability computed via simulation on a classical computer. For a given circuit, we collect the measured bitstrings {xi} and compute the linear cross-entropy benchmarking fidelity11,13,14 (see also Supplementary Information), which is the mean of the simulated probabilities of the bitstrings we measured:
$${ {\mathcal F} }_{{\rm{XEB}}}={2}^{n}{\langle P({x}_{i})\rangle }_{i}-1$$
where n is the number of qubits, P(xi) is the probability of bitstring xi computed for the ideal quantum circuit, and the average is over the observed bitstrings. Intuitively, \({ {\mathcal F} }_{{\rm{XEB}}}\) is correlated with how often we sample high-probability bitstrings. When there are no errors in the quantum circuit, the distribution of probabilities is exponential (see Supplementary Information), and sampling from this distribution will produce \({{\mathscr{F}}}_{{\rm{X}}{\rm{E}}{\rm{B}}}=1\). On the other hand, sampling from the uniform distribution will give ⟨P(xi)⟩i = 1/2n and produce \({{\mathscr{F}}}_{{\rm{X}}{\rm{E}}{\rm{B}}}=0\). Values of \({ {\mathcal F} }_{{\rm{XEB}}}\) between 0 and 1 correspond to the probability that no error has occurred while running the circuit. The probabilities P(xi) must be obtained from classically simulating the quantum circuit, and thus computing \({ {\mathcal F} }_{{\rm{XEB}}}\) is intractable in the regime of quantum supremacy. However, with certain circuit simplifications, we can obtain quantitative fidelity estimates of a fully operating processor running wide and deep quantum circuits.
Our goal is to achieve a high enough \({ {\mathcal F} }_{{\rm{XEB}}}\) for a circuit with sufficient width and depth such that the classical computing cost is prohibitively large. This is a difficult task because our logic gates are imperfect and the quantum states we intend to create are sensitive to errors. A single bit or phase flip over the course of the algorithm will completely shuffle the speckle pattern and result in close to zero fidelity11 (see also Supplementary Information). Therefore, in order to claim quantum supremacy we need a quantum processor that executes the program with sufficiently low error rates.
Building a high-fidelity processor
We designed a quantum processor named 'Sycamore' which consists of a two-dimensional array of 54 transmon qubits, where each qubit is tunably coupled to four nearest neighbours, in a rectangular lattice. The connectivity was chosen to be forward-compatible with error correction using the surface code26. A key systems engineering advance of this device is achieving high-fidelity single- and two-qubit operations, not just in isolation but also while performing a realistic computation with simultaneous gate operations on many qubits. We discuss the highlights below; see also the Supplementary Information.
In a superconducting circuit, conduction electrons condense into a macroscopic quantum state, such that currents and voltages behave quantum mechanically2,30. Our processor uses transmon qubits6, which can be thought of as nonlinear superconducting resonators at 5–7 GHz. The qubit is encoded as the two lowest quantum eigenstates of the resonant circuit. Each transmon has two controls: a microwave drive to excite the qubit, and a magnetic flux control to tune the frequency. Each qubit is connected to a linear resonator used to read out the qubit state5. As shown in Fig. 1, each qubit is also connected to its neighbouring qubits using a new adjustable coupler31,32. Our coupler design allows us to quickly tune the qubit–qubit coupling from completely off to 40 MHz. One qubit did not function properly, so the device uses 53 qubits and 86 couplers.
Fig. 1: The Sycamore processor.
a, Layout of processor, showing a rectangular array of 54 qubits (grey), each connected to its four nearest neighbours with couplers (blue). The inoperable qubit is outlined. b, Photograph of the Sycamore chip.
The processor is fabricated using aluminium for metallization and Josephson junctions, and indium for bump-bonds between two silicon wafers. The chip is wire-bonded to a superconducting circuit board and cooled to below 20 mK in a dilution refrigerator to reduce ambient thermal energy to well below the qubit energy. The processor is connected through filters and attenuators to room-temperature electronics, which synthesize the control signals. The state of all qubits can be read simultaneously by using a frequency-multiplexing technique33,34. We use two stages of cryogenic amplifiers to boost the signal, which is digitized (8 bits at 1 GHz) and demultiplexed digitally at room temperature. In total, we orchestrate 277 digital-to-analog converters (14 bits at 1 GHz) for complete control of the quantum processor.
We execute single-qubit gates by driving 25-ns microwave pulses resonant with the qubit frequency while the qubit–qubit coupling is turned off. The pulses are shaped to minimize transitions to higher transmon states35. Gate performance varies strongly with frequency owing to two-level-system defects36,37, stray microwave modes, coupling to control lines and the readout resonator, residual stray coupling between qubits, flux noise and pulse distortions. We therefore optimize the single-qubit operation frequencies to mitigate these error mechanisms.
We benchmark single-qubit gate performance by using the cross-entropy benchmarking protocol described above, reduced to the single-qubit level (n = 1), to measure the probability of an error occurring during a single-qubit gate. On each qubit, we apply a variable number m of randomly selected gates and measure \({ {\mathcal F} }_{{\rm{XEB}}}\) averaged over many sequences; as m increases, errors accumulate and average \({ {\mathcal F} }_{{\rm{XEB}}}\) decays. We model this decay by [1 − e1/(1 − 1/D2)]m where e1 is the Pauli error probability. The state (Hilbert) space dimension term, D = 2n, which equals 2 for this case, corrects for the depolarizing model where states with errors partially overlap with the ideal state. This procedure is similar to the more typical technique of randomized benchmarking27,38,39, but supports non-Clifford-gate sets40 and can separate out decoherence error from coherent control error. We then repeat the experiment with all qubits executing single-qubit gates simultaneously (Fig. 2), which shows only a small increase in the error probabilities, demonstrating that our device has low microwave crosstalk.
Fig. 2: System-wide Pauli and measurement errors.
a, Integrated histogram (empirical cumulative distribution function, ECDF) of Pauli errors (black, green, blue) and readout errors (orange), measured on qubits in isolation (dotted lines) and when operating all qubits simultaneously (solid). The median of each distribution occurs at 0.50 on the vertical axis. Average (mean) values are shown below. b, Heat map showing single- and two-qubit Pauli errors e1 (crosses) and e2 (bars) positioned in the layout of the processor. Values are shown for all qubits operating simultaneously.
We perform two-qubit iSWAP-like entangling gates by bringing neighbouring qubits on-resonance and turning on a 20-MHz coupling for 12 ns, which allows the qubits to swap excitations. During this time, the qubits also experience a controlled-phase (CZ) interaction, which originates from the higher levels of the transmon. The two-qubit gate frequency trajectories of each pair of qubits are optimized to mitigate the same error mechanisms considered in optimizing single-qubit operation frequencies.
To characterize and benchmark the two-qubit gates, we run two-qubit circuits with m cycles, where each cycle contains a randomly chosen single-qubit gate on each of the two qubits followed by a fixed two-qubit gate. We learn the parameters of the two-qubit unitary (such as the amount of iSWAP and CZ interaction) by using \({ {\mathcal F} }_{{\rm{XEB}}}\) as a cost function. After this optimization, we extract the per-cycle error e2c from the decay of \({ {\mathcal F} }_{{\rm{XEB}}}\) with m, and isolate the two-qubit error e2 by subtracting the two single-qubit errors e1. We find an average e2 of 0.36%. Additionally, we repeat the same procedure while simultaneously running two-qubit circuits for the entire array. After updating the unitary parameters to account for effects such as dispersive shifts and crosstalk, we find an average e2 of 0.62%.
For the full experiment, we generate quantum circuits using the two-qubit unitaries measured for each pair during simultaneous operation, rather than a standard gate for all pairs. The typical two-qubit gate is a full iSWAP with 1/6th of a full CZ. Using individually calibrated gates in no way limits the universality of the demonstration. One can compose, for example, controlled-NOT (CNOT) gates from 1-qubit gates and two of the unique 2-qubit gates of any given pair. The implementation of high-fidelity 'textbook gates' natively, such as CZ or \(\sqrt{{\rm{iSWAP}}}\), is work in progress.
Finally, we benchmark qubit readout using standard dispersive measurement41. Measurement errors averaged over the 0 and 1 states are shown in Fig. 2a. We have also measured the error when operating all qubits simultaneously, by randomly preparing each qubit in the 0 or 1 state and then measuring all qubits for the probability of the correct result. We find that simultaneous readout incurs only a modest increase in per-qubit measurement errors.
Having found the error rates of the individual gates and readout, we can model the fidelity of a quantum circuit as the product of the probabilities of error-free operation of all gates and measurements. Our largest random quantum circuits have 53 qubits, 1,113 single-qubit gates, 430 two-qubit gates, and a measurement on each qubit, for which we predict a total fidelity of 0.2%. This fidelity should be resolvable with a few million measurements, since the uncertainty on \({ {\mathcal F} }_{{\rm{XEB}}}\) is \(1/\sqrt{{N}_{{\rm{s}}}}\), where Ns is the number of samples. Our model assumes that entangling larger and larger systems does not introduce additional error sources beyond the errors we measure at the single- and two-qubit level. In the next section we will see how well this hypothesis holds up.
Fidelity estimation in the supremacy regime
The gate sequence for our pseudo-random quantum circuit generation is shown in Fig. 3. One cycle of the algorithm consists of applying single-qubit gates chosen randomly from \(\{\sqrt{X},\sqrt{Y},\sqrt{W}\}\) on all qubits, followed by two-qubit gates on pairs of qubits. The sequences of gates which form the 'supremacy circuits' are designed to minimize the circuit depth required to create a highly entangled state, which is needed for computational complexity and classical hardness.
Fig. 3: Control operations for the quantum supremacy circuits.
a, Example quantum circuit instance used in our experiment. Every cycle includes a layer each of single- and two-qubit gates. The single-qubit gates are chosen randomly from \(\{\sqrt{X},\sqrt{Y},\sqrt{W}\}\), where \(W=(X+Y)/\sqrt{2}\) and gates do not repeat sequentially. The sequence of two-qubit gates is chosen according to a tiling pattern, coupling each qubit sequentially to its four nearest-neighbour qubits. The couplers are divided into four subsets (ABCD), each of which is executed simultaneously across the entire array corresponding to shaded colours. Here we show an intractable sequence (repeat ABCDCDAB); we also use different coupler subsets along with a simplifiable sequence (repeat EFGHEFGH, not shown) that can be simulated on a classical computer. b, Waveform of control signals for single- and two-qubit gates.
Although we cannot compute \({ {\mathcal F} }_{{\rm{XEB}}}\) in the supremacy regime, we can estimate it using three variations to reduce the complexity of the circuits. In 'patch circuits', we remove a slice of two-qubit gates (a small fraction of the total number of two-qubit gates), splitting the circuit into two spatially isolated, non-interacting patches of qubits. We then compute the total fidelity as the product of the patch fidelities, each of which can be easily calculated. In 'elided circuits', we remove only a fraction of the initial two-qubit gates along the slice, allowing for entanglement between patches, which more closely mimics the full experiment while still maintaining simulation feasibility. Finally, we can also run full 'verification circuits', with the same gate counts as our supremacy circuits, but with a different pattern for the sequence of two-qubit gates, which is much easier to simulate classically (see also Supplementary Information). Comparison between these three variations allows us to track the system fidelity as we approach the supremacy regime.
We first check that the patch and elided versions of the verification circuits produce the same fidelity as the full verification circuits up to 53 qubits, as shown in Fig. 4a. For each data point, we typically collect Ns = 5 × 106 total samples over ten circuit instances, where instances differ only in the choices of single-qubit gates in each cycle. We also show predicted \({ {\mathcal F} }_{{\rm{XEB}}}\) values, computed by multiplying the no-error probabilities of single- and two-qubit gates and measurement (see also Supplementary Information). The predicted, patch and elided fidelities all show good agreement with the fidelities of the corresponding full circuits, despite the vast differences in computational complexity and entanglement. This gives us confidence that elided circuits can be used to accurately estimate the fidelity of more-complex circuits.
Fig. 4: Demonstrating quantum supremacy.
a, Verification of benchmarking methods. \({ {\mathcal F} }_{{\rm{XEB}}}\) values for patch, elided and full verification circuits are calculated from measured bitstrings and the corresponding probabilities predicted by classical simulation. Here, the two-qubit gates are applied in a simplifiable tiling and sequence such that the full circuits can be simulated out to n = 53, m = 14 in a reasonable amount of time. Each data point is an average over ten distinct quantum circuit instances that differ in their single-qubit gates (for n = 39, 42 and 43 only two instances were simulated). For each n, each instance is sampled with Ns of 0.5–2.5 million. The black line shows the predicted \({ {\mathcal F} }_{{\rm{XEB}}}\) based on single- and two-qubit gate and measurement errors. The close correspondence between all four curves, despite their vast differences in complexity, justifies the use of elided circuits to estimate fidelity in the supremacy regime. b, Estimating \({ {\mathcal F} }_{{\rm{XEB}}}\) in the quantum supremacy regime. Here, the two-qubit gates are applied in a non-simplifiable tiling and sequence for which it is much harder to simulate. For the largest elided data (n = 53, m = 20, total Ns = 30 million), we find an average \({ {\mathcal F} }_{{\rm{XEB}}}\) > 0.1% with 5σ confidence, where σ includes both systematic and statistical uncertainties. The corresponding full circuit data, not simulated but archived, is expected to show similarly statistically significant fidelity. For m = 20, obtaining a million samples on the quantum processor takes 200 seconds, whereas an equal-fidelity classical sampling would take 10,000 years on a million cores, and verifying the fidelity would take millions of years.
The largest circuits for which the fidelity can still be directly verified have 53 qubits and a simplified gate arrangement. Performing random circuit sampling on these at 0.8% fidelity takes one million cores 130 seconds, corresponding to a million-fold speedup of the quantum processor relative to a single core.
We proceed now to benchmark our computationally most difficult circuits, which are simply a rearrangement of the two-qubit gates. In Fig. 4b, we show the measured \({ {\mathcal F} }_{{\rm{XEB}}}\) for 53-qubit patch and elided versions of the full supremacy circuits with increasing depth. For the largest circuit with 53 qubits and 20 cycles, we collected Ns = 30 × 106 samples over ten circuit instances, obtaining \({ {\mathcal F} }_{{\rm{XEB}}}=(2.24\pm 0.21)\times {10}^{-3}\) for the elided circuits. With 5σ confidence, we assert that the average fidelity of running these circuits on the quantum processor is greater than at least 0.1%. We expect that the full data for Fig. 4b should have similar fidelities, but since the simulation times (red numbers) take too long to check, we have archived the data (see 'Data availability' section). The data is thus in the quantum supremacy regime.
The classical computational cost
We simulate the quantum circuits used in the experiment on classical computers for two purposes: (1) verifying our quantum processor and benchmarking methods by computing \({ {\mathcal F} }_{{\rm{XEB}}}\) where possible using simplifiable circuits (Fig. 4a), and (2) estimating \({ {\mathcal F} }_{{\rm{XEB}}}\) as well as the classical cost of sampling our hardest circuits (Fig. 4b). Up to 43 qubits, we use a Schrödinger algorithm, which simulates the evolution of the full quantum state; the Jülich supercomputer (with 100,000 cores, 250 terabytes) runs the largest cases. Above this size, there is not enough random access memory (RAM) to store the quantum state42. For larger qubit numbers, we use a hybrid Schrödinger–Feynman algorithm43 running on Google data centres to compute the amplitudes of individual bitstrings. This algorithm breaks the circuit up into two patches of qubits and efficiently simulates each patch using a Schrödinger method, before connecting them using an approach reminiscent of the Feynman path-integral. Although it is more memory-efficient, the Schrödinger–Feynman algorithm becomes exponentially more computationally expensive with increasing circuit depth owing to the exponential growth of paths with the number of gates connecting the patches.
To estimate the classical computational cost of the supremacy circuits (grey numbers in Fig. 4b), we ran portions of the quantum circuit simulation on both the Summit supercomputer as well as on Google clusters and extrapolated to the full cost. In this extrapolation, we account for the computation cost of sampling by scaling the verification cost with \({ {\mathcal F} }_{{\rm{XEB}}}\), for example43,44, a 0.1% fidelity decreases the cost by about 1,000. On the Summit supercomputer, which is currently the most powerful in the world, we used a method inspired by Feynman path-integrals that is most efficient at low depth44,45,46,47. At m = 20 the tensors do not reasonably fit into node memory, so we can only measure runtimes up to m = 14, for which we estimate that sampling three million bitstrings with 1% fidelity would require a year.
On Google Cloud servers, we estimate that performing the same task for m = 20 with 0.1% fidelity using the Schrödinger–Feynman algorithm would cost 50 trillion core-hours and consume one petawatt hour of energy. To put this in perspective, it took 600 seconds to sample the circuit on the quantum processor three million times, where sampling time is limited by control hardware communications; in fact, the net quantum processor time is only about 30 seconds. The bitstring samples from all circuits have been archived online (see 'Data availability' section) to encourage development and testing of more advanced verification algorithms.
One may wonder to what extent algorithmic innovation can enhance classical simulations. Our assumption, based on insights from complexity theory11,12,13, is that the cost of this algorithmic task is exponential in circuit size. Indeed, simulation methods have improved steadily over the past few years42,43,44,45,46,47,48,49,50. We expect that lower simulation costs than reported here will eventually be achieved, but we also expect that they will be consistently outpaced by hardware improvements on larger quantum processors.
Verifying the digital error model
A key assumption underlying the theory of quantum error correction is that quantum state errors may be considered digitized and localized38,51. Under such a digital model, all errors in the evolving quantum state may be characterized by a set of localized Pauli errors (bit-flips or phase-flips) interspersed into the circuit. Since continuous amplitudes are fundamental to quantum mechanics, it needs to be tested whether errors in a quantum system could be treated as discrete and probabilistic. Indeed, our experimental observations support the validity of this model for our processor. Our system fidelity is well predicted by a simple model in which the individually characterized fidelities of each gate are multiplied together (Fig. 4).
To be successfully described by a digitized error model, a system should be low in correlated errors. We achieve this in our experiment by choosing circuits that randomize and decorrelate errors, by optimizing control to minimize systematic errors and leakage, and by designing gates that operate much faster than correlated noise sources, such as 1/f flux noise37. Demonstrating a predictive uncorrelated error model up to a Hilbert space of size 253 shows that we can build a system where quantum resources, such as entanglement, are not prohibitively fragile.
Quantum processors based on superconducting qubits can now perform computations in a Hilbert space of dimension 253 ≈ 9 × 1015, beyond the reach of the fastest classical supercomputers available today. To our knowledge, this experiment marks the first computation that can be performed only on a quantum processor. Quantum processors have thus reached the regime of quantum supremacy. We expect that their computational power will continue to grow at a double-exponential rate: the classical cost of simulating a quantum circuit increases exponentially with computational volume, and hardware improvements will probably follow a quantum-processor equivalent of Moore's law52,53, doubling this computational volume every few years. To sustain the double-exponential growth rate and to eventually offer the computational volume needed to run well known quantum algorithms, such as the Shor or Grover algorithms25,54, the engineering of quantum error correction will need to become a focus of attention.
The extended Church–Turing thesis formulated by Bernstein and Vazirani55 asserts that any 'reasonable' model of computation can be efficiently simulated by a Turing machine. Our experiment suggests that a model of computation may now be available that violates this assertion. We have performed random quantum circuit sampling in polynomial time using a physically realizable quantum processor (with sufficiently low error rates), yet no efficient method is known to exist for classical computing machinery. As a result of these developments, quantum computing is transitioning from a research topic to a technology that unlocks new computational capabilities. We are only one creative algorithm away from valuable near-term applications.
The datasets generated and analysed for this study are available at our public Dryad repository (https://doi.org/10.5061/dryad.k6t1rj8).
Feynman, R. P. Simulating physics with computers. Int. J. Theor. Phys. 21, 467–488 (1982).
Article MathSciNet Google Scholar
Devoret, M. H., Martinis, J. M. & Clarke, J. Measurements of macroscopic quantum tunneling out of the zero-voltage state of a current-biased Josephson junction. Phys. Rev. Lett. 55, 1908 (1985).
Article CAS ADS Google Scholar
Nakamura, Y., Chen, C. D. & Tsai, J. S. Spectroscopy of energy-level splitting between two macroscopic quantum states of charge coherently superposed by Josephson coupling. Phys. Rev. Lett. 79, 2328 (1997).
Mooij, J. et al. Josephson persistent-current qubit. Science 285, 1036–1039 (1999).
Wallraff, A. et al. Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. Nature 431, 162–167 (2004).
Koch, J. et al. Charge-insensitive qubit design derived from the Cooper pair box. Phys. Rev. A 76, 042319 (2007).
Article ADS Google Scholar
You, J. Q. & Nori, F. Atomic physics and quantum optics using superconducting circuits. Nature 474, 589–597 (2011).
Preskill, J. Quantum computing and the entanglement frontier. Rapporteur Talk at the 25th Solvay Conference on Physics, Brussels https://doi.org/10.1142/8674 (World Scientific, 2012).
Aaronson, S. & Arkhipov, A. The computational complexity of linear optics. In Proc. 43rd Ann. Symp. on Theory of Computing https://doi.org/10.1145/1993636.1993682 (ACM, 2011).
Bremner, M. J., Montanaro, A. & Shepherd, D. J. Average-case complexity versus approximate simulation of commuting quantum computations. Phys. Rev. Lett. 117, 080501 (2016).
Article ADS MathSciNet Google Scholar
Boixo, S. et al. Characterizing quantum supremacy in near-term devices. Nat. Phys. 14, 595 (2018).
Bouland, A., Fefferman, B., Nirkhe, C. & Vazirani, U. On the complexity and verification of quantum random circuit sampling. Nat. Phys. 15, 159 (2019).
Aaronson, S. & Chen, L. Complexity-theoretic foundations of quantum supremacy experiments. In 32nd Computational Complexity Conf. https://doi.org/10.4230/LIPIcs.CCC.2017.22 (Schloss Dagstuhl–Leibniz Zentrum für Informatik, 2017).
Neill, C. et al. A blueprint for demonstrating quantum supremacy with superconducting qubits. Science 360, 195–199 (2018).
Article CAS ADS MathSciNet Google Scholar
Preskill, J. Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018).
Kechedzhi, K. et al. Efficient population transfer via non-ergodic extended states in quantum spin glass. In 13th Conf. on the Theory of Quantum Computation, Communication and Cryptography http://drops.dagstuhl.de/opus/volltexte/2018/9256/pdf/LIPIcs-TQC-2018-9.pdf (Schloss Dagstuhl–Leibniz Zentrum für Informatik, 2018).
Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008).
Farhi, E. & Neven, H. Classification with quantum neural networks on near term processors. Preprint at https://arxiv.org/abs/1802.06002 (2018).
McClean, J. R., Boixo, S., Smelyanskiy, V. N., Babbush, R. & Neven, H. Barren plateaus in quantum neural network training landscapes. Nat. Commun. 9, 4812 (2018).
Cong, I., Choi, S. & Lukin, M. D. Quantum convolutional neural networks. Nat. Phys. https://doi.org/10.1038/s41567-019-0648-8 (2019).
Bravyi, S., Gosset, D. & König, R. Quantum advantage with shallow circuits. Science 362, 308–311 (2018).
Aspuru-Guzik, A., Dutoi, A. D., Love, P. J. & Head-Gordon, M. Simulated quantum computation of molecular energies. Science 309, 1704–1707 (2005).
Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5, 4213 (2014).
Hempel, C. et al. Quantum chemistry calculations on a trapped-ion quantum simulator. Phys. Rev. X 8, 031022 (2018).
Shor, P. W. Algorithms for quantum computation: discrete logarithms and factoring proceedings. In Proc. 35th Ann. Symp. on Foundations of Computer Science https://doi.org/10.1109/SFCS.1994.365700 (IEEE, 1994).
Fowler, A. G., Mariantoni, M., Martinis, J. M. & Cleland, A. N. Surface codes: towards practical large-scale quantum computation. Phys. Rev. A 86, 032324 (2012).
Barends, R. et al. Superconducting quantum circuits at the surface code threshold for fault tolerance. Nature 508, 500–503 (2014).
Córcoles, A. D. et al. Demonstration of a quantum error detection code using a square lattice of four superconducting qubits. Nat. Commun. 6, 6979 (2015).
Ofek, N. et al. Extending the lifetime of a quantum bit with error correction in superconducting circuits. Nature 536, 441 (2016).
Vool, U. & Devoret, M. Introduction to quantum electromagnetic circuits. Int. J. Circuit Theory Appl. 45, 897–934 (2017).
Chen, Y. et al. Qubit architecture with high coherence and fast tunable coupling circuits. Phys. Rev. Lett. 113, 220502 (2014).
Yan, F. et al. A tunable coupling scheme for implementing high-fidelity two-qubit gates. Phys. Rev. Appl. 10, 054062 (2018).
Schuster, D. I. et al. Resolving photon number states in a superconducting circuit. Nature 445, 515 (2007).
Jeffrey, E. et al. Fast accurate state measurement with superconducting qubits. Phys. Rev. Lett. 112, 190504 (2014).
Chen, Z. et al. Measuring and suppressing quantum state leakage in a superconducting qubit. Phys. Rev. Lett. 116, 020501 (2016).
Klimov, P. V. et al. Fluctuations of energy-relaxation times in superconducting qubits. Phys. Rev. Lett. 121, 090502 (2018).
Yan, F. et al. The flux qubit revisited to enhance coherence and reproducibility. Nat. Commun. 7, 12964 (2016).
Knill, E. et al. Randomized benchmarking of quantum gates. Phys. Rev. A 77, 012307 (2008).
Magesan, E., Gambetta, J. M. & Emerson, J. Scalable and robust randomized benchmarking of quantum processes. Phys. Rev. Lett. 106, 180504 (2011).
Cross, A. W., Magesan, E., Bishop, L. S., Smolin, J. A. & Gambetta, J. M. Scalable randomised benchmarking of non-Clifford gates. npj Quant. Inform. 2, 16012 (2016).
Wallraff, A. et al. Approaching unit visibility for control of a superconducting qubit with dispersive readout. Phys. Rev. Lett. 95, 060501 (2005).
De Raedt, H. et al. Massively parallel quantum computer simulator, eleven years later. Comput. Phys. Commun. 237, 47–61 (2019).
Markov, I. L., Fatima, A., Isakov, S. V. & Boixo, S. Quantum supremacy is both closer and farther than it appears. Preprint at https://arxiv.org/abs/1807.10749 (2018).
Villalonga, B. et al. A flexible high-performance simulator for the verification and benchmarking of quantum circuits implemented on real hardware. npj Quant. Inform. (in the press); preprint at https://arxiv.org/abs/1811.09599 (2018).
Boixo, S., Isakov, S. V., Smelyanskiy, V. N. & Neven, H. Simulation of low-depth quantum circuits as complex undirected graphical models. Preprint at https://arxiv.org/abs/1712.05384 (2017).
Chen, J., Zhang, F., Huang, C., Newman, M. & Shi, Y. Classical simulation of intermediate-size quantum circuits. Preprint at https://arxiv.org/abs/1805.01450 (2018).
Villalonga, B. et al. Establishing the quantum supremacy frontier with a 281 pflop/s simulation. Preprint at https://arxiv.org/abs/1905.00444 (2019).
Pednault, E. et al. Breaking the 49-qubit barrier in the simulation of quantum circuits. Preprint at https://arxiv.org/abs/1710.05867 (2017).
Chen, Z. Y. et al. 64-qubit quantum circuit simulation. Sci. Bull. 63, 964–971 (2018).
Chen, M.-C. et al. Quantum-teleportation-inspired algorithm for sampling large random quantum circuits. Preprint at https://arxiv.org/abs/1901.05003 (2019).
Shor, P. W. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, R2493–R2496 (1995).
Devoret, M. H. & Schoelkopf, R. J. Superconducting circuits for quantum information: an outlook. Science 339, 1169–1174 (2013).
Mohseni, M. et al. Commercialize quantum technologies in five years. Nature 543, 171 (2017).
Grover, L. K. Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 79, 325 (1997).
Bernstein, E. & Vazirani, U. Quantum complexity theory. In Proc. 25th Ann. Symp. on Theory of Computing https://doi.org/10.1145/167088.167097 (ACM, 1993).
We are grateful to E. Schmidt, S. Brin, S. Pichai, J. Dean, J. Yagnik and J. Giannandrea for their executive sponsorship of the Google AI Quantum team, and for their continued engagement and support. We thank P. Norvig, J. Yagnik, U. Hölzle and S. Pichai for advice on the manuscript. We acknowledge K. Kissel, J. Raso, D. L. Yonge-Mallo, O. Martin and N. Sridhar for their help with simulations. We thank G. Bortoli and L. Laws for keeping our team organized. This research used resources from the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility (supported by contract DE-AC05-00OR22725). A portion of this work was performed in the UCSB Nanofabrication Facility, an open access laboratory. R.B., S.M., and E.G.R. appreciate support from the NASA Ames Research Center and from the Air Force Research (AFRL) Information Directorate (grant F4HBKC4162G001). T.S.H. is supported by the DOE Early Career Research Program. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of AFRL or the US government.
Google AI Quantum, Mountain View, CA, USA
Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, Sergio Boixo, Fernando G. S. L. Brandao, David A. Buell, Brian Burkett, Yu Chen, Zijun Chen, Roberto Collins, William Courtney, Andrew Dunsworth, Edward Farhi, Brooks Foxen, Austin Fowler, Craig Gidney, Marissa Giustina, Rob Graff, Keith Guerin, Steve Habegger, Matthew P. Harrigan, Michael J. Hartmann, Alan Ho, Markus Hoffmann, Trent Huang, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Paul V. Klimov, Sergey Knysh, Alexander Korotkov, Fedor Kostritsa, David Landhuis, Mike Lindmark, Erik Lucero, Jarrod R. McClean, Anthony Megrant, Xiao Mi, Masoud Mohseni, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Murphy Yuezhen Niu, Eric Ostby, Andre Petukhov, John C. Platt, Chris Quintana, Pedram Roushan, Nicholas C. Rubin, Daniel Sank, Kevin J. Satzinger, Vadim Smelyanskiy, Kevin J. Sung, Matthew D. Trevithick, Amit Vainsencher, Benjamin Villalonga, Theodore White, Z. Jamie Yao, Ping Yeh, Adam Zalcman, Hartmut Neven & John M. Martinis
Department of Electrical and Computer Engineering, University of Massachusetts Amherst, Amherst, MA, USA
Joseph C. Bardin
Quantum Artificial Intelligence Laboratory (QuAIL), NASA Ames Research Center, Moffett Field, CA, USA
Rupak Biswas, Salvatore Mandrà & Eleanor G. Rieffel
Institute for Quantum Information and Matter, Caltech, Pasadena, CA, USA
Fernando G. S. L. Brandao
Department of Physics, University of California, Santa Barbara, CA, USA
Ben Chiaro, Brooks Foxen, Matthew McEwen & John M. Martinis
Friedrich-Alexander University Erlangen-Nürnberg (FAU), Department of Physics, Erlangen, Germany
Michael J. Hartmann
Quantum Computing Institute, Oak Ridge National Laboratory, Oak Ridge, TN, USA
Travis S. Humble
Department of Electrical and Computer Engineering, University of California, Riverside, CA, USA
Alexander Korotkov
Scientific Computing, Oak Ridge Leadership Computing, Oak Ridge National Laboratory, Oak Ridge, TN, USA
Dmitry Lyakh
Stinger Ghaffarian Technologies Inc., Greenbelt, MD, USA
Salvatore Mandrà
Institute for Advanced Simulation, Jülich Supercomputing Centre, Forschungszentrum Jülich, Jülich, Germany
Kristel Michielsen
RWTH Aachen University, Aachen, Germany
Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, USA
Kevin J. Sung
Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL, USA
Benjamin Villalonga
Frank Arute
Kunal Arya
Ryan Babbush
Dave Bacon
Rami Barends
Rupak Biswas
Sergio Boixo
David A. Buell
Brian Burkett
Yu Chen
Zijun Chen
Ben Chiaro
Roberto Collins
William Courtney
Andrew Dunsworth
Edward Farhi
Brooks Foxen
Austin Fowler
Craig Gidney
Marissa Giustina
Rob Graff
Keith Guerin
Steve Habegger
Matthew P. Harrigan
Alan Ho
Markus Hoffmann
Trent Huang
Sergei V. Isakov
Evan Jeffrey
Dvir Kafri
Kostyantyn Kechedzhi
Julian Kelly
Paul V. Klimov
Sergey Knysh
Fedor Kostritsa
David Landhuis
Mike Lindmark
Erik Lucero
Jarrod R. McClean
Matthew McEwen
Anthony Megrant
Xiao Mi
Masoud Mohseni
Josh Mutus
Ofer Naaman
Matthew Neeley
Charles Neill
Murphy Yuezhen Niu
Eric Ostby
Andre Petukhov
John C. Platt
Chris Quintana
Eleanor G. Rieffel
Pedram Roushan
Nicholas C. Rubin
Daniel Sank
Kevin J. Satzinger
Vadim Smelyanskiy
Matthew D. Trevithick
Amit Vainsencher
Theodore White
Z. Jamie Yao
Ping Yeh
Adam Zalcman
Hartmut Neven
The Google AI Quantum team conceived the experiment. The applications and algorithms team provided the theoretical foundation and the specifics of the algorithm. The hardware team carried out the experiment and collected the data. The data analysis was done jointly with outside collaborators. All authors wrote and revised the manuscript and the Supplementary Information.
Correspondence to John M. Martinis.
Peer review information Nature thanks Scott Aaronson, Keisuke Fujii and William Oliver for their contribution to the peer review of this work.
This file contains Supplementary Information I–XI, which contains supplementary figures S1–S44 and Supplementary Tables I–X
Arute, F., Arya, K., Babbush, R. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019). https://doi.org/10.1038/s41586-019-1666-5
Issue Date: 24 October 2019
Scalable and robust quantum computing on qubit arrays with fixed coupling
N. H. Le
M. Cykiert
E. Ginossar
npj Quantum Information (2023)
Preparing random states and benchmarking with many-body quantum chaos
Joonhee Choi
Adam L. Shaw
Manuel Endres
Non-linear Boson Sampling
Nicolò Spagnolo
Daniel J. Brod
Fabio Sciarrino
Enhancing the coherence of superconducting quantum bits with electric fields
Jürgen Lisenfeld
Alexander Bilmes
Alexey V. Ustinov
Design of highly nonlinear confusion component based on entangled points of quantum spin states
Hafiz Muhammad Waseem
Seong Oun Hwang
Scientific Reports (2023)
Quantum computing takes flight
William D. Oliver
Nature News & Views 23 Oct 2019
Quantum supremacy: A three minute guide
Elizabeth Gibney
Nature Nature Video 05 Nov 2019
History of Nature
Nature (Nature) ISSN 1476-4687 (online) ISSN 0028-0836 (print) | CommonCrawl |
\begin{definition}[Definition:Conjunctive Normal Form]
A propositional formula $P$ is in '''conjunctive normal form''' {{iff}} it consists of a conjunction of:
:$(1):\quad$ disjunctions of literals
and/or:
:$(2):\quad$ literals.
\end{definition} | ProofWiki |
Evaluating medical education regulation changes in Brazil: workforce impact
Alexandre Medeiros Figueiredo ORCID: orcid.org/0000-0003-1433-088X1,2,
Danette Waller McKinley3,
Adriano Massuda4 &
George Dantas Azevedo5
Shortages and inequitable distribution of physicians is an obstacle to move towards Universal Health Coverage, especially in low-income and middle-income countries. In Brazil, expansion of medical school enrollment, curricula changes and recruitment programs were established to increase the number of physicians in underserved areas. This study seeks to analyze the impact of these measures in reduce inequities in access to medical education and physicians' distribution.
This is an observational study that analyzes changes in the number of undergraduate medical places and number of physicians per inhabitants in different areas in Brazil between the years 2010 and 2018. Data regarding the number of undergraduate medical places, number and the practice location of physicians were obtained in public databases. Municipalities with less than 20,000 inhabitants were considered underserved areas. Data regarding access to antenatal visits were analyzed as a proxy for impact in access to healthcare.
From 2010 to 2018, 19,519 new medical undergraduate places were created which represents an increase of 120.2%. The increase in the number of physicians engaged in the workforce throughout the period was 113,702 physicians, 74,771 of these physicians in the Unified Health System. The greatest increase in the physicians per 1000 inhabitants ratio in the municipalities with the smallest population, the lowest Gross Domestic Product per capita and in those located in the states with the lowest concentration of physicians occurred in the 2013–2015 period. Increase in physician supply improved access to antenatal care.
There was an expansion in the number of undergraduate medical places and medical workforce in all groups of municipalities assessed in Brazil. Medical undergraduate places expansion in the federal public schools was more efficient to reduce regional inequities in access to medical education than private sector expansion. The recruitment component of More Doctors for Brazil Program demonstrated effectiveness to increase the number of physicians in underserved areas. Our results indicate the importance of public policies to face inequities in access to medical education and physician shortages and the necessity of continuous assessment during the period of implementation, especially in the context of political and economic changes.
The availability of adequate health workforce to meet health needs is an important challenge for health systems to move towards Universal Health Coverage (UHC) [1,2,3]. Shortages and inequitable distribution of physicians is a global phenomenal that mainly affects low-income and middle-income countries [1,2,3,4]. In most countries, the health workforce is concentrated in larger and developed cities. Labor market factors such as better employment opportunities, practicing conditions and training opportunities make these regions more attractive [2, 4]. Even countries with well-established health systems have difficulty in attracting and retaining professionals in rural and underserved areas [5, 6]. This complex problem is influenced by economic, social and cultural factors [4]. Consistent and long-term public policies are necessary to promote the supply of physicians and their retention in underserved areas.
The main strategy to reach an adequate number of physicians in order to meet health needs has been the expansion of medical school enrollment [2, 4]. While essential, expansion alone is inadequate to guarantee the attraction and retention of physicians in underserved areas [4]. Interventions including professional regulation, financial incentives, and support activities for the education and work sectors are described in the literature as additional strategies to increase the number of physicians in underserved areas [7, 8]. Educational policies have proven to be effective and these policies include specific admission programs for students from underserved areas, expansion of undergraduate and postgraduate (residency) programs in regions with a low ratio of physicians per inhabitant, development of learning experiences in rural areas and curricula that prioritize primary health care (PHC) competencies [7, 8].
In Brazil, the Unified Health System (SUS) was established by the 1988 Constitution based on the principles of health as a citizen's right and the state's duty. The SUS aims to provide comprehensive health care through decentralized management and provision of health services [9]. Since the establishment of SUS inequalities in the number and regional distribution of physicians persisted as one of the main obstacles to universal and equitable access to healthcare [9, 10]. The areas with shortage of physicians are concentrated in rural and remote areas, mainly in the North and Northeast regions, which are the regions with the lowest economic development (Fig. 1) [9].
Gross domestic product per state, Brazil, 2010
The chronic underfunding of SUS and the high percentage of private health spending in Brazil aggravated the inequality in the distribution of these professionals [11]. Several regulatory mechanisms and programs to increase the number of physicians and improve their distribution have been implemented by the Brazilian Federal Government [12, 13]. Proposals for recruiting physicians to practice in PHC in underserved areas, such as the Program for the Interiorization of Health Work (Programa de Interiorização do Trabalho em Saúde, PITS-2001-2003) and the Program for Valuing Primary Care Professionals (Programa de Valorização dos Profissionais da Atenção Básica, PROVAB-2011) were also implemented nationally [12].
In parallel, educational policies were implemented. In 2001, publication of the National Curricular Guidelines was a milestone for emphasizing general training and expansion of practice scenarios to prioritize PHC settings [14]. In addition, there was an expansion of access to higher education in Brazil through the Program of Support for the Restructuration and Expansion of Federal Universities (Programa de Apoio a Planos de Reestruturação e Expansão das Universidades Federais, REUNI), created in 2007, directing the expansion of medical schools to municipalities located in regions with a lower physician ratio per inhabitant [2, 13]. We document the timeline of policies in Fig. 2.
Timeline of policies to promote the supply of physicians in Brazil
These policies, however, were not sufficient to address physician shortages especially in PHC [9, 13]. In 2013, the More Doctors for Brazil Program (Programa Mais Médicos para o Brasil, PMMB) was created to guarantee a medical workforce adequate to the health needs of the Brazilian population, reduce regional inequalities and increase the number of physicians practicing in PHC [15]. This program was defined as the main strategy to reach the proportion of 2.7 physicians per 1000 inhabitants by 2026 [13]. To achieve this result, the PMMB employs the following strategies: recruiting physicians to practice in PHC, expanding the number of undergraduate and medical residency enrollments and promoting curricula changes [13, 15].
Both Brazilian-educated and foreign-educated physicians are eligible under recruitment regulations [15]. The PMMB offers financial incentives with medical contracts of up to 6 years, grants for housing and food and a support program developed by public education institutions [9, 13, 15]. Professional regulation was modified to allow physicians with foreign undergraduate diplomas to be able to practice medicine in the PMMB without need to validate their diplomas via exams [15]. This measure allowed the incorporation of approximately 12,000 foreign physicians in PHC, made it possible to substantially increase the coverage of care in the country, especially in underserved areas [9]. The engagement of foreign physicians in the PMMB began in 2013 and increased until the end of 2015 [9]. Then, there was a gradual decline in foreign physician engagement that was highlighted in December 2018 when cooperation with the Cuban government ended [16].
The medical education component of the PMMB is the mainly strategy to increase physician's supply in Brazil [17]. PMMB set the goal of creating 11,500 new undergraduate places through 2017 [17]. The PMMB established a new regulatory framework for opening public and private medical schools, prioritizing the allocation of these schools in areas of physician shortages [13, 15]. The opening of new private schools was regulated by a new model in which the federal government defines a municipality in an underserved area for the establishment of the medical school, then a public call is made to choose the private institution responsible to implement the program [13, 15].
Additionaly, a new undergraduate medical curriculum was developed to strengthen PHC training [13, 18]. This included changes in the process of evaluating and accreditation of medical schools by adapting the evaluation tool for undergraduate education in the National Higher Education Assessment System (Sistema Nacional de Avaliação da Educação Superior; Sinaes). The new medical curriculum guidelines and objectives of the PMMB were used to implement these changes [13, 19]. This regulatory framework was expanded to award greater value to institutions integrated with the local health system, those with greater capacity to offer medical residency programs in priority specialties and those with greater social accountability and adaptation to local social health needs [13, 19]. Finally, the PMMB established a new paradigm for regulation of residency in Brazil with inclusion of an initial year of training in Family and Community Medicine for residency programs of most medical specialties [13, 15]. However, the latter strategy was never implemented.
This study analyzes whether changes in medical education regulation and physician recruitment and retention between 2010 and 2018 met the objectives of increasing the number of medical undergraduate places, number of physicians and reduction in inequalities in the access to medical education and in physician's distribution.
This is an observational study that analyzes changes in the number of medical undergraduate places and the increase in the number of physicians in Brazil between the years 2010 and 2018. In order to assess the changes in medical undergraduate places, the absolute number of places and the ratio of undergraduate places per 10,000 inhabitants were analyzed. The analysis of the evolution in the number of physicians in Brazil was carried out based on the total number of physicians and the number of physicians practicing in the SUS, aiming to understand the impact of the increase in undergraduate places and recruitment programs on the public and private workforce. In both cases, the ratio of physician per 1000 inhabitants was calculated considering the total number of physicians and the number of physicians practicing in SUS.
Data regarding the number of medical undergraduate places were obtained from the database on undergraduate education of the Higher Education Census of the National Institute of Studies and Research Anísio Teixeira (Instituto Nacional de Estudos e Pesquisas Anísio Teixeira, INEP) [20]. Data concerning the number and the practice location of physicians were obtained in January of 2020 from the National Register of Healthcare Establishments (Cadastro Nacional de Estabelecimentos de Saúde, CNES) [21]. This database registers all public and private health care facilities in Brazil and its health professionals monthly [21]. We extracted data for December of each year to calculate the study variables. The population data used in the study was based on population estimates developed by the Brazilian Institute of Geography and Statistics (Instituto Brasileiro de Geografia e Estatística, IBGE) [22].
The values found in 2009 were used as a baseline for the analysis of changes in the period. As described in the background, the period under analysis was characterized by the implementation of interventions with different beginnings and durations, as well as changes in federal government. Thus, we carried out the analysis in three distinct periods.
The first time period (2010–2012) represents the period prior to the PMMB, the second (2013–2015) represents the implementation of the program under the Dilma Rousseff administration and the third period (2016–2018), corresponded to a later phase of the PMMB during the Michel Temer administration. The variation in the number of physicians and the number of undergraduate places was calculated using data from the last year of each 3-year period. Thus, the number of medical undergraduate places created and the number of physicians engaged in the medical workforce in the first three years were obtained from the difference obtained by comparing the difference between the 2010 and 2012 data. The values for the second 3-years period were calculated from the difference between the values of 2013 and 2015 and the values of the third triennium of the difference between 2016 and 2018.
In order to identify changes in the distribution of new medical undergraduate medical places and the engagement of physician in the workforce, data were aggregated by macro-regions and by characteristics of the municipalities in 2009 (baseline) such as: municipal population size, physician inhabitant rate in the state and gross domestic product (GDP) per capita [23]. The categorization related to physician per inhabitant ratio in the state was obtained using the mean of the values derived from the physician per inhabitant ratio in each state of the federation in 2009. Data on medical undergraduate places were also aggregated by medical school ownership into: private, federal public schools, state public schools and municipal public schools. Schools classified in the Higher Education Census as a "special" ownership category were categorized as municipal, as they are governed by municipal laws and have approval flow from the State Education Councils.
The absolute variations in the medical undergraduate places ratio per 10,000 inhabitants and physicians per 1000 inhabitants for the entire period were calculated from the difference between the values found in 2009 and 2018. The relative variations, in turn, were calculated using the formula:
$$\frac{\mathrm{Value}\;\mathrm{in}\;2018}{\mathrm{Value}\;\mathrm{in}\;2009}-1\times100$$
Finally, data from the National Information System on Live Births (Sistema de informações de Nascidos Vivos, SINASC) were used to assess variations in access to antenatal visits during the study [24]. We analyze differences in percentage of pregnant women with more than 6 antenatal care visits between the years 2010 and 2018 in each state as a proxy of increase in healthcare access. We conducted Friedman test to determine whether there were statistically significant differences.
From 2010 to 2018, 19,519 new medical undergraduate places were created (variation from 16,236 to 35,755), which represents an increase of 120.2%. Of these, 18,014 (92.2%) undergraduate places were created after the PMMB implementation (Table 1).
Table 1 Evolution of absolute number of medical undergraduate places and ratio per inhabitants
The Midwest macro-region showed the highest relative growth in the medical undergraduate places ratio per 10,000 inhabitants (150.08%), followed by the Northeast (137.95%) and South (118.24%) macro-regions. When observing the absolute growth in the medical undergraduate places per 10,000 inhabitants, the largest increases occurred in the Midwest (1.03 medical undergraduate places per 10,000 inhabitants), South macro-region (0.95 medical undergraduate places per 10,000 inhabitants) and Southeast macro-region (0.86 medical undergraduate places per 10,000 inhabitants). As noted, the North and Northeast macro-regions remained the regions with the lowest medical undergraduate places ratio per inhabitant. States with an intermediate situation in relation of physician's shortages had the largest absolute and relative increase in the medical undergraduate place's ratio per inhabitant (Table 1). The greatest relative increase in the medical undergraduate places ratio per inhabitant occurred in municipalities with GDP below the national average in 2009. However, the absolute increase in the undergraduate medical places ratio per inhabitants occurred in the wealthiest municipalities.
In comparing the three time periods, it is observed that from 2010 to 2012, 1,505 medical undergraduate places were created (7.7% of the total), 6791 (34.8% of the total) in the triennium 2013–2015, and 11,223 (57.5% of the total) in the triennium 2016–2018 (Table 2). In these last three years, 5391 (48.0%) medical undergraduate places were opened in the Southeast region (Table 2). In addition, there was greater expansion in municipalities with more than 500,000 inhabitants and in states that already had more medical undergraduate places per 10,000 inhabitants (Table 2).
Table 2 Medical undergraduate places created and ratio per 10,000 inhabitants per triennium
Regarding the medical school ownership, 15,476 medical undergraduate places were created in private institutions, 10,002 (64.63%) of which were created in the 2016–2018 triennium (Table 2). The greatest public expansion took place within the scope of federal public medical schools with the creation of 2102 places, of which 1447 (68.84%) in the 2013–2015 period. In municipal public medical schools, 1291 places (51.12%) were created in the 2016–2018 period (Table 2).
The increase in the number of physicians engaged in the workforce throughout the period was 113,702 physicians, 74,771 of these physicians in the SUS (Table 3). Figure 3 shows the physician per inhabitant ratio by state in 2009 (baseline) and 2018.
Table 3 Evolution of absolute number of physicians (total and SUS) and physicians ratio per 1000 inhabitants
Physician per inhabitant ratio by state, 2009 and 2018, Brazil
Increase in the percentage of pregnant women with > 6 antenatal visits between 2010 and 2018
The municipalities with the largest population and with the highest GDP per capita had the highest physician per 1000 inhabitants ratio in 2018 (Table 3). These data reveal the importance of socioeconomic factors as determinants in the distribution of physicians. In municipalities below 20,000 inhabitants, almost the entire medical workforce is guaranteed by SUS.
The medical workforce increased by 23,833 doctors in the 2010–2012 period, 45,666 doctors in the 2013–2015 period and 44,203 doctors in the 2016–2018 period (Table 4). We observed that the increase in the workforce at SUS was 17,817 physicians (74.76% of total) in the period 2010–2012, 32,172 physicians (70.45% of total) in the period 2013–2015 and 24,838 (56.19% of total) in 2016–2018 (Table 5).
Table 4 Variation of physicians in the workforce and ratio per 1000 inhabitants per triennium
Table 5 Increase of physicians in SUS and ratio per 1000 inhabitants per triennium
The two macro-regions with the highest relative growth in the physicians per 1000 inhabitants ratio were the South and Northeast regions, respectively. However, the largest absolute increases occurred in the South and Southeast (Tables 4 and 5). The increase in the physicians per 1000 inhabitants ratio occurred in the largest and wealthiest municipalities in the three periods analyzed. The greatest increase in the physicians per 1000 inhabitants ratio in the municipalities with the smallest population, the lowest GDP per capita and in those located in the states with the lowest concentration of physicians occurred in the 2013–2015 period (Tables 4 and 5).
The percentage of pregnant women with more than 6 antenatal care visits was 60.6% in 2010 and 70.8% in 2018 in Brazil, an increase of 17%. This increase varied between 2.4% and 83.9% among Brazilian states. The greatest increases occurred in the poorest states and with the worst physician per inhabitant ratios. Differences in the percentage of pregnant women with more than 6 antenatal care visits between 2010 and 2018 in Brazilian states were statistically significant (p < 0.001) (Fig. 4).
From 2010 to 2018 there was an expansion in the medical workforce in all groups of municipalities assessed in Brazil. This increase can be attributed to the increase in the number of medical undergraduate places as well as to the recruitment of physicians trained abroad through the PMMB. As undergraduate training takes six years in Brazil, the results of the expansion of undergraduate places in the physician workforce is partial and reflect only the increase in enrollments that occurred until 2012.
The expansion planned by the PMMB was to create 11,500 new medical undergraduate places by 2017, reaching a ratio of 1.34 medical undergraduate places per 10,000 inhabitants [17]. According to data from the Higher Education Census, this target was reached in 2017. It was also noted that, in addition to the PMMB goal, just over 7500 additional medical undergraduate places were created by 2018.
The 2013–2015 period presented the greatest reductions in regional inequalities in the distribution of medical undergraduate places. In this triennium there was the greatest expansion of medical undergraduate places in the federal public schools, mostly in the establishment of new campuses in municipalities in Brazil´s countryside [2]. A previous analysis demonstrated that municipalities in countryside in which a medical school was established increased the capacity for attracting and retaining physicians as well expanding healthcare services [2]. The Northeast macro-region (the poorest macro-region in Brazil) had the highest increase in medical undergraduate places in public federal schools. These results demonstrate that expansion of federal medical schools was the most effective measure to reduce inequalities in access to medical education. The expansion of medical undergraduate places in private schools was effective to increase the number of medical undergraduate places but did not reduce regional inequalities.
In addition, it is important to consider that the increase in medical undergraduate places in private institutions does not represent an increase in access to medical education for the poorest population living in vulnerable regions due to the tuition and fees. This situation could be an obstacle to reduce healthcare access inequities, since students from vulnerable regions are more likely to practice in these regions after completing their studies [8, 25].
During the period analyzed, there was an expansion of 113,702 physicians in the workforce, which enabled an increase in the physicians per inhabitant's ratio in Brazil. The 2013–2015 was the period with the greatest increase in the number of doctors due to the incorporation of graduates from Brazilian medical schools and the recruitment of physicians trained outside Brazil [9]. Due to the regulatory framework of the PMMB regarding recruitment, all physicians trained abroad were incorporated to practice exclusively in primary care in the SUS [15]. In the other three-year periods, the incorporation of foreign doctors was small, and the physician supply was predominantly composed of physicians trained in Brazil. It is also observed that the percentage of physicians who were incorporated into the medical workforce in SUS declined over time. This phenomenon of greater incorporation of physicians in the private sector may pose a problem in ensuring a medical workforce in regions with less economic development in the future.
Despite this increase in the physician inhabitant ratio in Brazil, many municipalities still had less than one physician per 1,000 inhabitants. This situation is prevalent in municipalities with less than 50,000 inhabitants, where more than 65 million Brazilians lived in 2018 [22]. It is noteworthy that about 90% of the medical workforce living in municipalities between 20,000 and 50,000 inhabitants were guaranteed by SUS in 2018. This percentage reached 95% in municipalities with less than 20,000 inhabitants. In the 2013–2015 period, there was a substantial expansion in smaller municipalities, and the increase in the physician-inhabitant ratio in these locations were similar to that found in larger municipalities, especially when considering only physicians practicing in the SUS. These results reflect the importance of the PMMB's recruitment component and the government incentives to expand primary care in SUS to reduce inequalities in access to healthcare, although physicians graduated in Brazil and Brazilians trained in other countries have priority in the recruitment program, Cuban physicians, recruited as part of a Brazilian government cooperation with Cuba government, were an important contingent of professionals who practiced in rural and remote areas [9, 26]. This cooperation generated intense debate during its establishment.
The expansion of medical undergraduate places and the increase in the physician ratio per inhabitant show the importance of specific public policies to increase physician supply. Our results demonstrate that the regulatory mechanisms developed to expand medical undergraduate places in areas with the greatest need of physicians were only partially implemented. It is worth noting that only 20% of medical undergraduate places in private institutions were created through the regulatory framework that was defined by the PMMB. Recruitment of physicians from the PMMB has substantially increased the number of physicians practicing in underserved areas during the years 2013 to 2015 [9]. Growth in the 2016–2018 3-year period was lower, which could be due to the economic crisis and the fiscal austerity policies implemented in 2016 that reduced public investment to expand healthcare in underserved areas [11].
This challenge worsened with the end of cooperation with the Cuban government as these physicians represented the largest contingent of physician in the areas of greatest difficulty in supply [9, 16]. In December 2019, the federal government created the Doctors for Brazil Program [27]. This program established a new structure to expand the hiring of physicians in underserved areas and revoked the changes in the training of specialists defined by the PMMB [26]. However, none of the new measures proposed to increase the number of physicians in underserved areas of this program have been implemented.
The Brazilian experience brings important reflections to the challenge of facing physician shortages. This experience highlights the need for the development of integrated public policies throughout the lifecycle of the health worker (education, recruitment, retention). The implementation of educational policies in Brazil has been influenced by political and economic changes and part of the proposed regulatory mechanisms have not been implemented as planned. This situation demonstrates the importance of evaluating the implementation of public policies to identify problems, new demands, and challenges. The expansion of federal medical schools was effective to reduce inequities in access to medical education and to increase the supply of physicians. Our results indicate that the expansion of medical undergraduate places in the private sector did not meet the regulatory frameworks established in the PMMB and increased regional inequalities. Thus, it is recommended that the established regulatory framework be followed in the future planning of the offer of medical undergraduate places. Data regarding the origin of medical students must be analyzed to guide the formulation of admission programs for students from underserved areas. This is an effective measure to increase physicians supply and could be an important strategy to be included in medical education regulation in Brazil. Expanding public investment in health to expand and maintain SUS health services, especially in municipalities with a shortage of physicians, is another recommended measure to move towards Universal Health Coverage (UHC).
All data is disponible in public databases.
UHC:
Universal health coverage
PMMB:
More doctors for Brazil Program
PHC:
SUS:
Unified Health System
PITS:
Health Work Interiorization Program
REUNI:
Restructuring and Expansion Program of Federal Universities
PROVAB:
Program for Valuing Primary Care Professionals
SINAES:
National Higher Education Assessment System
INEP:
National Institute of Studies and Research Anísio Teixeira
CNES:
National Register of Healthcare Establishments
IBGE:
Brazilian Institute of Geography and Statistics
GDP:
Crisp N, Chen L. Global supply of health professionals. N Engl J Med. 2014;370(10):950–7.
Figueiredo AM, McKinley DW, Lima KC, Azeyedo GD. Medical school expansion policies: educational access and physician distribution. Med Educ. 2019;53(11):1121–31.
United nations. Transforming our world: the 2030 agenda for sustainable development. New York, 2015. https://www.un.org/ga/search/view_doc.asp?symbol=A/RES/70/1&Lang=E. Accessed 22 May 2020.
Sousa A, Scheffler Richard M, Nyoni J, Boerma T. A comprehensive health labour market framework for universal health coverage. Bull World Health Organ. 2013;91(11):892–4. https://doi.org/10.2471/BLT.13.118927.
Islam N. The dilemma of physician shortage and international recruitment in Canada. Int J Health Pol Manag. 2014;3(1):29–32.
Nair M, Webster P. Health professionals' migration in emerging market economies: patterns, causes and possible solutions. J Public Health. 2013;35(1):157–63.
Rourke J. How can medical schools contribute to the education, recruitment and retention of rural physicians in their region? Bull World Health Organ. 2010;88(5):395–6. https://doi.org/10.2471/BLT.09.073072.
World Health Organization. Increasing access to health workers in remote and rural areas through improved retention: global policy recommendations. 2010. https://apps.who.int/iris/bitstream/handle/10665/44369/9789241564014_eng.pdf;jsessionid=43FE194876C92E5463EEDD051E02EAC8?sequence=1. Accessed 22 Jan 2020.
Santos LM, Oliveira A, Trindade JS, Barreto IV, Palmeira PA, Comes Y, et al. Implementation research: towards universal health coverage with more doctors in Brazil. Bull World Health Organ. 2017;95(2):103–12. https://doi.org/10.2471/BLT.16.178236.
Andrade MV, Coelho AQ, Xavier Neto M, de Carvalho LR, Atun R, Castro MC. Transition to universal primary health care coverage in Brazil: Analysis of uptake and expansion patterns of Brazil's Family Health Strategy (1998–2012). PLoS ONE. 2018;13(8):e0201723. https://doi.org/10.1371/journal.pone.0201723.
Massuda A, Hone T, Leles FAG, de Castro MC, Atun R. The Brazilian health system at crossroads: progress, crisis and resilience. BMJ Glob Health. 2018;3(4):e000829. https://doi.org/10.1136/bmjgh-2018-000829.
Carvalho MS, Sousa MF. Como o Brasil tem enfrentado o tema provimento de médicos? Interface (Botucatu). 2013;17(47):913–26.
Oliveira FP, Vanni T, Pinto HA, Santos JTR, Figueiredo AM, Araújo SQ, Matos MFM, Cyrino EG. Mais Médicos: um programa brasileiro em uma perspectiva internacional. Interface (Botucatu). 2015;19(54):623–34. https://doi.org/10.1590/1807-57622014.1142.
Almeida MJ, Campos JJB, Turini B, Nicoletto SCS, Pereira LA, Rezende LR, et al. Implementation of National Guidelines in Medical Schools in Paraná. Rev Bras Educ Méd. 2007;31(2):156–65. https://doi.org/10.1590/S0100-55022007000200006.
Presidência da República (BR). Lei no 12.871, de 22 de Outubro de 2013. Institui o Programa Mais Médicos, altera a Lei 8.745, de 9 de Dezembro de 1993, e no 6.932, de 7 de Julho de 1981, e dá outras providências. Diário Oficial da União. 23 Out 2013. http://www.planalto.gov.br/ccivil_03/_Ato2011-2014/2013/Lei/L12871.htm. Accessed 15 May 2020.
Castro MC, Massuda A, Almeida G, et al. Brazil's unified health system: the first 30 years and prospects for the future. Lancet. 2019;394(10195):345–56. https://doi.org/10.1016/S0140-6736(19)31243-7.
Ministério da Saúde. Programa Mais Médicos—dois anos: mais saúde para os brasileiros. Brasília: Ministério da Saúde. 2015. http://maismedicos.gov.br/images/PDF/Livro_2_Anos_Mais_Medicos_Ministerio_da_Saude_2015.pdf. Accessed 26 Feb 2020.
Ministério da Educação. Resolução nº 3, de 20 de Junho de 2014. Diretrizes curriculares nacionais do curso de graduação em medicina. Brasília: Ministério da Educação. 2014. http://portal.mec.gov.br/index.php?option=com_docman&view=download&alias=15874-rces003-14&category_slug=junho-2014-pdf&Itemid=30192. Accessed 02 May 2020.
Instituto Nacional de Estudos e Pesquisas Educacionais Anísio Teixeira (INEP). Instrumento de Avaliação de Cursos de Graduação presencial e a distância. 2016. http://download.inep.gov.br/educacao_superior/avaliacao_cursos_graduacao/instrumentos/2016/instrumento_2016.pdf. Accessed 30 May 2020.
Instituto Nacional de Estudos e Pesquisas Educacionais Anísio Teixeira (INEP). Censo da Educação Superior 2009–2018. http://inep.gov.br/microdados. Accessed 02 Feb 2020.
Ministério da Saúde—DATASUS. CNES-Recursos Humanos-Profissionais-Indivíduos-segundo CBO 2002—Brasil. http://tabnet.datasus.gov.br/cgi/tabcgi.exe?cnes/cnv/prid02br.def. Accessed 12 Feb 2020.
Ministério da Saúde—DATASUS. População Residente-Estimativas para o TCU—Brasil. http://tabnet.datasus.gov.br/cgi/tabcgi.exe?ibge/cnv/poptuf.def. Accessed 12 Feb 2020.
Instituto Brasileira de Geografia e Estatística (IBGE). Sistema IBGE de Recuperação Automática—SIDRA. Produto interno bruto a preços correntes, impostos, líquidos de subsídios, sobre produtos a preços correntes e valor adicionado bruto a preços correntes total e por atividade econômica, e respectivas participações—Referência 2010. https://sidra.ibge.gov.br/tabela/5938. Accessed 12 Feb 2020.
Ministério da Saúde—DATASUS. Nascidos Vivos Brasil. Sistema de Informações de Nascidos Vivos. http://tabnet.datasus.gov.br/cgi/deftohtm.exe?sinasc/cnv/nvuf.def. Acessed 03 Feb 2021.
Ray RA, Woolley T, Sen GT. James Cook University's rurally orientated medical school selection process: quality graduates and positive workforce outcomes. Rural Rem Health. 2015;15(4):3424.
Pacheco Santos LM, Millett C, Rasella D, Hone T. The end of Brazil's More Doctors programme? BMJ. 2018;18(363):k5247. https://doi.org/10.1136/bmj.k5247.
Presidência da República (BR). Lei no 13.958, de 18 de dezembro de 2013. Institui o Programa Médicos pelo Brasil, no âmbito da atenção primária à saúde no Sistema Único de Saúde (SUS), e autoriza o Poder Executivo federal a instituir serviço social autônomo denominado Agência para o Desenvolvimento da Atenção Primária à Saúde (Adaps). Diário Oficial da União. http://www.planalto.gov.br/ccivil_03/_Ato2019-2022/2019/Lei/L13958.htm#art34. Accessed 12 Jul 2020.
This article was submitted to Human Resources for Health Accreditation of education and regulation of practice series sponsored by Foundation for Advancement of International Medical Education and Research.
Health Sciences Postgraduate Program, Universidade Federal do Rio Grande do Norte, Natal, Brazil
Alexandre Medeiros Figueiredo
Department of Health Promotion, Federal University of Paraíba, Campus I, Jardim Universitário, S/N, Castelo Branco, João Pessoa, PB, Brazil
Research and Data Resources, FAIMER, 3624 Market Street, Philadelphia, PA, 19104, USA
Danette Waller McKinley
School of Business Administration, Fundação Getulio Vargas (FGV EAESP), Av. 9 de julho, 2029, Bela Vista, São Paulo, SP, 01313-902, Brazil
Adriano Massuda
Multicampi School of Medical Sciences, Federal University of Rio Grande do Norte, Av. Cel. Martiniano, 541, Caicó, RN, 59300-000, Brazil
George Dantas Azevedo
AMF and GDA contributed to the conception of the study. AMF was responsible for data analysis. AMF, GDA, DWM and AM prepared the manuscript. All authors read and approved the final manuscript.
Correspondence to Alexandre Medeiros Figueiredo.
The research was approved by the Ethics and Research Committee of the Federal University of Paraíba (protocol no. 2.094.734 / 2017, CAAE 68502217.2.0000.8069).
The principal author was a director in the health ministry between 2013 and 2016.
Figueiredo, A.M., McKinley, D.W., Massuda, A. et al. Evaluating medical education regulation changes in Brazil: workforce impact. Hum Resour Health 19, 33 (2021). https://doi.org/10.1186/s12960-021-00580-5
Received: 20 December 2020
Accepted: 03 March 2021
Medically underserved area
Health workforce: Accreditation of education and regulation of practice | CommonCrawl |
\begin{document}
\def\spacingset#1{\renewcommand{\baselinestretch} {#1}\small\normalsize} \spacingset{1}
\if11 {
\title{\bf Addressing Detection Limits with Semiparametric Cumulative Probability Models}
\author{Yuqi Tian\thanks{
The authors gratefully acknowledge CCASAnet investigators for providing data for the HIV study. This study was supported by funding from the U.S. National Institutes of Health, grants \textit{R01 AI093234} and \textit{U01 AI069923}. }\hspace{.2cm}\\
Department of Biostatistics, Vanderbilt University,\\
Chun Li \\
Department of Population and Public Health Sciences, \\University of Southern California,\\
Shengxin Tu \\
Department of Biostatistics, Vanderbilt University,\\
Nathan T. James \\
Department of Biostatistics, Vanderbilt University,\\
Frank E. Harrell \\
Department of Biostatistics, Vanderbilt University,\\
and \\
Bryan E. Shepherd\\
Department of Biostatistics, Vanderbilt University}
\maketitle } \fi
\if01 {
\begin{center}
{\LARGE\bf Addressing Detection Limits with Semiparametric Cumulative Probability Models} \end{center}
} \fi
\begin{abstract} Detection limits (DLs), where a variable is unable to be measured outside of a certain range, are common in research. Most approaches to handle DLs in the response variable implicitly make parametric assumptions on the distribution of data outside DLs. We propose a new approach to deal with DLs based on a widely used ordinal regression model, the cumulative probability model (CPM). The CPM is a type of semiparametric linear transformation model. CPMs are rank-based and can handle mixed distributions of continuous and discrete outcome variables. These features are key for analyzing data with DLs because while observations inside DLs are typically continuous, those outside DLs are censored and generally put into discrete categories. With a single lower DL, the CPM assigns values below the DL as having the lowest rank. When there are multiple DLs, the CPM likelihood can be modified to appropriately distribute probability mass. We demonstrate the use of CPMs with simulations and two HIV data examples. The first example models a biomarker in which 15\% of observations are below a DL. The second uses multi-cohort data to model viral load, where approximately 55\% of observations are outside DLs which vary across sites and over time.
\end{abstract}
\noindent {\it Keywords:} Limit of detection; HIV; ordinal regression model; transformation model
\spacingset{1.9} \section{Introduction} \label{sec:intro}
Detection limits (DLs) are not uncommon in biomedical research and other fields. For example, radiation doses may only be detected above a certain threshold \citep{wing1991mortality}, antibody concentrations may not be measured below certain levels \citep{wu2001development}, and X-rays may have lower limits of detection \citep{pan2017cs}. In HIV research, viral load can only be detected above certain levels. To complicate matters, DLs often vary by assay and may change over time. For example, HIV viral load assays have had lower DLs at 400, 300, 200, 50, and 20 copies/mL depending on the commercial assay and year of application \citep{steegen2007evaluation}.
Different types of analysis methods to handle DLs have been proposed for different purposes. In this manuscript, we will focus on studying the association between an outcome variable and covariates, where the outcome variable is subject to DLs. This is typically achieved with some sort of regression model.
One common and simple method is to dichotomize the outcome as detectable or undetectable, and then to perform logistic regression \citep{jiamsakul2017hiv}. While it can be useful for some purposes, the dichotomization leads to information loss since the observed values inside the DLs are treated as if they are the same. Another common approach for handling DLs is substitution, where all nondetects are imputed with a single constant and a linear regression model is fit. The imputed constant may be, for example, the DL itself, DL/2, DL/$\sqrt{2}$ \citep{hornung1990estimation,lubin2004epidemiologic,helsel2011statistics}, or the expectation of the measurement conditional on being outside the DL under some assumed parametric model \citep{garland1993toenail}. For example, DL/2 corresponds to the expectation of a uniform distribution between $0$ and the DL. Although simple, these substitution approaches typically result in biased estimation, underestimated variances, and thus sometimes wrong conclusions \citep{baccarelli2005handling,fievet2010dealing}. In a third approach, one explicitly makes parametric assumptions on the distribution of the data, both within and outside the DLs. Parameters of interest can then be estimated by maximizing the censored data likelihood. Such a maximum likelihood approach is efficient and consistent when the distribution is correctly specified, but may perform poorly when distributional assumptions are incorrect. To compound the problem, there is typically no way to examine model fit outside the DLs; goodness-of-fit of a parametric model inside DLs does not ensure goodness-of-fit outside the DLs \citep{baccarelli2005handling, harel2014use}. A related, fourth approach for addressing DLs is to multiply impute values outside the DLs \citep{little2019statistical, harel2007multiple}. This approach may be computationally expensive, still requires parametric assumptions that can only be verified inside DLs, and may be particularly problematic with high rates of censoring or small sample sizes \citep{lubin2004epidemiologic,zhang2009nonparametric}.
To avoid strong parametric assumptions, nonparametric methods such as Kaplan–Meier, score and rank-based methods have been proposed in two-sample comparisons \citep{helsel2011statistics}. Zhang et al. (2009) explored the use of the Wilcoxon rank sum test, other weighted rank tests, Gehan and Peto-Peto tests, and a novel nonparametric method for location-shift inference with DLs. Although attractive for two-sample tests, these nonparametric methods do not permit the inclusion of covariates.
In this manuscript, we propose a new approach for analyzing data subject to detection limits. Data with DLs effectively follow a mixture distribution, where those below a lower DL can be thought of as belonging to a discrete category, those above an upper DL belonging to another discrete category, while those inside the DLs are continuous. Whether discrete or continuous, the values are orderable. In earlier work, \cite{liu2017modeling} showed that continuous response variables can be modeled using a popular model for ordinal outcomes, namely the cumulative probability model (CPM), also known as the \lq cumulative link model' \citep{agresti2003categorical}. CPMs are a type of semiparametric linear transformation model, in which the continuous response variable after some unspecified monotonic transformation is assumed to follow a linear model, and the transformation is nonparametrically estimated \citep{zeng2007maximum}. These models are very flexible and can handle a wide variety of outcomes, including variables with DLs. Importantly, when fitting CPMs to data with DLs, minimal assumptions are made on the distribution of the response variable outside the DLs as these models are based on ranks, and values below/above DLs are simply the lowest/highest rank values. Because of their relationship to the Wilcoxon rank sum test \citep{mccullagh1980regression}, the CPM can be thought of as a semiparametric extension to permit covariates to the approaches that \citet{zhang2009nonparametric} found effective for handling DLs in two-sample comparisons. Finally, as will be shown, because CPMs model the conditional cumulative distribution function (CDF), it is easy to extract many different measures of conditional association from a single fitted model, including conditional quantiles, conditional probabilities, odds ratios, and probabilistic indexes, which permits flexible and compatible interpretation.
In Section 2, we review the CPM, illustrate its use for simple settings where there is only a single set of DLs, and then show how CPMs can be extended to address multiple DLs. We also propose a new method for estimating the conditional quantile from a CPM. In Section 3, we illustrate and demonstrate the advantages of the proposed approach using real data from two studies. The first study aims to measure the association between covariates and a biomarker whose values are below a DL in approximately 15\% of observations. The second example is a large multi-cohort study of viral load (VL) after starting antiretroviral therapy among persons with HIV, where most observations are below DLs, but the DLs vary across sites and change over time. In Section 4, we demonstrate the performance of our method with simulations. The final section contains a discussion of the strengths and limitations of our method and future work.
\section{Methods} \label{sec:meth}
\subsection{Cumulative Probability Models}
Transformation is often needed for regression of a continuous outcome variable $Y$ to satisfy model assumptions, but specifying the correct transformation can be difficult. In a linear transformation model, the outcome is modeled as $Y=H(\beta^T X+\epsilon)$, where $H(\cdot)$ is an unknown monotonically increasing transformation, $X$ is a vector of covariates, and $\epsilon$ follows a known distribution with CDF $F_\epsilon$. This linear transformation model can be equivalently expressed in terms of the conditional CDF, \begin{align*}
F(y|X) \equiv \Pr(Y\le y |X)
=\Pr[\epsilon\le H^{-1}(y)-\beta^T X|X]
=F_\epsilon[H^{-1}(y)-\beta^T X]. \end{align*} Let $G=F_\epsilon^{-1}$ and $\alpha=H^{-1}$; $\alpha(\cdot)$ is monotonically increasing but otherwise unknown. Then \begin{equation}
G[F(y|X)]=\alpha(y)-\beta^T X,
\label{eq:CPM} \end{equation} where $G$ serves as a link function and the model becomes a cumulative probability model (CPM). The intercept function $\alpha(y)$ is the transformation of the response variable such that $\alpha(Y)=\beta^TX + \epsilon.$ The $\beta$ coefficients indicate the association between the response variable and covariates: fixing other covariates, a positive/negative $\beta_j$ means that an increase in $X_j$ is associated with a stochastic increase/decrease in the distribution of the response variable.
In the CPM (\ref{eq:CPM}), the intercept function $\alpha(y)$ can be nonparametrically estimated with a step function
\citep{zeng2007maximum, liu2017modeling}. This allows great model flexibility.
Consider an iid dataset $\{(y_i,x_i): i=1,\ldots, n\}$. The nonparametric likelihood is \begin{equation}
\prod_{i=1}^n \left[F (y_i|x_i)-F(y_i^-|x_i)\right],
\label{eq:nplikelihood} \end{equation}
where $F(y_i^-|x_i)=\lim_{t \uparrow y_i}F(t|x_i)$. In nonparametric maximum likelihood estimation, the probability mass given any $x$ will be distributed over the discrete set of observed outcome values. Thus we only need to consider functions for $\alpha(\cdot)$
such that $F(y|x_i)$ is a discrete distribution over the observed values. Let $J$ be the number of distinct outcome values, denoted as $a_{1}<\cdots<a_{J}$. Let $S=\{a_1,\ldots,a_J\}$. These serve as the anchor points for the nonparametric likelihood. Let $\alpha_j =\alpha(a_{j})$; then $\alpha_1<\cdots<\alpha_J$. The nonparametric likelihood (\ref{eq:nplikelihood}) can be written as \begin{align}
L(\beta, \bm{\alpha})=&\prod_{i:y_i=a_1} F_{\epsilon}(\alpha_1-\beta^T x_i) \times
\prod_{j=2}^{J-1}\prod_{i:y_i=a_j}\left[F_{\epsilon}(\alpha_j-\beta^T
x_i)-F_{\epsilon}(\alpha_{j-1}-\beta^Tx_i)\right] \times
\\
& \prod_{i:y_i=a_{J}}\left[1-F_{\epsilon}(\alpha_{J-1}-\beta^Tx_i)\right].
\label{eq:final_l} \end{align} Maximizing (\ref{eq:final_l}), we obtain the nonparametric maximum likelihood estimates (NPMLEs), $(\hat{\beta}, \bm{\hat{\alpha}})$, where $\bm{\hat\alpha} =(\hat \alpha_1, \dots, \hat \alpha_{J-1})$. Note the multinomial form of the likelihood (\ref{eq:final_l}); because the probabilities in a multinomial likelihood add to one, $\alpha_J$ is not estimated. Note also that the likelihood in (\ref{eq:final_l}) is identical to that of cumulative link models for ordinal data if the outcome $Y$ is treated as ordinal with categories $\{a_1,\ldots,a_J\}$. \citet{liu2017modeling} and \citet{tian2020empirical} have shown that CPMs can be fit to and work well for continuous and mixed types of responses. CPMs have also been shown to be consistent and asymptotically normal, with variance consistently estimated with the inverse of the information matrix under mild conditions including boundedness of the outcome variable \citep{li2022asymptotics}. The NPMLEs and their estimated variances can be efficiently computed with the \texttt{orm()} function in the \textbf{rms} package in \textsf{R} \citep{rms}, which takes advantage of the tridiagonal nature of the Hessian matrix using Cholesky decomposition \citep{liu2017modeling}.
CPMs have several nice features. Some widely used regression methods model only one aspect of the conditional distributions (e.g., conditional mean for linear regression and conditional quantile for quantile regression). With the NPMLEs $(\hat{\beta}, \bm{\hat{\alpha}})$, we can estimate the conditional CDFs as
$\hat{F}(y|x)=F_\epsilon(\hat{\alpha}_j - \hat{\beta}^T x)$ where $j$ is the index such that $a_j=\max\{a \in S: a \leq y\}$; standard errors can be obtained by the delta method. Since conditional CDFs are directly modeled, other characteristics of the distribution, such as the conditional quantiles and conditional expectations, can be easily derived
\citep{liu2017modeling}. Depending on the choice of link function, $\beta$ may be interpretable; for example, with the logit link function, exp$(\beta)$ is an odds ratio. Probabilistic indexes \citep{de2019semiparametric}, which are defined as $\Pr(Y_1<Y_2|X_1,X_2)$, can also be easily derived; for example, with the logit link, $P(Y_1 < Y_2|X_1, X_2)=\left[1+\exp\left(-(X_2-X_1)^T \beta \right)\right]^{-1}$. With the transformation $\alpha(\cdot)$ nonparametrically estimated, CPMs are invariant to any monotonic transformation of the outcome; therefore, no pre-transformation is needed. With a single binary covariate and the logit link function, the score test for the CPM is nearly identical to the Wilcoxon rank sum test \citep{mccullagh1980regression}; see Supplemental Material S1.1. Because only the order of the outcome values but not the specific values matter when estimating $\beta$ in the CPM, it can handle any ordinal, continuous, or mixture of ordinal and continuous distributions, which can be useful for analyzing data with DLs.
\subsection{Single Detection Limits}
In this subsection, we first present our method for the simple scenario that there is a single lower DL and/or a single upper DL. We will describe the general approach for multiple DLs in the next subsection.
Consider a dataset with a lower DL, $l$, and an upper DL, $u$. The outcome $Y$ is observed if it is inside the DLs (i.e., $l\le Y\le u$) or censored if it is outside the DLs. The $J$ distinct values of the observed outcomes are denoted as $l\le a_{1}<\cdots<a_{J}\le u$. When there are no observations outside the DLs, these values are treated as ordered categories in CPMs and they are the anchor points in the nonparametric likelihood (\ref{eq:final_l}), and correspondingly there are $J-1$ alpha parameters, $\alpha_1<\cdots<\alpha_{J-1}$. With observations outside the DLs, the likelihood (\ref{eq:final_l}) needs to be modified accordingly.
When there are observations below the lower DL, we do not know their values except that they are $<l$. As there is no way to distinguish them, we treat them as a single category, denoted as $a_{0}$. Note that $a_{0}$ is not a value but a symbol for the additional category below $a_{1}$. The nonparametric likelihood for a subject outcome censored at the lower DL $l$ is \begin{equation*}
\Pr(Y_i<l|X_i=x_i) = F_\epsilon(\alpha_0-\beta^Tx_i), \end{equation*} where $\alpha_0$ is the extra alpha parameter corresponding to category $a_{0}$ such that $\alpha_0<\alpha_1$. Because $a_{1}$, the previously lowest category, now has a category below it, the nonparametric likelihood for a subject with $y_i=a_{1}$ becomes \begin{equation*}
F_\epsilon(\alpha_1-\beta^Tx_i) - F_\epsilon(\alpha_0-\beta^Tx_i). \end{equation*}
Similarly, when there are observations above the upper DL, they are also treated as a single category, denoted as $a_{J+1}$, which is a symbol for the additional category above $a_{J}$. The nonparametric likelihood for a subject censored at the upper DL $u$ is \begin{equation*}
\Pr(Y_i>u|X_i=x_i) = 1-F_\epsilon(\alpha_J-\beta^Tx_i), \end{equation*} Because $a_{J}$ is no longer the highest category, $\alpha_{J}$ will need to be estimated, and the likelihood for a subject with $y_i=a_{J}$ is now \begin{equation*}
F_\epsilon(\alpha_J-\beta^Tx_i) -F_\epsilon(\alpha_{J-1}-\beta^Tx_i). \end{equation*}
Put together, with observed data subject to a single lower DL and a single upper DL, the CPM likelihood is \begin{align}
L(\beta, \bm{\alpha})=&\prod_{i:y_i=a_0} F_{\epsilon}(\alpha_0-\beta^T x_i) \times
\prod_{j=1}^{J}\prod_{i:y_i=a_j}\left[F_{\epsilon}(\alpha_j-\beta^T
x_i)-F_{\epsilon}(\alpha_{j-1}-\beta^T x_i)\right] \times \nonumber\\
&\quad \prod_{i:y_i=a_{J+1}}\left[1-F_{\epsilon}(\alpha_{J}-\beta^T x_i)\right], \label{eq:final_singleDL} \end{align} which is equivalent to (\ref{eq:final_l}) except with two new anchor points, $a_0$ and $a_{J+1}$. Therefore, (\ref{eq:final_singleDL}) is maximized in an identical manner to (\ref{eq:final_l}), with outcomes below the lower DL and outcomes above the upper DL simply assigned to categories $a_0$ and $a_{J+1}$, respectively.
In summary, when there are data censored below the lower DL, we add a new anchor point $a_{0}<a_{1}$ and a new parameter $\alpha_0$; when there are data censored above the upper DL, we add a new anchor point $a_{J+1}>a_{J}$ and a new parameter $\alpha_{J}$. The alpha parameters to be estimated are $(\alpha_1,\ldots,\alpha_{J-1})$ when there are no DLs or no data censored at DLs, $(\alpha_0,\alpha_1,\ldots,\alpha_{J-1},\alpha_J)$ when both categories $a_{0}$ and $a_{J+1}$ are added, $(\alpha_0,\alpha_1,\ldots,\alpha_{J-1})$ when only $a_{0}$ is added, and $(\alpha_1,\ldots,\alpha_{J-1},\alpha_J)$ when only $a_{J+1}$ is added.
In practice, one can fit the NPMLE in these settings using the \texttt{orm()} function by setting outcomes below the lower DL to some arbitrary number $<l$ and outcomes above the upper DL to some arbitrary number $>u$. Note that unlike single imputation approaches for dealing with DLs, the CPM estimation procedure is invariant to the choice of these numbers assigned to values outside the DLs. The CPM (\ref{eq:CPM}) assumes that after some unspecified transformation, the outcome follows a linear model both within and outside the DLs. In contrast, parametric approaches to deal with DLs assume the full distribution of the outcome conditional on covariates is known, both within and outside DLs. Hence, CPMs make much weaker assumptions than fully parametric approaches.
\subsection{Multiple Detection Limits}
We now consider the general situation where data may be collected from multiple study sites. A site may have no DL, only one DL, or both lower and upper DLs. Each site may have different lower DLs and different upper DLs, which may change over time.
Every subject has a vector $X$ of covariates and three underlying random variables $(Y, C_L, C_U)$, where $Y$ is the true outcome and $C_L<C_U$ are the lower and upper DLs. When there is no upper DL, $C_U=\infty$, and when there is no lower DL, $C_L=-\infty$. $C_L$ and $C_U$ are assumed to be independent of $Y$ conditional on $X$; the vector $X$ may contain variables for study sites or calendar time. This non-informative censoring assumption is typically plausible as DLs are determined by available equipment/assays.
We assume the CPM (\ref{eq:CPM}) holds for all subjects. Due to DLs, we may not always observe $Y$. Instead, we only observe $(Z,\Delta)$, where $Z=\max(\min(Y,C_U),C_L)$ and $\Delta$ is a variable indicating whether $Y$ is observed or censored at a DL: $\Delta=1$ and $Z=Y$ if $Y$ is observed, $\Delta=L$ and $Z=C_L$ if $Y<C_L$, and $\Delta=U$ and $Z=C_U$ if $Y>C_U$.
Given a dataset $\{(z_i,\delta_i;x_i)\}$ ($i=1,\ldots,n$), we first determine how many anchor points are needed to support the nonparametric likelihood of the CPM. Let $J$ be the number of distinct values of $z_i$ among those with $\delta_i=1$; they are denoted as $a_{1}<\cdots<a_{J}$. For data without any DLs, these points are the anchor points for the nonparametric likelihood, and they are effectively treated as ordered categories in a CPM. Let $S=\{a_{1},\cdots,a_{J}\}$ be the set of these values. When there are data with $\delta_i=L$, let $l$ be the smallest $z_i$ with $\delta_i=L$. Similarly, when there are data with $\delta_i=U$, let $u$ be the largest $z_i$ with $\delta_i=U$. If $l\le a_{1}$, we add a category into $S$ below $a_{1}$, denoted as $a_{0}$; note that it is not a value but a symbol for the additional category in $S$ below $a_{1}$. Similarly, if $u\ge a_{J}$, we add $a_{J+1}$ into $S$, which is a symbol for the additional category above $a_{J}$. Depending on the data, the number of ordered categories can be $J$, $J+1$, or $J+2$.
Consider the situation where both $a_{0}$ and $a_{J+1}$ have been added to $S$ (i.e., $S=\{a_{0},a_{1},\ldots,a_{J},a_{J+1}\}$). When $\delta_i=1$, the nonparametric likelihood for $(z_i,1)$ is \begin{equation}
F_\epsilon(\alpha_{j}-\beta^Tx_i) - F_\epsilon(\alpha_{j-1}-\beta^Tx_i), \end{equation} where $j$ is the index such that $a_{j}=z_i$. When $\delta_i=L$, the nonparametric likelihood for $(z_i,L)$ is \begin{equation}
\Pr(Y<z_i|x_i)= \left\{
\begin{array}{lr}
F_\epsilon(\alpha_{0}-\beta^Tx_i), & (z_i= l) \\
F_\epsilon(\alpha_{j}-\beta^Tx_i), & (z_i\ne l)
\end{array} \right.
\label{eq:likelihoodL} \end{equation} where $j$ is the index such that $a_{j}=\max\{a\in S:a<z_i\}$ when $z_i\ne l$. When $\delta_i=U$, the nonparametric likelihood for $(z_i,U)$ is \begin{equation}
\Pr(Y>z_i|x_i)= \left\{
\begin{array}{lr}
1-F_\epsilon(\alpha_{J}-\beta^Tx_i), & (z_i= u) \\
1-F_\epsilon(\alpha_{j-1}-\beta^Tx_i), & (z_i\ne u)
\end{array} \right.
\label{eq:likelihoodU} \end{equation} where $j$ is the index such that $a_{j}=\min\{a\in S:a>z_i\}$ when $z_i\ne u$. The overall nonparametric likelihood is the product of these individual likelihoods over all subjects. Note that if there are no uncensored observations between two lower (or upper) DLs, the two DLs are effectively treated as the same DL. A toy example to illustrate our definition is provided in Table S1 of the Supplementary Material.
Slight modifications will be applied when no or only one additional category is added to $S$. When there is no need to add $a_{0}$ to $S$ (i.e., when $l>a_1$ or there are no lower DLs), only the second row in the likelihood (\ref{eq:likelihoodL}) for $(z_i,L)$ will be employed, and the likelihood for $(z_i,1)$ with $z_i=a_{1}$ is $F_\epsilon(\alpha_{1}-\beta^Tx_i)$. When there is no need to add $a_{J+1}$ to $S$ (i.e., when $u<a_J$ or there are no upper DLs), only the second row in the likelihood (\ref{eq:likelihoodU}) for $(z_i,U)$ will be employed, and the likelihood for $(z_i,1)$ with $z_i=a_{J}$ is $1-F_\epsilon(\alpha_{J-1}-\beta^Tx_i)$.
Similar to the likelihood of CPM for data without DLs, the individual likelihoods presented above involve either one alpha parameter or two adjacent alpha parameters. As a result, the Hessian matrix continues to be tridiagonal, allowing us to use Cholesky decomposition to solve for the NPMLEs and efficiently estimate their variances.
We have developed an \textsf{R} package, \textbf{multipleDL} available at https://github.com/YuqiTian35/multipleDL, which uses the \texttt{optimizing()} function in the \textbf{rstan} package to maximize the likelihood \citep{rstan}.
\subsection{Interpretable Quantities and Conditional Quantiles}
Interpretation of results after fitting CPMs to outcomes with DLs is similar to settings without DLs. Depending on the link function, $\beta$ may be directly interpretable. The conditional CDF, probabilistic indexes, and conditional quantiles are also easily derived. Note, however, that without additional assumptions on the distribution of the outcome outside DLs, conditional expectations cannot be estimated.
\begin{figure}\label{fig:quantile}
\end{figure}
We now describe how to infer conditional quantiles from a CPM fitted on data with DLs. The conditional CDF from a CPM for a given $x$ can be computed as $\hat F(y|x) =F_\epsilon(\hat\alpha_j-\hat\beta^Tx)$ where $j$ is the index such that $a_{j}=\max\{a\in S:a\le y\}$; if there is no $a\in S$ such that $a\le y$, then $\hat F(y|x)=0$. For ease of presentation, we fix $x$ and let $P_j=\hat F(a_j|x)$ ($j=0,1,\ldots,J,J+1$); for convenience, let $P_{-1}=0$. Our goal is to define a quantile function $\hat Q(p)$, where $0<p<1$, for the conditional distribution given $x$.
The quantile function for a CDF $F(\cdot)$ is typically defined as $Q(p)=\inf\{z:F(z) \ge p\}$. A plug-in estimator for an estimated CDF, $\hat F$, is $\hat{Q}_0(p)=\inf\{z:\hat{F}(z) \ge p\}$. When applied to our setting, $\hat Q_0(p)=a_j$ when $P_{j-1}<p\le P_j$. This estimator may not be suitable for CPMs because $\hat F(\cdot)$ is a step function and therefore $\hat Q_0(p)$ only takes values at the anchor points, which can be undesirable for continuous outcomes, especially when there is a large gap between adjacent anchor points.
\citet{liu2017modeling} proposed to estimate quantiles for CPMs with linear interpolation. Specifically, given a fixed $p$, let $j=j(p)$ be the index such that $P_{j-1}<p\le P_j$. When $p>P_0$, $j\ge 1$ and define $\hat Q_1(p) =a_{j-1} + (p-P_{j-1})/ (P_{j}-P_{j-1})\times (a_{j} - a_{j-1})$, which is a linear interpolation between $a_{j-1}$ and $a_{j}$. When $p\le P_0$, $\hat Q_1(p)$ is set to be $a_0$. Recall that $a_0$ is not a value but a symbol for being below the lower DL, $l$; we thus relabel it as `${<}l$', so when $p\le P_0$, $\hat Q_1(p)=$ `${<}l$'. For the linear interpolation between $a_0$ and $a_1$, we set $a_0$ to be $l$. Similarly, $a_{J+1}$ is labeled `${>}u$' and assigned the value $u$ for the linear interpolation between $a_J$ and $a_{J+1}$. $\hat Q_1(p)$ is illustrated as the dashed lines in Figure \ref{fig:quantile}. An alternative definition is to interpolate between $a_{j}$ and $a_{j+1}$: $\hat Q_2(p)=a_{j}+ (p- P_{j-1}) / (P_{j}-P_{j-1}) \times (a_{j+1} - a_{j})$ when $p<P_J$ and $\hat Q_2(p)=a_{J+1}=$ `${>}u$' when $p\ge P_J$. $\hat Q_2(p)$ is illustrated as the dotted lines in Figure \ref{fig:quantile}. For continuous data without DLs, $\hat Q_1(p)$ and $\hat Q_2(p)$ converge as the sample size increases. However, they are problematic for continuous data with DLs because $\hat Q_1(p)<a_{J+1}$ for all $p<1$ and $\hat Q_2(p)>a_{0}$ for all $p>0$ even though there are non-zero estimated probabilities at the lower DL $a_0$ and upper DL $a_{J+1}$.
We propose a new quantile estimator as a weighted average between $\hat Q_1(p)$ and $\hat Q_2(p)$, \begin{equation}
\hat Q(p)=(1-w) \hat Q_1(p) + w \hat Q_2(p), \end{equation} where $w=w(p) = (p-P_{0}) / (P_{J}-P_{0})$ when $P_0<p<P_J$, $0$ when $p\le P_0$, and $1$ when $p\ge P_J$. This definition is shown as the black curve in Figure \ref{fig:quantile}. Note that $\hat Q(p)$ equals $\hat Q_1(p)=$ `${<}l$' when $p\le P_0$, and equals $\hat Q_2(p)=$ `${>}u$' when $p\ge P_J$. It can be shown that similar to $\hat Q_1(p)$ and $\hat Q_2(p)$, $\hat Q(p)$ is also piecewise linear with transition points at $P_j$ ($j=0,1,\ldots,J$).
In situations where there is only a lower DL or an upper DL, our definition of $\hat Q(p)$ is similar. Confidence intervals for the conditional quantiles can be estimated by applying a weighted linear interpolation to the confidence intervals of the conditional CDF similar to the above procedure \citep{liu2017modeling}.
\section{Applications}
In this section, we illustrate our method with two datasets, one from a biomarker study with a single lower DL and the other from a multi-center study with multiple DLs varying within and across centers.
\subsection{Single Detection Limit} Our first example uses data from a study investigating the relationship between HIV, diabetes, obesity, and various biomarkers. Data were collected on 161 adults, some of whom were highly overweight (body mass index (BMI) ranged from 22 to 58 kg/m\textsuperscript{2}). Several biomarkers were measured. Here, we focus on interleuken 4 (IL-4), a cytokine that is related to T-cell production and metabolism and has been seen to limit lipid accumulation in mice \citep{tsao2014interleukin}. We examine the association between IL-4 and BMI, controlling for age, sex, HIV status, and diabetes status. Our measures of IL-4 had a single lower DL of 0.019 pg/ml, and 24 subjects (15\%) had IL-4 values below the DL. The distribution of IL-4, which is right-skewed, is shown in Figure S1(A) in Supplemental Material S2.1.
We fit a CPM as described in Section 2.2, using the logit link; full results are in Table S2 in the Supplementary Material S2.1. No transformation of IL-4 was needed. With the logit link function, the $\beta$ parameters can be interpreted as log odds ratios. BMI was found to be negatively associated with IL-4 (p-value 0.023). Holding other covariates constant, a 5 kg/m$^2$ increase in BMI corresponded to a 22\% decrease in the odds of having a higher IL-4 value (adjusted odds ratio 0.78, 95\% confidence interval (CI) of $(0.62, 0.97)$). The corresponding probabilistic index was 0.264, meaning that holding other variables constant, a 5 kg/m$^2$ increase in BMI was associated with a 0.736 ($=1-0.264$) probability of having a lower IL-4. The median IL-4 conditional on BMI and controlling for all other covariates at their median/mode levels were estimated from the CPM and is shown in Figure \ref{fig:median_cpm}. The conditional median decreased as BMI increased, with the 95\% CI including the category `$<$0.019' for those with a very large BMI. Note that `$<$0.019' is the smallest ordered category indicating values below the DL. Other quantiles and quantities can also be easily derived from the CPM; for example, Figure S2 in the Supplementary Material shows the 90th percentile of IL-4 as a function of BMI, and the probabilities of IL-4 being greater than 0.019 (the DL) and greater than 0.05 as functions of BMI.
\begin{figure}
\caption{The conditional median obtained by CPMs varying BMI while fixing other covariates at median/mode levels}
\label{fig:median_cpm}
\end{figure}
It is worth comparing results from the CPM to other potential analysis approaches. (i) The most common approach in practice would be to singly impute those values below the DL; given the skewed nature of the data, one would then likely log-transform the data and fit a linear regression model. The result can vary depending on the choice of the imputed number: if one imputes with the DL itself (0.019) vs.\ $0.001$, the log-transformed IL-4 is estimated to decrease $0.013$ pg/ml vs.\ $0.032$ pg/ml, respectively, per 5 kg/m$^2$ increase in BMI, with different statistical significance (p-value 0.020 vs.\ 0.073). (ii) A more sophisticated approach might be to assume the data are log-normally distributed and perform a likelihood-based analysis, which results in an estimated change on the log-scale of $-0.015$ pg/ml per 5 kg/m$^2$ BMI increase (p-value 0.018). The conditional median as a function of BMI could also be extracted from this analysis and is in Figure S3(B). The curve of conditional median as a function of BMI is similar to what was estimated with the CPM (Figure S3(A)), but it is slightly lower and its confidence bands are tighter than those of the CPM. The tighter bands reflect the parametric assumption that the data are truly log-normally distributed. In contrast, the CPM does not require transformation of the data, and it non-parametrically estimates the best transformation. (iii) One could also directly estimate the conditional median as a function of BMI using quantile regression \citep{koenker2001quantile}. This estimated curve is in Figure S3(C), which closely matches that estimated from the CPM. However, the confidence bands for median regression are wider than those of the CPM, and the 95\% CI for the slope contains 0. One could argue that the CPM is assuming more than median regression (which only assumes a linear relationship between the median on the original outcome scale and the covariates); hence the narrower confidence bands. However, the CPM is able to yield several additional quantities (e.g., other quantiles, odds ratios, exceedance probabilities) from a single model that cannot be obtained from median regression. Also, the confidence bands obtained by the CPM do not go below the DL. (iv) Finally, one could dichotomize IL-4 into ``undetectable'' and ``detectable'' and fit a logistic regression model. However, logistic regression was not able to provide a stable estimation for this dichotomization. One could consider other dichotomizations, but the choice is arbitrary. In fact, a beta coefficient in the CPM can be thought of as a weighted average of the log-odds ratios for logistic regression models that consider all possible orderable dichotomizations of the outcome.
\subsection{Multiple Detection Limits} We illustrate our approach to handle multiple DLs with data from a multi-center HIV study. The data include 5301 adults living with HIV starting antiretroviral therapy (ART) at one of 5 study centers in Latin America between 2000 and 2018. Viral load (VL) measures the amount of virus circulating in a person with HIV. A high VL after ART initiation may indicate non-adherence or an ineffective ART regimen that should be switched. We study the association between VL at approximately 6 months after ART initiation and variables measured at ART initiation (baseline). The DLs for the outcome VL differed by site and calendar time. Figure \ref{fig:dl_change} shows the most frequent lower DL values for each year and at each site. There are five distinct lower DLs in this database: 20, 40, 50, 80, and 400 copies/mL. A total of 2992 (56\%) patients had 6-month VL censored at a DL: 45\%, 54\%, 52\%, 65\%, and 57\% at study sites in Argentina, Brazil, Chile, Mexico, and Peru, respectively. More study details are in Supplemental Material S2.2.
\begin{figure}
\caption{The changes of most frequent DL values every year at each study site over time.}
\label{fig:dl_change}
\end{figure}
A traditional analysis in the HIV literature would dichotomize VL as detectable and undetectable and perform logistic regression \citep{jiamsakul2017hiv}. There are a few issues that make this analysis less than ideal. First, all VLs above the DL (nearly half of all observations) would be collapsed into a \lq\lq detectable" category resulting in well-known loss of information due to dichotomizing continuous variables \citep{fedorov2009}. Second, because the DL varies with time and by site, the analyst is forced to dichotomize at the largest DL (in this case 400 copies/mL) or else perform an analysis where values above the DL at one site are treated differently than they would be treated at another site. For example, a VL of 300 copies/mL measured in Mexico in 2005 would be measured as `${<}400$' that same year in Peru; assigning this value as `${<}400$' results in lost information but leaving it as \lq\lq detectable" would make the outcome variable different across time and sites. A more parametric analysis might assume that the VL follows a specified distribution (e.g., log-normal distribution) and fit the censored data likelihood or multiply impute values below the DL from the assumed distribution to obtain estimated regression coefficients. However, distributional assumptions for values below the DL are strong and untestable, and given that over half of the response variables are below the DL, these assumptions would have a large impact on results.
In contrast, the CPM uses all available information (i.e., does not dichotomize the response variable) and makes much weaker assumptions than the fully parametric approaches. Similar to the parametric approaches, the CPM assumes non-informative censoring conditional on covariates (which is reasonable, given that DLs are determined by equipment/assays independent of true values) and that all observations follow a common distribution conditional on covariates, which permits borrowing information across sites and time. Unlike the fully parametric approach, however, the CPM does not fully specify this distribution. Rather, the CPM assumes that response variables follow a linear model with known error distribution after some unspecified transformation.
\begin{table}[]
\centering
\caption{The $\beta$ coefficients in CPMs can be interpreted as log odds ratios. We show the odds ratio (95\% confidence interval) and p-value for the predictors included in the model.}
\label{tab:odds_ratio}
\renewcommand{0.53}{0.53}
\begin{tabular}{lcc}
\hline
Predictor & Odds Ratio (95\% CI) & P-value \\
\hline
\textbf{Age} (per 10 years) & 0.98 (0.93, 1.03) & 0.418 \\
\textbf{Sex} & & 0.201 \\
\hspace{3mm} Male (reference) & 1 & \\
\hspace{3mm} Female & 0.90 (0.76, 1.06) & \\
\textbf{Study center} & & ${<}0.001$ \\
\hspace{3mm} Peru (reference) & 1 & \\
\hspace{3mm} Argentina & 1.26 (0.98, 1.61) & \\
\hspace{3mm} Brazil & 1.07 (0.91, 1.26) & \\
\hspace{3mm} Chile & 1.07 (0.90, 1.26) & \\
\hspace{3mm} Mexico & 0.59 (0.49, 0.70) & \\
\textbf{Route of infection} & & ${<}0.001$ \\
\hspace{3mm} Homosexual/Bisexual (reference) & 1 & \\
\hspace{3mm} Heterosexual & 0.96 (0.83, 1.10) & \\
\hspace{3mm} Other/Unknown & 0.79 (0.62, 1.01) & \\
\textbf{Prior AIDS event} & & 0.001 \\
\hspace{3mm} No (reference) & 1 & \\
\hspace{3mm} Yes & 1.24 (1.09, 1.41) & \\
\textbf{Baseline CD4} (per 1 square root cells/$\mu$L) & 1.09 (1.08, 1.10) & ${<}0.001$ \\
\textbf{Baseline VL} (per 1 $\log_{10}$ copies/mL) & 1.44 (1.34, 1.54) & ${<}0.001$ \\
\textbf{ART regimen} & & 0.034 \\
\hspace{3mm} NNRTI-based (reference) & 1 & \\
\hspace{3mm} INSTI-based & 0.55 (0.40, 0.75) & \\
\hspace{3mm} PI-based & 1.10 (0.95, 1.29) & \\
\hspace{3mm} Other & 2.57 (1.28, 5.16) & \\
\textbf{Months to VL measure} & 0.95 (0.92, 0.98) & 0.002 \\
\textbf{Calendar year} & 0.89 (0.88, 0.91) & ${<}0.001$\\
\hline
\end{tabular} \end{table}
\begin{figure}
\caption{The estimated conditional 50th and 90th percentiles of 6-month VL and the conditional probability of 6-month VL being greater than 1000 and 20 as functions of age (top row), prior AIDS events (middle row), and baseline VL (bottom row) while keeping other covariates at their medians (for continuous variables) or modes (for categorical variables) based on our method.}
\label{fig:age_aids_vl}
\end{figure}
We applied our method in Section 2.3 to fit a CPM of the 6-month VL on baseline variables with the logit link. Results are shown in Table \ref{tab:odds_ratio}. With the logit link, the $\beta$ parameters can be interpreted as log odds ratios and are presented as odds ratios in the table along with 95\% CIs. P-values are likelihood ratio test p-values. The results suggest that study center, route of infection, prior AIDS event, baseline CD4 count, baseline VL, ART regimen, the time from ART initiation until the VL measurement, and calendar year are all associated with VL at 6 months. Holding other variables fixed, a 10-fold increase in VL at baseline is associated with a 44\% increase in the odds of having a higher VL at 6 months (95\% CI 34\% to 54\%).
Quantiles and cumulative probabilities are also easily extracted from the CPM. The first row of Figure \ref{fig:age_aids_vl} are the estimated conditional 50th and 90th percentiles of 6-month VL and the conditional probabilities for 6-month VL being greater than 1000 and 20 copies/mL as functions of age. The plots show that VL at 6 months is fairly similar across ages after fixing the other covariates. The smallest DL is 20 copies/mL, and all VL less than this DL belong to the smallest ordered category, which we label as `${<}20$'. The second row of Figure \ref{fig:age_aids_vl} contains the estimated conditional quantiles and probabilities as functions of whether a patient had an AIDS event prior to starting ART. People with a prior AIDS event (36\%) tended to have a higher VL at 6 months. The third row of Figure \ref{fig:age_aids_vl} is the estimated conditional quantiles and probabilities as functions of baseline VL. People with a higher baseline VL tended to have a higher VL at 6 months.
Supplementary Material S2.2 contains results from a similar CPM, except with continuous covariates expanded using restricted cubic splines to relax linearity assumptions and increase model flexibility. The results are fairly similar.
For comparisons, we also analyzed the data using competing approaches described earlier. First, we fit logistic regression to 6-month VL values dichotomized as ${<}400$ vs. ${\ge}400$ copies/mL, corresponding to the highest DL. Results are in Table S4 of the Supplementary Material S2.2. The CPM and the logistic regression model gave similar estimates of the beta coefficients (which are log odds ratios), although there were some differences in the estimates and the CIs from CPMs tend to be narrower, as expected. In logistic regression, the log odds ratios are based on the single undetectable vs. detectable dichotomization, while those in CPMs are based on dichotomizations at each response value. Second, we fit a full likelihood-based model assuming the outcome variable was normally distributed after $\log_{10}(\cdot)$ transformation. Note that even the log-transformed 6-month VL was still quite skewed (shown in Figure S4), and hence the assumptions of this fully parametric approach were questionable. The parameters in this approach and those from the CPM are not directly comparable because they are on different scales, however, the directions of associations were similar.
\section{Simulations}
Extensive simulations of CPMs with continuous data have been reported elsewhere \citep{liu2017modeling, tian2020empirical}. Here we present a limited set of simulations investigating the performance of CPMs with data subject to single and multiple DLs.
\subsection{Single Detection Limits}
Data were generated for sample sizes of $n=100$ and $n=500$ such that the outcome $Y$ followed a normal linear model after log transformation in the following manner: \begin{align*}
Y = \exp(Y^*), \textrm{ where } Y^* &= X\beta + \epsilon,
\beta =1,
X \sim N(0,1), \textrm{ and }
\epsilon \sim N(0,1).
\end{align*} Various scenarios of DLs of $Y$ were considered: 1. No DL. 2. One lower DL at 0.25 (censoring rate 16.3\%). 3. One upper DL at 4 (censoring rate 16.3\%). 4. One lower DL at 0.25 and one upper DL at 4 (censoring rate 32.7\%). 5. One lower DL at 4 (censoring rate 83.7\%). In addition, we considered a scenario with a more complicated transformation: 6. One lower DL at 0.0625 and \begin{equation*}
Y = \left\{\begin{array}{lr}
\exp(2Y^*) \textrm{ if } Y^*<\log(0.25)\\
\sqrt{\exp(Y^*)}\textrm{ if } \log(0.25) \le Y^* <\log(2)\\
\exp(Y^*)\textrm{ if } Y^*\ge \log(2).
\end{array} \right. \end{equation*} Note that the $Y$ in scenario 6 is a monotonic transformation of that in scenario 2 with exactly the same censoring rate.
CPMs were fit to the observed data $\{X,Y\}$ without any knowledge of the correct transformation or $Y^*$. We simulated 10,000 replications under each scenario. Percent bias, root mean squared error (RMSE), and coverage of 95\% CIs were estimated with respect to $\beta$, conditional medians for $X=\{0,1\}$, and conditional CDFs at $y=1.5$ for $X=\{0,1\}$.
Table \ref{tab:one_dl} shows results under correctly specified models (i.e., probit link function and $X$ correctly included). CPMs resulted in nearly unbiased estimation and good CI coverage. As the sample size increased, both the bias and RMSE decreased. Note that estimation of the condition medians was ``perfect'' in scenario 5 because the true conditional medians were below the lower DL due to the high censoring rate and the estimated conditional medians were always `${<}4$', the lowest outcome category corresponding to below the DL. The estimate of $\beta$ was more variable in scenario 5 because of the high censoring rate. The estimation of $\beta$ in scenario 6, where data were generated from the complicated transformation, was exactly the same as that in scenario 2 because the same seed was used in all simulation scenarios and the order information above the DL was identical between scenarios 2 and 6. However, the conditional medians and CDFs depend on the scale of the outcome, and their estimates differed between scenarios 2 and 6.
\begin{table} \begin{threeparttable}
\centering
\caption{Simulation results for single DLs}
\label{tab:one_dl}
\renewcommand{0.53}{0.53}
\begin{tabular}{@{\extracolsep{5pt}}cccccccc@{}}
\hline
& & \multicolumn{3}{c}{n=100} & \multicolumn{3}{c}{n=500}\\
\cline{3-5}\cline{6-8}
Parameter & Truth & Bias(\%) & RMSE & Coverage & Bias(\%)
& RMSE & Coverage\\
\hline
\textbf{Scenario 1} & & & & & & & \\
$\beta$ & 1 & 2.803 & 0.133 & 0.944 & 0.638 & 0.057 & 0.945\\
$Q(0.5|X=0)$ & 1 & -0.388 & 0.140 & 0.951 & -0.124 & 0.063 & 0.951\\
$Q(0.5|X=1)$ & 2.718 & 1.552 & 0.494 & 0.949 & 0.321 & 0.218 & 0.951\\
$F(1.5|X=0)$ & 0.658 & 0.117 & 0.054 & 0.949 & 0.059 & 0.024 & 0.951\\
$F(1.5|X=1)$ & 0.276 & -1.429 & 0.060 & 0.949 & -0.383 & 0.026 & 0.945\\
\hline
\textbf{Scenario 2}& & & & & & & \\
$\beta$ & 1& 2.665 & 0.138 & 0.945 & 0.585 & 0.057 & 0.948\\
$Q(0.5|X=0)$ & 1 & -0.240 & 0.142 & 0.953 & 0.028 & 0.063 & 0.948\\
$Q(0.5|X=1)$ & 2.718 & 1.445 & 0.498 & 0.953 & 0.406 & 0.222 & 0.946\\
$F(1.5|X=0)$ & 0.658 & 0.005 & 0.054 & 0.946 & -0.085 & 0.024 & 0.950\\
$F(1.5|X=1)$ & 0.276 & -0.479 & 0.061 & 0.948 & 0.368 & 0.028 & 0.943\\
\hline
\textbf{Scenario 3}& & & & & & &\\
$\beta$ & 1 & 2.710 & 0.139 & 0.943 & 0.581 & 0.058 & 0.948\\
$Q(0.5|X=0)$ & 1 & -0.460 & 0.141 & 0.951 & -0.020 & 0.063 & 0.949\\
$Q(0.5|X=1)$ & 2.718 & 0.803 & 0.477 & 0.954 & 0.310 & 0.223 & 0.945\\
$F(1.5|X=0)$ & 0.658 & 0.0147 & 0.054 & 0.946 & -0.083 & 0.024 & 0.951\\
$F(1.5|X=1)$ & 0.276 & -0.487 & 0.062 & 0.948 & 0.381 & 0.028 & 0.941\\
\hline
\textbf{Scenario 4}& & & & & & &\\
$\beta$ & 1 & 2.544 & 0.139 & 0.945 & 0.538 & 0.058 & 0.951\\
$Q(0.5|X=0)$ & 1 & -0.243 & 0.141 & 0.954 & 0.028 & 0.063 & 0.949\\
$Q(0.5|X=1)$ & 2.718 & 1.017 & 0.477 & 0.953 & 0.358 & 0.223 & 0.947\\
$F(1.5|X=0)$ & 0.658 & 0.004 & 0.054 & 0.947 & -0.086 & 0.024 & 0.950 \\
$F(1.5|X=1)$ & 0.276 & -0.285 & 0.062 & 0.948 & 0.432 & 0.028 & 0.943\\
\hline
\textbf{Scenario 5}& & & & & & \\
$\beta$ & 1 & 7.315 & 0.276 & 0.946 & 1.330 & 0.101 & 0.948\\
$Q(0.5|X=0)$ & 1 & 0\tnote{*} & 0 & 1 & 0 & 0 & 1\\
$Q(0.5|X=1)$ & 2.718 & 0 & 0 & 1 & 0 & 0 & 1\\
$F(1.5|X=0)$ & 0.658 & 0.183 & 0.026 & 0.954 & -0.029 & 0.010 & 0.949 \\
$F(1.5|X=0)$ & 0.276 &-0.189 & 0.069 & 0.952 & -0.169 & 0.030 & 0.949 \\
\hline
\textbf{Scenario 6} & & & & & & \\
$\beta$ & 1 & 2.665 & 0.138 & 0.945 & 0.585 & 0.057 & 0.948 \\
$Q(0.5|X=0)$ & 1 & -0.841 & 0.071 & 0.951 & -0.503 & 0.032 & 0.945\\
$Q(0.5|X=1)$ & 0.368 & -0.312 & 0.542 & 0.953 & -0.529 & 0.222 & 0.946\\
$F(1.5|X=0)$ & 0.654 & 0.254 & 0.048 & 0.947 & 0.056 & 0.022 & 0.953\\
$F(1.5|X=1)$ & 0.500 & 0.061 & 0.069 & 0.949 & 0.536 & 0.032 & 0.946\\
\hline
\end{tabular}
\begin{tablenotes}\footnotesize \item[*] The results of zero bias and RMSE when there is a high censoring rate are because the true conditional medians are below the lower DL and the estimated conditional medians were always `${<}4$', the lowest outcome category corresponding to below the DL. \end{tablenotes}
\end{threeparttable} \end{table}
Table S5 in the Supplementary Material shows results under scenario 2 with $n=1000$ comparing CPMs with some widely used methods for handling DLs, specifically single imputation with $l/2$, single imputation with $l/\sqrt{2}$, multiple imputation, and fully parametric maximum likelihood estimation (MLE). For all non-CPM approaches, we first correctly assumed that the outcome variable followed a log-normal distribution. With the imputation approaches, unobserved values were imputed, then a linear regression model was fit on the log-transformed outcome to obtain the $\beta$ estimate, and median regression was used to estimate conditional medians. In multiple imputation, the correct tail distribution was used for imputing data and 10 iterations were performed for each data set. As expected, the MLE performed the best with the lowest bias and RMSE, and highest efficiency because the distributional assumptions matched the true distribution. The performance of multiple imputation was similar to that of the MLE, but with higher RMSE. As a semiparametric method, the CPM, also resulted in minimal bias and correct coverage but had slightly larger variance and RMSE. In contrast, the single imputation estimators were biased and tended to have poor coverage, especially for estimating $\beta$. We also evaluated the comparator methods under misspecification of the transformation. We simulated datasets with $X\sim N(5,1)$, $Y^*=X\beta + \epsilon$, $\beta=1$, $\epsilon \sim N(0,1)$, $Y = {Y^*}^2$, $n=1000$, and $l=13.12$ (approximately 17\% censored). The non-CPM approaches assumed a normal linear model after an incorrectly specified log-transformation. As shown in the bottom half of Table S5, only the CPM was able to properly estimate $\beta$ and the conditional medians, because pre-transformation and strict distributional assumptions are not needed for fitting CPMs.
Finally, the Supplementary Material Table S6 shows the performance of CPMs for the data generated in scenario 2 under moderate and severe link function misspecification (i.e., fitting CPMs with logit and loglog link functions, respectively). Link function misspecification is equivalent to misspecification of the distribution of $\epsilon$ because $F_\epsilon=G^{-1}$. The performance of CPMs was reasonable with moderate link function misspecification with bias under 6\% and coverage of 95\% CI close to 0.95 with $n=100$, although as low as 0.91 with $n=500$. With severe link function misspecification, the performance of CPMs was noticeably worse, with bias as high as 12\% and coverage as low as 0.60 for the conditional median at $X=1$.
\subsection{Multiple Detection Limits}
To illustrate the use of CPMs with multiple detection limits, we simulated data from 3 study sites. The data were generated in a similar way as in Section 4.1, but different DLs were applied at different sites and the distribution of the covariate $X$ was allowed to vary across sites in some scenarios. Specifically, we considered the following 5 scenarios: \begin{enumerate}
\item[1.] Lower DLs 0.16, 0.30, and 0.50 for the 3 sites (about 10\%, 20\%, and 30\% censored), and $X$ is independent of DLs/sites.
\item[2.] Upper DLs 0.16, 0.30, and 0.50 for the 3 sites (about 90\%, 80\%, and 70\% censored), and $X$ is independent of DLs/sites.
\item[3.] Lower DLs 0.16, 0.30, and 0.50 for the 3 sites (about 17\%, 20\%, and 20\% censored), and $X \sim N(\mu_x,1)$ where $\mu_x=-0.5, 0,$ and 0.5 for site 1, 2, and 3, respectively.
\item[4.] Upper DLs 0.16, 0.30, and 0.50 for the 3 sites (about 83\%, 80\%, and 80\% censored), and $X \sim N(\mu_x,1)$ where $\mu_x=-0.5, 0,$ and 0.5 for site 1, 2, and 3, respectively.
\item[5.] Lower DLs 0.2, 0.3, and -$\infty$ (13\%, 20\%, and 0\% censored) and upper DLs at $\infty$, 4, and 3.5 (0\%, 19\%, and 16\% censored) for the 3 sites, and $X$ is independent of DLs/sites. \end{enumerate} We considered two sample sizes, $n=150$ and $n=900$, with the sample sizes distributed equally across sites. In scenarios 2 and 4, because of the high censoring rates, we estimated the quantiles at $p=0.03$ (i.e., 3rd percentile) and CDFs at $y=0.05$. Results from fitting the CPM based on 10,000 replications are shown in Supplementary Material S3. In summary, estimates had very low bias and confidence intervals had proper coverage in all simulation scenarios.
\section{Discussion}
In this paper, we have described an approach to address detection limits in response variables using CPMs. CPMs have several advantages over existing methods for addressing DLs. They make minimal distributional assumptions, they yield interpretable parameters, and they are invariant to the value assigned to measures outside DLs. Any values outside the lowest/highest DLs are simply assigned to the lowest/highest ordinal categories, and estimation proceeds naturally. CPMs are also easily extended to handle multiple DLs. From simulation studies, we saw that CPMs performed well, even with high censoring rates and relatively small sample sizes. We also illustrated the use of CPMs with two quite different HIV datasets with censored response data. Similar datasets with limits of detection are quite common in biomedical research; the CPM is an effective analysis tool in these settings.
CPMs have some limitations. Although CPMs do not make distributional assumptions on the response variable, the link function must still be specified, which corresponds to making an assumption on the distribution of the response variable after an unspecified transformation. Performance can be poor with severe link function misspecification; however, CPMs appear to be fairly robust to moderate misspecification. In addition, because we do not make distributional assumptions outside DLs, we are not able to estimate conditional expectations after fitting a CPM; however, with DLs, conditional quantiles are probably more reasonable statistics to report anyway. The codes for applications and simulations are available at https://github.com/YuqiTian35/DetectionLimitCode.
Further research could consider extensions of CPMs to handle clustered or longitudinal data with DLs. It may be of interest to study the use of these models with right-censored failure time data (i.e., survival data), where each observation is potentially subject to a different censoring time; the current manuscript only considered situations with a relatively small number of potential censoring times (i.e., upper DLs).
\end{document} | arXiv |
\begin{document}
\title{Resonant semilinear Robin problems with a general potential}
\pagestyle{plain}
\begin{center} \noindent \begin{minipage}{0.85\textwidth}\parindent=15.5pt
{\small{ \noindent {\bf Abstract.} We consider a semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential. The reaction term is a Carath\'eodory function which is resonant with respect to any nonprincipal eigenvalue both at $\pm \infty$ and 0. Using a variant of the reduction method, we show that the problem has at least two nontrivial smooth solutions.}
\noindent {\bf{Keywords:}} resonant problem, reduction method, regularity theory, indefinite and unbounded potential, local linking.
\noindent{\bf{2010 Mathematics Subject Classification:}} 35J20, 35J60. }
\end{minipage} \end{center}
\section{Introduction}
Let $\Omega\subseteq\mathbb R^N$ be a bounded domain with a $C^2$-boundary $\partial\Omega$. In this paper we study the following semilinear Robin problem:
\begin{equation}\label{eq1}
\left\{\begin{array}{l}
-\Delta u(z)+\xi(z)u(z) = f(z,u(z)) \ \mbox{in}\ \Omega, \ \\
\displaystyle\frac{\partial u}{\partial n} + \beta(z)u = 0 \ \mbox{in}\ \partial\Omega\,.
\end{array}\right\} \end{equation}
In this problem, the potential function $\xi(\cdot)$ is unbounded and indefinite (that is, sign-changing). So, in problem (\ref{eq1}) the differential operator (on the left-hand side of the equation), is not coercive. The reaction term $f(z,x)$ is a Carath\'eodory function (that is, for all $x\in\mathbb R$, $z\mapsto f(z,x)$ is measurable and for almost all $z\in\Omega$, $x\mapsto f(z,x)$ is continuous) and $f(z,\cdot)$ exhibits linear growth as $x\rightarrow\pm\infty$. In fact, we can have resonance with respect to any nonprincipal eigenvalue of $-\Delta+\xi(z)I$ with the Robin boundary condition. This general structure of the reaction term, makes the use of variational methods problematic. To overcome these difficulties, we develop a variant of the so-called ``reduction method", originally due to Amann \cite{1} and Castro \& Lazer \cite{3}. However, in contrast to the aforementioned works, the particular features of our problem lead to a reduction on an infinite dimensional subspace and this
is a source of additional technical difficulties. In the boundary condition, $\frac{\partial u}{\partial n}$ is the normal derivative defined by extension of the continuous linear map $$u\mapsto\frac{\partial u}{\partial n}=(Du,n)_{\mathbb R^N}\ \mbox{for all}\ u\in C^1(\overline{\Omega}),$$ with $n(\cdot)$ being the outward unit normal on $\partial\Omega$. The boundary coefficient $\beta\in W^{1,\infty}(\partial\Omega)$ satisfies $\beta(z)\geq 0$ for all $z\in\partial\Omega$. We can have $\beta\equiv 0$, which corresponds to the Neumann problem.
Recently there have been existence and multiplicity results for semilinear elliptic problems with general potential. We mention the works of Hu \& Papageorgiou \cite{9}, Kyritsi \& Papageorgiou \cite{10}, Papageorgiou \& Papalini \cite{12}, Qin, Tang \& Zhang \cite{17} (Dirichlet problems), Gasinski \& Papageorgiou \cite{6}, Papageorgiou \& R\u adulescu \cite{13, 14} (Neumann problems) and for Robin problems there are the works of Shi \& Li \cite{18} (superlinear reaction), D'Agui, Marano \& Papageorgiou \cite{4} (asymmetric reaction), Hu \& Papageorgiou (logistic reaction) and Papageorgiou \& R\u adulescu \cite{15} (reaction with zeros). In all the aforementioned works the conditions are in many respects more restrictive or different and consequently the mathematical tools are different. It seems that our work here is the first to use this variant of the reduction method on Robin problems.
\section{Mathematical background}
Let $X$ be a Banach space and let $X^*$ be its topological dual. By $\left\langle \cdot,\cdot\right\rangle$ we denote the duality brackets for the pair $(X^*,X)$. Given $\varphi\in C^1(X,\mathbb R)$, we say that $\varphi$ satisfies the ``Cerami condition" (the ``C-condition" for short), if the following property holds \begin{center} ``Every sequence $\{u_n\}_{n\geq 1}\subseteq X$ such that $\{\varphi(u_n)\}_{n\geq 1}\subseteq\mathbb R$ is bounded and
$$(1+||u_n||)\varphi'(u_n)\rightarrow 0\ \mbox{in}\ X^*,$$ admits a strongly convergent subsequence". \end{center}
This is a compactness-type condition on the functional $\varphi$ and is more general that the usual Palais-Smale condition. The two notions are equivalent when $\varphi$ is bounded below (see Motreanu, Motreanu \& Papageorgiou \cite[p. 104]{11}).
Our multiplicity result will use the following abstract ``local linking" theorem of Brezis \& Nirenberg \cite{2}. \begin{theorem}\label{th1}
Let $X$ be a Banach space such that $X=Y\oplus V$ with ${\rm dim}\, Y<+\infty$. Assume that $\varphi\in C^1(X,\mathbb R)$ satisfies the C-condition, it is bounded below, $\varphi(0)=0$, $\inf\limits_{X}\varphi<0$ and there exists $\rho>0$ such that
\begin{eqnarray*}
&&\varphi(y)\leq 0\ \mbox{for all}\ y\in Y\ \mbox{with}\ ||y||\leq\rho,\\
&&\varphi(v)\geq 0\ \mbox{for all}\ v\in V\ \mbox{with}\ ||v||\leq\rho
\end{eqnarray*}
(that is, $\varphi$ has a local linking at $u=0$ with respect to the direct sum $Y\oplus V$). Then $\varphi$ has at least two nontrivial critical points. \end{theorem} \begin{remark}
The result is true even if one of the component subspaces $Y$ or $V$ is trivial. Moreover, if ${\rm dim}\,V=0$, then we can allow $Y$ to be infinite dimensional. \end{remark}
We will use the following spaces:
\begin{itemize}
\item the Sobolev space $H^1(\Omega)$;
\item the Banach space $C^1(\overline{\Omega})$; and
\item the ``boundary" Lebesgue spaces $L^r(\partial\Omega)\ 1\leq r\leq\infty$.
\end{itemize}
The Sobolev space $H^1(\Omega)$ is a Hilbert space with the following inner product
$$(u,v)=\int_{\Omega}uvdz+\int_{\Omega}(Du,Dv)_{\mathbb R^N}dz\ \mbox{for all}\ u,v\in H^1(\Omega).$$
By $||\cdot||$ we denote the norm corresponding to this inner product, that is,
$$||u||=[||u||^2_2+||Du||_2^2]^{1/2}\ \mbox{for all}\ u,v\in H^1(\Omega).$$
On $\partial\Omega$ we consider the $(N-1)$-dimensional Hausdorff (surface) measure denoted by $\sigma(\cdot)$. Using this measure on $\partial\Omega$, we can define in the usual way the Lebesgue spaces $L^r(\partial\Omega)$, $1\leq r\leq\infty$. From the theory of Sobolev spaces we know that there exists a unique continuous linear map $\gamma_0:H^1(\Omega)\rightarrow L^2(\partial\Omega)$, known as the ``trace map", which satisfies
$$\gamma_0(u)=u|_{\partial\Omega}\ \mbox{for all}\ u\in H^1(\Omega)\cap C(\overline{\Omega}).$$
So, the trace map assigns ``boundary values" to any Sobolev function (not just to the regular ones). This map is compact into $L^r(\partial\Omega)$ for all $r\in\left[1,\frac{2(N-1)}{N-2}\right)$ if $N\geq 3$ and into $L^r(\partial\Omega)$ for all $r\geq 1$ if $N=1,2$. Also, we have
$${\rm ker}\, \gamma_0=H^1_0(\Omega)\ \mbox{and}\ {\rm im}\, \gamma_0=H^{\frac{1}{2},2}(\partial\Omega).$$
In what follows, for the sake of notational simplicity, we shall drop the use the trace map $\gamma_0$. The restrictions of all Sobolev functions on $\partial\Omega$, are understood in the sense of traces.
Next, we recall some basic facts about the spectrum of the differential operator $-\Delta+\xi(z)I$ with the Robin boundary condition. So, we consider the following linear eigenvalue problem:
\begin{eqnarray}\label{eq2}
\left\{\begin{array}{ll}
-\Delta u(z)+\xi(z)u(z)=\hat{\lambda}u(z)&\mbox{in}\ \Omega,\\
\displaystyle\frac{\partial u}{\partial n}+\beta(z)u=0&\mbox{on}\ \partial\Omega
\end{array}\right\}
\end{eqnarray}
Our conditions on the data of (\ref{eq2}) are the following: \begin{itemize}
\item $\xi\in L^s(\Omega)$ with $s>N$; and
\item $\beta\in W^{1,\infty}(\partial\Omega)$ with $\beta(z)\geq 0$ for all $z\in\partial\Omega$. \end{itemize}
Let $\gamma:H^1(\Omega)\rightarrow \mathbb R$ be the $C^1$-functional defined by
$$\gamma(u)=||Du||_2^2+\int_{\Omega}\xi(z)u^2dz+\int_{\partial\Omega}\beta(z)u^2d\sigma\ \mbox{for all}\ u\in H^1(\Omega).$$
By D'Agui, Marano \& Papageorgiou \cite{4}, we know that there exists $\mu>0$ such that \begin{equation}\label{eq3}
\gamma(u)+\mu||u||^2_2\geq c_0||u||^2\ \mbox{for all}\ u\in H^1(\Omega),\ \mbox{and some}\ c_0>0. \end{equation}
Using (\ref{eq3}) and the spectral theorem for compact self-adjoint operators on a Hilbert space, we produce the spectrum $\sigma_0(\xi)$ of (\ref{eq2}) and we have that $\sigma_0(\xi)=\{\hat{\lambda}_k\}_{k\geq 1}$ a sequence of distinct eigenvalues with $\hat{\lambda}_k\rightarrow+\infty$ as $k\rightarrow+\infty$. By $E(\hat{\lambda}_k)$ (for all $k\in\mathbb N$), we denote the eigenspace corresponding to the eigenvalue $\hat{\lambda}_k$. We know that $E(\hat{\lambda}_k)$ is finite dimensional. Moreover, the regularity theory of Wang \cite{19} implies that $E(\hat{\lambda}_k)\subseteq C^1(\overline{\Omega})$ for all $k\in\mathbb N$. The Sobolev space $H^1(\Omega)$ admits the following orthogonal direct sum decomposition $$H^1(\Omega)=\overline{{\underset{\mathrm{k\geq 1}}\oplus}E(\hat{\lambda}_k)}.$$
The elements of $\sigma_0(\xi)$ have the following properties: \begin{itemize}
\item $\hat{\lambda}_1$ is simple (that is, ${\rm dim}\, E(\hat{\lambda}_1)=1$).\\
\begin{eqnarray}
&&\bullet\ \ \hat{\lambda}_1=\inf\left[\frac{\gamma(u)}{||u||^2_2}:u\in H^1(\Omega),u\neq 0\right].\hspace{7cm}\label{eq4}\\
&&\bullet\ \ \hat{\lambda}_m=\inf\left[\frac{\gamma(u)}{||u||^2_2}:u\in\overline{{\underset{\mathrm{k\geq m}}\oplus}E(\hat{\lambda}_k)},u\neq 0\right]\hspace{7cm}\nonumber\\
&&\hspace{1cm}=\sup\left[\frac{\gamma(u)}{||u||^2_2}:u\in\overset{m}{\underset{\mathrm{k=1}}\oplus}E(\hat{\lambda}_k),u\neq 0\right]\ \mbox{for}\ m\geq 2.\label{eq5}
\end{eqnarray} \end{itemize}
The infimum in (\ref{eq4}) is realized on $E(\hat{\lambda}_1)$, while both the infimum and supremum in (\ref{eq5}) are realized on $E(\hat{\lambda}_m)$. It follows that the elements of $E(\hat{\lambda}_1)$ have fixed sign, while those of $E(\hat{\lambda}_m)$ ($m\geq 2$) are nodal (sign-changing). The eigenspaces have the so-called ``Unique Continuation Property" (UCP for short) which says that if $u\in E(\hat{\lambda}_k)$ and $u(\cdot)$ vanishes on a set of positive Lebesgue measure, then $u\equiv 0$. As a consequence of the UCP, we have the following useful inequalities (see D'Agui, Marano \& Papageorgiou \cite{4}). \begin{lemma}\label{lem2}
\begin{itemize}
\item[(a)] If $\eta\in L^{\infty}(\Omega),\ \eta(z)\geq\hat{\lambda}_m$ for almost all $z\in\Omega,\ m\in\mathbb N$ and $\eta\neq \hat{\lambda}_m$, then there exists $c_1>0$ such that
$$\gamma(u)-\int_{\Omega}\eta(z)u^2dz\leq-c_1||u||^2\ \mbox{for all}\ u\in\overset{m}{\underset{\mathrm{k=1}}\oplus}E(\hat{\lambda}_k).$$
\item[(b)] If $\eta\in L^{\infty}(\Omega),\ \eta(z)\leq\hat{\lambda}_m$ for almost all $z\in\Omega$, $m\in\mathbb N$ and $\eta\neq\hat{\lambda}_m$ then there exists $c_2>0$ such that
$$\gamma(u)-\int_{\Omega}\eta(z)u^2dz\geq c_2||u||^2\ \mbox{for all}\ u\in\overline{{\underset{\mathrm{k\geq m}}\oplus E(\hat{\lambda}_k)}}.$$
\end{itemize} \end{lemma}
Given $m\in\mathbb N$, let $H_-=\overset{m}{\underset{\mathrm{k=1}}\oplus}E(\hat{\lambda}_k)$, $H^0=E(\hat{\lambda}_{m+1})$, $H_+=\overline{{\underset{\mathrm{k\geq m+2}}\oplus E(\hat{\lambda}_k)}}$. We have the following orthogonal direct sum decomposition $$H^1(\Omega)=H_-\oplus H^0\oplus H_+.$$
So, every $u\in H^1(\Omega)$ admits a unique sum decomposition of the form $$u=\bar{u}+u^0+\hat{u}\ \mbox{with}\ \bar{u}\in H_-,\ u^0\in H^0,\ \hat{u}\in H_+.$$
Also, we set $$V=H^0\oplus H_+.$$
Finally, let us fix our notation. By $|\cdot|_N$ we denote the Lebesgue measure on $\mathbb R^N$ and by $A\in\mathcal{L}(H^1(\Omega),H^1(\Omega)^*)$ the linear operator defined by $$\left\langle A(u),h\right\rangle=\int_{\Omega}(Du,Dh)_{\mathbb R^N}dz\ \mbox{for all}\ u,h\in H^1(\Omega)$$ (by $\left\langle \cdot,\cdot\right\rangle$ we denote the duality brackets for the pair $(H^1(\Omega)^*,H^1(\Omega))$). Also, given a measurable function $f:\Omega\times\mathbb R\rightarrow\mathbb R$ (for example a Carath\'eodory function), we set $$N_f(u)(\cdot)=f(\cdot,u(\cdot))\ \mbox{for all}\ u\in H^1(\Omega)$$ (the Nemytski map corresponding to $f$). Evidently, $z\mapsto N_f(u)(z)$ is measurable. For $\varphi\in C^1(X,\mathbb R)$, we set $$K_{\varphi}=\{u\in X:\varphi'(u)=0\}$$ (the critical set of $\varphi$).
\section{Pair of nontrivial solutions}
The hypotheses on the data of (\ref{eq1}) are the following:
\begin{itemize}
\item $H(\xi)$: $\xi\in L^s(\Omega)$ with $s>N$; and
\item $H(\beta)$: $\beta\in W^{1,\infty}(\partial\Omega)$ with $\beta(z)\geq 0$ for all $z\in\partial\Omega$. \end{itemize}
\begin{remark}
We can have $\beta\equiv 0$ and this case corresponds to the Neumann problem. \end{remark}
$H(f)$: $f:\Omega\times\mathbb R\rightarrow\mathbb R$ is a Carath\'eodory function such that $f(z,0)=0$ for almost all $z\in\Omega$ and \begin{itemize}
\item[(i)] $|f(z,x)|\leq a(z)(1+|x|)$ for almost all $z\in\Omega$, and all $x\in\mathbb R$ with $a\in L^{\infty}(\Omega)_+$;
\item[(ii)] there exist $m\in\mathbb N$ and $\eta\in L^{\infty}(\Omega)$ such that
\begin{eqnarray*}
&&\eta(z)\geq\hat{\lambda}_m\ \mbox{for almost all}\ z\in\Omega,\eta\not\equiv\hat{\lambda}_m,\\
&&(f(z,x)-f(z,x'))(x-x')\geq\eta(z)(x-x')^2\ \mbox{for almost all}\ z\in\Omega,\ \mbox{and all}\ x,x'\in\mathbb R;
\end{eqnarray*}
\item[(iii)] if $F(z,x)=\int^x_0f(z,s)ds$, then $\limsup\limits_{x\rightarrow\pm\infty}\frac{2F(z,x)}{x^2}\leq \hat{\lambda}_{m+1}$ and
$$\lim\limits_{x\rightarrow\pm\infty}[f(z,x)x-2F(z,x)]=+\infty\ \mbox{uniformly for almost all}\ z\in\Omega;$$
\item[(iv)] there exist $l\in\mathbb N$, $l\geq m+2$, a function $\vartheta\in L^{\infty}(\Omega)$ and $\delta>0$ such that
\begin{eqnarray*}
&&\vartheta(z)\leq\hat{\lambda}_l\ \mbox{for almost all}\ z\in\Omega,\ \vartheta\not\equiv\hat{\lambda}_l,\\
&&\hat{\lambda}_{l-1}x^2\leq f(z,x)x\leq\vartheta(z)x^2\ \mbox{for almost all}\ z\in\Omega,\ \mbox{and all}\ |x|\leq\delta.
\end{eqnarray*} \end{itemize}
Let $\varphi:H^1(\Omega)\rightarrow\mathbb R$ be the energy (Euler) functional for problem (\ref{eq1}) defined by $$\varphi(u)=\frac{1}{2}\gamma(u)-\int_{\Omega}F(z,u)dz\ \mbox{for all}\ u\in H^1(\Omega).$$
Evidently, $\varphi\in C^1(H^1(\Omega))$.
Recall that $$H^1(\Omega)=H_-\oplus H^0\oplus H_+$$ with $H_-=\overset{m}{\underset{\mathrm{k=1}}\oplus}E(\hat{\lambda}_k).\ H^0=E(\hat{\lambda}_{m+1}),\ H_+=\overline{{\underset{\mathrm{k\geq m+2}}\oplus}E(\hat{\lambda}_k)}$ and $$V=H^0\oplus H_+.$$
The next proposition is crucial in the implementation of the reduction method. \begin{proposition}\label{prop3}
If hypotheses $H(\xi),H(\beta),H(f)$ hold, then there exists a continuous map $\tau:V\rightarrow H_-$ such that
$$\varphi(v+\tau(v))=\max[\varphi(v+y):y\in H_-]\ \mbox{for all}\ v\in V.$$ \end{proposition} \begin{proof}
We fix $v\in V$ and consider the $C^1$-functional $\varphi_v:H^1(\Omega)\rightarrow\mathbb R$ defined by
$$\varphi_v(u)=\varphi(v+u)\ \mbox{for all}\ u\in H^1(\Omega).$$
By $i_{H_-}:H_-\rightarrow H^1(\Omega)$ we denote the embedding of $H_-$ into $H^1(\Omega)$. Let
$$\hat{\varphi}_v=\varphi_v\circ i_{H_-}.$$
From the chain rule, we have
\begin{equation}\label{eq6}
\hat{\varphi}'_v=p_{H^*_-}\circ\varphi'_v,
\end{equation}
with $p_{H^*_-}$ being the orthogonal projection of the Hilbert space $H^1(\Omega)$ onto $H^*_-$. By $\left\langle \cdot,\cdot\right\rangle_{H_-}$ we denote the duality brackets for the pair $(H^*_-,H_-)$. For $y,\ y'\in H_-$, we have
\begin{eqnarray}\label{eq7}
&&\left\langle \hat{\varphi}'_v(y)-\hat{\varphi}'_v(y'),y-y'\right\rangle_{H_-}\nonumber\\
&=&\left\langle \varphi'_v(y)-\varphi'_v(y'),y-y'\right\rangle\ (\mbox{see (\ref{eq6})})\nonumber\\
&=&\gamma(y-y')-\int_{\Omega}(f(z,v+y)-f(z,v+y'))(y-y')dz\nonumber\\
&\leq&\gamma(y-y')-\int_{\Omega}\eta(z)(y-y')^2dz\ (\mbox{see hypothesis}\ H(f)(ii))\nonumber\\
&\leq&-c_1||y-y'||^2\ (\mbox{see Lemma \ref{lem2}}).
\end{eqnarray}
This implies that
\begin{equation}\label{eq8}
-\hat{\varphi}'_v\ \mbox{is strongly monotone and therefore}\ -\hat{\varphi}_v\ \mbox{is strictly convex}.
\end{equation}
We have
\begin{eqnarray}\label{eq9}
\left\langle \hat{\varphi}'_v(y),y\right\rangle_{H_-}&=&\left\langle \varphi'_v(y),y\right\rangle\nonumber\\
&=&\left\langle \varphi'_v(y)-\varphi'_v(0),y\right\rangle+\left\langle \varphi'_v(0),y\right\rangle\nonumber\\
&\leq&-c_1||y||^2+c_3||y||\ \mbox{for some}\ c_3>0\ (\mbox{see (\ref{eq7})}),\nonumber\\
\Rightarrow-\hat{\varphi}'_v\ \mbox{is coercive}&&
\end{eqnarray}
The continuity and monotonicity of $-\hat{\varphi}'_v$ (see (\ref{eq8})), imply that
\begin{equation}\label{eq10}
-\hat{\varphi}'_v\ \mbox{is maximal monotone}.
\end{equation}
However, a maximal monotone and coercive map is surjective (see, for example, Hu \& Papageorgiou \cite[p. 322]{8}). So, we infer from (\ref{eq9}) and (\ref{eq10}) that there is a unique $y_0\in H_-$ such that
\begin{equation}\label{eq11}
\hat{\varphi}'_v(y_0)=0\ (\mbox{see (\ref{eq8})}).
\end{equation}
Moreover, $y_0$ is the unique maximizer of the function $\hat\varphi_v$. So, we can define the map $\tau:V\rightarrow H_-$ by setting $\tau(v)=y_0$. Then we have
\begin{eqnarray}
&&\varphi(v+\tau(v))=\max[\varphi(v+y):y\in H_-],\label{eq12}\\
&\Rightarrow&p_{H^*_-}\varphi'(v+\tau(v))=0\ (\mbox{see (\ref{eq11}) and (\ref{eq6})}).\label{eq13}
\end{eqnarray}
We need to show that the map $\tau:V\rightarrow H_-$ is continuous. To this end, let $v_n\rightarrow v$ in $V$. First, note that if $\bar{u}\in H_-$, then
\begin{eqnarray*}
\varphi(\bar{u})&=&\frac{1}{2}\gamma(\bar{u})-\int_{\Omega}F(z,\bar{u})dz\\
&\leq&\frac{1}{2}\gamma(\bar{u})-\frac{1}{2}\int_{\Omega}\eta(z)\bar{u}^2dz\ (\mbox{see hypothesis}\ H(f)(ii))\\
&\leq&-c_1||\bar{u}||^2\ (\mbox{see Lemma \ref{lem2}}),\\
\Rightarrow\tau(0)=0.
\end{eqnarray*}
Since $\varphi\in C^1(H^1(\Omega))$ and $\varphi'(u)=\gamma'(u)-N_f(u)$, it follows that $\varphi'$ is bounded on bounded sets of $H^1(\Omega)$. Therefore
$$||\varphi'(v_n)||_*\leq c_4$$ with $c_4>0$ independent of $n\in\mathbb N$ (recall that $v_n\rightarrow v$ in $H^1(\Omega)$).
Then we have
\begin{eqnarray*}
0&=&\left\langle \varphi'(v_n+\tau(v_n)),\tau(v_n)\right\rangle\ (\mbox{see (\ref{eq13})})\\
&=&\left\langle \varphi'(v_n+\tau(v_n))-\varphi'(v_n+\tau(0)),\tau(v_n)\right\rangle+\left\langle \varphi'(v_n+\tau(0)),\tau(v_n)\right\rangle\\
&\leq&-c_1||\tau(v_n)||^2+c_4||\tau(v_n)||,\ \mbox{for all}\ n\in\mathbb N\ (\mbox{see (\ref{eq7})})\\
\Rightarrow&&\{\tau(v_n)\}_{n\geq 1}\subseteq H_-\ \mbox{is bounded}.
\end{eqnarray*}
By passing to a suitable subsequence if necessary and using the finite dimensionality of $H_-$, we can infer that
\begin{equation}\label{eq14}
\tau(v_n)\rightarrow\hat{y}\ \mbox{in}\ H^1(\Omega),\ \hat{y}\in H_-.
\end{equation}
We have
\begin{eqnarray*}
&&\varphi(v_n+\tau(v_n))\leq\varphi(v_n+y)\ \mbox{for all}\ y\in H_-,\ \mbox{all}\ n\in\mathbb N\ (\mbox{see (\ref{eq12})}),\\
&\Rightarrow&\varphi(v+\hat{y})\leq\varphi(v+y)\ \mbox{for all}\ y\in H_-\ (\mbox{see (\ref{eq14}) and recall that}\ \varphi\ \mbox{is continuous}),\\
&\Rightarrow&\hat{y}=\tau(v).
\end{eqnarray*}
By the Urysohn convergence criterion (see, for example, Gasinski \& Papageorgiou \cite[p. 33]{7}), we have for the original sequence
\begin{eqnarray*}
&&\tau(v_n)\rightarrow \tau(v)\ \mbox{in}\ H_-,\\
&\Rightarrow&\tau(\cdot)\ \mbox{is continuous.}
\end{eqnarray*} \end{proof}
Consider the functional $\tilde{\varphi}:V\rightarrow\mathbb R$ defined by $$\tilde{\varphi}(v)=\varphi(v+\tau(v))\ \mbox{for all}\ v\in V.$$ \begin{proposition}\label{prop4}
If hypotheses $H(\xi), H(\beta), H(f)$ hold, then $\tilde{\varphi}\in C^1(V,\mathbb R)$ and $\tilde{\varphi}'(v)=p_{V^*}\varphi'(v+\tau(v))$ for all $v\in V$ (here $p_{V^*}$ denotes the orthogonal projection of the Hilbert space $H^1(\Omega)^*$ onto $V^*$). \end{proposition} \begin{proof}
Let $v,h\in V$ and $t>0$. We have
\begin{eqnarray}\label{eq15}
&&\frac{1}{t}\left[\tilde{\varphi}(v+th)-\tilde{\varphi}(v)\right]\nonumber\\
&\geq&\frac{1}{t}\left[\varphi(v+th+\tau(v))-\varphi(v+\tau(v))\right]\ (\mbox{see (\ref{eq12})}),\nonumber\\
&\Rightarrow&\liminf\limits_{t\rightarrow 0^+}\frac{1}{t}\left[\tilde{\varphi}(v+th)-\tilde{\varphi}(v)\right]\geq\left\langle \varphi'(v+\tau(v)),h\right\rangle.
\end{eqnarray}
Also, we have
\begin{eqnarray}\label{eq16}
&&\frac{1}{t}\left[\tilde{\varphi}(v+th)-\tilde{\varphi}(v)\right]\nonumber\\
&\leq&\frac{1}{t}\left[\varphi(v+th+\tau(v+th))-\varphi(v+\tau(v+th))\right]\nonumber\\
&\Rightarrow&\limsup\limits_{t\rightarrow 0^+}\frac{1}{t}\left[\tilde{\varphi}(v+th)-\tilde{\varphi}(v)\right]\leq\left\langle \varphi'(v+\tau(v)),h\right\rangle\\
&&(\mbox{recall that}\ \tau(\cdot)\ \mbox{is continuous, see Proposition \ref{prop3} and that}\ \varphi\in C^1(H^1(\Omega),\mathbb R)).\nonumber
\end{eqnarray}
From (\ref{eq15}) and (\ref{eq16}) it follows that
\begin{equation}\label{eq17}
\lim\limits_{t\rightarrow 0^+}\frac{1}{t}\left[\tilde{\varphi}(v+th)-\tilde{\varphi}(v)\right]=\langle \varphi'(v+\tau(v)),h\rangle\ \mbox{for all}\ v,h\in V.
\end{equation}
Similarly we show that
\begin{equation}\label{eq18}
\lim\limits_{t\rightarrow 0^-}\frac{1}{t}\left[\tilde{\varphi}(v+th)-\tilde{\varphi}(v)\right]=\left\langle \varphi'(v+\tau(v)),h\right\rangle\ \mbox{for all}\ v,h\in V.
\end{equation}
From (\ref{eq17}) and (\ref{eq18}) we conclude that
$$\tilde{\varphi}\in C^1(V,\mathbb R)\ \mbox{and}\ \tilde{\varphi}'(v)=p_{V^*}\varphi'(v+\tau(v))\ \mbox{for all}\ v\in V.$$ \end{proof} \begin{proposition}\label{prop5}
If hypotheses $H(\xi),H(\beta),H(f)$ hold, then $v\in K_{\tilde{\varphi}}$ if and only if $v+\tau(v)\in K_{\varphi}$. \end{proposition} \begin{proof}
$\Leftarrow$ Follows from Proposition \ref{prop4}.
$\Rightarrow$ Let $v\in K_{\tilde{\varphi}}$. Then
\begin{eqnarray}\label{eq19}
&&0=\tilde{\varphi}'(v)=p_{V^*}\varphi'(v+\tau(v))\ (\mbox{see Proposition \ref{prop4}}),\nonumber\\
&\Rightarrow&\varphi'(v+\tau(v))\in H^*_-\ (\mbox{recall that}\ H^1(\Omega)^*=H^*_-\oplus V^*).
\end{eqnarray}
On the other hand from (\ref{eq13}) we have
\begin{eqnarray}\label{eq20}
&&p_{H^*_-}\varphi'(v+\tau(v))=0,\nonumber\\
&\Rightarrow&\varphi'(v+\tau(v))\in V^*.
\end{eqnarray}
But $H^*_-\cap V^*=\{0\}$. So, it follows from (\ref{eq19}) and (\ref{eq20}) that
\begin{eqnarray*}
&&\varphi'(v+\tau(v))=0,\\
&\Rightarrow&v+\tau(v)\in K_{\varphi}.
\end{eqnarray*} \end{proof} \begin{proposition}\label{prop6}
If hypotheses $H(\xi),H(\beta),H(f)$ hold, then $\tilde{\varphi}$ is coercive. \end{proposition} \begin{proof}
Let $\psi=\varphi|_V$. Evidently, $\psi\in C^1(V,\mathbb R)$ and by the chain rule we have
\begin{equation}\label{eq21}
\psi'=p_{V^*}\circ\varphi'.
\end{equation}
\begin{claim}\label{cl1}
$\psi$ satisfies the C-condition.
\end{claim}
Let $\{v_n\}_{n\geq 1}\subseteq V$ be a sequence such that
\begin{eqnarray}
&&|\psi(v_n)|\leq M_1\ \mbox{for some}\ M_1>0,\ \mbox{and all}\ n\in\mathbb N,\label{eq22}\\
&&(1+||v_n||)\psi'(v_n)\rightarrow 0\ \mbox{in}\ V^*\ \mbox{as}\ n\rightarrow\infty\label{eq23}.
\end{eqnarray}
From (\ref{eq23}) we have
\begin{eqnarray}\label{eq24}
&&|\left\langle \psi'(v_n),h\right\rangle_V|\leq\frac{\epsilon_n||h||}{1+||v_n||}\ \mbox{for all}\ h\in V,\ n\in\mathbb N,\ \mbox{with}\ \epsilon_n\rightarrow 0^+,\nonumber\\
&\Rightarrow&|\left\langle \varphi'(v_n),h\right\rangle|\leq\frac{\epsilon_n||h||}{1+||v_n||}\ \mbox{for all}\ h\in V,\ n\in\mathbb N\ (\mbox{see (\ref{eq21})}).
\end{eqnarray}
In (\ref{eq24}) we choose $h=v_n\in V$ and obtain
\begin{equation}\label{eq25}
\gamma(v_n)-\int_{\Omega}f(z,v_n)v_ndz\leq\epsilon_n\ \mbox{for all}\ n\in\mathbb N.
\end{equation}
We show that $\{v_n\}_{n\geq 1}\subseteq V$ is bounded. Arguing by contradiction, suppose that
\begin{equation}\label{eq26}
||v_n||\rightarrow\infty\,.
\end{equation}
Let $\hat{w}_n=\frac{v_n}{||v_n||},\ n\in\mathbb N$. Then $\hat{w}_n\in V,\ ||\hat{w}_n||=1$ for all $n\in\mathbb N$. By passing to a suitable subsequence if necessary, we may assume that
\begin{equation}\label{eq27}
\hat{w}_n\stackrel{w}{\rightarrow}\hat{w}\ \mbox{in}\ H^1(\Omega)\ \mbox{and}\ \hat{w}_n\rightarrow\hat{w}\ \mbox{in}\ L^2(\Omega)\ \mbox{and}\ L^2(\partial\Omega).
\end{equation}
Hypotheses $H(f)$ imply that
\begin{equation}\label{eq28}
|f(z,x)|\leq c_5|x|\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in\mathbb R,\ \mbox{and some}\ c_5>0.
\end{equation}
By (\ref{eq24}) we have
\begin{equation}\label{eq29}
\left|\left\langle \gamma'(\hat{w}_n),h\right\rangle-\int_{\Omega}\frac{N_f(v_n)}{||v_n||}hdz\right|
\leq\frac{\epsilon_n||h||}{(1+||v_n||)||v_n||}\ \mbox{for all}\ n\in\mathbb N,\ h\in H^1(\Omega).
\end{equation}
From (\ref{eq28}) and (\ref{eq27}) we see that
\begin{equation}\label{eq30}
\left\{\frac{N_f(v_n)}{||v_n||}\right\}_{n\geq 1}\subseteq L^2(\Omega)\ \mbox{is bounded}.
\end{equation}
So, if in (\ref{eq29}) we choose $h=\hat{w}_n-\hat{w}\in H^1(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (\ref{eq26}), (\ref{eq27}) and (\ref{eq30}), then
\begin{eqnarray*}
&&\lim\limits_{n\rightarrow\infty}\left\langle A(\hat{w}_n),\hat{w}_n-\hat{w}\right\rangle=0,\\
&\Rightarrow&||D\hat{w}_n||_2\rightarrow||D\hat{w}||_2,\\
&\Rightarrow&\hat{w}_n\rightarrow\hat{w}\ \mbox{in}\ H^1(\Omega)\\
&&(\mbox{by the Kadec-Klee property, see Gasinski \& Papageorgiou \cite[p. 911]{5}}),\\
&\Rightarrow&||\hat{w}||=1\ \mbox{and so}\ \hat{w}\neq 0.
\end{eqnarray*}
Let $\Omega_0=\{z\in\Omega:\hat{w}(z)\neq 0\}$. Then $|\Omega_0|_N>0$ and
$$v_n(z)\rightarrow\pm\infty\ \mbox{for almost all}\ z\in\Omega_0\ (\mbox{see (\ref{eq26})}).$$
Hypothesis $H(f)(iii)$ implies that
\begin{equation}\label{eq31}
f(z,v_n(z))v_n(z)-2F(z,v_n(z))\rightarrow+\infty\ \mbox{for almost all}\ z\in\Omega_0.
\end{equation}
From (\ref{eq31}) via Fatou's lemma (hypothesis $H(f)(iii)$ permits its use), we have
\begin{equation}\label{eq32}
\int_{\Omega_0}[f(z,v_n)v_n-2F(z,v_n)]dz\rightarrow+\infty.
\end{equation}
Using hypothesis $H(f)(iii)$ we see that we can find $M_2>0$ such that
\begin{equation}\label{eq33}
f(z,x)x-2F(z,x)\geq 0\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ |x|\geq M_2.
\end{equation}
So, we have
\begin{eqnarray*}
&&\int_{\Omega\backslash\Omega_0}[f(z,v_n)v_n-2F(z,v_n)]dz\\
&=&\int_{(\Omega\backslash\Omega_0)\cap\{|v_n|\geq M_2\}}[f(z,v_n)v_n-2F(z,v_n)]dz+\\
&&\hspace{1cm}\int_{(\Omega\backslash\Omega_0)\cap\{|v_n|<M_2\}}[f(z,v_n)v_n-2F(z,v_n)]dz\\
&\geq&\int_{(\Omega\backslash\Omega_0)\cap\{|v_n|<M_2\}}[f(z,v_n)v_n-2F(z,v_n)]dz\ (\mbox{see (\ref{eq33})})\\
&\geq&-M_3\ \mbox{for some}\ M_3>0,\ \mbox{all}\ n\in\mathbb N\ (\mbox{see hypothesis}\ H(f)(i)).
\end{eqnarray*}
Then
\begin{eqnarray}\label{eq34}
&&\int_{\Omega}[f(z,v_n)v_n-2F(z,v_n)]dz\nonumber\\
&=&\int_{\Omega_0}[f(z,v_n)v_n-2F(z,v_n)]dz+\int_{\Omega\backslash\Omega_0}[f(z,v_n)v_n-2F(z,v_n)]dz\nonumber\\
&\geq&\int_{\Omega_0}[f(z,v_n)v_n-2F(z,v_n)]dz-M_3\ \mbox{for all}\ n\in\mathbb N\nonumber\\
\Rightarrow&&\int_{\Omega}[f(z,v_n)v_n-2F(z,v_n)]dz\rightarrow+\infty\ \mbox{as}\ n\rightarrow\infty\ (\mbox{see (\ref{eq32})}).
\end{eqnarray}
From (\ref{eq24}) with $h=v_n\in H^1(\Omega)$, we have
\begin{equation}\label{eq35}
-\gamma(v_n)+\int_{\Omega}f(z,v_n)v_ndz\leq\epsilon_n\ \mbox{for all}\ n\in\mathbb N.
\end{equation}
Also, from (\ref{eq22}) we have
\begin{equation}\label{eq36}
\gamma(v_n)-\int_{\Omega}2F(z,v_n)dz\leq 2M_1\ \mbox{for all}\ n\in\mathbb N.
\end{equation}
We add (\ref{eq35}) and (\ref{eq36}) and obtain
\begin{equation}\label{eq37}
\int_{\Omega}[f(z,v_n)v_n-2F(z,v_n)]dz\leq M_4\ \mbox{for some}\ M_4>0,\ \mbox{and all}\ n\in\mathbb N.
\end{equation}
Comparing (\ref{eq34}) and (\ref{eq37}), we get a contradiction. This proves that $\{v_n\}_{n\geq 1}\subseteq V$ is bounded. So, we may assume that
\begin{equation}\label{eq38}
v_n\stackrel{w}{\rightarrow}u\ \mbox{in}\ H^1(\Omega)\ \mbox{and}\ v_n\rightarrow u\ \mbox{in}\ L^2(\Omega)\ \mbox{and}\ L^2(\partial\Omega).
\end{equation}
In (\ref{eq24}) we choose $h=v_n-u\in H^1(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (\ref{eq38}). Then
\begin{eqnarray*}
&&\lim\limits_{n\rightarrow\infty}\left\langle A(v_n),v_n-u\right\rangle=0,\\
&\Rightarrow& v_n\rightarrow u\ \mbox{in}\ H^1(\Omega)\ (\mbox{as before via the Kadec-Klee property}).
\end{eqnarray*}
This proves Claim \ref{cl1}.
\begin{claim}\label{cl2}
$\hat{\lambda}_{m+1}x^2-2F(z,x)\rightarrow+\infty$ as $x\rightarrow +\infty$ uniformly for almost all $z\in\Omega$.
\end{claim}
Hypothesis $H(f)(iii)$ implies that given any $\lambda>0$, we can find $M_5=M_5(\lambda)>0$ such that
\begin{equation}\label{eq39}
f(z,x)x-2F(z,x)\geq\lambda\ \mbox{for almost all}\ z\in\Omega,\ \mbox{and all}\ |x|\geq M_5.
\end{equation}
For almost all $z\in\Omega$, we have
\begin{eqnarray}\label{eq40}
&&\frac{d}{dx}\left(\frac{F(z,x)}{|x|^2}\right)=\frac{f(z,x)x-2F(z,x)}{|x|^2x}\left\{\begin{array}{ll}
\geq\frac{\lambda}{x^2}&\mbox{if}\ x\geq M_5\\
\leq\frac{\lambda}{|x|^2x}&\mbox{if}\ x\leq -M_5
\end{array}\right.\ (\mbox{see (\ref{eq39})}),\nonumber\\
&\Rightarrow&\frac{F(z,y)}{|y|^2}-\frac{F(z,v)}{|v|^2}\geq\frac{\lambda}{2}\left[\frac{1}{|v|^2}-\frac{1}{|y|^2}\right]\ \mbox{for all}\ |y|\geq|v|\geq M_5.
\end{eqnarray}
We let $|y|\rightarrow\infty$ and use hypothesis $H(f)(iii)$. Then
$$\hat{\lambda}_{m+1}|v|^2-2F(z,v)\geq\lambda\ \mbox{for almost all}\ z\in\Omega,\ \mbox{and all}\ |v|\geq M_5.$$
Since $\lambda>0$ is arbitrary, we conclude that
$$\hat{\lambda}_{m+1}|v|^2-2F(z,v)\rightarrow+\infty\ \mbox{as $v\rightarrow +\infty$ uniformly for almost all}\ z\in\Omega.$$
This proves Claim \ref{cl2}.
For every $v\in V$, we have
\begin{eqnarray}\label{eq41}
\psi(v)=\varphi(v)&=&\frac{1}{2}\gamma(v)-\int_{\Omega}F(z,v)dz\nonumber\\
&\geq&\int_{\Omega}\left[\frac{1}{2}\hat{\lambda}_{m+1}v^2-F(z,v)\right]dz\ (\mbox{see (\ref{eq5})})\nonumber\\
&\geq&-c_6\ \mbox{for some}\ c_6>0\ (\mbox{see Claim \ref{cl2} and hypothesis H(f)(i)})\nonumber\\
&\Rightarrow&\psi\ \mbox{is bounded below}.
\end{eqnarray}
From (\ref{eq41}) and Claim \ref{cl1} it follows that
$$\psi\ \mbox{is coercive}$$
(see Motreanu, Motreanu \& Papageorgiou \cite[p. 103]{11}).
We have
\begin{eqnarray*}
&&\psi(v)=\varphi(v)\leq\varphi(v+\tau(v))=\tilde{\varphi}(v)\ \mbox{for all}\ v\in V\ (\mbox{see (\ref{eq12})}),\\
&\Rightarrow&\tilde{\varphi}\ \mbox{is coercive}.
\end{eqnarray*} \end{proof}
From Proposition \ref{prop5}, we deduce that: \begin{corollary}\label{cor6}
If hypotheses $H(\xi),H(\beta),H(f)$ hold, then $\tilde{\varphi}$ is bounded below and satisfies the C-condition. \end{corollary}
Next we show that $\tilde{\varphi}$ admits a local linking (see Theorem \ref{th1}) with respect to the orthogonal direct sum decomposition $V=W\oplus\hat{E}$ where $W=\overset{l-1}{\underset{\mathrm{i=m+1}}\oplus}E(\hat{\lambda}_i),\hat{E}=\overline{{\underset{\mathrm{i\geq l}}\oplus}E(\lambda_i)}$. \begin{proposition}\label{prop7}
If hypotheses $H(\xi),H(\beta),H(f)$ hold, then $\tilde{\varphi}$ has a local linking at $u=0$ with respect to the decomposition
$$V=W\oplus\hat{E}.$$ \end{proposition} \begin{proof}
From hypotheses $H(f)(i),(iv)$, we see that given $r>2$, we can find $c_7=c_7(r)>0$ such that
\begin{equation}\label{eq42}
F(z,x)\leq\frac{\vartheta(z)}{2}x^2+c_7|x|^r\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in\mathbb R.
\end{equation}
For $\hat{v}\in\hat{E}$ we have
\begin{eqnarray*}
\tilde{\varphi}(\hat{v})&=&\varphi(\hat{v}+\tau(\hat{v}))\\
&\geq&\varphi(\hat{v})\ \mbox{(see Proposition \ref{prop3})}\\
&=&\frac{1}{2}\gamma(\hat{v})-\int_{\Omega}F(z,\hat{v})dz\\
&\geq&\frac{1}{2}\gamma(\hat{v})-\frac{1}{2}\int_{\Omega}\vartheta(z)\hat{v}^2dz-c_8||\hat{v}||^r\ \mbox{for some}\ c_8>0\ (\mbox{see (\ref{eq42})})\\
&\geq&c_9||\hat{v}||^2-c_8||\hat{v}||^r\ \mbox{for some}\ c_9>0\ (\mbox{see Lemma \ref{lem2}(b)}).
\end{eqnarray*}
Since $r>2$, we see that we can find $\rho_1\in(0,1)$ small such that
\begin{equation}\label{eq43}
\tilde{\varphi}(\hat{v})>0\ \mbox{for all}\ \hat{v}\in\hat{E}\ \mbox{with}\ 0<||\hat{v}||\leq\rho_1.
\end{equation}
The space $Y=H_-\oplus W$ is finite dimensional and so all norms are equivalent. Hence we can find $\epsilon_0>0$ such that
\begin{equation}\label{eq44}
y\in Y\ \mbox{and}\ ||y||\leq\epsilon_0\Rightarrow|y(z)|\leq\delta\ \mbox{for all}\ z\in\overline{\Omega}\ (\mbox{recall that}\ Y\subseteq C^1(\overline{\Omega})).
\end{equation}
By Proposition \ref{prop3} we know that $\tau(\cdot)$ is continuous. So, we can find $\rho_2>0$ such that
\begin{equation}\label{eq45}
\tilde{u}\in W\ \mbox{and}\ ||\tilde{u}||\leq\rho_2\Rightarrow||\tilde{u}+\tau(\tilde{u})||\leq\epsilon_0.
\end{equation}
From (\ref{eq44}) and (\ref{eq45}) it follows that
\begin{eqnarray*}
\tilde{\varphi}(\tilde{u})&=&\varphi(\tilde{u}+\tau(\tilde{u}))\\
&=&\frac{1}{2}\gamma(\tilde{u}+\tau(\tilde{u}))-\int_{\Omega}F(z,\tilde{u}+\tau(\tilde{u}))dz\\
&\leq&\frac{1}{2}\hat{\lambda}_{l-1}||\tilde{u}+\tau(\tilde{u})||^2_2-\frac{1}{2}\hat{\lambda}_{l-1}||\tilde{u}+\tau(\tilde{u})||^2_2\ (\mbox{see hypothesis}\ H(f)(iv))\\
&=&0.
\end{eqnarray*}
So, we have that
\begin{equation}\label{eq46}
\tilde{\varphi}(\tilde{u})\leq 0\ \mbox{for all}\ \tilde{u}\in W\ \mbox{with}\ ||\tilde{u}||\leq\rho_2.
\end{equation}
If $\rho=\min\{\rho_1,\rho_2\}$, then from (\ref{eq43}) and (\ref{eq46}) we conclude that $\varphi$ has a local linking at $u=0$ with respect to the decomposition $V=W\oplus\hat{E}$. \end{proof}
Now we are ready for proving our multiplicity theorem. \begin{theorem}\label{th8}
If hypotheses $H(\xi),H(\beta),H(f)$ hold, then problem (\ref{eq1}) admits at least two nontrivial solutions
$$u_0,\hat{u}\in C^1(\overline{\Omega}).$$ \end{theorem} \begin{proof}
From Proposition \ref{prop7} we know that
$$\inf\limits_{V}\tilde{\varphi}\leq 0.$$
If $\inf\limits_{V}\tilde{\varphi}=0$, then by Proposition \ref{prop7} all $\tilde{u}\in W$ with $0<||\tilde{u}||\leq\rho$ are nontrivial critical points of $\tilde{\varphi}$. Hence $\tilde{u}+\tau(\tilde{u})$ are nontrivial critical points of $\varphi$ (see Proposition \ref{prop5}).
If $\inf\limits_{V}\tilde{\varphi}<0$, then we can apply Theorem \ref{th1} (see Corollary \ref{cor6}) and produce two nontrivial critical points $\tilde{u}_0$ and $\tilde{u}_*$ of $\tilde{\varphi}$. Then
$$u_0=\tilde{u}_0+\tau(\tilde{u}_0)\ \mbox{and}\ \hat{u}=\tilde{u}_*+\tau(\hat{u}_*)$$
are two nontrivial critical points of $\varphi$ (see Proposition \ref{prop5}).
For $u=u_0$ or $u=\hat{u}$, we have
\begin{eqnarray}\label{eq47}
&&-\Delta u(z)+\xi(z)u(z)=f(z,u(z))\ \mbox{for almost all}\ z\in\Omega,\\
&&\frac{\partial u}{\partial n}+\beta(z)u=0\ \mbox{on}\ \partial\Omega\ (\mbox{see Papageorgiou \& R\u adulescu \cite{16,15}}).\nonumber
\end{eqnarray}
Evidently, hypotheses $H(f)$ imply that
\begin{equation}\label{eq48}
|f(z,x)|\leq c_{10}|x|\ \mbox{for almost all}\ x\in\mathbb R,\ \mbox{and some}\ c_{10}>0.
\end{equation}
We set
$$b(z)=\left\{\begin{array}{ll}
\frac{f(z,u(z))}{u(z)}&\mbox{if}\ u(z)\neq 0\\
0&\mbox{if}\ u(z)=0.
\end{array}\right.$$
From (\ref{eq48}) it follows that $b\in L^{\infty}(\Omega)$. From (\ref{eq47}) we have
$$\left\{\begin{array}{ll}
-\Delta u(z)=(b-\xi)(z)u(z)\ \mbox{for almost all}\ z\in\Omega,&\\
\frac{\partial u}{\partial n}+\beta(z)u=0\ \mbox{on}\ \partial\Omega.&
\end{array}\right.$$
Note that $b-\xi\in L^s(\Omega),\ s>N$ (see hypothesis $H(\xi)$). Then Lemmata 5.1 and 5.2 of Wang \cite{19} imply that
\begin{eqnarray*}
&&u\in W^{2,s}(\Omega),\\
&\Rightarrow&u\in C^{1,\alpha}(\overline{\Omega})\ \mbox{with}\ \alpha=1-\frac{N}{s}>0\ (\mbox{by the Sobolev embedding theorem}).
\end{eqnarray*}
Therefore $u_0,\hat{u}\in C^1(\overline{\Omega})$. \end{proof}
{\bf Acknowledgments.} This research was supported by the Slovenian Research Agency grants P1-0292, J1-8131, J1-7025. V.D. R\u adulescu acknowledges the support through a grant of the Romanian Ministry of Research and Innovation, CNCS - UEFISCDI, project number PN-III-P4-ID-PCE-2016-0130, within PNCDI III.
\end{document} | arXiv |
17 Jan MATLAB Benchmark Code for WiDS Datathon 2020
20 Dec Walking Robot Modeling and Simulation
10 Dec What's New in Automated Driving in MATLAB and Simulink?
20 Nov Where Are They Now? - Nathan Gyger, Speedgoat
6 Nov Trajectory Planning for Robot Manipulators
MATLAB 25
Automotive 33
Automated driving 3
Live script 4
Simscape 10
Simulink 35
Skills 38
Stateflow 2
Team achievements 27
Where-Are-They-Now 3
Workflow 26
Sebastian Castro on Walking Robot Control: From PID to Reinforcement Learning
guessam ali on Walking Robot Control: From PID to Reinforcement Learning
Sebastian Castro on Walking Robot Modeling and Simulation
KKamiya on Walking Robot Modeling and Simulation
Sebastian Castro on Introduction to Contact Modeling
The racing lounge blog focuses on student success stories. Winning student competition teams share their knowledge and the MathWorks competition team shares best practices and workflows using MATLAB and Simulink
Best practices and teamwork for student competitions
< MATLAB and Simulink for Autonomous... < Previous
Designing Object Detectors in MATLAB >Next >
Modeling Robotic Boats in Simulink
Posted by Christoph Hahn, March 18, 2019
39 views | 0 likes | 0 comment
Connell D'Souza our co-blogger has worked with a team that develops robotic boats. The outcome is clearly impressive.
For today's post, I would like to introduce you to Alejandro Gonzalez. Alex is a member of the RoboBoat team – VantTec of Tecnológico de Monterrey in Monterrey, Mexico. I met Alex at RoboBoat 2018 where I got a chance to see his team's innovative solution to the tasks at the competition – an Unmanned Aerial Vehicle (UAV) guiding an Unmanned Surface Vehicle (USV) through the course, and I was glad when Alex offered to write about his team's work with MATLAB and Simulink for the Racing Lounge Blog! Alex will talk about using the Robotics System Toolbox to develop a path planning algorithm and the Aerospace Blockset to build a dynamic model of their boat to tune controllers. So, let me hand it off to Alex to take it away – Alex, the stage is yours!
I lead VantTec, a student robotics group and we build an autonomous robotic boat for RoboBoat. The competition encourages collaboration between unmanned aerial vehicles (UAVs) and unmanned surface vehicles (USVs) for the docking task – the UAV tells the USV where to dock. We decided to take this collaboration further and use the UAV to develop a path for the USV to follow through the course.
The challenges with autonomous navigation of a robot are threefold:
Creating a map of the environment
Choosing/developing a path planning algorithm
Developing a robust controller to follow the desired path
So, how do we tackle these challenges?
To create a map of the environment we use our UAV to click a bird's-eye-view photo of the course. This photo can create a grid to work on. Here, computer vision and artificial intelligence algorithms can give a relative position of obstacles, referenced to the picture dimensions or the vehicle itself.
For this to work, first we take a picture from above with the aerial vehicle. We use a DJI Phantom 4 with a mobile application we developed for autonomous waypoint navigation; the UAV takes the picture and sends it to a mobile phone. Next, we send this picture to our central ground station, where a neural network we developed detects the buoys and creates bounding boxes around them. From these bounding boxes, we obtain the center of each buoy, which we arrange in a matrix to create the map.
Using the Robotics System Toolbox's binary occupancy grid, the data gathered creates a map where the robot can navigate. Here, the obstacles' relative coordinates will set their location inside the grid.
Below, is an example on how to create the grid; the values 50 and 10 should be changed to the dimensions in meters that the UAV camera frames on the taken picture. Then, the variable xy is the set of obstacles taken from their centers. The sample code is an example of the kind of matrix that should be introduced. Our computer vision module creates a similar matrix with a corresponding vector of coordinates for each obstacle.
robotics.BinaryOccupancyGrid(50,10,30);
xy = [3 2; 8 5; 13 7; 20 1; 25 8; 32 6; 38 3; 40 9; 42 4; 23 2; 28 5; 33 7];
setOccupancy(map, xy, 1);
Then, the function inflate can change the obstacle dimensions by a known or obtained radius.
inflate(map,0.3);
Choosing/ Developing a Path Planning Algorithm
The Robotics Systems Toolbox presents another solution, this time using a sampling-based path planner algorithm called Probabilistic Roadmap. In this case, a tag on the vehicle can help for its aerial recognition, resulting in the start location coordinates and end location coordinates and number of nodes to set are required to get the route the vehicle needs to follow.
prm = robotics.PRM
prm.Map = map;
startLocation = [3 3];
endLocation = [47 7];
prm.NumNodes = 25;
% Search for a solution between start and end location.
path = findpath(prm, startLocation, endLocation);
while isempty(path)
prm.NumNodes = prm.NumNodes + 25;
update(prm);
Developing a Robust Controller
The challenge of developing a robust controller is easier with a model of the vehicle to reference it. The better your model, the better your controller. A kinematic model serves as a start, but a dynamic model of the robot is better suited to create a simulation environment. For an underactuated USV, a 3 DOF dynamic model can achieve the environment needed to work with.
Simulink is a great tool to develop these kinds of model, even more using the toolboxes available. Aerospace Blockset presents Utilities blocks, which includes math operations with 3×3 matrices, needed for 3 DOF dynamic models.
Building the Model
The equation for the dynamic model is:
$ \tau = M \dot{\nu} + C(\nu)\nu + D(\nu)\nu $
or rewritten:
$ \dot{\nu} = M^{-1} [\tau – C(\nu) – D(\nu)] $
The first matrix in the equation is the inertia tensor. This M matrix is constructed using the 3×3 Matrix utility block from the Aerospace Blockset.
$M = \begin{pmatrix} m – X_{\dot{u}} & 0 & -m y_G \\ 0 & m – Y_{\dot{u}} & m x_{G} – Y_{\dot{r}} \\ -my_{G} & m x_{G} – N_{\dot{\nu}} & I_{Z} – N_{\dot{r}} \end{pmatrix} $
Then, it was made a subsystem for the overall dynamic model, having the vehicle physical constants (m, X_G, Y_G, I_Z) and hydrodynamic coefficients needed as inputs and the matrix (M) as output.
The second matrix in the system is a vector of forces ($\tau $-matrix) which is programmed as shown below. Then, it was inserted into a subsystem for the overall model, with the boat beam (B) and individual thrust (Tport & Tstbd) as inputs and the vector of forces (T) as output.
$ \tau = \begin{pmatrix} \tau_{x} \\ \tau_{y} \\ \tau_{z} \end{pmatrix} = \begin{pmatrix} (T_{port} + T_{stbd}) \\ 0 \\ 0.5*B (T_{port} – T_{stbd}) \end{pmatrix} $
Then, it was inserted into a subsystem for the overall model, with the boat beam and individual thrust as inputs and the vector of forces as output.
Similarly, the next matrix is the Coriolis matrix (C matrix). As shown below, the sum of two 3×3 matrices is needed and hence the matrix sum block was used. Then, a subsystem was created which has, as inputs, physical parameters (X_G, Y_G, m), hydrodynamic coefficients and the values of the surge and sway speed as well as the yaw rate (V local) and the Coriolis matrix as the output:
$ C(\nu) = \begin{pmatrix} 0 & 0& -m(x_G r + \nu) \\ 0 & 0& -m(y_G r – u) \\ m(x_G r + \nu) & m(y_G r – u) & 0 \end{pmatrix} + \begin{pmatrix} 0 & 0& \frac{Y_{\dot{\nu}} \nu +\frac{Y_{\dot{r}} + N_{\dot{\nu}}}{2}r}{200}\\ 0 & 0 & -X_{\dot{u}} u \\ \frac{-Y_{\dot{\nu}} \nu -\frac{Y_{\dot{r}} + N_{\dot{\nu}}}{2}r}{200} & X_{\dot{u}} u & 0 \end{pmatrix} $
The next matrix is the drag matrices. Like the Coriolis matrix, the Drag matrix is a sum of two matrices, but this time with a negative sign. Again, a subsystem was created, with all the hydrodynamic coefficients required, surge and sway speeds, and the yaw rate as inputs and the matrix as output.
$D(\nu) = \begin{pmatrix} Y_u & 0 & 0 \\ 0 & Y_{\nu} & Y_r \\ 0 & N_{\nu} & N_r \end{pmatrix} – \begin{pmatrix} X_{u\mid u \mid}\mid u \mid & 0 & 0 \\ 0 & Y_{\nu \mid \nu \mid} \mid \nu \mid + Y_{\nu \mid r \mid} \mid r \mid & Y_{r \mid \nu \mid} \mid \nu \mid + Y_{r \mid r \mid} \mid r \mid \\ 0 & Y_{\nu \mid \nu \mid} \mid \nu \mid + Y_{\nu \mid r \mid} \mid r \mid & Y_{r \mid \nu \mid} \mid \nu \mid + Y_{r \mid r \mid} \mid r \mid \end{pmatrix} $
Afterwards, a matrix sum was used for the first algebraic part of the equation.
Then, the resultant matrix is multiplied with the inverted M matrix. The result is the derivative of the local reference frame velocity vector, and it is subsequently integrated.
The transformation matrix is represented as shown below and is used to relate the local reference frame with the global reference frame:
$ J(\eta) = \begin{pmatrix} cos \psi & -sin \psi & 0 \\ sin \psi & cos \psi & 0\\ 0 & 0 & 1 \\ \end{pmatrix} $
The local velocity vector represented by V-local is transformed to the global reference frame and then integrated to obtain the x,y and orientation or heading of the boat and is stored in the vector defined by "n_global" as shown below. You can use a demux block to index into the individual elements of the vector.
Finally, a subsystem was created with the equations necessary to obtain the hydrodynamic coefficients, after introducing parameters that can be measured or estimated. These hydrodynamic coefficients are collected into a Simulink Bus to enable data transfer to other subsystems of the model.
Developing a Model-Based Controller
The programmed equation creates a dynamic boat model to base a controller on. Here the body-fixed frame (v) and North-East-Down -fixed frame (n) are the outputs and the thruster values or control commands as inputs to the model. You can also set up the boat parameters to be accepted as mask variables, this will give you a parameterized model that can be modified as you make physical changes to your boat. With this parameterized model, you can use Control System Toolbox and Simulink Control Design to design a controller that can follow the desired path generated earlier.
Here I show you an example surge speed and heading controller that we developed. To test this controller, we used the Signal Builder block to create an example sinusoidal trajectory that represents the desired heading. As you can imagine, in our complete system this trajectory is generated from the map as we discussed earlier, but we are showing a test input for now.
From the plots below, we can see that our controller is able to track the heading fairly well. This can be improved by tuning the controller gains and the XY Graph below shows the trajectory of our boat with our test control inputs.
Robotics,
Skills,
Team achievements,
Trajectory Planning for Robot Manipulators
What do drones and one of Alfred Hitchcock's best-ever movies have in common?
Tuning Waypoint Follower for Fixed-Wing UAV
(sp) matrix
Quadcopter Project | CommonCrawl |
How to prove floor function inequality $\sum\limits_{k=1}^{n}\frac{\{kx\}}{\lfloor kx\rfloor }<\sum\limits_{k=1}^{n}\frac{1}{2k-1}$ for $x>1$
Let $x>1$ be a real number. Show that for any positive $n$ $$\sum_{k=1}^{n}\dfrac{\{kx\}}{\lfloor kx\rfloor }<\sum_{k=1}^{n}\dfrac{1}{2k-1}\tag{1}$$ where $\{x\}=x-\lfloor x\rfloor$
My attempt: I try use induction prove this inequality.
It is clear for $n=1$, because $\{x\}<1\le \lfloor x\rfloor$.
Now if assume that $n$ holds, in other words: $$\sum_{k=1}^{n}\dfrac{\{kx\}}{\lfloor kx\rfloor }<\sum_{k=1}^{n}\dfrac{1}{2k-1}$$ Consider the case $n+1$. We have
$$\sum_{k=1}^{n+1}\dfrac{\{kx\}}{\lfloor kx\rfloor }=\sum_{k=1}^{n}\dfrac{\{kx\}}{\lfloor kx\rfloor }+\dfrac{\{(n+1)x\}}{\lfloor (n+1)x\rfloor}<\sum_{k=1}^{n}\dfrac{1}{2k-1}+\dfrac{\{(n+1)x\}}{\lfloor (n+1)x\rfloor}$$ It suffices to prove that $$\dfrac{\{(n+1)x\}}{\lfloor (n+1)x\rfloor}<\dfrac{1}{2n+1}\tag{2}$$ But David gives an example showing $(2)$ is wrong, so how to prove $(1)$?
inequality ceiling-and-floor-functions
math110math110
$\begingroup$ if $x=1.6$ and $n=2$, then $\dfrac{\{(n+1)x\}}{\lfloor (n+1)x\rfloor}<\dfrac{1}{2n+1}$ means $\frac{4.8}{4}<\frac{1}{5}$, but it is not true. $\endgroup$
– xunitc
$\begingroup$ Indeed it is quite apparent that (1) is much stronger than the desired inequality (and, as it happens, too strong). $\endgroup$
– Did
$\begingroup$ I think the inequality should be true in general for x>2 see desmos.com/calculator/ldz4ewl7m8 $\endgroup$
– Navin
$\begingroup$ @navinstudent Well, of course it is since, for every $x>2$, $(n+1)x>2n+2$ hence $$\frac{\{(n+1)x\}}{\lfloor(n+1)x\rfloor}<\frac1{2n+2}$$ The question asks for every $x>1$. $\endgroup$
$\begingroup$ Source: China Team Selection Test 2017 TST 1 Day Problem 2. (2017.03.06) artofproblemsolving.com/community/… $\endgroup$
– River Li
To start the proof, first it's proven that the inequality is true for all $x \geq 2$ so we are interested in the case $ 1 \leq x \leq 2$.
also its true for $n=1$ because in that case its just $\frac{x}{\lfloor x \rfloor} < 2$ which is $ x < 2$ because $\lfloor x \rfloor = 1$ since $1 \leq x \leq 2$ exclusion.
also from now on we will let $x=2-\epsilon$ such that $0 < \epsilon <1$
First Case : $0 < \epsilon < \frac{1}{n}$ so $\lfloor (2-\epsilon) k \rfloor$ is less than or equal to $2k-1$ so it becomes $\sum \limits_{k=1}^{n} \frac{(2-\epsilon)k}{2k-1} < \sum \limits_{k=1}^{n} \frac{2k}{2k-1}$ which is clearly true (even just be looking at it).
Second Case : $\frac{1}{n} < \epsilon < \frac{2}{n}$ so $\lfloor (2-\epsilon) k\rfloor$ is less than or equal to $2k-1$ when $1 \leq k \leq \frac{n}{2}$ and is less than or equal to $2k-2$ when $1+\frac{n}{2} \leq k \leq n$ so it becomes
$\sum \limits_{k=1}^{\frac{n}{2}} \frac{(2-\frac{2}{n})k}{2k-1}+\sum \limits_{k=1+\frac{n}{2}}^{n} \frac{(2-\frac{2}{n})k}{2k-2} < \sum \limits_{k=1}^{n} \frac{2k}{2k-1} $ evaluating this summation and moving all terms to the right side we arrive at : $$0<-\frac{n^2+n^2 \psi ^{(0)}\left(\frac{n}{2}+1\right)-n^2 \psi ^{(0)}\left(\frac{n}{2}\right)-3 n-n \psi ^{(0)}\left(\frac{n}{2}+1\right)-n \psi ^{(0)}\left(\frac{n}{2}\right)+2 n \psi ^{(0)}(n)+2 \psi ^{(0)}\left(\frac{n}{2}\right)-2 \psi ^{(0)}(n)+2}{2 n}-\frac{(n-1) \left(n+\psi ^{(0)}\left(\frac{n}{2}+\frac{1}{2}\right)-\psi ^{(0)}\left(\frac{1}{2}\right)\right)}{2 n}+\frac{1}{2} \left(2 n+\psi ^{(0)}\left(n+\frac{1}{2}\right)-\psi ^{(0)}\left(\frac{1}{2}\right)\right)$$
simplifying this expression we arrive at results : $$\frac{n H_{n-\frac{1}{2}}-(n-1) H_{\frac{n-1}{2}}+2 n+2 (n-1) \psi ^{(0)}\left(\frac{n}{2}\right)-2 (n-1) \psi ^{(0)}(n)+\log (4)}{2 n} >0$$
just to state before continuing in the proof :
1) $\psi^{(k)}(n)$ is the poly gamma function, a special case to this function is $\psi^{(0)}(n)$ which is equal to $H_{n-1}-\gamma$
2) $H_n = \sum \limits_{k=1}^{n} \frac{1}{k}$ is the $n$-th harmonic number and Euler proved that $H_n \geq \ln(n) +\gamma$ and its also true that $H_n \leq \ln(n)+1$
3) $\gamma \approx 0.5772156649$ which is Euler–Mascheroni constant.
to return to the proof, with some arithmetic manipulation and lower and upper bound for $H_n$ as stated above we reach at : $$\frac{1}{2} n \left(3 \gamma n-n+n (-\log (n-1))-2 n \log (n)+n \log (2 n-1)+2 (n-1) \log \left(\frac{n-2}{2}\right)+2 \log (n)+\log (2 n-2)-2 \gamma +3\right)>0$$ multiply by $2n$ which will not effect the inequality because $n$ is positive. we arrive at : $$3 \gamma n-n+n (-\log (n-1))-2 n \log (n)+n \log (2 n-1)+2 (n-1) \log \left(\frac{n-2}{2}\right)+2 \log (n)+\log (2 n-2)-2 \gamma +3>0$$ solving it in Wolfram we get that its true for all $n>2.37646$ and we check for $n=1,2$ and by this we conclude the proof for the second case.
General Case : $\frac{m}{n} \leq \epsilon \leq \frac{m+1}{n}$ for any $1 \leq m \leq n$ so $\lfloor (2-\epsilon) k \rfloor$ is less or equal to $2k-1$ when $1 \leq k \leq \frac{n}{m}$ and $\lfloor (2-\epsilon) k \rfloor$ is less or equal to $2k-2$ when $ 1+\frac{n}{m} \leq k \leq \frac{2 n}{m}$ and in general $\lfloor (2-\epsilon) k \rfloor$ is less or equal to $2k-1-j$ when $1+\frac{j n}{m} \leq k \leq \frac{(j+1)n}{m}$ for $0 \leq j \leq m-1$.
so it becomes : $$\sum _{j=0}^{m-1} \left(\sum _{k=1+\frac{n j}{m}}^{\frac{n (j+1)}{m}} \frac{\left(2-\frac{m}{n}\right) k}{2 k-j-1}\right) < \sum \limits_{k=1}^{n} \frac{2k}{2k-1}$$ evaluating both sides in the inequality we get:
$$ \sum _{j=0}^{m-1} \frac{2 m^2 \psi ^{(0)}\left(\frac{n j}{m}-\frac{j}{2}+\frac{1}{2}\right)+j m^2 \psi ^{(0)}\left(\frac{n j}{m}-\frac{j}{2}+\frac{3}{2}\right)-m^2 \psi ^{(0)}\left(\frac{n j}{m}-\frac{j}{2}+\frac{3}{2}\right)-j m^2 \psi ^{(0)}\left(\frac{n j}{m}-\frac{j}{2}+\frac{n}{m}+\frac{1}{2}\right)-m^2 \psi ^{(0)}\left(\frac{n j}{m}-\frac{j}{2}+\frac{n}{m}+\frac{1}{2}\right)-4 j n^2 \psi ^{(0)}\left(\frac{n j}{m}-\frac{j}{2}+\frac{1}{2}\right)+4 j n^2 \psi ^{(0)}\left(\frac{n j}{m}-\frac{j}{2}+\frac{3}{2}\right)+2 j m n \psi ^{(0)}\left(\frac{n j}{m}-\frac{j}{2}+\frac{1}{2}\right)-4 m n \psi ^{(0)}\left(\frac{n j}{m}-\frac{j}{2}+\frac{1}{2}\right)-4 j m n \psi ^{(0)}\left(\frac{n j}{m}-\frac{j}{2}+\frac{3}{2}\right)+2 m n \psi ^{(0)}\left(\frac{n j}{m}-\frac{j}{2}+\frac{3}{2}\right)+2 j m n \psi ^{(0)}\left(\frac{n j}{m}-\frac{j}{2}+\frac{n}{m}+\frac{1}{2}\right)+2 m n \psi ^{(0)}\left(\frac{n j}{m}-\frac{j}{2}+\frac{n}{m}+\frac{1}{2}\right)+2 m^2-6 m n+4 n^2}{4 m n} <\frac{1}{2} \left(2 n+\psi ^{(0)}\left(n+\frac{1}{2}\right)-\psi ^{(0)}\left(\frac{1}{2}\right)\right) $$ , Wolfram was not able to evaluate the upper summation, (no problem) because the inner summation was increasing function with respect to $j$ (easy to see : will not prove),then by the summation bounded by integration law for increasing function $f$ we get that : $$ \sum \limits_{i=a}^{b} f(i) \leq \int \limits_{a}^{b+1} f(t)dt $$
so the above inequality becomes : $$ \int_0^m \frac{2 m^2-6 m n+4 n^2+2 m^2 \psi ^{(0)}\left(\frac{1}{2}-\frac{j}{2}+\frac{j n}{m}\right)-4 m n \psi ^{(0)}\left(\frac{1}{2}-\frac{j}{2}+\frac{j n}{m}\right)+2 j m n \psi ^{(0)}\left(\frac{1}{2}-\frac{j}{2}+\frac{j n}{m}\right)-4 j n^2 \psi ^{(0)}\left(\frac{1}{2}-\frac{j}{2}+\frac{j n}{m}\right)-m^2 \psi ^{(0)}\left(\frac{3}{2}-\frac{j}{2}+\frac{j n}{m}\right)+j m^2 \psi ^{(0)}\left(\frac{3}{2}-\frac{j}{2}+\frac{j n}{m}\right)+2 m n \psi ^{(0)}\left(\frac{3}{2}-\frac{j}{2}+\frac{j n}{m}\right)-4 j m n \psi ^{(0)}\left(\frac{3}{2}-\frac{j}{2}+\frac{j n}{m}\right)+4 j n^2 \psi ^{(0)}\left(\frac{3}{2}-\frac{j}{2}+\frac{j n}{m}\right)-m^2 \psi ^{(0)}\left(\frac{1}{2}-\frac{j}{2}+\frac{n}{m}+\frac{j n}{m}\right)-j m^2 \psi ^{(0)}\left(\frac{1}{2}-\frac{j}{2}+\frac{n}{m}+\frac{j n}{m}\right)+2 m n \psi ^{(0)}\left(\frac{1}{2}-\frac{j}{2}+\frac{n}{m}+\frac{j n}{m}\right)+2 j m n \psi ^{(0)}\left(\frac{1}{2}-\frac{j}{2}+\frac{n}{m}+\frac{j n}{m}\right)}{4 m n} \, dj < \frac{1}{2} \left(2 n+\psi ^{(0)}\left(n+\frac{1}{2}\right)-\psi ^{(0)}\left(\frac{1}{2}\right)\right)$$ now before evaluating the left side we want to find the values of $m$ that produces the maximum value, which mean the derivative of the left side is equal to $0$, assume that the left side integrate result is $F(m)-F(0)$ so the derivative is $F'(m)-F'(0)$ which is equal to $\sum _{k=1+\frac{m n}{m}}^{\frac{(m+1) n}{m}} \frac{\left(2-\frac{m}{n}\right) k}{2 k-m-1}-\sum _{k=1}^{\frac{n}{m}} \frac{\left(2-\frac{m}{n}\right) k}{2 k-1}$ which evaluate to $\frac{(m-2 n) \left((-m-1) H_{-\frac{(m+1) (m-2 n)}{2 m}}+(m+1) H_{-\frac{m}{2}+n-\frac{1}{2}}+H_{\frac{n}{m}-\frac{1}{2}}+\log (4)\right)}{4 n}$ we want this derivative to equal to $0$, one simple answer is when $m=\frac{n}{2}$ another answer which is a bit harder to see but also simple is $m=n$ (we know that one of the answer is minimum and one is maximum, calculation and experimentation suggest that $m=\frac{n}{2}$ is the maximum and $m=n$ is the minimum, assuming that we don't know which is which) we will substitute both values.
now back to were we left, we will evaluate the new left side, we arrive at:
$$ \frac{m^2 \log \left(32 \pi ^{12} A^{36}\right)-12 m \left((m+1) (m-2 n) \text{log$\Gamma $}\left(-\frac{m}{2}+n+\frac{1}{2}\right)+(m-2 n) \text{log$\Gamma $}\left(\frac{n}{m}+\frac{1}{2}\right)-(m+1) (m-2 n) \text{log$\Gamma $}\left(-\frac{m}{2}+n+\frac{1}{2}+\frac{n}{m}\right)+2 m \psi ^{(-2)}\left(-\frac{m}{2}+n+\frac{1}{2}\right)+2 m \psi ^{(-2)}\left(\frac{n}{m}+\frac{1}{2}\right)-2 m \psi ^{(-2)}\left(-\frac{m}{2}+n+\frac{1}{2}+\frac{n}{m}\right)\right)-12 n \left((m-2 n)^2+m \log (\pi )\right)}{12 n (m-2 n)} < \frac{1}{2} \left(2 n+\psi ^{(0)}\left(n+\frac{1}{2}\right)-\psi ^{(0)}\left(\frac{1}{2}\right)\right)$$
(note : the $A$ written after the evaluation is the Glaisher–Kinkelin constant,$A \approx 1.282427129$, and the function $Log\Gamma(x)$ is the log-gamma function which is just $\ln(\Gamma(t))$).
first we prove the inequality when $m=n$ we arrive at :
$$-(n+1) \text{log$\Gamma $}\left(\frac{n+1}{2}\right)+(n+1) \text{log$\Gamma $}\left(\frac{n+3}{2}\right)+n+n \left(-\log \left(\frac{n+1}{2}\right)\right)-\log (n+1)+\log (2)<\frac{1}{2} \left(2 n+\psi ^{(0)}\left(n+\frac{1}{2}\right)-\psi ^{(0)}\left(\frac{1}{2}\right)\right) $$
so giving the proper uppers bound and lower bounds and basic arithmetic manipulation we arrive at :
$$ n+n \left(-\log \left(\frac{n+1}{2}\right)\right)-\log (n+1)+(n+1) \log \left(\frac{n+3}{2}\right)+\log (2)< \frac{1}{2} \left(2 n+\log \left(n-\frac{1}{2}\right)+\gamma -\psi ^{(0)}\left(\frac{1}{2}\right)\right)$$ solving for $n$ we get that its true for all $n > 2.29577$ and we solved for $n=1,2$
so we are left with the last part of the proof,we prove the inequality when $m=\frac{n}{2}$ we arrive at :
$$ \frac{1}{24} \left(-3 (3 n+2) \text{log$\Gamma $}\left(\frac{3 n}{4}+\frac{1}{2}\right)+(9 n+6) \text{log$\Gamma $}\left(\frac{3 (n+2)}{4}\right)+24 n-9 n \log \left(\frac{1}{4} (3 n+2)\right)-2 \log (3 n+2)+\log (256)\right) < \frac{1}{2} \left(2 n+\psi ^{(0)}\left(n+\frac{1}{2}\right)-\psi ^{(0)}\left(\frac{1}{2}\right)\right)$$ we will do the same tricks of upper and lower bound and basic arithmetic manipulation,we get to : $$\frac{1}{24} \left(24 n-9 n \log \left(\frac{1}{4} (3 n+2)\right)+(9 n+6) \log \left(\frac{3 (n+2)}{4}\right)-2 \log (3 n+2)+\log (256)\right) < \frac{1}{2} \left(2 n+\log \left(n-\frac{1}{2}\right)+\gamma -\psi ^{(0)}\left(\frac{1}{2}\right)\right) $$ and solving this inequality we get that its true for all $n > 0.701281$ and thus concluding that the inequality is true for all $x \geq 1$ and $n$ positive integers.
note : please don't down vote, it took me 3 hours to finish the proof so if there is any poor language, or anything else cut me some slack.
hope its what your are looking for.
AhmadAhmad
$\begingroup$ At the Second Case, how it becomes to $\sum \limits_{k=1}^{\frac{n}{2}} \frac{(2-\frac{2}{n})k}{2k-1}+\sum \limits_{k=1+\frac{n}{2}}^{n} \frac{(2-\frac{2}{n})k}{2k-2} < \sum \limits_{k=1}^{n} \frac{2k}{2k-1}$ please? If $1 \le k \le \frac{n}{2}$ or $1+\frac{n}{2} \le k \le n$, I think when $n=3$ it misses $k=2$, maybe it is $1 \le k \le \frac{n}{2}$ or $\frac{n}{2} < k \le n$. and then, when $\lfloor (2-\epsilon) k\rfloor \le 2k-2$, by reciprocal, $\frac{1}{\lfloor (2-\epsilon) k\rfloor} \ge \frac{1}{2k-2}$, so inequality reverse? I can not understand, could you explain it in detail please? $\endgroup$
$\begingroup$ @xunitc you are right it should be $1 \leq k \leq \frac{n}{2}+1$ and $\frac{n}{2}+2 \leq k \leq n$, and doing the same to the general case, but i checked and it does not effect that much and the argument holds true. $\endgroup$
– Ahmad
$\begingroup$ well, let $n$ is even. then at $\frac{1}{n} < \epsilon < \frac{2}{n}$, we have $\lfloor (2-\epsilon) k \rfloor \le 2k-2$, but not $\lfloor (2-\epsilon) k \rfloor \color{red}{=} 2k-2$, so by inequality, I think we just can write$$\sum_{k=1}^{n}\frac{kx}{[kx]} = \sum_{k=1}^{n/2}\frac{kx}{[kx]}+\sum_{k=n/2+1}^{n}\frac{kx}{[kx]} \le \sum_{k=1}^{n/2}\frac{kx}{2k-1}+\sum_{k=n/2+1}^{n}\frac{kx}{\color{green}{2k-1}}$$ the green part can not be $2k-2$ by '$\le$'. and $kx$ on numerator, I think it is $kx = k(2-\varepsilon) < k(2-\frac{1}{n})$, not $< k(2-\frac{2}{n})$. what is my problem please? $\endgroup$
$\begingroup$ @xunitc here you are relaxing my argument because $\sum \limits_{k=n/2+1}^{n} \frac{k x}{2k-2} > \sum \limits_{k=n/2+1}^{n} \frac{k x}{2k-1}$, thus i proved it to the extreme point and what you did is just a relaxation, because you are right $\lfloor (2-\epsilon)k \rfloor =2k-1$ for most of the cases but even if it did not hold even for one $k$ then the proof will collapse in the contrary, in my argument i took it to the extreme. $\endgroup$
$\begingroup$ Thank you. I know your means now. Thank you for your patience. $\endgroup$
Looking at several plots indicates that $$f_n(x):=\sum_{k=1}^n{\{kx\}\over\lfloor kx\rfloor}$$ is largest immediately to the left of $x=2$. Now for $x=2-\epsilon$ with $0<\epsilon\ll1$ one has $$\lfloor kx\rfloor=2k-1,\quad\{kx\}=1-2k\epsilon$$ and therefore $$f_n(x)=\sum_{k=1}^n{1-2k\epsilon\over2k-1}<\sum_{k=1}^n{1\over2k-1}\ .$$ Maybe you want to take a look at the following graph of $f_{250}$:
Christian BlatterChristian Blatter
$\begingroup$ Thanks,This inequality prove by $x\in (1,2)$ is key,so How prove it? $\endgroup$
– math110
Source: China Team Selection Test 2017 TST Day 1 Problem 2. (2017.03.06) https://artofproblemsolving.com/community/c422484_2017_china_team_selection_test
AoPS user hutu683 gave a solution. I put it here for people to check the proof.
hutu683's solution: Clearly, we only need to prove the case when $x \in (1, 2)$. Let $x = 1 + \alpha$ with $\alpha \in (0, 1)$. We need to prove that $$\sum_{k=1}^n \frac{\{k\alpha \}}{k + \lfloor k\alpha \rfloor} < \sum_{k=1}^n \frac{1}{2k-1}. \tag{1}$$
To proceed, we need the following lemma. The proof is given later.
Lemma 1: For positive integers $a\le b$ and $m$ satisfying $\lfloor k \alpha\rfloor = m, \forall k \in [a, b]\cap\ \mathbb{N}$ and $\lfloor (a-1)\alpha\rfloor < m$, we have $$\sum_{k=a}^b \frac{\{k\alpha \}}{k + \lfloor k\alpha \rfloor} < \sum_{k=a}^b \frac{1}{2k-1}.$$
From Lemma 1, the inequality in (1) holds. This completes the proof.
$\phantom{2}$
Remarks: Here I give some explanation about what hutu683's proof did.
Let $I_m = \{k: \ \lfloor k\alpha\rfloor = m, \quad k\in \{1, 2, \cdots, n\}\}$ for $m = 0, 1, 2, \cdots, \lfloor n\alpha \rfloor$. Then $\{I_0, I_1, \cdots, I_{\lfloor n\alpha \rfloor}\}$ is a partition of $\{1, 2, \cdots, n\}$. We have $$\sum_{k=1}^n \frac{\{k\alpha \}}{k + \lfloor k\alpha \rfloor} = \sum_{m=0}^{\lfloor n\alpha \rfloor} \sum_{k\in I_m} \frac{\{k\alpha \}}{k + \lfloor k\alpha \rfloor}, \quad\quad \sum_{k=1}^n \frac{1}{2k-1} = \sum_{m=0}^{\lfloor n\alpha \rfloor} \sum_{k\in I_m} \frac{1}{2k-1}. $$ It suffices to prove that $$\sum_{k\in I_m} \frac{\{k\alpha \}}{k + \lfloor k\alpha \rfloor} < \sum_{k\in I_m} \frac{1}{2k-1}$$ for $m = 0, 1, 2, \cdots, \lfloor n\alpha \rfloor$.
For $m=0$, clearly we have $$\sum_{k\in I_0} \frac{\{k\alpha \}}{k + \lfloor k\alpha \rfloor} = \sum_{k\in I_0} \frac{k\alpha}{k} = |I_0|\alpha < 1 \le \sum_{k\in I_0} \frac{1}{2k-1}.$$
For $m \in \{1, 2, \cdots, \lfloor n\alpha \rfloor\}$ (only if $\lfloor n\alpha \rfloor \ge 1$), we need to prove that $$\sum_{k\in I_m} \frac{k+m - (2k-1)\{k\alpha\}}{(2k-1)(k+m)} > 0. \tag{2}$$ From hutu683's Lemma 1, the inequality in (2) is true.
Proof of Lemma 1: From the conditions, we have $(a-1)\alpha < m \le a\alpha$ and $b\alpha < m + 1$. We need to prove that $$\sum_{k=a}^b \frac{k+m - (2k-1)\{k\alpha\}}{(2k-1)(k+m)} > 0.$$
There are two possible cases as follows.
1st Case $\alpha \ge \frac{1}{b-a + 1}$: For $k\in [a, b]\cap\ \mathbb{N}$, since $\lfloor k\alpha \rfloor = \lfloor b\alpha \rfloor$, we have $\{k\alpha\} = \{b\alpha\} - (b-k)\alpha < 1 - (b-k)\alpha$. Combining this with $m > b\alpha - 1$, we have \begin{align} k + m - (2k-1)\{k\alpha\} &> k + b\alpha - 1 - (2k-1)(1-(b-k)\alpha)\\ &= k((2b-2k+1)\alpha - 1). \end{align} Thus, we have $$\sum_{k=a}^b \frac{k+m - (2k-1)\{k\alpha\}}{(2k-1)(k+m)} > \sum_{k=a}^b \frac{(2b-2k+1)\alpha - 1}{(2-\frac{1}{k})(k+m)}.$$ Note that $(2b-2k+1)\alpha - 1$ and $\frac{1}{(2-\frac{1}{k})(k+m)}$ both decrease when $k$ increases. Thus, by Chebyshev's sum inequality, we have \begin{align} \sum_{k=a}^b \frac{(2b-2k+1)\alpha - 1}{(2-\frac{1}{k})(k+m)} &\ge \frac{1}{b-a+1}\sum_{k=a}^b ((2b-2k+1)\alpha - 1) \sum_{k=a}^b \frac{1}{(2-\frac{1}{k})(k+m)}\\ &= \frac{1}{b-a+1}(b-a+1)((b-a+1)\alpha - 1) \sum_{k=a}^b \frac{1}{(2-\frac{1}{k})(k+m)}\\ &\ge 0. \end{align} The desired result follows.
2nd Case $\alpha < \frac{1}{b-a + 1}$: For $k\in [a, b]\cap\ \mathbb{N}$, since $\lfloor k\alpha \rfloor = \lfloor a\alpha \rfloor$, we have $\{k\alpha\} = \{a\alpha\} + (k-a)\alpha < \alpha + (k-a)\alpha$ where $\{a\alpha\} < \alpha$ follows from $(a-1)\alpha < m \le a\alpha$. Combining this with $m > (a-1)\alpha$, we have \begin{align} k+ m - (2k-1)\{k\alpha\} &> k+ (a-1)\alpha - (2k-1)(\alpha + (k-a)\alpha)\\ &= k(1 - (2k-2a+1)\alpha). \end{align} Thus, we have $$\sum_{k=a}^b \frac{k+m - (2k-1)\{k\alpha\}}{(2k-1)(k+m)} > \sum_{k=a}^b \frac{1 - (2k-2a+1)\alpha}{(2-\frac{1}{k})(k+m)}.$$ Note that $1 - (2k-2a+1)\alpha$ and $\frac{1}{(2-\frac{1}{k})(k+m)}$ both decrease when $k$ increases. Thus, by Chebyshev's sum inequality, we have \begin{align} \sum_{k=a}^b \frac{1 - (2k-2a+1)\alpha}{(2-\frac{1}{k})(k+m)} &\ge \frac{1}{b-a+1}\sum_{k=a}^b ( 1 - (2k-2a+1)\alpha) \sum_{k=a}^b \frac{1}{(2-\frac{1}{k})(k+m)}\\ &= \frac{1}{b-a+1}(b-a+1)(1-(b-a+1)\alpha)\sum_{k=a}^b \frac{1}{(2-\frac{1}{k})(k+m)}\\ &\ge 0. \end{align} The desired result follows. This completes the proof of Lemma 1.
edited Nov 9 '19 at 6:15
River LiRiver Li
Equation (1) is not true in general, in fact, for every $n$ one can find an $x$ for which it is false. Specifically, given $n\ge1$, choose $$x=\frac{n+\frac74}{n+1}>1\ .$$ Then $2n>1$, so $4n+4=4n+3+1<6n+3$, so $$\frac{\{(n+1)x\}}{\lfloor(n+1)x\rfloor} =\frac{\frac34}{n+1}=\frac3{4n+4}>\frac3{6n+3}=\frac1{2n+1}\ .$$
DavidDavid
$\begingroup$ Thanks,that said my indution is not right.so How to prove $(2)$ $\endgroup$
$\begingroup$ @HazemOrabi If $x>2$ one can even replace each $\frac1{2k-1}$ by $\frac1{2k}$ (and shorten considerably your approach, see my comment on main). $\endgroup$
$\begingroup$ @Did yes, if $x>2$. $\{kx\}=kx-[kx]$, so just need to proof $$\sum_{k=1}^{n}\frac{kx-[kx]}{[kx]}<\sum_{k=1}^{n}\frac{1}{2k-1}$$ as $$\sum_{k=1}^{n}\frac{kx}{[kx]}<\sum_{k=1}^{n}(\frac{1}{2k-1}+1)=\sum_{k=1}^{n}\frac{2k}{2k-1}(*)$$ when $x \ge 2$, then $\frac{x}{2} \ge 1 > kx-[kx]$, so $[kx] > kx-\frac{x}{2}$ and $\frac{1}{[kx]} < \frac{1}{kx-\frac{x}{2}}$, so $\frac{kx}{[kx]} < \frac{kx}{kx-\frac{x}{2}} = \frac{2k}{2k-1}$. (*) is true. $\endgroup$
$\begingroup$ @xunitc Please see previous comment to Hazem and/or comment on main, for a simpler approach to a stronger result when $x>2$. $\endgroup$
I tried all day and couldn't prove it but I made a little progress:
Let's define $\{x\}'$ to be 1 if $x$ is an integer and $\{x\}$ otherwise, and note that the LHS of the original inequality satisfies $$\sum_{k=1}^{n}\dfrac{\{kx\}}{\lfloor kx\rfloor} \leq \sum_{k=1}^{n}\dfrac{\{kx\}'}{\lceil kx\rceil-1}\tag{1}$$
If $a=\lceil nx\rceil$ then $\lceil kx\rceil=\lceil k\frac an\rceil$ for $k=1,2,... n$ (can be proved by contradiction), and the modified fractional part is non-decreasing, so it suffices to prove that $$\sum_{k=1}^{n}\dfrac{\{k\frac an\}'}{\lceil k\frac an\rceil-1}<\sum_{k=1}^{n}\dfrac{1}{2k-1}$$ for integers $a\in (n,2n)$ (since we can assume $1<x<2$).
The RHS of (1) can be rewritten as $$\sum_{k=1}^{n}\dfrac{\{kx\}'}{kx-\{kx\}'}=\sum_{k=1}^{n}\dfrac{1}{kx/\{kx\}'-1}$$ since $\lceil kx\rceil=kx+(1-\{kx\}')$.
Letting $x=\frac an$, if $\gcd(a,n)=1$ then $\{\{kx\}':k=1,2,...n\}=\{\frac 1n,\frac 2n,...\frac nn\}$. For $k\in[1,n-1]$, let unique $t\in[1,n]$ such that $t\equiv ak\pmod{n}$. Then $\{kx\}'=\frac tn$ and $k=[a^{-1}t]_n$ so we can write our summation with index $t$: $$\frac1{a-1}+\sum_{k=1}^{n-1}\dfrac{1}{kx/\{kx\}'-1}=\frac1{a-1}+\sum_{t=1}^{n-1}\dfrac{1}{k\frac an/\frac tn-1}$$ $$=\frac1{a-1}+\sum_{t=1}^{n-1}\dfrac{1}{k\frac at-1}$$ Now since $ka\equiv t\pmod n$ we have $ka=u_tn+t$ for some $u_t\in[1,a]$, so this then becomes $$=\frac1{a-1}+\sum_{t=1}^{n-1}\dfrac{1}{\frac {u_tn+t}t-1}$$ $$=\frac1{a-1}+\frac 1n\sum_{t=1}^{n-1}\dfrac{t}{u_t}$$
I stopped at this point but I'll try to see if I can turn it into a proof tomorrow. Comment if you have any ideas!
AkababaAkababa
This problem blew up on Reddit, due to the massively machine-aided proof in the (as of now) top answer. Since this is a competition problem, one expects a more human solution. I'll complete the idea presented by DVDthe1st in the AoPS post for this problem.
We will need the following form of Abel summation: given real numbers $a_1,a_2,\ldots,a_m$ and $b_1\geq b_2\geq\cdots\geq b_m\geq b_{m+1}=0$, we have $$\sum_{1\leq i\leq m}a_ib_i=\sum_{1\leq i\leq m}a_i\sum_{j\geq i}(b_j-b_{j+1})=\sum_{1\leq j\leq m}(b_j-b_{j+1})\sum_{1\leq i\leq j}a_i.$$ Here we used telescoping series, then swapped the order of summation.
Main idea: we want to choose something like $b_i\overset?=\frac1i$ and $a_{\lfloor kx\rfloor}\overset?=\{kx\}$; then we can bound the original LHS by bounding partial sums of $\{kx\}$, which is much easier to handle.
First, we fix some notation to be used throughout the solution. We will be considering functions of the form $f(x)=\sum_j\lambda_j\{\mu_jx\}$ for positive constants $\lambda_j,\mu_j$; these functions are increasing except at integer multiples of $\frac1{\mu_j}$, where they jump downwards. At such a discontinuity $x=x_0$, we will say that $$f\rightsquigarrow C\text{ at }x=x_0^-$$ to mean that $$\lim_{x\to x_0^-}f(x)=C>\lim_{x\to x_0^+}f(x).$$ For example, in the original inequality, we have LHS $\rightsquigarrow$ RHS at $x=2^-$: if we take $x$ approaching $2$ from below (ie. $x=2-\varepsilon$ for small $\varepsilon>0$) then each term in LHS approaches the corresponding term in RHS, but equality is not achieved at either $x=2-\varepsilon$ or $x=2$, due to the jump discontinuity. (This also show that RHS is the best possible constant.)
Lemma: For any real $x\geq1$ and any integer $c\geq1$, we have $$\sum_{0<ix<c}\{ix\}<\frac c2.$$
Proof: We first consider the special case where $x=(\frac cd)^-$ for some integer $\frac c2<d<c$ coprime to $c$. Since the numbers $c,2c,\ldots,(d-1)c$ cover all nonzero residue classes (mod $d$), the numbers $\{x\},\{2x\},\ldots,\{(d-1)x\}$ form some permutation of $\frac1d,\frac2d,\ldots,\frac{d-1}d$. Now $x=(\frac cd)^-$ means that the term $\{dx\}$ is also counted in the sum, with value $\rightsquigarrow1$. Hence $$\sum_{0<ix<c}\{ix\}\rightsquigarrow\sum_{0<j\leq d}\frac jd=\frac1d\frac{d(d+1)}2=\frac{d+1}2\leq\frac c2.$$
For the general case, note that LHS is increasing in $x$ except at discontinuities, which occur whenever $bx$ hits an integer for some $k=b$ in the sum. Hence we may increase $x$ to $x=(\frac ab)^-$ for some integer $a\leq c$ and $a,b$ coprime. In this case we have $$\begin{aligned} \sum_{0<ix<c}\{ix\}&=\sum_{0<ix<a}\{ix\}+\sum_{a\leq ix<c}\{ix\}\\ &\rightsquigarrow\frac{b-1}2+\sum_{0<i'x<c-a}\{(i'+b)x\}\\ &\leq\frac a2+\sum_{0<i'x<c-a}\{i'x\}, \end{aligned}$$ and so we may finish by induction on $c$. (The base case $c=1$ is clearly true since LHS = 0.) $\square$
It turns out that the above lemma is too weak to be used with Abel summation directly. This is because the "equality case" of the original inequality, namely $x=2^-$, is not preserved in the lemma; specifically, the RHS of the lemma is $\frac12$ bigger than LHS when $x=2^-$ and $c$ is odd. As such, we need a stronger result which is tight at $x=2^-$. After a few more hours, we come up with the following:
Main Lemma: For any real $x\geq1$ and any integer $c\geq1$, we have $$\sum_{0<ix<c-1}\{ix\}+\frac12\sum_{c-1\leq ix<c}\{ix\}<\frac{c-1}2.$$
Note that this is equivalent to $$\sum_{0<ix<c-1}\{ix\}+\sum_{0<ix<c}\{ix\}<c-1,$$ while our previous lemma can only give $c-\frac12$ on the RHS.
Proof: In fact, the proof of the previous lemma carries over almost verbatim. The only difference is in the calculation for the special case, where we halve the last term: if $x=(\frac cd)^-$ then $$\sum_{0<ix<c}\{ix\}\rightsquigarrow\sum_{0<j\leq d-1}\frac jd+\frac12=\frac1d\frac{d(d-1)}2+\frac12=\frac d2\leq\frac{c-1}2.\quad\square$$
We are now ready to finish the proof. Remember that as long as we only use the Lemma for even $c$ and the Main Lemma for any $c$, we will preserve the equality case at $2^-$, so the bounds are guaranteed to work out (even if the calculations look intimidating).
Proof of original inequality: First, take $$a_i=\begin{cases}\{kx\},&\text{if }i=\lfloor kx\rfloor\text{ for some }k\geq1,\\0,&\text{else}.\end{cases}$$ This is well-defined since for each $i\geq1$ there is at most one $k\geq1$ such that $\lfloor kx\rfloor=i$ (because $x\geq1$). We also write $$A_k=\sum_{1\leq i\leq k}a_i=\sum_{0<ix<k+1}\{ix\},$$ so the Lemma states $A_k<\frac{k+1}2$, and the Main Lemma states that $A_{k-1}+A_k<k$.
Note that $$\sum_{1\leq k\leq n}\frac{\{kx\}}{\lfloor kx\rfloor}=\sum_{1\leq i\leq\lfloor nx\rfloor}\frac{a_i}i\leq\sum_{1\leq i\leq2n-1}\frac{a_i}i.$$ For even $i$ we will use the inequality $\frac1i\leq\frac12(\frac1{i-1}+\frac1{i+1})$, so we can rewrite the whole sum as $$\begin{aligned} \sum_{1\leq i\leq2n-1}\frac{a_i}i&\leq\frac12\left(a_1+a_2+\frac13(a_3+a_4)+\cdots+\frac1{2n-3}(a_{2n-3}+a_{2n-2})+\frac1{2n-1}a_{2n-1}\right)\\ &\qquad+\frac12\left(a_1+\frac13(a_2+a_3)+\cdots+\frac1{2n-1}(a_{2n-2}+a_{2n-1})\right)\\ &\overset{\text{Abel}}=\frac12\left[\left(1-\frac13\right)A_2+\left(\frac13-\frac15\right)A_4+\cdots+\left(\frac1{2n-3}-\frac1{2n-1}\right)A_{2n-2}+\frac1{2n-1}A_{2n-1}\right]\\ &\qquad+\frac12\left[\left(1-\frac13\right)A_1+\left(\frac13-\frac15\right)A_3+\cdots+\left(\frac1{2n-3}-\frac1{2n-1}\right)A_{2n-3}+\frac1{2n-1}A_{2n-1}\right]\\ &=\left(1-\frac13\right)\frac{A_1+A_2}2+\left(\frac13-\frac15\right)\frac{A_3+A_4}2+\cdots+\left(\frac1{2n-3}-\frac1{2n-1}\right)\frac{A_{2n-3}+A_{2n-2}}2+\frac1{2n-1}A_{2n-1}\\ &<\left(1-\frac13\right)(1)+\left(\frac13-\frac15\right)(2)+\cdots+\left(\frac1{2n-3}-\frac1{2n-1}\right)(n-1)+\frac1{2n-1}n\\ &=1+\frac13+\frac15+\cdots+\frac1{2n-1}, \end{aligned}$$ and we are done. $\square$
Spent a couple of days on this problem. I really enjoyed the statement and proof of the Lemma, which is essentially combinatorial in nature. Strengthening it to the Main Lemma is pretty crazy though; kudos to DVDthe1st for the idea.
chronondecaychronondecay
Not the answer you're looking for? Browse other questions tagged inequality ceiling-and-floor-functions or ask your own question.
Prove that: $\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$, for $n > 1$
How prove this $\{a\}\cdot\{b\}\cdot\{c\}=0$ if $\lfloor na\rfloor+\lfloor nb\rfloor=\lfloor nc\rfloor$
Prove this Floor function indentity $\sum_{k=0}^{n-1} \bigl\lfloor \frac{ak+b}{c} \bigr\rfloor$
How to prove ceiling and floor inequality more 'formally'?
Inequality involving floor function
Prove that $\sum_{k=0}^{n-1}\left \lfloor x+\dfrac{k}{n}\right \rfloor=\lfloor nx\rfloor$ | CommonCrawl |
\begin{document}
\title{\huge \bf Precise rates in the law of the iterated logarithm \footnote{Research supported by National Natural Science Foundation of China }} \author{ {\sc Li-Xin Zhang \footnote{Department of Mathematics, Zhejiang University, Hangzhou 310027, China,\newline E-mail: [email protected]}} \\ {\em Department of Mathematics, Zhejiang University, Hangzhou 310027, China }}
\date{ }
\maketitle
{\rm
{\sc Abstract.} \quad Let $X$, $X_1$, $X_2$, $\ldots$ be i.i.d. random variables, and set $S_n=X_1+\ldots + X_n$, $M_n=\max_{k\le n}|S_k|$, $n\ge 1$. Let $a_n=o(\sqrt{n/\log\log n})$. By using the strong approximation, we prove that, if $\textsf{E} X^2I\{|X|\ge t\}=o((\log\log t)^{-1})$ as $t\to \infty$, then for $a>-1$ and $b>-1$, \begin{eqnarray*} & & \lim_{\epsilon\searrow \sqrt{1+a}} (\epsilon^2-a-1)^{b+1/2} \sum_{n=1}^{\infty} \frac{(\log n)^a (\log\log n)^b}{n} \textsf{P} \Big\{M_n \ge \sigma \phi(n) \epsilon +a_n\Big\} \\ & &= 2\sqrt{\frac 1{\pi (a+1)} } \Gamma(b+1/2) \end{eqnarray*} holds if and only if $$ \textsf{E} X= 0,\;\; \textsf{E} X^2=\sigma^2<\infty \; \text{ and }\;
\textsf{E}\big[ X^2(\log |X|)^a(\log\log |X|)^{b-1}\big]<\infty.$$
We also show that the condition $\textsf{E} X^2I\{|X|\ge t\}=o((\log\log t)^{-1})$ is sharp. The results of Gut and Sp\u ataru (2000) are special cases of ours.
{\bf Keywords:} Tail probabilities of sums of i.i.d. random variables, \quad the law of the iterated logarithm, \quad strong approximation.
{\bf AMS 1991 subject classification:} Primary 60F15, Secondary 60G50.
\vskip 0.2in
}
\section{Introduction and main results.} \setcounter{equation}{0}
Let $\{X, X_n; n\ge 1\}$ be a sequence of i.i.d random variables with common distribution function $F$, mean $0$ and positive, finite variance $\sigma^2$, and set $S_n=\sum_{k=1}^n X_k$,
$M_n=\max_{k\le n}|S_k|$, $n\ge 1$. Also let $\log x=\ln(x\vee e)$, $\log\log x=\log(\log x)$ and $\phi(x)=\sqrt{2x\log\log x}$.
Then by the well-known law of the iterated logarithm
(LIL) we have
\begin{equation} \label{eq1.1}
\limsup_{n\to \infty}\frac{M_n}{\phi(n)}
=\limsup_{n\to \infty}\frac{|S_n|}{\phi(n)} =\sigma \quad a.s..
\end{equation} Gut and Sp\u ataru (2000) proved the following two results on its precise asymptotics.
\proclaim{Theorem A} Suppose that $\textsf{E} X=0$, $\textsf{E} X^2=\sigma^2$ and $\textsf{E}[X^2(\log\log |X|)^{1+\delta}]<\infty$ for some $\delta>0$, and let $a_n=O(\sqrt{n}/ (\log\log n)^{\gamma})$ for some $\gamma>1/2$. Then $$\lim_{\epsilon\searrow 1}\sqrt{\epsilon^2-1}\sum_{n=1}^{\infty}\frac 1n
\textsf{P}(|S_n|\ge \epsilon \sigma \phi(n)+a_n)=1 . $$ \egroup\par
\proclaim{Theorem B} Suppose that $\textsf{E} X=0$ and $\textsf{E} X^2=\sigma^2<\infty$. Then $$\lim_{\epsilon\searrow 0}\epsilon^2 \sum_{n=1}^{\infty}\frac 1{n\log n}
\textsf{P}(|S_n|\ge \epsilon\sqrt{ n\log\log n})=\sigma^2. $$ \egroup\par
The main purpose of this paper is to show general results under the {\it minimal} conditions by using an Feller's (1945) and Einmahl's (1989) truncation method. The following two theorems are our main results.
\begin{theorem} \label{th1} Let $a>-1$ and $b>-1/2$ and let $a_n(\epsilon)$ be a function of $\epsilon$ such that \begin{eqnarray} \label{co1.1} a_n(\epsilon) \log \log n \to \tau \; \text{ as } \; n\to \infty \text{ and } \epsilon \searrow \sqrt{1+a}. \end{eqnarray} Suppose that \begin{eqnarray} \label{co1.2} \textsf{E} X=0, \; \textsf{E} X^2=\sigma^2<\infty \; \text{ and } \;
\textsf{E}\big[ X^2(\log |X|)^a(\log\log |X|)^{b-1}\big]<\infty \end{eqnarray} and \begin{eqnarray} \label{co1.3}
\textsf{E} X^2I\{|X|\ge t\}=o((\log\log t)^{-1}) \; \text{ as }\; t\to \infty. \end{eqnarray} Then \begin{eqnarray}\label{eqT1.1} & & \lim_{\epsilon\searrow \sqrt{1+a}} (\epsilon^2-a-1)^{b+1/2} \sum_{n=1}^{\infty} \frac{(\log n)^a (\log\log n)^b}{n} \textsf{P} \Big\{M_n \ge \sigma \phi(n) (\epsilon +a_n(\epsilon))\Big\} \nonumber \\ & & \qquad \qquad \qquad=2\sqrt{\frac 1{\pi (a+1)} }\exp\{-2\tau\sqrt{1+a}\} \Gamma(b+1/2) \end{eqnarray} and \begin{eqnarray}\label{eqT1.2} & & \lim_{\epsilon\searrow \sqrt{1+a}} (\epsilon^2-a-1)^{b+1/2} \sum_{n=1}^{\infty} \frac{(\log n)^a (\log\log n)^b}{n}
\textsf{P} \Big\{|S_n| \ge \sigma \phi(n) (\epsilon +a_n(\epsilon))\Big\} \nonumber \\ & & \qquad \qquad \qquad=\sqrt{\frac 1{\pi (a+1)} }\exp\{-2\tau\sqrt{1+a}\} \Gamma(b+1/2). \end{eqnarray} Here, $\Gamma(\cdot)$ is a gamma function. Conversely, if (\ref{eqT1.1}) or (\ref{eqT1.2}) holds for $a>-1$, $b>-1/2$ and some $0<\sigma<\infty$, then (\ref{co1.2}) holds and \begin{equation}\label{eqT1.3}
\liminf_{t\to \infty}(\log\log t)\textsf{E} X^2I\{|X|\ge t\}=0. \end{equation} \end{theorem}
\begin{theorem} \label{th2} Suppose that $\textsf{E} X=0$ and $\textsf{E} X^2=\sigma^2<\infty$, and let $a_n=O(1/\log\log n)$. For $b>-1$, we have \begin{eqnarray}\label{eqT2.1} & & \lim_{\epsilon\searrow 0} \epsilon^{2(b+1)} \sum_{n=1}^{\infty} \frac{(\log\log n)^b}{n\log n} \textsf{P} \Big\{M_n \ge \sigma\phi(n) (\epsilon+a_n) \Big\} \nonumber \\ & & \qquad \qquad \qquad= \frac 2{(b+1)\sqrt{\pi}}\Gamma(b+3/2)\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^{2b+2}} \end{eqnarray} and \begin{eqnarray}\label{eqT2.2} & & \lim_{\epsilon\searrow 0} \epsilon^{2(b+1)} \sum_{n=1}^{\infty}
\frac{(\log\log n)^b}{n\log n} \textsf{P} \Big\{|S_n| \ge \sigma\phi(n) (\epsilon+a_n) \Big\} \nonumber \\ & & \qquad \qquad \qquad= \frac 1{(b+1)\sqrt{\pi}}\Gamma(b+3/2). \end{eqnarray} Conversely, if (\ref{eqT2.1}) or (\ref{eqT2.2}) holds for some $b>-1$ and $0<\sigma<\infty$, then $\textsf{E} X=0$ and $\textsf{E} X^2=\sigma^2$. \end{theorem}
\begin{remark} Note that the condition (\ref{co1.3}) is sharp. A sufficient condition for it is given by
$$ \textsf{E} X^2 \log\log |X|<\infty. $$ So, (\ref{co1.3}) is weaker than Gut and Sp\u ataru's condition in Theorem A (see also their Remark 1.1). When $a>0$ (or $a=0$ and $b\ge 2$), the condition (\ref{co1.3}) is implied by (\ref{co1.2}). \end{remark}
\begin{remark} The condition that $\textsf{E} X=0$ and $\textsf{E} X^2<\infty$ is obviously sufficient and necessary for the conclusion of Theorem B to hold, by Theorem \ref{th2}. (see also Remark 1.2 of Gut and Sp\u ataru, 2000). \end{remark}
The proofs of Theorem \ref{th1} and \ref{th2} are given in Section 4. Before that, we first verify (\ref{eqT1.1}), (\ref{eqT1.2}), (\ref{eqT2.1}) and (\ref{eqT2.2}) under the assumption that $F$ is the normal distribution in Section 2, after which, by using the truncation and approximation method, we then show that the probabilities in (\ref{eqT1.1}), (\ref{eqT1.2}), (\ref{eqT2.1}) and (\ref{eqT2.2}) can be replaced by those for normal random variables in Section 3. Throughout this paper, we let $K(\alpha,\beta,\cdots)$, $C(\alpha,\beta,\cdots)$ etc denote positive constants which depend on $\alpha,\beta, \cdots$ only, whose values can differ in different places. $a_n\sim b_n$ means that $a_n/b_n\to 1$.
\section{Normal cases.} \setcounter{equation}{0}
In this section, we prove Theorems \ref{th1} and \ref{th2} in the case that $\{X, X_n; n\ge 1\}$ are normal random variables. Let $\{W(t); t\ge 0\}$ be a standard Wiener process and $N$ a standard normal variable. Our results are as follows.
\begin{proposition}\label{prop2.1} Let $a>-1$ and $b>-1/2$ and let $a_n(\epsilon)$ be a function of $\epsilon$ satisfying (\ref{co1.1}). Then \begin{eqnarray}\label{eqprop2.1.1} & & \lim_{\epsilon\searrow \sqrt{1+a}} (\epsilon^2-a-1)^{b+1/2} \sum_{n=1}^{\infty} \frac{(\log n)^a (\log\log n)^b}{n} \nonumber \\ & & \qquad \qquad \qquad \qquad\cdot
\textsf{P} \Big\{\sup_{0\le s\le 1}|W(s)| \ge \sqrt{2\log\log n} (\epsilon +a_n(\epsilon))\Big\} \nonumber \\ & & \qquad \qquad \qquad=2\sqrt{\frac 1{\pi (a+1)} }\exp\{-2\tau\sqrt{1+a}\} \Gamma(b+1/2) \end{eqnarray} and \begin{eqnarray}\label{eqprop2.1.2} & & \lim_{\epsilon\searrow \sqrt{1+a}} (\epsilon^2-a-1)^{b+1/2} \sum_{n=1}^{\infty} \frac{(\log n)^a (\log\log n)^b}{n}
\textsf{P} \Big\{|N| \ge \sqrt{2\log\log n} (\epsilon +a_n(\epsilon))\Big\} \nonumber \\ & & \qquad \qquad \qquad=\sqrt{\frac 1{\pi (a+1)} }\exp\{-2\tau\sqrt{1+a}\} \Gamma(b+1/2) \end{eqnarray} \end{proposition}
\begin{proposition}\label{prop2.2} Let $a_n=O(1/\log\log n)$. For any $b>-1$, we have \begin{eqnarray*} & & \lim_{\epsilon\searrow 0} \epsilon^{2(b+1)} \sum_{n=1}^{\infty}
\frac{(\log\log n)^b}{n\log n} \textsf{P} \Big\{\sup_{0\le s\le 1}|W(s)|
\ge
(\epsilon+a_n)\sqrt{2\log\log n} \Big\} \\ & & \qquad \qquad \qquad= \frac 2{(b+1)\sqrt{\pi}}\Gamma(b+3/2)\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^{2b+2}} \end{eqnarray*} and \begin{eqnarray*} & & \lim_{\epsilon\searrow 0} \epsilon^{2(b+1)} \sum_{n=1}^{\infty}
\frac{(\log\log n)^b}{n\log n} \textsf{P} \Big\{|N| \ge
(\epsilon +a_n)\sqrt{2\log\log n} \Big\} \\ & & \qquad \qquad \qquad= \frac 1{(b+1)\sqrt{\pi}}\Gamma(b+3/2). \end{eqnarray*} \end{proposition}
The following lemma will be used in the proofs.
\begin{lemma}\label{lem2.1} Let $\{W(t); t\ge 0\}$ be a standard Wiener process. Then for all $x>0$, \begin{eqnarray}\label{eqL2.1.1}
\textsf{P} \big(\sup_{0\le s\le 1}|W(s)| \ge x\big) &=& 1-\sum_{k=-\infty}^{\infty} (-1)^k\textsf{P}\big((2k-1)x\le N\le (2k+1)x\big) \nonumber \\ &=&4\sum_{k=0}^{\infty} (-1)^k\textsf{P}\big( N\ge (2k+1)x\big) \nonumber \\ &=&2\sum_{k=0}^{\infty}
(-1)^k\textsf{P}\big( |N|\ge (2k+1)x\big). \end{eqnarray} In particular, $$
\textsf{P}\big( \sup_{0\le s\le 1}|W(s)|\ge x \big)
\sim 2\textsf{P}\big( |N|\ge x\big)\sim \frac 4{\sqrt{2\pi}x}e^{-x^2/2} \; \text{ as } \; x\to +\infty. $$ \end{lemma} {\bf Proof.} It is well known. See Billingsley (1968).
Now, we turn to prove the propositions.
{\noindent\bf Proof Proposition \ref{prop2.1}:} First, note that the limit in (\ref{eqprop2.1.1}) does not depend on any finite terms of the infinite series. Secondly, by Lemma \ref{lem2.1} and the condition (\ref{co1.1}) we have \begin{eqnarray*}
& &\textsf{P} \Big\{\sup_{0\le s\le 1}|W(s)|
\ge \sqrt{2\log\log n} (\epsilon +a_n(\epsilon))\Big\}
\sim 2\textsf{P} \Big\{|N|
\ge \sqrt{2\log\log n} (\epsilon +a_n(\epsilon))\Big\} \\ & & \qquad \sim \frac 4{\sqrt{2\pi}(\epsilon+a_n(\epsilon))
\sqrt{2\log\log n}} \exp\Big\{-(\epsilon+a_n(\epsilon))^2 \log\log n \Big\} \\ & & \qquad \sim \frac 2{\sqrt{\pi}\epsilon } \frac 1{\sqrt{\log\log n}} \exp\Big\{-\epsilon^2\log\log n \Big\} \exp\Big\{-2\epsilon a_n(\epsilon)\log\log n \Big\} \end{eqnarray*} as $n\to \infty$, uniformly in $\epsilon\in (\sqrt{1+a},\sqrt{1+a}+\delta)$ for some $\delta>0$. So, for any $0<\theta<1$, there exist $\delta>0$ and $n_0$ such that for all $n\ge n_0$ and $\epsilon\in (\sqrt{1+a},\sqrt{1+a}+\delta)$, $$\begin{array}{ll} & \frac 2{\sqrt{\pi(1+a)} }\frac 1{\sqrt{\log\log n}} \exp\Big\{-\epsilon^2\log\log n \Big\} \exp\Big\{-2\tau\sqrt{1+a}-\theta\Big\} \\
\le &\textsf{P} \Big\{\sup_{0\le s\le 1}|W(s)|
\ge \sqrt{2\log\log n} (\epsilon +a_n(\epsilon))\Big\} \\ \le & \frac 2{\sqrt{\pi(1+a)} }\frac 1{\sqrt{\log\log n}} \exp\Big\{-\epsilon^2\log\log n \Big\} \exp\Big\{-2\tau\sqrt{1+a}+\theta\Big\} \end{array} $$ and $$\begin{array}{ll} & \frac 1{\sqrt{\pi(1+a)} }\frac 1{\sqrt{\log\log n}} \exp\Big\{-\epsilon^2\log\log n\Big\} \exp\Big\{-2\tau\sqrt{1+a}-\theta\Big\} \\
\le &\textsf{P} \Big\{|N|
\ge \sqrt{2\log\log n} (\epsilon +a_n(\epsilon))\Big\} \\ \le & \frac 1{\sqrt{\pi(1+a)} }\frac 1{\sqrt{\log\log n}} \exp\Big\{-\epsilon^2\log\log n \Big\} \exp\Big\{-2\tau\sqrt{1+a}+\theta\Big\}, \end{array} $$ by the condition (\ref{co1.1}) again. Also, \begin{eqnarray*} & & \lim_{\epsilon\searrow \sqrt{1+a}} (\epsilon^2-a-1)^{b+1/2} \sum_{n=1}^{\infty}\frac{(\log n)^a (\log\log n)^b}{n} \frac 1{\sqrt{\log\log n}} \exp\Big\{-\epsilon^2 \log\log n\Big\} \\ &=& \lim_{\epsilon\searrow \sqrt{1+a} } (\epsilon^2-a-1)^{b+1/2} \int_{e^e}^{\infty} \frac{(\log x)^a (\log\log x)^{b-1/2}}{x} \exp\Big\{-\epsilon^2\log\log x\Big\} dx \\ &=&
\lim_{\epsilon\searrow \sqrt{1+a}} (\epsilon^2-a-1)^{b+1/2} \int_1^{\infty} y^{b-1/2} \exp\Big\{-y\big(\epsilon^2-1-a\big)\Big\} dy \\ &=& \lim_{\epsilon\searrow \sqrt{1+a}} \int_{\epsilon^2-1-a}^{\infty} y^{b-1/2} e^{-y} dy = \int_0^{\infty} y^{b-1/2}e^{-y} dy
=\Gamma(b+1/2). \end{eqnarray*} (\ref{eqprop2.1.1}) and (\ref{eqprop2.1.2}) are now proved.
{\noindent\bf Proof Proposition \ref{prop2.2}:} Observe (\ref{eqL2.1.1}),
$$ \textsf{P}(|N|\ge x)=2\textsf{P}(N\ge x), \; \forall x>0, $$ and for any $m\ge 1$ and $x>0$, \begin{eqnarray*}
4\sum_{k=0}^{2m+1} (-1)^k\textsf{P}\big( N\ge (2k+1)x\big)
&\le& \textsf{P} \big(\sup_{0\le s\le 1}|W(s)| \ge x\big) \\ &\le&4\sum_{k=0}^{2m} (-1)^k\textsf{P}\big( N\ge (2k+1)x\big). \end{eqnarray*} It is sufficient to show that for any $q>0$, \begin{eqnarray*} & & \lim_{\epsilon\searrow 0} \epsilon^{2(b+1)} \sum_{n=1}^{\infty} \frac{(\log\log n)^b}{n\log n} \textsf{P}\Big(N\ge q(\epsilon+a_n)\sqrt{2\log\log n}\Big) \\ & & \qquad \qquad =q^{-2(b+1)}\frac 1{2(b+1)\sqrt{\pi}}\Gamma(b+3/2). \end{eqnarray*} Obviously, \begin{eqnarray*} & & \lim_{\epsilon\searrow 0} \epsilon^{2(b+1)} \sum_{n=1}^{\infty} \frac{(\log\log n)^b}{n\log n} \textsf{P}\Big(N\ge q(\epsilon+a_n)\sqrt{2\log\log n}\Big) \\ &=& q^{-2(b+1)}
\lim_{\epsilon\searrow 0} \epsilon^{2(b+1)} \sum_{n=1}^{\infty} \frac{(\log\log n)^b}{n\log n} \textsf{P}\Big(N\ge (\epsilon+q a_n)\sqrt{2\log\log n}\Big). \end{eqnarray*} So, it is sufficient to show that \begin{eqnarray*}
\lim_{\epsilon\searrow 0} \epsilon^{2(b+1)} \sum_{n=1}^{\infty} \frac{(\log\log n)^b}{n\log n} \textsf{P}\Big(N\ge (\epsilon+a_n)\sqrt{2\log\log n}\Big) =\frac 1{2(b+1)\sqrt{\pi}}\Gamma(b+3/2). \end{eqnarray*}
Without losing of generality, we assume that $|a_n|\le \tau/\log\log n$. Notice that \begin{eqnarray*}
& & \Big|\textsf{P}\Big(N\ge (\epsilon+a_n)\sqrt{2\log\log n}\Big)
-\textsf{P}\Big(N\ge \epsilon\sqrt{2\log\log n}\Big)\Big|\\ &\le& \frac 1{\sqrt{2\pi}} \exp\big\{-\frac{2\log\log n(\epsilon-\tau/\log\log n)^2}{2}\big\}
|a_n|\sqrt{2\log\log n} \\ &\le&\frac{\tau }{\sqrt{\log\log n}} \exp\big\{-\epsilon^2\log\log n+2\epsilon\tau \big\}
\end{eqnarray*} and \begin{eqnarray*}
& &\lim_{\epsilon\searrow 0} \epsilon^{2(b+1)} \sum_{n=1}^{\infty} \frac{(\log\log n)^b}{n\log n}\cdot \frac{1}{\sqrt{\log\log n}} \exp\big\{-\epsilon^2\log\log n \big\} \\ &=& \lim_{\epsilon\searrow 0} \epsilon^{2(b+1)} \int_{e^e}^{\infty} \frac{(\log\log x)^{b-1/2}}{x\log x} \exp\big\{-\epsilon^2\log\log x \big\} dx \\ &=& \lim_{\epsilon\searrow 0} \epsilon^{2(b+1)} \int_1^{\infty} y^{b-1/2}\exp\{-\epsilon^2 y \} dy = \lim_{\epsilon\searrow 0} \epsilon \int_{\epsilon^2}^{\infty} y^{b-1/2}e^{-y} dy\\ &=&\lim_{\epsilon\searrow 0} \epsilon \int_{\epsilon^2}^1 y^{b-1/2}e^{-y} dy +\lim_{\epsilon\searrow 0} \epsilon \int_1^{\infty} y^{b-1/2}e^{-y} dy \\ &\le& \lim_{\epsilon\searrow 0} \epsilon \int_{\epsilon^2}^1 y^{b-1/2} dy=0. \end{eqnarray*} Thus, it follows that \begin{eqnarray*} & & \lim_{\epsilon\searrow 0} \epsilon^{2(b+1)} \sum_{n=1}^{\infty} \frac{(\log\log n)^b}{n\log n} \textsf{P}\Big(N\ge (\epsilon+a_n)\sqrt{2\log\log n}\Big) \\ &=&\lim_{\epsilon\searrow 0} \epsilon^{2(b+1)} \sum_{n=1}^{\infty} \frac{(\log\log n)^b}{n\log n} \textsf{P}\Big(N\ge \epsilon \sqrt{2\log\log n}\Big)\\
&= &\lim_{\epsilon\searrow 0} \epsilon^{2(b+1)} \int_{e^e}^{\infty} \frac{(\log\log x)^b}{x\log x} \textsf{P}\Big(N\ge \epsilon\sqrt{2\log\log x}\Big)dx \\ &=&\lim_{\epsilon\searrow 0}
\int_{\epsilon^2}^{\infty} y^b\textsf{P}\Big(N\ge \sqrt{2y}\Big)dy = \frac 1{b+1}\int_0^{\infty} \textsf{P}\Big(N\ge \sqrt{2y}\Big)d y^{b+1} \\
&=&\frac 1{b+1}\textsf{P}\Big(N\ge \sqrt{2y}\Big) y^{b+1}\big|_0^{\infty} +\frac 1{2(b+1)\sqrt{\pi} }\int_0^{\infty} y^{b+1/2}e^{-y}dy\\ &=&\frac 1{2(b+1)\sqrt{\pi}}\Gamma(b+3/2). \end{eqnarray*} The proposition is now proved.
\section{Truncation and Approximation.} \setcounter{equation}{0}
The purpose of this section is to use Feller's (1945)
and Einmahl's (1989) truncation methods to show that the probabilities in
(\ref{eqT1.1}), (\ref{eqT2.1}) for $M_n$ can be approximated by those for $\sqrt{n}\sup_{0\le s\le 1}|W(s)|$ and the probabilities in (\ref{eqT1.2}), (\ref{eqT2.2}) for $S_n$ can be approximated by those for $\sqrt{n}N$.
Suppose that $\textsf{E} X=0$ and $\textsf{E} X^2=\sigma^2<\infty$. Without losing of generality, we assume that $\sigma=1$ throughout this section.
Let $p>1/2$. For each $n$ and $1\le j\le n$, we let \begin{eqnarray*}
X_{nj}^{\prime}=X_{nj}I\{|X_j|\le \sqrt{n}/(\log\log n)^p \}, & &\overline X_{nj}^{\prime}=X_{nj}^{\prime}-\textsf{E} [X_{nj}^{\prime}], \\ S_{nj}^{\prime}=\sum_{i=1}^jX_{nj}^{\prime}, & & \overline S_{nj}^{\prime}=\sum_{i=1}^j \overline X_{nj}^{\prime}, \\
\overline M_n^{\prime}=\max_{k\le n}|\overline S_{nk}^{\prime}|, & &B_n=\sum_{k=1}^n \textsf{Var}(\overline X_{nk}^{\prime}) \end{eqnarray*} and \begin{eqnarray*} X_{nj}^{\prime\prime}=
X_{nj}I\{ \sqrt{n}/(\log\log n)^p<|X_j|\le \phi(n) \}, & &\overline X_{nj}^{\prime\prime}=X_{nj}^{\prime\prime}
-\textsf{E} [X_{nj}^{\prime\prime}], \\ X_{nj}^{\prime\prime\prime}=
X_{nj}I\{ |X_j|> \phi(n) \}, & &\overline X_{nj}^{\prime\prime\prime} =X_{nj}^{\prime\prime\prime}-\textsf{E} [X_{nj}^{\prime\prime\prime}]. \end{eqnarray*} And also define $S_{nj}^{\prime\prime}$, $S_{nj}^{\prime\prime\prime}$, $\overline S_{nj}^{\prime\prime}$, $\overline S_{nj}^{\prime\prime\prime}$, $\overline M_n^{\prime\prime}$ and $\overline M_n^{\prime\prime\prime}$ similarly.
The following two propositions are the main results of this section.
\begin{proposition}\label{prop3.1} Let $a>-1$, $b>-1$ and $2\ge p>p^{\prime}>1/2$. Suppose that the condition (\ref{co1.2}) is satisfied. Then there exist $\delta>0$ and a sequence of positive numbers $\{q_n \}$ such that \begin{eqnarray} \label{eqprop3.1.1}
&&\textsf{P}\Big\{\sup_{0\le s\le 1}|W(s)|
\ge \epsilon \sqrt{2\log\log n }
+ \frac{3}{ (\log\log n)^{p^{\prime}} } \Big\} -q_n \nonumber \\ &\le&
\textsf{P}\Big\{M_n \ge \epsilon \sqrt{2B_n\log\log n } \Big\} \nonumber \\
&\le&\textsf{P}\Big\{\sup_{0\le s\le 1}|W(s)|
\ge \epsilon \sqrt{2\log\log n }
- \frac{3}{ (\log\log n)^{p^{\prime}} } \Big\} +q_n, \end{eqnarray} \begin{eqnarray} \label{eqprop3.1.1b}
&&\textsf{P}\Big\{|N| \ge \epsilon \sqrt{2\log\log n }
+ \frac{3}{ (\log\log n)^{p^{\prime}} } \Big\} -q_n \nonumber \\ &\le&
\textsf{P}\Big\{|S_n| \ge \epsilon \sqrt{2B_n\log\log n } \Big\} \nonumber \\
&\le&\textsf{P}\Big\{|N| \ge \epsilon
\sqrt{2\log\log n }
-\frac{3}{(\log\log n)^{p^{\prime}} } \Big\} +q_n, \nonumber \\ & & \qquad \forall \epsilon\in (\sqrt{1+a}-\delta, \sqrt{1+a}+\delta), \quad n\ge 1 \end{eqnarray} and \begin{eqnarray}\label{eqprop3.1.2} \sum_{n=1}^{\infty}\frac{(\log n)^a(\log\log n)^b}{n} q_n \le K(a,b,p,p^{\prime},\delta)<\infty. \end{eqnarray} \end{proposition}
\begin{proposition}\label{prop3.2} Let $b$ be a real number and $2\ge p>p^{\prime}>1/2$. Suppose that $\textsf{E} X=0$ and $\textsf{E} X^2=1$. Then \begin{eqnarray}\label{eqprop3.2.1} & &
\textsf{P}\big(\sup_{0\le s\le 1}|W(s)|\ge x+3/(\log\log n)^{p^{\prime}}\big)-q_n^{\ast} \le \textsf{P}\big(M_n\ge x\sqrt{B_n}\big) \nonumber \\ &\le&
\textsf{P}\big(\sup_{0\le s\le 1}|W(s)|\ge x-3/(\log\log n)^{p^{\prime}}\big)+q_n^{\ast}, \quad \forall x>0, \end{eqnarray} \begin{eqnarray}\label{eqprop3.2.1b} & &
\textsf{P}\big(|N|\ge x+3/(\log\log n)^{p^{\prime}}\big)-q_n^{\ast}
\le \textsf{P}\big(|S_n|\ge x\sqrt{B_n}\big) \nonumber \\ &\le&
\textsf{P}\big(|N|\ge x-3/(\log\log n)^{p^{\prime}}\big)+q_n^{\ast}, \quad \forall x>0, \end{eqnarray} where $q_n^{\ast}\ge 0$ satisfies \begin{equation} \label{eqprop3.2.2} \sum_{n=1}^{\infty} \frac{(\log\log n)^b}{n\log n} q_n^{\ast} \le K(b,p, p^{\prime})<\infty. \end{equation} \end{proposition}
To show this two results, we need some lemmas.
\begin{lemma}\label{lem1} For any sequence of independent random variables $\{\xi_n; n\ge 1\}$ with mean zero and finite variance, there exists a sequence of independent normal variables $\{\eta_n; n\ge 1\}$ with $\textsf{E} \eta_n=0$ and $\textsf{E} \eta_n^2=\textsf{E} \xi_n^2$ such that, for all $Q>2$ and $y>0$,
$$ \textsf{P}\Big(\max_{k\le n}|\sum_{i=1}^k \xi_i-\sum_{i=1}^k \eta_i|\ge y\Big)
\le (AQ)^Qy^{-Q}\sum_{i=1}^n \textsf{E} |\xi_i|^Q, $$
whenever $\textsf{E}|\xi_i|^Q<\infty$, $i=1,\ldots,n$. Here, $A$ is a universal constant. \end{lemma}
{\bf Proof.} See Sakhaneko (1980,1984, 1985).
\begin{lemma}\label{lem2} Let $Q\ge 2$, $\xi_1, \xi_2,\ldots, \xi_n$ be independent random variables with $\textsf{E} \xi_k=0$ and $\textsf{E} |\xi_k|^Q<\infty$, $k=1,\ldots, n$. Then for all $y>0$,
$$ \textsf{P}\Big(\max_{k\le n}|\sum_{i=1}^k \xi_i|\ge y \Big) \le 2\exp\Big\{-\frac{y^2}{8\sum_{k=1}^n \textsf{Var}(\xi_k) }\Big\}
+(2AQ)^Qy^{-Q}\sum_{i=1}^n \textsf{E} |\xi_i|^Q, $$ where $A$ is a universal constant as in Lemma \ref{lem1}. \end{lemma}
{\bf Proof.} It follows from Lemma \ref{lem1} easily. See also Petrov (1995, Page 78).
\begin{lemma}\label{lem3}
Define $\Delta_n=\max_{k\le n}|\overline S_{nk}^{\prime}-S_k|$. Let $a>-1$, $b>-1$ and $p>1/2$. Suppose that the condition (\ref{co1.2}) is satisfied and $\textsf{E} X^2=1$. Then for any $\lambda>0$ there exist a constant $K=K(a,b,p,\lambda)$ such that \begin{eqnarray}\label{eqL3.1} \sum_{n=1}^{\infty} \frac{(\log n)^a(\log\log n)^b}{n}I_n
\le K \textsf{E} \Big[X^2(\log |X|)^a(\log\log |X|)^{b-1} \Big]
<\infty, \end{eqnarray} where $$ I_n=\textsf{P}\Big(\Delta_n\ge \sqrt{n}/(\log\log n)^2,
\overline M_n^{\prime}\ge \lambda \phi(n)\Big). $$ \end{lemma} {\bf Proof.}
Let $\beta_n=n \textsf{E}[|X|I\{|X|>\sqrt{n}/(\log\log n)^p\}]$. Then
$|\textsf{E}\sum_{i=1}^jX_{ni}^{\prime}|\le \beta_n$, $1\le j\le n$. Setting $$\Cal L=\{n:\beta_n\le \frac 18 \sqrt{n}/(\log\log n)^2\},$$ we have $$\{\Delta_n\ge \sqrt{n}/(\log\log n)^2\} \subset \bigcup_{j=1}^n \{ X_j\ne X_{nj}^{\prime}\}, \quad n\in \Cal L. $$ So for $n\in \Cal L$, \begin{eqnarray*} I_n&\le& \sum_{j=1}^n \textsf{P}\Big(X_j\ne X_{nj}^{\prime}, \overline M_n^{\prime}\ge \lambda\phi(n) \Big). \end{eqnarray*} Observer that $X_{nj}^{\prime}=0$ whenever $X_j\ne X_{nj}^{\prime}$, $j\le n$, so that we have for $n$ large enough and all $1\le j\le n$, \begin{eqnarray*} & &\textsf{P}\Big(X_j\ne X_{nj}^{\prime}, \overline M_n^{\prime}\ge \lambda\phi(n) \Big)\\ &=&\textsf{P}\Big(X_j\ne X_{nj}^{\prime},
\max_{k\le j-1}|\overline S_{nk}^{\prime}|\vee
\max_{j<k\le n}|\overline S_{nk}^{\prime}-X_{nj}^{\prime}| \ge \lambda\phi(n) \Big)\\ &=&\textsf{P}\Big(X_j\ne X_{nj}^{\prime}\Big)
\textsf{P}\Big(\max_{k\le j-1}|\overline S_{nk}^{\prime}|\vee
\max_{j<k\le n}|\overline S_{nk}^{\prime}-X_{nj}^{\prime}| \ge \lambda\phi(n) \Big)\\ &\le &\textsf{P}\Big(X_j\ne X_{nj}^{\prime}\Big) \textsf{P}\Big(\overline M_n^{\prime}\ge \lambda\phi(n)
-|X_{nj}^{\prime}|\Big)\\
&\le &\textsf{P}\Big(|X|>\sqrt{n}/(\log\log n)^p \Big) \textsf{P}\Big(\overline M_n^{\prime}\ge \lambda\phi(n) -\sqrt{n}/(\log\log n)^p\Big)\\
&\le &\textsf{P}\Big(|X|>\sqrt{n}/(\log\log n)^p \Big) \textsf{P}\Big(\overline M_n^{\prime}\ge \frac{\lambda}{2}\phi(n) \Big). \end{eqnarray*} A straightforward application of the inequalities of Ottaviani and Bernstein yields: \begin{eqnarray*}
\textsf{P}\Big(\overline M_n^{\prime}\ge \frac{\lambda}{2}\phi(n) \Big)
&\le& 2\textsf{P}\Big(|\overline S_n^{\prime}|\ge \frac{\lambda}{4}\phi(n) \Big) \le (\log n)^{-\eta} \\ & & \quad \text{ for some } \eta=\eta(\lambda)>0. \end{eqnarray*}
So, \begin{eqnarray*} & & \sum_{n\in \Cal L} \frac{(\log n)^a (\log\log n)^b }{n} I_n \\ &\le& C\sum_{n=1}^{\infty}\frac{(\log n)^a (\log\log n)^b }{n}
\cdot n \textsf{P}\Big(|X|>\frac{\sqrt{n}}{(\log\log n)^p} \Big)(\log n)^{-\eta}\\ &\le& \sum_{n=1}^{\infty}\sum_{j=n}^{\infty}
\textsf{P}\Big(\frac{\sqrt{j}}{(\log \log j)^p}<|X|
\le \frac{\sqrt{j+1}}{(\log\log (j+1))^p}\Big) (\log n)^{a-\eta}(\log\log n)^b \\ &\le& \sum_{j=1}^{\infty}
\textsf{P}\Big(\frac{\sqrt{j}}{(\log \log j)^p}<|X|
\le \frac{\sqrt{j+1}}{(\log\log (j+1))^p}\Big) \sum_{n=1}^j (\log n)^{a-\eta }(\log\log n)^b \\ &\le& \sum_{j=1}^{\infty}
\textsf{P}\Big(\frac{\sqrt{j}}{(\log \log j)^p}<|X|
\le \frac{\sqrt{j+1}}{(\log\log (j+1))^p}\Big) j (\log j)^{a-\eta }(\log\log j)^b \\
&\le& C\textsf{E}\Big[X^2(\log |X|)^{a-\eta }(\log\log |X|)^{b+2p} \Big]
\le C\textsf{E} \Big[X^2(\log |X|)^a(\log\log |X|)^{b-1} \Big]. \end{eqnarray*} If $n\not\in \Cal L$, then we have \begin{eqnarray*}
I_n \le \textsf{P}\Big(\overline M_n^{\prime}
\ge \lambda \phi(n) \Big) \le (\log n)^{-\eta}. \end{eqnarray*} It follows that \begin{eqnarray*} & & \sum_{n\not\in \Cal L} \frac{(\log n)^a(\log\log n)^b}{n} I_n \le \sum_{n\not\in \Cal L} \frac{(\log n)^{a-\eta}(\log\log n)^b}{n} \\ &\le& 8\sum_{n\not\in \Cal L} \frac{(\log n)^{a-\eta}(\log\log n)^{b+2}}{n^{3/2}}
\beta_n \\ &\le& 8\sum_{n=1}^{\infty} \frac{(\log n)^{a-\eta}(\log\log n)^{b+2}}{n^{1/2}} \\ & & \qquad \cdot \sum_{j=n}^{\infty}
\textsf{E}\Big[|X|I\big\{\frac{\sqrt{j}}{(\log \log j)^p}<|X|
\le \frac{\sqrt{j+1}}{(\log\log (j+1))^p}\big\}\Big]\\ &=& 8\sum_{j=1}^{\infty}
\textsf{E}\Big[|X|I\big\{\frac{\sqrt{j}}{(\log \log j)^p}<|X|
\le \frac{\sqrt{j+1}}{(\log\log (j+1))^p}\big\}\Big] \\ & & \qquad \cdot \sum_{n=1}^j \frac{(\log n)^{a-\eta}(\log\log n)^{b+2}}{n^{1/2}}\\ &\le& C\sum_{j=1}^{\infty}
\textsf{E}\Big[|X|I\big\{\frac{\sqrt{j}}{(\log \log j)^p}<|X|
\le \frac{\sqrt{j+1}}{(\log\log (j+1))^p}\big\}\Big]\\ & & \qquad \cdot \sqrt{j}(\log j)^{a-\eta}(\log\log j)^{b+2}\\
&\le& C\textsf{E}\Big[X^2(\log |X|)^{a-\eta}
(\log\log |X|)^{b+2+p}\Big]
\le C\textsf{E} \Big[X^2(\log |X|)^a(\log\log |X|)^{b-1} \Big]. \end{eqnarray*} (\ref{eqL3.1}) is proved.
\begin{lemma}\label{lem4} Let $a>-1$, $b>-1$ and $p>1/2$. Suppose the condition (\ref{co1.2}) is satisfied and $\textsf{E} X^2=1$. Then for any $\lambda>0$ there exist a constant $K=K(a,b,p,\lambda)$ such that $$\sum_{n=1}^{\infty} \frac{(\log n)^a(\log\log n)^b}{n}II_n
\le K \textsf{E} \Big[X^2(\log |X|)^a(\log\log |X|)^{b-1} \Big]
<\infty, $$ where $$II_n=\textsf{P}\Big(\Delta_n\ge \sqrt{n}/(\log\log n)^2,
M_n \ge \lambda\phi(n)\Big). $$ \end{lemma} {\bf Proof.} Obviously, $$ II_n\le \textsf{P}\Big(\Delta_n\ge \sqrt{n}/(\log\log n)^2,
\overline M_n^{\prime} \ge \frac{\lambda}{3}\phi(n)\Big) +\textsf{P}\Big(\overline M_n^{\prime\prime} \ge \frac{\lambda}{3}\phi(n)\Big) +\textsf{P}\Big(\overline M_n^{\prime\prime\prime} \ge \frac{\lambda}{3}\phi(n)\Big). $$ Observe that
$\max_{k\le n}|\textsf{E} S_{nk}^{\prime\prime\prime}| \le n \textsf{E} X^2 /\phi(n)=o(\sqrt n)$. We have \begin{eqnarray*} & & \sum_{n=1}^{\infty} \frac{(\log n)^a(\log\log n)^b}{n} \textsf{P}\Big(\overline M_n^{\prime\prime\prime} \ge \frac{\lambda}{3}\phi(n)\Big) \\ &\le& C\sum_{n=1}^{\infty} \frac{(\log n)^a(\log\log n)^b}{n} \sum_{j=1}^n \textsf{P}\Big(X_j^{\prime\prime\prime}\ne 0\Big)\\
&\le&\sum_{n=1}^{\infty} (\log n)^a (\log\log n)^b\textsf{P}\big(|X|\ge \phi(n)\big) \\
&\le& K \textsf{E} \Big[X^2(\log |X|)^a(\log\log |X|)^{b-1} \Big]. \end{eqnarray*} Also, notice that
$\sum_{k=1}^n \textsf{Var}(\overline X_{nk}^{\prime\prime})\le n \textsf{E} \big[X^2I\big\{\sqrt{n}/(\log\log n)^p<|X|\le \phi(n)\big\}\big] =o(n)$. By Lemma \ref{lem2} we have for $Q>2$, \begin{eqnarray*} & & \sum_{n=1}^{\infty} \frac{(\log n)^a(\log\log n)^b}{n} \textsf{P}\Big(\overline M_n^{\prime\prime} \ge \frac{\lambda}{3}\phi(n)\Big) \\ &\le& C \sum_{n=1}^{\infty} \frac{(\log n)^a(\log\log n)^b}{n} \exp\Big\{ - \frac{\lambda^2\phi^2(n)}{3^2 8\cdot o(n)}\Big\} \\ & & \quad + C \sum_{n=1}^{\infty} \frac{(\log n)^a(\log\log n)^b}{n}
\cdot \frac {3^Q}{\lambda^Q \phi^Q(n)}n \textsf{E}\big[|X|^QI\{|X|\le \phi(n)\}\big] \\ &\le& K+C\sum_{n=1}^{\infty} \frac{(\log n)^a(\log\log n)^b}{\phi^Q(n)}
\sum_{j=1}^n\textsf{E}\big[|X|^QI\{\phi(j-1)<|X|\le \phi(j)\}\big] \\
&\le& K+C\sum_{j=1}^{\infty}\textsf{E}\big[|X|^QI\{\phi(j-1)<|X|\le \phi(j)\}\big] \sum_{n=j}^{\infty} \frac{(\log n)^a(\log\log n)^b}{\phi^Q(n)} \\
&\le& K+C\sum_{j=1}^{\infty}\textsf{E}\big[|X|^QI\{\phi(j-1)<|X|\le \phi(j)\}\big] j\frac{(\log j)^a(\log\log j)^b}{\phi^Q(j)}\\
&\le& K+C\sum_{j=1}^{\infty}\textsf{E}\big[|X|^2I\{\phi(j-1)<|X|\le \phi(j)\}\big] (\log j)^a(\log\log j)^{b-1} \\
&\le& K+C \textsf{E} \Big[X^2(\log |X|)^a(\log\log |X|)^{b-1} \Big]<\infty. \end{eqnarray*} Finally, by noticing Lemma \ref{lem3}, we compete the proof of Lemma \ref{lem4}.
\begin{lemma}\label{lem6} Suppose that the condition (\ref{co1.2}) is satisfied. Then for any $1/2<p^{\prime}<p$ we have \begin{eqnarray}\label{eqL6.1}
& &\textsf{P}\big(\sup_{0\le s\le 1}|W(s)|
\ge x+1/(\log\log n)^{p^{\prime}}\big)-p_n \le\textsf{P}\big(\overline M_n^{\prime}\ge x\sqrt{B_n}\big) \nonumber \\ &\le&
\textsf{P}\big(\sup_{0\le s\le 1}|W(s)|
\ge x- 1/(\log\log n)^{p^{\prime}}\big)+p_n, \quad \forall x>0 \end{eqnarray} and \begin{eqnarray}\label{eqL6.1b}
& &\textsf{P}\big(|N|\ge x+1/(\log\log n)^{p^{\prime}}\big)-p_n
\le\textsf{P}\big(|\overline S_n^{\prime}|\ge x\sqrt{B_n}\big) \nonumber \\ &\le&
\textsf{P}\big(|N|\ge x-1/(\log\log n)^{p^{\prime}} \big)+p_n, \quad \forall x>0, \end{eqnarray}
where $p_n\ge 0$ satisfies \begin{equation}\label{eqL6.2} \sum_{n=1}^{\infty}\frac{(\log n)^a(\log\log n)^b}{n} p_n \le K(a,b,p,p^{\prime})<\infty. \end{equation} \end{lemma}
{\bf Proof.} By Lemma \ref{lem1}, there exist a universal constant $A>0$ and a sequence of standard Wiener processes $\{W_n(\cdot)\}$ such that for all $Q>2$, \begin{eqnarray*}
& & \textsf{P}\Big( \max_{k\le n}|\overline S_{nk}^{\prime}-W_n(\frac kn B_n)| \ge \frac 12 \sqrt{B_n}/(\log\log n)^{p^{\prime}} \Big) \\ &\le& (AQ)^Q\Big(\frac{ (\log\log n)^{p^{\prime}} }{\sqrt{B_n}}\Big)^Q
\sum_{k=1}^n \textsf{E}\big|\overline X_{nk}^{\prime}\big|^Q \\ &\le& C n \Big(\frac{(\log\log n)^{p^{\prime}}}{\sqrt{n}}\Big)^Q
\textsf{E}\big[|X|^QI\{|X|\le \sqrt{n}/(\log \log n)^p\}\big]. \end{eqnarray*} On the other hand, by Lemma 1.1.1 of Cs\"org\H o and R\'ev\'esz (1981), \begin{eqnarray*}
& & \textsf{P}\Big(|\max_{0\le s\le 1}|W_n(s B_n)-W_n(\frac{[ns]}{n} B_n)| \ge \frac 12 \sqrt{B_n}/(\log\log n)^{ p^{\prime} }\Big) \\
&=& \textsf{P}\Big(\max_{0\le s\le 1}|W_n(s)-W_n(\frac{[ns]}{n} )| \ge \frac 12 \sqrt{\frac 1n} \frac{\sqrt n}{ (\log\log n)^{p^{\prime}} }\Big) \\ &\le& Cn \exp\Big\{ -\frac{(\frac 12 \sqrt{n}/(\log\log n)^{p^{\prime}} )^2}{3}\Big\} \le C n\exp\Big\{-\frac 1{12}n/(\log\log n)^{2p^{\prime}}\Big\}. \end{eqnarray*} Let \begin{equation} \label{eqL6.3} p_n= \textsf{P}\Big(\sup_{0\le s\le 1}
\Big| \frac{\overline S_{n,[ns]}^{\prime}}{\sqrt{B_n}}
-\frac{W_n(sB_n)}{\sqrt{B_n}}\Big| \ge \frac 1{(\log\log n)^{p^{\prime}} } \Big). \end{equation} Then $p_n$ satisfies (\ref{eqL6.1}) and (\ref{eqL6.1b}), since $\{W_n(t B_n)/\sqrt{B_n}; t\ge 0\}\overset{\Cal D}=\{W(t); t\ge 0\}$ for each $n$. And also, $$p_n\le C n \Big(\frac{ (\log\log n)^{p^{\prime}} }{\sqrt{n}}\Big)^Q
\textsf{E}\big[|X|^QI\{|X|\le \sqrt{n}/(\log\log n)^p \}\big] +C n\exp\Big\{-\frac 1{12}n/(\log\log n)^{2p^{\prime}}\Big\}. $$ It follows that \begin{eqnarray*} & &\sum_{n=1}^{\infty}\frac{(\log n)^a(\log\log n)^b}{n} p_n \\ &\le& K_1+ C\sum_{n=1}^{\infty} \frac{(\log n)^a(\log\log n)^{ b+ p^{\prime}Q } }{n^{Q/2}}
\textsf{E}\big[|X|^QI\{|X|\le \sqrt{n}/(\log\log n)^p\}\big] \\ &\le& K_1+ C\sum_{n=1}^{\infty} \frac{(\log n)^a(\log\log n)^{b+ p^{\prime}Q } }{n^{Q/2}}\\ & & \quad \cdot
\sum_{j=1}^n\textsf{E}\big[|X|^Q
I\big\{\frac{\sqrt{j-1}}{(\log\log(j-1) )^p}<|X|\le
\frac{\sqrt{j}}{(\log\log j)^p}\big\} \big] \\ &\le& K_1+ C\sum_{j=1}^{\infty}
\textsf{E}\big[|X|^Q
I\big\{\frac{\sqrt{j-1}}{(\log\log(j-1) )^p}<|X|\le
\frac{\sqrt{j}}{(\log\log j)^p}\big\} \big] j\frac{(\log j)^a(\log\log j)^{ b+ p^{\prime}Q } }{j^{Q/2}} \\ &\le& K_1+
C\textsf{E}\big[|X|^2(\log|X|)^a(\log\log |X|)^{b+ (p^{\prime}-p)Q +2p}\big] \le K<\infty, \end{eqnarray*} whenever $(p^{\prime}-p)Q +2p<-1$. So, (\ref{eqL6.2}) is satisfied.
Now, we turn to prove Propositions \ref{prop3.1} and \ref{prop3.2}.
{\noindent\bf Proof of Proposition \ref{prop3.1}:}
Let $0<\delta<\frac 14\sqrt{1+a}$. Observe that, if $n$ is large enough, \begin{eqnarray*} & & \textsf{P}\Big\{M_n \ge \epsilon \sqrt{2 B_n\log\log n}
\Big\}\\ &=&\textsf{P}\Big\{M_n \ge \epsilon \sqrt{2 B_n\log\log n}, \Delta_n\le \frac{\sqrt{n}}{(\log\log n)^2} \Big\} \\ & & \quad +\textsf{P}\Big\{M_n \ge \epsilon \sqrt{2 B_n\log\log n}, \Delta_n> \frac{\sqrt{n}}{(\log\log n)^2} \Big\}\\ &\le&\textsf{P}\Big\{\overline M_n^{\prime} \ge \epsilon \sqrt{2 B_n\log\log n} - \frac{\sqrt{n}}{(\log\log n)^2} \Big\} \\ & & \quad +\textsf{P}\Big\{M_n \ge \frac {\sqrt{1+a}}{4} \phi(n), \Delta_n> \frac{\sqrt{n}}{(\log\log n)^2} \Big\}\\ &\le&\textsf{P}\Big\{\overline M_n^{\prime} \ge \sqrt{B_n}\big[\epsilon \sqrt{2 \log\log n}- \frac 2{(\log\log n)^2}\big] \Big\} +II_n \\
&\le&\textsf{P}\Big\{\sup_{0\le s\le 1}|W(s)| \ge \epsilon \sqrt{2 \log\log n}- \frac {2}{(\log\log n)^2} -\frac{1}{(\log\log n)^{p^{\prime}} } \Big\} +p_n +II_n \\
&\le&\textsf{P}\Big\{\sup_{0\le s\le 1}|W(s)| \ge \epsilon \sqrt{2 \log\log n} -\frac{3}{(\log\log n)^{p^{\prime}} } \Big\} +p_n +II_n \end{eqnarray*} for all $\epsilon\in(\sqrt{1+a}-\delta,\sqrt{1+a}+\delta)$, where $II_n$ is defined in Lemmas \ref{lem4} with $\lambda=\sqrt{1+a}/4$ and $p_n$ is defined in \ref{lem6}. Also, if $n$ is large enough, \begin{eqnarray*} & & \textsf{P}\Big\{M_n \ge \epsilon \sqrt{2 B_n\log\log n} \Big\}\\ &\ge&\textsf{P}\Big\{M_n \ge \epsilon \sqrt{2 B_n\log\log n}, \Delta_n\le \frac{\sqrt{n}}{(\log\log n)^2} \Big\} \\ &\ge&\textsf{P}\Big\{\overline M_n^{\prime}
\ge \epsilon \sqrt{2 B_n\log\log n}
+ \frac{\sqrt{n}}{(\log\log n)^2}, \Delta_n\le \frac{\sqrt{n}}{(\log\log n)^2} \Big\} \\ &\ge&\textsf{P}\Big\{\overline M_n^{\prime} \ge \sqrt{B_n} \big[\epsilon \sqrt{2\log\log n} + \frac{2}{(\log\log n)^2}\big] \Big\} \\ & &\quad -\textsf{P}\Big\{\overline M_n^{\prime}
\ge \frac {\sqrt{1+a}}{4} \phi(n),
\Delta_n> \frac{\sqrt{n}}{(\log\log n)^2}\Big\} \\
&\ge&\textsf{P}\Big\{\sup_{0\le s\le 1}|W(s)| \ge \epsilon \sqrt{2\log\log n} + \frac{3}{(\log\log n)^{p^{\prime}} }\Big\} -p_n -I_n \end{eqnarray*} for all $\epsilon\in(\sqrt{1+a}-\delta,\sqrt{1+a}+\delta)$, where $I_n$ is defined in Lemma \ref{lem3}
with $\lambda=\sqrt{1+a}/4$.
Similarly, if $n$ is large enough, \begin{eqnarray*}
&&\textsf{P}\Big\{|N| \ge \epsilon \sqrt{2\log\log n }
+ \frac{3}{ (\log\log n)^{p^{\prime}} } \Big\} -p_n-I_n \\ &\le&
\textsf{P}\Big\{|S_n| \ge \epsilon \sqrt{2B_n\log\log n } \Big\} \\
&\le&\textsf{P}\Big\{|N| \ge \epsilon
\sqrt{2\log\log n }
-\frac{3}{(\log\log n)^{p^{\prime}} } \Big\} +p_n+II_n. \end{eqnarray*} Letting $q_n=p_n+I_n+II_n$ completes the proof by Lemmas \ref{lem3}, \ref{lem4} and \ref{lem6}.
{\noindent\bf Proof Proposition \ref{prop3.2}:} Let $\{W_n(\cdot)\}$ be a sequence of standard Wiener processes being defined in the proof of Lemma \ref{lem6}, and let $p_n$ be defined in (\ref{eqL6.3}). And set $$q_n^{\ast}=
\textsf{P}\Big(\sup_{0\le s\le 1}\big|M_{[ns]}/\sqrt{B_n}-W_n(sB_n)/\sqrt{B_n}\big|\ge 3/(\log\log n)^{p^{\prime}}\Big). $$ Then $q_n^{\ast}$ satisfies (\ref{eqprop3.2.1}) and (\ref{eqprop3.2.1b}),
and also $$q_n^{\ast}\le \textsf{P}\big(\Delta_n\ge \sqrt{n}/(\log\log n)^2\big)+p_n. $$ By Lemma \ref{lem6}, $$\sum_{n=1}^{\infty} \frac{(\log\log n)^b}{n\log n} p_n \le K_1(b,p, p^{\prime})<\infty. $$ Also, following the lines in the proof of (\ref{eqL3.1}) we have \begin{eqnarray*} & &\sum_{n=1}^{\infty}\frac{(\log\log n)^b}{n\log n}
\textsf{P}\big(\Delta_n\ge \sqrt{n}/(\log\log n)^2\big) \\ &\le&\sum_{n\in \Cal L}\frac{(\log\log n)^b}{n\log n}\cdot n
\textsf{P}\big(|X|> \sqrt{n}/(\log \log n)^p \big) +\sum_{n\not\in \Cal L}\frac{(\log\log n)^{b+2}}{n^{3/2}\log n}
\beta_n\\ &\le&\sum_{n=1}^{\infty}\frac{(\log\log n)^b}{\log n}
\textsf{P}\big(|X|> \sqrt{n}/(\log \log n)^p \big) \\ & & \quad +\sum_{n=1}^{\infty}\frac{(\log\log n)^{b+2}}{\sqrt{n}\log n}
\textsf{E}\big[|X|I\{|X|> \sqrt{n}/(\log \log n)^p \}\big] \\
&\le&C\textsf{E}\big[X^2(\log|X|)^{-1}(\log\log |X|)^{b+2p}\big]
+C\textsf{E}\big[X^2(\log|X|)^{-1}(\log\log |X|)^{b+2+p}\big]\\ &\le& C\textsf{E} X^2<\infty. \end{eqnarray*} So, $q_n^{\ast}$ satisfies (\ref{eqprop3.2.2}).
\section{Proofs of the Theorems.} \setcounter{equation}{0}
\subsection{Proofs of the direct parts.}
Without losing of generality, we assume that $\textsf{E} X=0$ and $\textsf{E} X^2=1$.
{\noindent\bf Proof of the direct part of Theorem \ref{th1}:} Let $\delta>0$ small enough and $\{q_n\}$ be such that (\ref{eqprop3.1.1}), (\ref{eqprop3.1.1b}) and (\ref{eqprop3.1.2}) hold. Then \begin{eqnarray*}
\lim_{\epsilon\searrow \sqrt{1+a}} (\epsilon^2-a-1)^{b+1/2} \sum_{n=1}^{\infty} \frac{(\log n)^a (\log\log n)^b}{n} q_n=0, \end{eqnarray*} by (\ref{eqprop3.1.2}). Notice that $a_n(\epsilon)\to 0$. By (\ref{eqprop3.1.1}), we have that for $n$ large enough, \begin{eqnarray*}
&&\textsf{P}\Big\{\sup_{0\le s\le 1}|W(s)| \ge \sqrt{2\log\log n} (\epsilon+a_n(\epsilon))
+ \frac{3}{(\log\log n)^{p^{\prime}} } \Big\} -q_n \\ &\le& \textsf{P}\Big\{M_n \ge \sqrt{2 B_n \log\log n} (\epsilon+a_n(\epsilon)) \Big\} \\
&\le&\textsf{P}\Big\{\sup_{0\le s\le 1}|W(s)| \ge \sqrt{2\log\log n} (\epsilon+a_n(\epsilon)) - \frac{3}{ (\log\log n)^{p^{\prime}} } \Big\} +q_n, \\ & & \qquad \forall \epsilon\in (\sqrt{1+a}-\delta/2, \sqrt{1+a}+\delta/2). \end{eqnarray*} On the other hand, by Proposition \ref{prop2.1}, \begin{eqnarray*} & & \lim_{\epsilon\searrow \sqrt{1+a}} (\epsilon^2-a-1)^{b+1/2} \sum_{n=1}^{\infty} \frac{(\log n)^a (\log\log n)^b}{n} \nonumber \\ & & \qquad \qquad \qquad \qquad\cdot
\textsf{P} \Big\{\sup_{0\le s\le 1}|W(s)|
\ge \sqrt{2\log\log n} (\epsilon +a_n(\epsilon)) \pm \frac{3}{(\log\log n)^{p^{\prime}} }\Big\} \\ & & \qquad \qquad \qquad=2\sqrt{\frac 1{\pi (a+1)} }\exp\{-2\tau\sqrt{1+a}\} \Gamma(b+1/2). \end{eqnarray*} It follows that \begin{eqnarray}\label{eqP1.1} & & \lim_{\epsilon\searrow \sqrt{1+a}} (\epsilon^2-a-1)^{b+1/2} \sum_{n=1}^{\infty} \frac{(\log n)^a (\log\log n)^b}{n} \nonumber \\ & & \qquad \qquad \qquad \qquad\cdot \textsf{P} \Big\{M_n \ge \sqrt{2B_n \log\log n} (\epsilon +a_n(\epsilon))\Big\} \nonumber \\ & & \qquad \qquad \qquad=2\sqrt{\frac 1{\pi (a+1)} }\exp\{-2\tau\sqrt{1+a}\} \Gamma(b+1/2). \end{eqnarray} Similarly, \begin{eqnarray}\label{eqP1.1b} & & \lim_{\epsilon\searrow \sqrt{1+a}} (\epsilon^2-a-1)^{b+1/2} \sum_{n=1}^{\infty} \frac{(\log n)^a (\log\log n)^b}{n} \nonumber \\ & & \qquad \qquad \qquad \qquad\cdot
\textsf{P} \Big\{|S_n| \ge \sqrt{2B_n \log\log n} (\epsilon +a_n(\epsilon))\Big\} \nonumber \\ & & \qquad \qquad \qquad=\sqrt{\frac 1{\pi (a+1)} }\exp\{-2\tau\sqrt{1+a}\} \Gamma(b+1/2). \end{eqnarray}
Finally, noticing the condition (\ref{co1.3}), we have
$$0\le n-B_n\le 2 n \textsf{E}[X^2I\{|X|\ge \sqrt{n}/(\log\log n)^p \}] =o(n(\log\log n)^{-1}). $$ Let $a_n^{\prime}(\epsilon)=\sqrt{n/B_n}(\epsilon+a_n(\epsilon))-\epsilon$. Then $$\textsf{P}\Big\{M_n \ge \phi(n)(\epsilon +a_n(\epsilon))\Big\} =\textsf{P}\Big\{M_n \ge \sqrt{2 B_n\log\log n} (\epsilon +a_n^{\prime}(\epsilon))\Big\}, $$
$$\textsf{P}\Big\{|S_n| \ge \phi(n)(\epsilon +a_n(\epsilon))\Big\}
=\textsf{P}\Big\{ |S_n| \ge \sqrt{2 B_n\log\log n} (\epsilon +a_n^{\prime}(\epsilon))\Big\}, $$ and, $$a_n^{\prime}(\epsilon)\log\log n =\epsilon \frac{(n-B_n)\log\log n}{\sqrt{B_n}(\sqrt n+\sqrt{B_n})}+\sqrt{\frac{n}{B_n}}a_n(\epsilon)\log\log n \to \tau $$ as $n\to \infty$ and $\epsilon\searrow \sqrt{1+a}$. Now, (\ref{eqT1.1}) and (\ref{eqT1.2}) follow from (\ref{eqP1.1}) and (\ref{eqP1.1b}), respectively.
{\noindent \bf Proof of the direct part of Theorem \ref{th2}:} We show (\ref{eqT2.1}) only, since the proof of (\ref{eqT2.2}) is similar. Noticing $n\ge B_n\sim n$ and Proposition \ref{prop3.2}, for any $0<\delta<1$ we have for $n$ large enough and all $\epsilon>0$, \begin{eqnarray*}
& & \textsf{P}\Big\{\sup_{0\le s\le 1}|W(s)|
\ge [\epsilon(1+\delta)+2|a_n|+3/\log\log n] \sqrt{2\log\log n}\Big\}-q_n^{\ast} \\
&\le& \textsf{P}\Big\{\sup_{0\le s\le 1}|W(s)|\ge (\epsilon+a_n)(1+\delta) \sqrt{2\log\log n} + 3/(\log\log n)^{p^{\prime}}\Big\}
-q_n^{\ast} \\ &\le&\textsf{P}\Big\{M_n\ge (\epsilon +a_n)(1+\delta)\sqrt{2 B_n \log\log n} \Big\}\\ &\le&\textsf{P}\Big\{M_n\ge (\epsilon+a_n)\phi(n)\Big\} \le \textsf{P}\Big\{M_n\ge
(\epsilon+a_n)
\sqrt{2 B_n \log\log n} \Big\} \\
&\le&\textsf{P}\Big\{\sup_{0\le s\le 1}|W(s)|\ge (\epsilon+a_n) \sqrt{2\log\log n}
- 3/(\log\log n)^{p^{\prime}} \Big\}
+q_n^{\ast} \\
&\le&\textsf{P}\Big\{\sup_{0\le s\le 1}|W(s)|\ge (\epsilon+a_n-3/\log\log n) \sqrt{2\log\log n} \Big\}
+q_n^{\ast}. \end{eqnarray*} So, by Propositions \ref{prop2.2} and \ref{prop3.2}, \begin{eqnarray*} & &(1+\delta)^{-2(b+1)} \frac 2{(b+1)\sqrt{\pi}}\Gamma(b+3/2)\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^{2b+2}} \\ &\le& \liminf_{\epsilon\searrow 0} \epsilon^{2(b+1)} \sum_{n=1}^{\infty} \frac{(\log\log n)^b}{n\log n} \textsf{P}\Big\{M_n\ge (\epsilon+a_n)\phi(n) \Big\} \\ &\le& \limsup_{\epsilon\searrow 0} \epsilon^{2(b+1)} \sum_{n=1}^{\infty} \frac{(\log\log n)^b}{n\log n} \textsf{P}\Big\{M_n\ge (\epsilon+a_n)\phi(n) \Big\} \\ &\le& \frac 2{(b+1)\sqrt{\pi}}\Gamma(b+3/2)\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^{2b+2}}. \end{eqnarray*} Letting $\delta\to 0$ completes the proof.
\subsection{Proofs of the converse parts.} Now, we turn to prove the converse parts of Theorem \ref{th1} and \ref{th2}. First, we show that each of (\ref{eqT1.1}), (\ref{eqT1.2}), (\ref{eqT2.1}) and (\ref{eqT2.2}) implies \begin{eqnarray} \label{eqpf4.3} \textsf{E} X^2<\infty,\; \; \textsf{E} X=0 \; \; \text{ and } \;
\textsf{E}\big[ X^2(\log |X|)^a(\log\log |X|)^{b-1}\big]<\infty, \end{eqnarray} where $a=-1$ in Theorem \ref{th2}. We only give the proof that (\ref{eqT1.2}) implies (\ref{eqpf4.3}), since other proofs are similar. Let $\{\widetilde X, \widetilde X_n; n\ge 1\}$ be the symmetrization of $\{X, X_n; n\ge 1\}$, and let $\widetilde S_n=\sum_{k=1}^n\widetilde X_k$. Then by (\ref{eqT1.2}), \begin{eqnarray*}
\limsup_{\epsilon\searrow \sqrt{1+a}} (\epsilon^2-a-1)^{b+1/2} \sum_{n=1}^{\infty} \frac{(\log n)^a (\log\log n)^b}{n}
\textsf{P} \Big\{|\widetilde S_n| \ge 2\sigma \phi(n) (\epsilon +a_n(\epsilon))\Big\} \le K<\infty. \end{eqnarray*}
For $M>0$, define $Y=Y(M)=\widetilde XI\{|\widetilde X|<M\}$ and
$Y_n=Y_n(M)=\widetilde X_nI\{|\widetilde X_n|<M\}$. Observing that
$\widetilde XI\{|\widetilde X|<M\}-\widetilde XI\{|\widetilde X|\ge M\}\overset{\Cal D}=\widetilde X$ and
$\widetilde XI\{|\widetilde X|<M\}-\widetilde XI\{|\widetilde X|\ge M\}+\widetilde X=2 Y$, we obtain that \begin{eqnarray}\label{eqpf4.4} & & \limsup_{\epsilon\searrow \sqrt{1+a}} (\epsilon^2-a-1)^{b+1/2} \sum_{n=1}^{\infty} \frac{(\log n)^a (\log\log n)^b}{n}
\textsf{P} \Big\{|\sum_{k=1}^n Y_k| \ge 2\sigma \phi(n) (\epsilon +a_n(\epsilon))\Big\} \nonumber \\ & &\le
2\limsup_{\epsilon\searrow \sqrt{1+a}} (\epsilon^2-a-1)^{b+1/2} \sum_{n=1}^{\infty} \frac{(\log n)^a (\log\log n)^b}{n}
\textsf{P} \Big\{|\widetilde S_n| \ge 2\sigma \phi(n) (\epsilon +a_n(\epsilon))\Big\} \nonumber \\ & &\le 2K<\infty. \end{eqnarray} However, since $Y$ is a bounded random variable which satisfies conditions (\ref{co1.2}) and (\ref{co1.3}), by the direct part of Theorem \ref{th1} we have \begin{eqnarray}\label{eqpf4.5} & & \lim_{\epsilon\searrow \sqrt{1+a}} (\epsilon^2-a-1)^{b+1/2} \sum_{n=1}^{\infty} \frac{(\log n)^a (\log\log n)^b}{n}
\textsf{P} \Big\{|\sum_{k=1}^n Y_k| \ge \sqrt{\textsf{E} Y^2} \phi(n) (\epsilon +a_n(\epsilon))\Big\} \nonumber \\ & & \qquad \qquad \qquad=\sqrt{\frac 1{\pi (a+1)} }\exp\{-2\tau\sqrt{1+a}\} \Gamma(b+1/2)>0. \end{eqnarray} Putting (\ref{eqpf4.4}) and (\ref{eqpf4.5}) together yields
$\sqrt{\textsf{E} \widetilde X^2I\{|\widetilde X|<M\}}=\sqrt{\textsf{E} Y^2}\le 2\sigma$. Then, letting $M\to \infty$ yields $\textsf{E} X^2<\infty$.
$\textsf{E} X=0$ is obvious when $\textsf{E} X^2<\infty$, for otherwise we have
$$\textsf{P}\{|S_n|\ge \epsilon \sigma \phi(n)\}\to 1, \quad \forall \epsilon>0, $$ which implies that $$\sum_{n=1}^{\infty} \frac{(\log n)^a (\log\log n)^b}{n}
\textsf{P} \{|S_n| \ge \epsilon\sigma \phi(n) \}=\infty, \quad \forall \epsilon>0,\; a\ge -1 \; \text{ and } \; b\ge -1. $$
Now, by (\ref{eqT1.2}) and the L\'evy inequality we obtain that for some $\epsilon>0$, \begin{eqnarray*} & &\sum_{n=1}^{\infty} \frac{(\log n)^a (\log\log n)^b}{n}
\textsf{P}\{\max_{k\le n}|X_k|\ge 3\epsilon \sigma \phi(n) \} \\ & &\le C\sum_{n=1}^{\infty} \frac{(\log n)^a (\log\log n)^b}{n}
\textsf{P}\{\max_{k\le n}|X_k|\ge 2\epsilon \sigma \phi(n) +2\sqrt{n\textsf{E} X^2} \} \\ & & \le C\sum_{n=1}^{\infty} \frac{(\log n)^a (\log\log n)^b}{n}
\textsf{P} \{\max_{k\le n}|S_k| \ge\epsilon \sigma \phi(n)+\sqrt{n\textsf{E} X^2} \} \\ & & \le C\sum_{n=1}^{\infty} \frac{(\log n)^a (\log\log n)^b}{n}
\textsf{P} \{|S_n| \ge\epsilon \sigma \phi(n) \} <\infty. \end{eqnarray*} Observe that
$$\textsf{P}\{\max_{k\le n}|X_k|\ge 3\epsilon \sigma \phi(n) \} \le \frac{\textsf{E} X^2}{18\epsilon^2\log\log n}\to 0. $$ We conclude that \begin{eqnarray*} & &\sum_{n=1}^{\infty} (\log n)^a (\log\log n)^b
\textsf{P}\{|X|\ge 3\epsilon \sigma \phi(n) \} \\ &\le &C\sum_{n=1}^{\infty} \frac{(\log n)^a (\log\log n)^b}{n}
\textsf{P}\{\max_{k\le n}|X_k|\ge 3\epsilon \sigma \phi(n) \} <\infty, \end{eqnarray*} which implies
$$ \textsf{E}\big[ X^2(\log |X|)^a(\log\log |X|)^{b-1}\big]<\infty. $$ (\ref{eqpf4.3}) is proved.
Next, we show that $\textsf{E} X^2=\sigma^2$.
By the direct part of Theorem \ref{th2}, (\ref{eqT2.1}) and (\ref{eqT2.2}) shall hold with $\textsf{E} X^2$ taking the place of $\sigma^2$, which are obviously contradictory to (\ref{eqT2.1}) and (\ref{eqT2.2}) themselves, respectively, if $\textsf{E} X^2\ne \sigma^2$. Notice that (\ref{eqP1.1}) and (\ref{eqP1.1b}) hold whenever (\ref{eqpf4.3}) is satisfied. However, if $\textsf{E} X^2\ne \sigma^2$, (\ref{eqT1.1}) and (\ref{eqT1.2}) are contradictory to (\ref{eqP1.1}) and (\ref{eqP1.1b}), respectively, since $B_n\sim n \textsf{E} X^2$.
Finally, we show (\ref{eqT1.3}). Suppose that
(\ref{eqT1.3}) fails. Without losing of generality, we can assume that $\sigma^{-2}\textsf{E} [X^2I\{|X|\ge \sqrt{n}/(\log\log n)^p\}]\ge \tau_0/\log\log n$ for some $\tau_0>0$ and all $n\ge 1$. Then
$n\sigma^2-B_n\ge n\textsf{E} [X^2I\{|X|\ge \sqrt{n}/(\log\log n)^p\}]\ge n\sigma^2\tau_0/\log\log n$. Let $a_n^{\prime}(\epsilon)=\sqrt{1+\tau_0/\log\log n}\big(\epsilon+a_n(\epsilon)\big)-\epsilon$. Then $$ a_n^{\prime}(\epsilon)\log\log n\to \tau+\tau_0\sqrt{1+a}/2,$$ and $$\textsf{P}\Big\{M_n\ge \sigma\phi(n)\big(\epsilon+a_n(\epsilon)\big)\Big\}
\le \textsf{P}\Big\{M_n\ge \sqrt{2B_n\log\log n}
\big(\epsilon+a_n^{\prime}(\epsilon)\big)\Big\}, $$
$$\textsf{P}\Big\{|S_n|\ge \sigma\phi(n)\big(\epsilon+a_n(\epsilon)\big)\Big\}
\le \textsf{P}\Big\{|S_n|\ge \sqrt{2B_n\log\log n}
\big(\epsilon+a_n^{\prime}(\epsilon)\big)\Big\}, $$ It follows that (\ref{eqT1.1}) and (\ref{eqT1.2}) are contradictory to (\ref{eqP1.1}) and (\ref{eqP1.1b}), respectively. The proof is now completed.
\end{document} | arXiv |
\begin{document}
\title[Homogeneous spaces of finite volume]{Simply connected indefinite homogeneous spaces of finite volume}
\author[Baues]{Oliver Baues} \address{Oliver Baues, Department of Mathematics, Chemin du Mus\'ee 23, University of Fribourg, CH-1700 Fribourg, Switzerland} \email{[email protected]}
\author[Globke]{Wolfgang Globke} \address{Wolfgang Globke, Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria} \email{[email protected]}
\author[Zeghib]{Abdelghani Zeghib} \address{Abdelghani Zeghib, \'Ecole Normale Sup\'erieure de Lyon, Unit\'e de Math\'ematiques Pures et Appliqu\'ees, 46 All\'ee d'Italie, 69364 Lyon, France} \email{[email protected]}
\date{\today}
\subjclass[2010]{Primary 53C50; Secondary 53C30, 57S20}
\begin{abstract} Let $M$ be a simply connected pseudo-Riemannian homogeneous space of finite volume with isometry group $G$. We show that $M$ is compact and that the solvable radical of $G$ is abelian and the Levi factor is a compact semisimple Lie group acting transitively on $M$. For metric index less than three, we find that the isometry group of $M$ is compact itself. Examples demonstrate that $G$ is not necessarily compact for higher indices. To prepare these results, we study Lie algebras with abelian solvable radical and a nil-invariant symmetric bilinear form. For these, we derive an orthogonal decomposition into three distinct types of metric Lie algebras. \end{abstract}
\maketitle
\tableofcontents
\section{Introduction and main results} \label{sec:intro}
In this article we are interested in the isometry groups of simply connected homogeneous pseudo-Riemannian manifolds of finite volume.
D'Ambra \cite[Theorem 1.1]{dambra} showed that a simply connected compact analytic Lorentzian manifold (not necessarily homogeneous) has compact isometry group, and she also gave an example of a simply connected compact analytic manifold of metric signature $(7,2)$ that has a non-compact isometry group.
Here we study homogeneous spaces for arbitrary metric signature. Our main tool is the structure theory of the isometry Lie algebras developed by the authors in \cite{BGZ}. The metric on the homogeneous space induces a symmetric bilinear form on the isometry Lie algebra, and as shown in \cite{BG,BGZ}, the existence of a finite invariant measure then implies that this bilinear form is nil-invariant.
The first main result is the following theorem:
\begin{mthm}\label{mthm:geometric} Let $M$ be a connected and simply connected pseudo-Riemannian homogeneous space of finite volume, $G=\mathrm{Iso}(M)^\circ$, and let $H$ be the stabilizer subgroup in $G$ of a point in $M$. Let $G=KR$ be a Levi decomposition, where $R$ is the solvable radical of $G$. Then: \begin{enumerate} \item $M$ is compact. \item $K$ is compact and acts transitively on $M$. \item $R$ is abelian. Let $A$ be the maximal compact subgroup of $R$. Then $A=\mathrm{Z}(G)^\circ$. More explicitely, $R=A\times V$ where $V\cong\mathbbm{R}^n$ and $V^{K}=\mathbf{0}$. \item $H$ is connected. If $\dim R>0$, then $H=(H\cap K) E$, where $E$ and $H\cap K$ are normal subgroups in $H$, $(H\cap K)\cap E$ is finite, and $E$ is the graph of a non-trivial homomorphism
$\varphi:R\to K$, where the restriction $\varphi|_A$ is injective. \end{enumerate} \end{mthm}
In Section \ref{sec:simplyconn} we give examples of isometry groups of compact simply connected homogeneous $M$ with non-compact radical. However, for metric index $1$ or $2$ the isometry group of a simply connected $M$ is always compact:
\begin{mthm}\label{mthm:index2} The isometry group of any simply connected pseudo-Riemannian homogeneous manifold of finite volume with metric index $\ell\leq2$ is compact. \end{mthm}
As follows from Theorem \ref{mthm:geometric}, the isometry Lie algebra of a simply connected pseudo-Riemannian homogeneous space of finite volume has abelian radical. This motivates a closer investigation of Lie algebras with abelian radical that admit nil-invariant symmetric bilinear forms in Section \ref{sec:abelianradical}. Our main result is the following algebraic theorem:
\begin{mthm}\label{mthm:abelian_rad1} Let $\liealgebra{G}$ be a Lie algebra whose solvable radical $\liealgebra{R}$ is abelian. Suppose $\liealgebra{G}$ is equipped with a nil-invariant symmetric bilinear form $\langle\cdot,\cdot\rangle$ such that the kernel $\liealgebra{G}^\perp$ of $\langle\cdot,\cdot\rangle$ does not contain a non-trivial ideal of $\liealgebra{G}$. Let $\liealgebra{K}\times\liealgebra{S}$ be a Levi subalgebra of $\liealgebra{G}$, where $\liealgebra{K}$ is of compact type and $\liealgebra{S}$ has no simple factors of compact type. Then $\liealgebra{G}$ is an orthogonal direct product of ideals \[ \liealgebra{G} = \liealgebra{G}_1 \times \liealgebra{G}_2 \times \liealgebra{G}_3, \] with \[ \liealgebra{G}_1=\liealgebra{K}\ltimes\liealgebra{A}, \quad \liealgebra{G}_2=\liealgebra{S}_0, \quad \liealgebra{G}_3=\liealgebra{S}_1\ltimes\liealgebra{S}_1^*, \] where $\liealgebra{R}=\liealgebra{A}\times\liealgebra{S}_1^*$ and $\liealgebra{S}=\liealgebra{S}_0\times\liealgebra{S}_1$ are orthogonal direct products, and $\liealgebra{G}_3$ is a metric cotangent algebra. The restrictions of $\langle\cdot,\cdot\rangle$ to $\liealgebra{G}_2$ and $\liealgebra{G}_3$ are invariant and non-degenerate. In particular, $\liealgebra{G}^\perp\subseteq\liealgebra{G}_1$. \end{mthm}
For the definition of metric cotangent algebra, see Section \ref{sec:nilinvariant}. We call an algebra $\liealgebra{G}_1=\liealgebra{K}\ltimes\liealgebra{A}$ with $\liealgebra{K}$ semisimple of compact type and $\liealgebra{A}$ abelian a Lie algebra of \emph{Euclidean type}. By Theorem \ref{mthm:geometric}, isometry Lie algebras of compact simply connected pseudo-Riemannian homogeneous spaces are of Euclidean type. However, not every Lie algebra of Euclidean type appears as the isometry Lie algebra of a compact pseudo-Riemannian homogeneous space. In fact, this is the case for the Euclidean Lie algebras $\liealgebra{E}_n=\liealgebra{SO}_n\ltimes\mathbbm{R}^n$ with $n\neq 3$.
\begin{mthm}\label{thm:noSOnRn2} The Euclidean group $\mathrm{E}_n=\mathrm{O}_n\ltimes\mathbbm{R}^n$, $n\neq 1,3$, does not have compact quotients with a pseudo-Riemannian metric such that $\mathrm{E}_n$ acts isometrically and almost effectively. \end{mthm}
Note that $\mathrm{E}_n$ acts transitively and effectively on compact manifolds with finite fundamental group, as we remark at the end of Section \ref{sec:abelianradical}.
\subsection*{Notations and conventions}
For a Lie group $G$, we let $G^\circ$ denote the connected component of the identity. For a subgroup $H$ of $G$, we write $\mathrm{Ad}_{\liealgebra{G}}(H)$ for the adjoint representation of $H$ on the Lie algebra $\liealgebra{G}$ of $G$, to distinguish it from the adjoint representation $\mathrm{Ad}(H)$ on its own Lie algebra $\liealgebra{H}$.
The \emph{solvable radical} $R$ of $G$ is the maximal connected solvable normal subgroup of $G$. The \emph{solvable radical} $\liealgebra{R}$ of $\liealgebra{G}$ is the maximal solvable ideal of $\liealgebra{G}$. The semisimple Lie algebra $\liealgebra{F}=\liealgebra{G}/\liealgebra{R}$ is a direct product $\liealgebra{F}=\liealgebra{K}\times\liealgebra{S}$, where $\liealgebra{K}$ is a semisimple Lie algebra of \emph{compact type}, meaning its Killing form is definite, and $\liealgebra{S}$ is semisimple without factors of compact type.
The center of a group $G$, or a Lie algebra $\liealgebra{G}$, is denoted by $\mathrm{Z}(G)$, or $\liealgebra{Z}(\liealgebra{G})$, respectively. Similarly, the centralizer of a subgroup $H$ in $G$ (or a subalgebra $\liealgebra{H}$ in $\liealgebra{G}$) is denoted by $\mathrm{Z}_G(H)$ (or $\liealgebra{Z}_\liealgebra{G}(\liealgebra{H})$).
The action of a Lie group $G$ on a homogeneous space $M$ is \emph{(almost) effective} if the stabilizer of any point in $M$ does not contain a non-trivial (connected) normal subgroup of $G$.
If $V$ is a $G$-module, then we write $V^G=\{v\in V\mid gv=v \text{ for all } g\in G\}$ for the module of \emph{$G$-invariants}. Similary, $V^\liealgebra{G}=\{v\in V\mid xv=0 \text{ for all } x\in\liealgebra{G}\}$ for a $\liealgebra{G}$-module.
For direct products of Lie algebras $\liealgebra{G}_1$, $\liealgebra{G}_2$ we write $\liealgebra{G}_1\times\liealgebra{G}_2$, whereas $\liealgebra{G}_1+\liealgebra{G}_2$ or $\liealgebra{G}_1\oplus\liealgebra{G}_2$ refers to sums as vector spaces.
\section{Nil-invariant bilinear forms} \label{sec:nilinvariant}
Let $\liealgebra{G}$ be a finite-dimensional real Lie algebra, let $\mathrm{Inn}(\liealgebra{G})$ denote the inner auto\-morphism group of $\liealgebra{G}$ and $\ac{\mathrm{Inn}(\liealgebra{G})}$ its Zariski closure in $\mathrm{Aut}(\liealgebra{G})$. A symmetric bilinear form $\langle\cdot,\cdot\rangle$ on $\liealgebra{G}$ is called \emph{nil-invariant} if for all $x_1,x_2\in\liealgebra{G}$, \begin{equation} \langle \varphi x_1, x_2\rangle = -\langle x_1,\varphi x_2\rangle \label{eq:nilinvariant} \end{equation} for all nilpotent elements $\varphi$ of the Lie algebra of $\ac{\mathrm{Inn}(\liealgebra{G})}$. For a subalgebra $\liealgebra{H}$ of $\liealgebra{G}$, we say $\langle\cdot,\cdot\rangle$ is \emph{$\liealgebra{H}$-invariant} if for all $x\in\liealgebra{H}$, $\mathrm{ad}_\liealgebra{G}(x)$ is skew-symmetric for $\langle\cdot,\cdot\rangle$.
The \emph{kernel} of $\langle\cdot,\cdot\rangle$ is the subspace \[ \liealgebra{G}^\perp = \{x\in\liealgebra{G}\mid \langle x,y\rangle=0\text{ for all }y\in\liealgebra{G}\}. \] We use a Levi decomposition of $\liealgebra{G}$, \[ \liealgebra{G} = (\liealgebra{K}\times\liealgebra{S})\ltimes\liealgebra{R}, \] where $\liealgebra{K}$ is semisimple of compact type, $\liealgebra{S}$ is semisimple without factors of compact type, and $\liealgebra{R}$ is the solvable radical of $\liealgebra{G}$. Let further $\frg_{\rm s}=\liealgebra{S}\ltimes\liealgebra{R}$.
\begin{thm}[\mbox{\cite[Theorem A]{BGZ}}]\label{thm:BGZ_invariance} Let $\liealgebra{G}$ be a finite-dimensional real Lie algebra with nil-invariant symmetric bilinear form $\langle\cdot,\cdot\rangle$. Let $\langle\cdot,\cdot\rangle_{\frg_{\rm s}}$ denote the restriction of $\langle\cdot,\cdot\rangle$ to $\frg_{\rm s}$. Then: \begin{enumerate} \item $\langle\cdot,\cdot\rangle_{\frg_{\rm s}}$ is invariant by the adjoint action of $\liealgebra{G}$ on $\frg_{\rm s}$. \item $\langle\cdot,\cdot\rangle$ is invariant by the adjoint action of $\frg_{\rm s}$. \end{enumerate} \end{thm}
This implies some orthogonality relations that will be useful later on: \begin{equation} \liealgebra{S}\perp[\liealgebra{K},\liealgebra{G}], \quad \liealgebra{K}\perp[\liealgebra{S},\liealgebra{G}]. \label{eq:orthogonal} \end{equation}
\begin{thm}[\mbox{\cite[Corollary C]{BGZ}}]\label{thm:BGZ_kernel} Let $\liealgebra{G}$ be a finite-dimensional real Lie algebra with nil-invariant symmetric bilinear form $\langle\cdot,\cdot\rangle$, where we further assume that $\liealgebra{G}^\perp$ does not contain any non-zero ideal of $\liealgebra{G}$. Let $\liealgebra{Z}(\frg_{\rm s})$ denote the center of $\frg_{\rm s}$. Then \[ \liealgebra{G}^\perp\subseteq\liealgebra{K}\ltimes\liealgebra{Z}(\frg_{\rm s}) \quad\text{ and }\quad [\liealgebra{G}^\perp,\frg_{\rm s}]\subseteq\liealgebra{Z}(\frg_{\rm s})\cap\liealgebra{G}^\perp. \] \end{thm}
We say that $\langle\cdot,\cdot\rangle$ has \emph{relative index} $\ell$ if the induced scalar product on $\liealgebra{G}/\liealgebra{G}^\perp$ has index $\ell$. For relative index $\ell\leq 2$, we have a general structure theorem for $\liealgebra{G}$.
\begin{thm}[\mbox{\cite[Theorem D]{BGZ}}]\label{thm:BGZ_index2} Let $\liealgebra{G}$ be a finite-dimensional real Lie algebra with nil-invariant symmetric bilinear form $\langle\cdot,\cdot\rangle$ of relative index $\ell\leq 2$, and assume that $\liealgebra{G}^\perp$ does not contain any non-zero ideal of $\liealgebra{G}$. Then: \begin{enumerate} \item The Levi decomposition of $\liealgebra{G}$ is a direct sum of ideals $\liealgebra{G}=\liealgebra{K}\times\liealgebra{S}\times\liealgebra{R}$. \item $\liealgebra{G}^\perp$ is contained in $\liealgebra{K}\times\liealgebra{Z}(\liealgebra{R})$ and $\liealgebra{G}^\perp\cap\liealgebra{R}=\mathbf{0}$. \item $\liealgebra{S}\perp(\liealgebra{K}\times\liealgebra{R})$ and $\liealgebra{K}\perp[\liealgebra{R},\liealgebra{R}]$. \end{enumerate} \end{thm}
\subsection{Cotangent algebras} \label{subsec:cotangent}
Let $\liealgebra{L}$ be a Lie algebra. A \emph{cotangent algebra} constructed from $\liealgebra{L}$ is a Lie algebra $\liealgebra{G}=\liealgebra{L}\ltimes\liealgebra{L}^*$ where $\liealgebra{L}$ acts on its dual space $\liealgebra{L}^*$ by its coadjoint representation. We call $\liealgebra{G}$ a \emph{metric cotangent algebra} if it has a non-degenerate invariant scalar product $\langle\cdot,\cdot\rangle$ such that $\liealgebra{L}^*$ is totally isotropic.
\subsection[Invariance by $\liealgebra{G}^\perp$]{Invariance by $\boldsymbol{\liealgebra{G}^\perp}$}
We are mainly interested in nil-invariant bi\-linear forms $\langle\cdot,\cdot\rangle$ on $\liealgebra{G}$ induced by pseudo-Riemannian metrics on homogeneous spaces. In this case, $\langle\cdot,\cdot\rangle$ is invariant by the stabilizer subalgebra $\liealgebra{G}^\perp$. We can then further sharpen the statement of Theorem \ref{thm:BGZ_kernel}.
\begin{prop} \label{prop:gperp_inv} Let $\liealgebra{G}$ and $\langle\cdot,\cdot\rangle$ be as in Theorem \ref{thm:BGZ_kernel}. If in addition $\langle\cdot,\cdot\rangle$ is $\liealgebra{G}^{\perp}$-invariant,
then \[ [\liealgebra{G}^{\perp}, \frg_{\rm s}] = \mathbf{0}. \] \end{prop}
The proof is based on the following immediate observations:
\begin{lem}\label{lem0} Suppose $\langle\cdot,\cdot\rangle$ is $\liealgebra{G}^{\perp}$-invariant. Then $ [ [\liealgebra{K}, \liealgebra{G}^{\perp}], \frg_{\rm s}] \subseteq \liealgebra{G}^{\perp} \cap \frg_{\rm s}$. \end{lem}
and
\begin{lem} \label{lemA} Let $\liealgebra{H}$ be any Lie algebra and $V$ a module for $\liealgebra{H}$. Suppose that the subalgebra $\liealgebra{Q}$ of $\liealgebra{H}$ is generated by the subspace $\liealgebra{M}$ of $\liealgebra{H}$. Then $\liealgebra{Q} \cdot V = \liealgebra{M} \cdot V$. \end{lem}
Together with
\begin{lem} \label{lemB} Let $\liealgebra{K}$ be semisimple of compact type and $\liealgebra{K}_{0}$ a subalgebra of $\liealgebra{K}$. Then the subalgebra $\liealgebra{Q}$ generated $\liealgebra{M} = \liealgebra{K}_{0} + [\liealgebra{K}, \liealgebra{K}_{0}]$ is an ideal of $\liealgebra{K}$. \end{lem} \begin{proof} Put $\liealgebra{Z}= \liealgebra{Z}_{\liealgebra{K}}(\liealgebra{K}_{0})$. Then $[\liealgebra{Z}, \liealgebra{M}] \subseteq \liealgebra{M}$ and $[ [\liealgebra{K}, \liealgebra{K}_{0}], \liealgebra{M}] \subseteq \liealgebra{M} + [\liealgebra{M}, \liealgebra{M}]$. Since $\liealgebra{K} = [\liealgebra{K}, \liealgebra{K}_{0}] + \liealgebra{Z}$, this shows $[\liealgebra{K}, \liealgebra{M}] \subseteq \liealgebra{Q}$. Since $\liealgebra{Q}$ is linearly spanned by the iterated commutators of elements of $\liealgebra{M}$, $[\liealgebra{K}, \liealgebra{Q}] \subseteq \liealgebra{Q}$. \end{proof}
\begin{proof}[Proof of Proposition \ref{prop:gperp_inv}] Let $\liealgebra{K}_{0}$ be the image of $\liealgebra{G}^{\perp}$ under the projection homomorphism $\liealgebra{G} \to \liealgebra{K}$. Note that by Theorem \ref{thm:BGZ_kernel} above, $[\liealgebra{G}^{\perp}, \frg_{\rm s}] = [\liealgebra{K}_{0}, \frg_{\rm s}]$. Let $\liealgebra{Q} \subseteq \liealgebra{K}$ be the subalgebra generated by $\liealgebra{M} = \liealgebra{K}_{0} + [\liealgebra{K}, \liealgebra{K}_{0}]$ and consider $V = \frg_{\rm s}$ as a module for $\liealgebra{Q}$.
Since $\liealgebra{Q}$ is an ideal of $\liealgebra{K}$, $[\liealgebra{Q}, V]$ is a submodule for $\liealgebra{K}$, that is, $[\liealgebra{K}, [\liealgebra{Q}, V]] \subseteq [\liealgebra{Q}, V]$. By Lemmas \ref{lem0}, \ref{lemA} and Theorem \ref{thm:BGZ_kernel} we have $[\liealgebra{Q}, V] = [\liealgebra{M}, V] \subseteq \liealgebra{G}^{\perp} \cap \liealgebra{Z}(\frg_{\rm s})$. Hence, $\liealgebra{J} = [\liealgebra{M}, V] \subseteq \liealgebra{G}^{\perp}$ is an ideal in $\liealgebra{G}$, with $\liealgebra{J} \supseteq [\liealgebra{G}^{\perp}, \frg_{\rm s}] = [\liealgebra{K}_{0}, \frg_{\rm s}]$. Since $\liealgebra{G}^\perp$ contains no non-trivial ideals of $\liealgebra{G}$ by assumption, we conclude that $\liealgebra{J}=\mathbf{0}$. \end{proof}
\section[Metric Lie algebras with abelian radical]{Metric Lie algebras with abelian radical} \label{sec:abelianradical}
In this section we study finite-dimensional real Lie algebras $\liealgebra{G}$ whose solvable radical $\liealgebra{R}$ is abelian and which are equipped with a nil-invariant symmetric bilinear form $\langle\cdot,\cdot\rangle$.
\subsection{An algebraic theorem} The Lie algebras with abelian radical and a nil-invariant symmetric bilinear form decompose into three distinct types of metric Lie algebras.
{ \renewcommand{\ref{mthm:index2}}{\ref{mthm:abelian_rad1}} \begin{mthm} Let $\liealgebra{G}$ be a Lie algebra whose solvable radical $\liealgebra{R}$ is abelian. Suppose $\liealgebra{G}$ is equipped with a nil-invariant symmetric bilinear form $\langle\cdot,\cdot\rangle$ such that the kernel $\liealgebra{G}^\perp$ of $\langle\cdot,\cdot\rangle$ does not contain a non-trivial ideal of $\liealgebra{G}$. Let $\liealgebra{K}\times\liealgebra{S}$ be a Levi subalgebra of $\liealgebra{G}$, where $\liealgebra{K}$ is of compact type and $\liealgebra{S}$ has no simple factors of compact type. Then $\liealgebra{G}$ is an orthogonal direct product of ideals \[ \liealgebra{G} = \liealgebra{G}_1 \times \liealgebra{G}_2 \times \liealgebra{G}_3, \] with \[ \liealgebra{G}_1=\liealgebra{K}\ltimes\liealgebra{A}, \quad \liealgebra{G}_2=\liealgebra{S}_0, \quad \liealgebra{G}_3=\liealgebra{S}_1\ltimes\liealgebra{S}_1^*, \] where $\liealgebra{R}=\liealgebra{A}\times\liealgebra{S}_1^*$ and $\liealgebra{S}=\liealgebra{S}_0\times\liealgebra{S}_1$ are orthogonal direct products, and $\liealgebra{G}_3$ is a metric cotangent algebra. The restrictions of $\langle\cdot,\cdot\rangle$ to $\liealgebra{G}_2$ and $\liealgebra{G}_3$ are invariant and non-degenerate. In particular, $\liealgebra{G}^\perp\subseteq\liealgebra{G}_1$. \end{mthm} \addtocounter{mthm}{-1} }
We split the proof into several lemmas. Consider the submodules of invariants $\liealgebra{R}^\liealgebra{S}, \liealgebra{R}^\liealgebra{K}\subseteq\liealgebra{R}$. Since $\liealgebra{S}$, $\liealgebra{K}$ act reductively, we have \[ [\liealgebra{S},\liealgebra{R}]\oplus\liealgebra{R}^\liealgebra{S}=\liealgebra{R}=[\liealgebra{K},\liealgebra{R}]\oplus\liealgebra{R}^\liealgebra{K}. \] Then $\liealgebra{A}=\liealgebra{R}^\liealgebra{S}$, $\liealgebra{B}=[\liealgebra{S},\liealgebra{R}^\liealgebra{K}]$ and $\liealgebra{C}=[\liealgebra{S},\liealgebra{R}]\cap[\liealgebra{K},\liealgebra{R}]$ are ideals in $\liealgebra{G}$ and $\liealgebra{R} = \liealgebra{A}\oplus\liealgebra{B}\oplus\liealgebra{C}$. Recall from Theorem \ref{thm:BGZ_invariance} that $\langle\cdot,\cdot\rangle$ is in particular $\liealgebra{S}$- and $\liealgebra{R}$-invariant.
\begin{lem}\label{lem:R=AxB} $\liealgebra{C}=\mathbf{0}$ and $\liealgebra{R}$ is an orthogonal direct sum of ideals in $\liealgebra{G}$ \[ \liealgebra{R} = \liealgebra{A}\oplus\liealgebra{B} \] where $[\liealgebra{K},\liealgebra{R}]\subseteq\liealgebra{A}$ and $[\liealgebra{S},\liealgebra{R}]=\liealgebra{B}$. \end{lem} \begin{proof} The $\liealgebra{S}$-invariance of $\langle\cdot,\cdot\rangle$ immediately implies $\liealgebra{A}\perp\liealgebra{B}$. Since $\liealgebra{R}$ is abelian, $\liealgebra{R}$-invariance implies $\liealgebra{C}\perp\liealgebra{R}$. Since $\liealgebra{C}\perp(\liealgebra{S}\times\liealgebra{K})$ by \eqref{eq:orthogonal}, this shows $\liealgebra{C}$ is an ideal contained in $\liealgebra{G}^\perp$, hence $\liealgebra{C}=\mathbf{0}$. Now $[\liealgebra{K},\liealgebra{R}]\subseteq\liealgebra{A}$ and $[\liealgebra{S},\liealgebra{R}]=\liealgebra{B}$ by definition of $\liealgebra{A}$ and $\liealgebra{B}$. \end{proof}
\begin{lem}\label{lem:G=KAxSB} $\liealgebra{G}$ is a direct product of ideals \[ \liealgebra{G} = (\liealgebra{K}\ltimes\liealgebra{A})\times(\liealgebra{S}\ltimes\liealgebra{B}), \] where $(\liealgebra{K}\ltimes\liealgebra{A})\perp(\liealgebra{S}\ltimes\liealgebra{B})$. \end{lem} \begin{proof} The splitting as a direct product of ideals follows from Lemma \ref{lem:R=AxB}. The orthogonality follows together with \eqref{eq:orthogonal} and the fact that the $\liealgebra{S}$-invariance of $\langle\cdot,\cdot\rangle$ implies $\liealgebra{S}\perp\liealgebra{A}$ and $\liealgebra{K}\perp\liealgebra{B}$. \end{proof}
\begin{lem}\label{lem:SB}
$\liealgebra{G}^\perp\subseteq \liealgebra{K}\ltimes\liealgebra{A}$ and $\liealgebra{S}\ltimes\liealgebra{B}$ is a non-degenerate ideal of $\liealgebra{G}$. \end{lem} \begin{proof} $\liealgebra{Z}(\frg_{\rm s})=\liealgebra{A}$, therefore $\liealgebra{G}^\perp\subseteq\liealgebra{K}\ltimes\liealgebra{A}$ by Theorem \ref{thm:BGZ_kernel}. Since also $(\liealgebra{S}\ltimes\liealgebra{B})\perp(\liealgebra{K}\ltimes\liealgebra{A})$, we have $(\liealgebra{S}\ltimes\liealgebra{B})\cap(\liealgebra{S}\ltimes\liealgebra{B})^\perp\subseteq\liealgebra{G}^\perp\subseteq\liealgebra{K}\ltimes\liealgebra{A}$. It follows that $(\liealgebra{S}\ltimes\liealgebra{B})\cap(\liealgebra{S}\ltimes\liealgebra{B})^\perp=\mathbf{0}$.
\end{proof}
To complete the proof of Theorem \ref{mthm:abelian_rad1}, it remains to understand the structure of the ideal $\liealgebra{S}\ltimes\liealgebra{B}$, which by Theorem \ref{thm:BGZ_invariance} and the preceding lemmas is a Lie algebra with an invariant non-degenerate scalar product given by the restriction of $\langle\cdot,\cdot\rangle$.
\begin{lem}\label{lem:trivial} $\liealgebra{B}$ is totally isotropic. Let $\liealgebra{S}_0$ be the kernel of the $\liealgebra{S}$-action on $\liealgebra{B}$. Then $\liealgebra{S}_0=\liealgebra{B}^\perp\cap\liealgebra{S}$. \end{lem} \begin{proof} Since $\langle\cdot,\cdot\rangle$ is $\liealgebra{R}$-invariant and $\liealgebra{R}$ is abelian, $\liealgebra{B}$ is totally isotropic. For the second claim, use $\liealgebra{B}\cap\liealgebra{S}^\perp=\mathbf{0}$ and the invariance of $\langle\cdot,\cdot\rangle$. \end{proof}
\begin{lem}\label{lem:coadjoint} $\liealgebra{S}$ is an orthogonal direct product of ideals $\liealgebra{S}=\liealgebra{S}_0\times\liealgebra{S}_1$ with the following properties: \begin{enumerate} \item $\liealgebra{S}_1\ltimes\liealgebra{B}$ is a metric cotangent algebra. \item $[\liealgebra{S}_0,\liealgebra{B}]=\mathbf{0}$ and $\liealgebra{S}_0=\liealgebra{B}^\perp\cap\liealgebra{S}$. \end{enumerate} \end{lem} \begin{proof} The kernel $\liealgebra{S}_0$ of the $\liealgebra{S}$-action on $\liealgebra{B}$ is an ideal in $\liealgebra{S}$, and by Lemma \ref{lem:trivial} orthogonal to $\liealgebra{B}$. Let $\liealgebra{S}_1$ be the ideal in $\liealgebra{S}$ such that $\liealgebra{S}=\liealgebra{S}_0\times\liealgebra{S}_1$. Then $\liealgebra{S}_0\perp\liealgebra{S}_1$ by invariance of $\langle\cdot,\cdot\rangle$.
$\liealgebra{S}_1$ acts faithfully on $\liealgebra{B}$ and so $\liealgebra{S}_1\cap\liealgebra{B}^\perp=\mathbf{0}$ by Lemma \ref{lem:trivial}. Moreover, $\liealgebra{S}_1\ltimes\liealgebra{B}$ is non-degenerate since $\liealgebra{S}\ltimes\liealgebra{B}$ is. But $\liealgebra{B}$ is totally isotropic by Lemma \ref{lem:trivial}, so non-degeneracy implies $\dim\liealgebra{S}_1=\dim\liealgebra{B}$. Therefore $\liealgebra{B}$ and $\liealgebra{S}_1$ are dually paired by $\langle\cdot,\cdot\rangle$.
Now identify $\liealgebra{B}$ with $\liealgebra{S}_1^*$ and write $\xi(s)=\langle \xi,s\rangle$ for $\xi\in\liealgebra{S}_1^*$, $s\in\liealgebra{S}_1$. Then, once more by invariance of $\langle\cdot,\cdot\rangle$, \[ [s,\xi](s') = \langle [s,\xi],s'\rangle = \langle \xi,-[s,s']\rangle = \xi(-\mathrm{ad}(s)s') = (\mathrm{ad}^*(s)\xi)(s') \] for all $s,s'\in\liealgebra{S}_1$. So the action of $\liealgebra{S}_1$ on $\liealgebra{S}_1^*$ is the coadjoint action. This means $\liealgebra{S}\ltimes\liealgebra{B}$ is a metric cotangent algebra (cf.~Subsection \ref{subsec:cotangent}). \end{proof}
\begin{proof}[Proof of Theorem \ref{mthm:abelian_rad1}] The decomposition into the desired orthogonal ideals follows from Lemmas \ref{lem:G=KAxSB} to \ref{lem:coadjoint}. The structure of the ideals $\liealgebra{G}_2$ and $\liealgebra{G}_3$ is Lemma \ref{lem:coadjoint}. \end{proof}
The algebra $\liealgebra{G}_1$ in Theorem \ref{mthm:abelian_rad1} is of Euclidean type. Let $\liealgebra{G}=\liealgebra{K}\ltimes V$, with $V\cong\mathbbm{R}^n$, be an algebra of Euclidean type and let $\liealgebra{K}_0$ be the kernel of the $\liealgebra{K}$-action on $V$. Proposition \ref{prop:gperp_inv} and the fact that the solvable radical $V$ is abelian immediately give the following:
\begin{prop}\label{prop:Hcompact} Let $\liealgebra{G}=\liealgebra{K}\ltimes V$ be a Lie algebra of Euclidean type, and suppose $\liealgebra{G}$ is equipped with a symmetric bilinear form that is nil-invariant and $\liealgebra{G}^\perp$-invariant, such that $\liealgebra{G}^\perp$ does not contain a non-trivial ideal of $\liealgebra{G}$. Then \begin{equation} \liealgebra{G}^\perp \subseteq \liealgebra{K}_0\times V. \label{eq:GperpK0V0V1} \end{equation} \end{prop}
The following Examples \ref{ex:not_easy} and \ref{ex:not_easy2} show that in general a metric Lie algebra of Euclidean type cannot be further decomposed into orthogonal direct sums of metric Lie algebras. Both examples will play a role in Section \ref{sec:simplyconn}.
\begin{example}\label{ex:not_easy} Let $\liealgebra{K}_1=\liealgebra{SO}_3$, $V_1=\mathbbm{R}^3$, $V_0=\mathbbm{R}^3$ and $\liealgebra{G}=(\liealgebra{SO}_3\ltimes V_1)\times V_0$ with the natural action of $\liealgebra{SO}_3$ on $V_1$. Let $\varphi:V_1\to V_0$ be an isomorphism and put \[ \liealgebra{H} = \{(0,v,\varphi(v))\mid v\in V_0\}\subset(\liealgebra{K}_0\ltimes V_1) \times V_0. \] We can define a nil-invariant symmetric bilinear form on $\liealgebra{G}$ by identifying $V_1\cong\liealgebra{SO}_3^*$ and requiring for $k\in\liealgebra{K}_1$, $v_0\in V_0$, $v_1\in V_1$, \[ \langle k,v_0+v_1\rangle = v_1(k)-\varphi^{-1}(v_0)(k), \] and further $\liealgebra{K}_1\perp\liealgebra{K}_1$, $(V_0\oplus V_1)\perp(V_0\oplus V_1)$. Then $\langle\cdot,\cdot\rangle$ has signature $(3,3,3)$ and kernel $\liealgebra{H}=\liealgebra{G}^\perp$, which is not an ideal in $\liealgebra{G}$. Note that $\langle\cdot,\cdot\rangle$ is not invariant. Moreover, $\liealgebra{K}_1\ltimes V_1$ is not orthogonal to $V_0$. A direct factor $\liealgebra{K}_0$ can be added to this example at liberty. \end{example}
\begin{example}\label{ex:not_easy2} We can modify the Lie algebra $\liealgebra{G}$ from Example \ref{ex:not_easy} by embedding the direct summand $V_0\cong\mathbbm{R}^3$ in a torus subalgebra in a semisimple Lie algebra $\liealgebra{K}_0$ of compact type, say $\liealgebra{K}_0=\liealgebra{SO}_6$, to obtain a Lie algebra $\liealgebra{F}=(\liealgebra{K}_1\ltimes V_1)\times\liealgebra{K}_0$. We obtain a nil-invariant symmetric bilinear form of signature $(15,3,3)$ on $\liealgebra{F}$ by extending $\langle\cdot,\cdot\rangle$ by a definite form on a vector space complement of $V_0$ in $\liealgebra{K}_0$. The kernel of the new form is still $\liealgebra{G}^\perp=\liealgebra{H}$. \end{example}
\subsection{Nil-invariant bilinear forms on Euclidean algebras}
A \emph{Euclidean algebra} is a Lie algebra $\liealgebra{E}_n=\liealgebra{SO}_n\ltimes\mathbbm{R}^n$, where $\liealgebra{SO}_n$ acts on $\mathbbm{R}^n$ by the natural action.
By a \emph{skew pairing} of a Lie algebra $\liealgebra{L}$ and an $\liealgebra{L}$-module $V$ we mean a bilinear map $\langle\cdot,\cdot\rangle:\liealgebra{L}\times V \to \mathbbm{R}$ such that $\langle x, y v \rangle = -\langle y, x v\rangle$ for all $x,y\in\liealgebra{L}$, $v\in V$. Note that any nil-invariant symmetric bilinear form on $\liealgebra{G}=\liealgebra{K}\ltimes\mathbbm{R}^n$ yields a skew pairing of $\liealgebra{K}$ and $\mathbbm{R}^n$.
\begin{prop}[\mbox{\cite[Proposition A.5]{BGZ}}]\label{prop:so3_skew_module} Let $\langle\cdot,\cdot\rangle: \liealgebra{SO}_3 \times V \to \mathbbm{R}$ be a skew pairing for the (non-trivial) irreducible module $V$. If the skew pairing is non-zero, then $V$ is isomorphic to the adjoint representation of $\liealgebra{SO}_3$ and $\langle\cdot,\cdot\rangle$ is proportional to the Killing form. \end{prop}
\begin{example}\label{ex:so3R3} Consider $\liealgebra{G}=\liealgebra{SO}_3\ltimes \mathbbm{R}^n$ with a nil-invariant symmetric bilinear form $\langle\cdot,\cdot\rangle$, and assume that the action of $\liealgebra{SO}_3$ is irreducible. By Proposition \ref{prop:so3_skew_module}, either $\liealgebra{SO}_3\perp\mathbbm{R}^n$, or $n=3$ and $\liealgebra{SO}_3$ acts by its coadjoint representation on $\mathbbm{R}^3\cong\liealgebra{SO}_3^*$, and $\langle\cdot,\cdot\rangle$ is the dual pairing. In the first case, $\mathbbm{R}^n$ is an ideal in $\liealgebra{G}^\perp$, and in the second case, $\langle\cdot,\cdot\rangle$ is invariant and non-degenerate. \end{example}
\begin{example}\label{ex:so4R4} Let $\liealgebra{G}$ be the Euclidean Lie algebra $\liealgebra{SO}_4\ltimes \mathbbm{R}^4$ with a nil-invariant symmetric bilinear form $\langle\cdot,\cdot\rangle$. Since $\liealgebra{SO}_4\cong\liealgebra{SO}_3\times\liealgebra{SO}_3$, and here both factors $\liealgebra{SO}_3$ act irreducibly on $\mathbbm{R}^4$, it follows from Example \ref{ex:so3R3} that in $\liealgebra{G}$, $\mathbbm{R}^4$ is orthogonal to both factors $\liealgebra{SO}_3$, hence to all of $\liealgebra{SO}_4$. In particular, $\mathbbm{R}^4$ is an ideal contained in $\liealgebra{G}^\perp$. \end{example}
\begin{thm}\label{thm:noSOnRn} Let $\langle\cdot,\cdot\rangle$ be a nil-invariant symmetric bilinear form on the Euclidean Lie algebra $\liealgebra{SO}_n\ltimes\mathbbm{R}^n$ for $n\geq 4$. Then the ideal $\mathbbm{R}^n$ is contained in $\liealgebra{G}^\perp$. \end{thm} \begin{proof} For $n=4$, this is Example \ref{ex:so4R4}. So assume $n>4$. Consider an embedding of $\liealgebra{SO}_4$ in $\liealgebra{SO}_n$ such that $\mathbbm{R}^n=\mathbbm{R}^4\oplus\mathbbm{R}^{n-4}$, where $\liealgebra{SO}_4$ acts on $\mathbbm{R}^4$ in the standard way and trivially on $\mathbbm{R}^{n-4}$. By Example \ref{ex:so4R4}, $\liealgebra{SO}_4\perp\mathbbm{R}^4$. Since $\mathbbm{R}^{n-4}\subseteq[\liealgebra{SO}_n,\mathbbm{R}^n]$, the nil-invariance of $\langle\cdot,\cdot\rangle$ implies $\liealgebra{SO}_4\perp\mathbbm{R}^{n-4}$. Hence $\mathbbm{R}^n\perp\liealgebra{SO}_4$.
The same reasoning shows that $\mathrm{Ad}(g)\liealgebra{SO}_4\perp\mathbbm{R}^n$, where $g\in\mathrm{SO}_n$. Then $\liealgebra{B} = \sum_{g\in\mathrm{SO}_n}\mathrm{Ad}(g)\liealgebra{SO}_4$ is orthogonal to $\mathbbm{R}^n$. But $\liealgebra{B}$ is clearly an ideal in $\liealgebra{SO}_n$, so $\liealgebra{B}=\liealgebra{SO}_n$ by simplicity of $\liealgebra{SO}_n$ for $n>4$. \end{proof}
{ \renewcommand{\ref{mthm:index2}}{\ref{thm:noSOnRn2}} \begin{mthm} The Euclidean group $\mathrm{E}_n=\mathrm{O}_n\ltimes\mathbbm{R}^n$, $n\neq 1, 3$, does not have compact quotients with a pseudo-Riemannian metric such that $\mathrm{E}_n$ acts isometrically and almost effectively. \end{mthm} \addtocounter{mthm}{-1} } \begin{proof} For $n>3$, such an action of $\mathrm{E}_n$ would induce a nil-invariant symmetric bilinear form on the Lie algebra $\liealgebra{SO}_n\ltimes\mathbbm{R}^n$ without non-trivial ideals in its kernel, contradicting Theorem \ref{thm:noSOnRn}.
For $n=2$, the Lie algebra $\liealgebra{E}_2$ is solvable, and hence by Baues and Globke \cite{BG}, any nil-invariant symmetric bilinear form must be invariant. For such a form, the ideal $\mathbbm{R}^2$ of $\liealgebra{E}_2$ must be contained in $\liealgebra{E}_2^\perp$, and therefore action cannot be effective.
Note that $\liealgebra{E}_3$ is an exception, as it is the metric cotangent algebra of $\liealgebra{SO}_3$. \end{proof}
\begin{remark} Clearly the Lie group $\mathrm{E}_n$ admits compact quotient manifolds on which $\mathrm{E}_n$ acts almost effectively. For example take the quotient by a subgroup $F\ltimes\mathbbm{Z}^n$, where $F\subset\mathrm{O}_n$ is a finite subgroup preserving $\mathbbm{Z}^n$. Compact quotients with finite fundamental group also exist. For example, for any non-trivial homomorphism $\varphi:\mathbbm{R}^n\to\mathrm{O}_n$, the graph $H$ of $\varphi$ is a closed subgroup of $\mathrm{E}_n$ isomorphic to $\mathbbm{R}^n$, and the quotient $M=\mathrm{E}_n/H$ is compact (and diffeomorphic to $\mathrm{O}_n$). Since $H$ contains no non-trivial normal subgroup of $\mathrm{E}_n$, the $\mathrm{E}_n$-action on $M$ is effective. Theorem \ref{thm:noSOnRn2} tells us that none of these quotients admit $\mathrm{E}_n$-invariant pseudo-Riemannian metrics. \end{remark}
\section[Simply connected spaces]{Simply connected compact homogeneous spaces with indefinite metric} \label{sec:simplyconn}
Let $M$ be a connected and simply connected pseudo-Riemannian homogeneous space of finite volume. Then we can write \begin{equation} M=G/H \label{eq:MGH} \end{equation} for a connected Lie group $G$ acting almost effectively and by isometries on $M$, and $H$ is a closed subgroup of $G$ that contains no non-trivial connected normal subgroup of $G$ (for example, $G=\mathrm{Iso}(M)^\circ$). Note that $H$ is connected since $M$ is simply connected.
Let $\liealgebra{G}$, $\liealgebra{H}$ denote the Lie algebras of $G$, $H$, respectively. Recall that the pseudo-Riemannian metric on $M$ induces an $\liealgebra{H}$-invariant and nil-invariant symmetric bilinear form $\langle\cdot,\cdot\rangle$ on $\liealgebra{G}$, and the kernel of $\langle\cdot,\cdot\rangle$ is precisely $\liealgebra{G}^\perp=\liealgebra{H}$ and contains no non-trivial ideal of $\liealgebra{G}$.
We decompose $G=KSR$, where $K$ is a compact semi\-simple subgroup, $S$ is a semisimple subgroup without compact factors, $R$ the solvable radical of $G$
\begin{prop}\label{prop:S_trivial_M_compact} The subgroup $S$ is trivial and $M$ is compact. \end{prop} \begin{proof} As $M$ is simply connected, $H=H^\circ$. Now $H\subseteq K R$ by Theorem \ref{thm:BGZ_kernel}. On the other hand, since $M$ has finite invariant volume, the Zariski closure of $\mathrm{Ad}_\liealgebra{G}(H)$ contains $\mathrm{Ad}_\liealgebra{G}(S)$, see Mostow \cite[Lemma 3.1]{mostow3}. Therefore $S$ must be trivial. It follows from Mostow's result \cite[Theorem 6.2]{mostow4} on quotients of solvable Lie groups that $M=(KR)/H$ is compact. \end{proof}
We can therefore restrict ourselves in \eqref{eq:MGH} to groups $G = K R$ and connected uniform subgroups $H$ of $G$.
The structure of a general compact homogeneous manifold with finite fundamental group is surveyed in Onishchik and Vinberg \cite[II.5.\S 2]{OV1}. In our context it follows that \begin{equation} G = L\ltimes V \label{eq:OV} \end{equation} where $V$ is a normal subgroup isomorphic to $\mathbbm{R}^n$ and $L=KA$ is a maximal compact subgroup of $G$. The solvable radical is $R=A\ltimes V$. Moreover, $V^L=\mathbf{0}$. By a theorem of Montgomery \cite{montgomery} (also \cite[p.~137]{OV1}), $K$ acts transitively on $M$. \\
The existence of a $G$-invariant metric on $M$ further restricts the structure of $G$.
\begin{prop}\label{prop:rad_abelian} The solvable radical $R$ of $G$ is abelian. In particular, $R=A\times V$, $V^K=\mathbf{0}$ and $A=\mathrm{Z}(G)^\circ$. \end{prop} \begin{proof} Let $\mathrm{Z}(R)$ denote the center of $R$ and $N$ its nilradical. Since $H$ is connected, $H\subseteq K\mathrm{Z}(R)^\circ$ by Theorem \ref{thm:BGZ_kernel}. Hence there is a surjection $G/H\to G/(K\mathrm{Z}(R)^\circ)=R/\mathrm{Z}(R)^\circ$. It follows that $\mathrm{Z}(R)^\circ$ is a connected uniform subgroup. Therefore the nilradical $N$ of $R$ is $N=T\mathrm{Z}(R)^\circ$ for some compact torus $T$. But a compact subgroup of $N$ must be central in $R$, so $T\subseteq\mathrm{Z}(R)$. Hence $N\subseteq\mathrm{Z}(R)$, which means $R=N$ is abelian. \end{proof}
Combined with \eqref{eq:OV}, we obtain \begin{equation} G = K R = (K_0 A)\times(K_1\ltimes V), \label{eq:GKR} \end{equation} with $K=K_0\times K_1$, $R=A\times V$, where $K_0$ is the kernel of the $K$-action on $V$.\\
For any subgroup $Q$ of $G$ we write $H_{Q}=H\cap Q$.
\begin{lem}\label{lem:HKHLnormal} $[H,H]\subseteq H_K$. In particular, $H_K$ is a normal subgroup of $H$. \end{lem} \begin{proof} By Proposition \ref{prop:Hcompact} and the connectedness of $H$, we have $H\subseteq K_0 R$. Since $R$ is abelian, $[H,H]\subseteq H_{K_0}$. \end{proof}
If $G$ is simply connected, we have $K\cap R=\{e\}$. Then let $\mathsf{p}_K$, $\mathsf{p}_R$ denote the projection maps from $G$ to $K$, $R$.
\begin{lem}\label{lem:surjective_R} Suppose $G$ is simply connected. Then $\mathsf{p}_R(H)=R$. \end{lem} \begin{proof} Since $K$ acts transitively on $M$, we have $G=KH$. Then $R=\mathsf{p}_R(G)=\mathsf{p}_R(H)$. \end{proof}
\begin{prop}\label{prop:HKE} Suppose $G$ is simply connected. Then the stabilizer $H$ is a semidirect product $H=H_K\times E$, where $E$ is the graph of a homomorphism $\varphi:R\to K$ that is non-trivial if $\dim R>0$. Moreover, $\varphi(R\cap H)=\{e\}$. \end{prop} \begin{proof} The subgroup $H_{K}$ is a maximal compact subgroup of the stabilizer $H$. By Lemma \ref{lem:HKHLnormal}, $H=H_K\times E$ for some normal subgroup $E$ diffeomorphic to a vector space. By Lemma \ref{lem:surjective_R}, $H$ projects onto $R$ with kernel $H_K$, so that $E\cong R$. Then $E$ is necessarily the graph of a homomorphism $\varphi:R\to K$. If $\dim R>0$, then $\varphi$ is non-trivial, for otherwise $R\subseteq H$, in contradiction to the almost effectivity of the action.
Observe that $R\cap H\subseteq E$. Therefore $\varphi(R\cap H)\subseteq H_K\cap E=\{e\}$. \end{proof}
Now we can state our main result:
{\renewcommand{\ref{mthm:index2}}{\ref{mthm:geometric}} \begin{mthm} Let $M$ be a connected and simply connected pseudo-Riemannian homogeneous space of finite volume, $G=\mathrm{Iso}(M)^\circ$, and let $H$ be the stabilizer subgroup in $G$ of a point in $M$. Let $G=KR$ be a Levi decomposition, where $R$ is the solvable radical of $G$. Then: \begin{enumerate} \item $M$ is compact. \item $K$ is compact and acts transitively on $M$. \item $R$ is abelian. Let $A$ be the maximal compact subgroup of $R$. Then $A=\mathrm{Z}(G)^\circ$. More explicitely, $R=A\times V$ where $V\cong\mathbbm{R}^n$ and $V^{K}=\mathbf{0}$. \item $H$ is connected. If $\dim R>0$, then $H=(H\cap K) E$, where $E$ and $H\cap K$ are normal subgroups in $H$, $(H\cap K)\cap E$ is finite, and $E$ is the graph of a non-trivial homomorphism
$\varphi:R\to K$, where the restriction $\varphi|_A$ is injective. \end{enumerate} \end{mthm} \addtocounter{mthm}{-1} } \begin{proof} Claims (1), (2) and (3) follow from Proposition \ref{prop:S_trivial_M_compact}, Proposition \ref{prop:rad_abelian} and \eqref{eq:OV}, applied to $G=\mathrm{Iso}(M)^\circ$.
For claim (4), let $\widetilde{G}$ be the universal cover of $G$. Since $G$ acts effectively on $M$, $\widetilde{G}$ acts almost effectively on $M$ with stabilizer $\widetilde{H}$, the preimage of $H$ in $\widetilde{G}$. Let $\widetilde{\varphi}:\widetilde{R}\to\widetilde{K}$ be the homomorphism given by Proposition \ref{prop:HKE} for $\widetilde{G}$. Then $\widetilde{R}=\widetilde{A}\oplus V$, with $\widetilde{A}\cong\mathbbm{R}^k$ for some $k$, and $R=\widetilde{R}/Z$ for some central discrete subgroup $Z\subset\widetilde{A}\cap\widetilde{H}$. Since $Z\subset\widetilde{R}\cap\widetilde{H}$ we have $Z\subseteq\ker\widetilde{\varphi}$. Therefore $\widetilde{\varphi}$ descends to a homomorphism $R\to\widetilde{K}$, and by composing with the canonical projection
$\widetilde{K}\to K$, we obtain a homomorphism $\varphi:R\to K$ with the desired properties. Observe that $\ker\varphi|_A\subset A\cap H$ is a normal subgroup in $G$. Hence it is trivial, as $G$ acts effectively. \end{proof}
Now assume further that the index of the metric on $M$ is $\ell\leq 2$. Theorem \ref{thm:BGZ_index2} has strong consequences in the simply connected case.
{\renewcommand{\ref{mthm:index2}}{\ref{mthm:index2}} \begin{mthm} The isometry group of any simply connected pseudo-Riemannian homogeneous manifold of finite volume and metric index $\ell\leq2$ is compact. \end{mthm} } \begin{proof} We know from Theorem \ref{mthm:geometric} that $M$ is compact. Let $G=\mathrm{Iso}(M)^\circ$, with $G=KR$ as before.
By Theorem \ref{thm:BGZ_index2}, $R$ commutes with $K$ and thus $R=A$ by part 3 of Theorem \ref{mthm:geometric}.
It follows that $G=K A$ is compact.
Then $K$ is a characteristic subgroup of $G$ which also acts transitively on $M$. Therefore we may identify $\mathrm{T}_x M$ at $x\in M$ with $\liealgebra{K}/(\liealgebra{H}\cap\liealgebra{K})$, where $\liealgebra{K}$ is the Lie algebra of $K$. Hence the isotropy representation of the stabilizer $\mathrm{Iso}(M)_x$ factorizes over a closed subgroup of the automorphism group of $\liealgebra{K}$. As this latter group is compact, the isotropy representation has compact closure in $\mathrm{GL}(\mathrm{T}_x M)$. If follows that there exists a Riemannian metric on $M$ that is preserved by $\mathrm{Iso}(M)$. Hence $\mathrm{Iso}(M)$ is compact. \end{proof}
\begin{remark} Note that in fact the isometry group of every compact analytic simply connected pseudo-Riemannian manifold has finitely many connected components (Gromov \cite[Theorem 3.5.C]{gromov}). \end{remark}
For indices higher than two, the identity component of the iso\-metry group of a simply connected $M$ can be non-compact. This is demonstrated by the examples below in which we construct some $M$ on which a non-compact group acts isometrically. The following Lemma \ref{lem:no_compact} then ensures that these groups cannot be contained in any compact Lie group.
\begin{lem}\label{lem:no_compact} Assume that the action of $K$ on $V$ in the semidirect product $G=K\ltimes V$ is non-trivial. Then $G$ cannot be immersed in a compact Lie group. \end{lem} \begin{proof} Suppose there is a compact Lie group $C$ that contains $G$ as a subgroup. As the action of $K$ on $V$ is non-trivial, there exists an element $v\in V\subseteq C$ such that $\mathrm{Ad}_{\liealgebra{C}}(v)$ has non-trivial unipotent Jordan part. But by compactness of $C$, every $\mathrm{Ad}_{\liealgebra{C}}(g)$, $g\in C$, is semisimple, a contradiction. \end{proof}
\begin{example}\label{ex:not_easy_group} Start with $G_1=(\widetilde{\mathrm{SO}}_3\ltimes\mathbbm{R}^3)\times\mathrm{T}^3$, where $\widetilde{\mathrm{SO}}_3$ acts on $\mathbbm{R}^3$ by the coadjoint action, and let $\varphi:\mathbbm{R}^3\to\mathrm{T}^3$ be a homomorphism with discrete kernel. Put \[ H = \{(\mathrm{I}_3,v,\varphi(v))\mid v\in\mathbbm{R}^3\}. \] The Lie algebras $\liealgebra{G}_1$ of $G_1$ and $\liealgebra{H}$ of $H$ are the corresponding Lie algebras from Example \ref{ex:not_easy}. We can extend the nil-invariant scalar product $\langle\cdot,\cdot\rangle$ on $\liealgebra{G}_1$ from Example \ref{ex:not_easy} to a left-invariant tensor on $G_1$, and thus obtain a $G_1$-invariant pseudo-Riemannian metric of signature $(3,3)$ on the quotient $M_1=G_1/H=\widetilde{\mathrm{SO}}_3\times\mathrm{T}^3$. Here, $M_1$ is a non-simply connected manifold with a non-compact connected stabilizer.
In order to obtain a simply connected space, embed $\mathrm{T}^3$ in a simply connected compact semisimple group $K_0$, for example $K_0=\widetilde{\mathrm{SO}}_6$, so that $G_1$ is embedded in $G=(\widetilde{\mathrm{SO}}_3 \ltimes \mathbbm{R}^3) \times K_0$. As in Example \ref{ex:not_easy2}, we can extend $\langle\cdot,\cdot\rangle$ from $\liealgebra{G}_1$ to $\liealgebra{G}$, and thus obtain a compact simply connected pseudo-Riemannian homogeneous manifold $M=G/H=\widetilde{\mathrm{SO}}_3\times K_0$.
\end{example}
\begin{example} Example \ref{ex:not_easy_group} can be generalized by replacing $\widetilde{\mathrm{SO}}_3$ by any simply connected compact semi\-simple group $K$, acting by the coadjoint representation on $\mathbbm{R}^d$, where $d=\dim K$, and trivially on $\mathrm{T}^d$. Define $H$ similarly as a graph in $\mathbbm{R}^d\times\mathrm{T}^d$, and embed $\mathrm{T}^d$ in a simply connected compact semisimple Lie group $K_0$. \end{example}
\end{document} | arXiv |
Delayed payment policy in multi-product single-machine economic production quantity model with repair failure and partial backordering
JIMO Home
Continuous-time mean-variance portfolio selection with no-shorting constraints and regime-switching
doi: 10.3934/jimo.2018190
A primal-dual interior-point method capable of rapidly detecting infeasibility for nonlinear programs
Yu-Hong Dai 1,2, , Xin-Wei Liu 3,, and Jie Sun 4,5,6,
LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
School of Mathematical Sciences, University of Chinese Academy of Sciences
Institute of Mathematics, Hebei University of Technology, Tianjin 300401, China
Institute of Mathematics, Hebei University of Technology, Tianjin China
School of Science, Curtin University, Perth, Australia
School of Business, National University of Singapore, Singapore
Received July 2018 Revised August 2018 Published December 2018
Fund Project: The first draft of this paper was completed on December 2, 2014. The first author is supported by the Chinese NSF grants (nos. 11631013, 11331012 and 81173633) and the National Key Basic Research Program of China (no. 2015CB856000). The second author is supported by the Chinese NSF grants (nos. 11671116 and 11271107) and the Major Research Plan of the NSFC (no. 91630202). The third author is supported by Grant DP-160101819 of Australia Research Council
Full Text(HTML)
With the help of a logarithmic barrier augmented Lagrangian function, we can obtain closed-form solutions of slack variables of logarithmic-barrier problems of nonlinear programs. As a result, a two-parameter primal-dual nonlinear system is proposed, which corresponds to the Karush-Kuhn-Tucker point and the infeasible stationary point of nonlinear programs, respectively, as one of two parameters vanishes. Based on this distinctive system, we present a primal-dual interior-point method capable of rapidly detecting infeasibility of nonlinear programs. The method generates interior-point iterates without truncation of the step. It is proved that our method converges to a Karush-Kuhn-Tucker point of the original problem as the barrier parameter tends to zero. Otherwise, the scaling parameter tends to zero, and the method converges to either an infeasible stationary point or a singular stationary point of the original problem. Moreover, our method has the capability to rapidly detect the infeasibility of the problem. Under suitable conditions, the method can be superlinearly or quadratically convergent to the Karush-Kuhn-Tucker point if the original problem is feasible, and it can be superlinearly or quadratically convergent to the infeasible stationary point when the problem is infeasible. Preliminary numerical results show that the method is efficient in solving some simple but hard problems, where the superlinear convergence to an infeasible stationary point is demonstrated when we solve two infeasible problems in the literature.
Keywords: Nonlinear programming, constrained optimization, infeasibility, interior-point method, global and local convergence.
Mathematics Subject Classification: Primary: 90C30, 90C51; Secondary: 90C26.
Citation: Yu-Hong Dai, Xin-Wei Liu, Jie Sun. A primal-dual interior-point method capable of rapidly detecting infeasibility for nonlinear programs. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018190
R. Andreani, E. G. Birgin, J. M. Martinez and M. L. Schuverdt, Augmented Lagrangian methods under the constant positive linear dependence constraint qualification, Math. Program., 111 (2008), 5-32. doi: 10.1007/s10107-006-0077-1. Google Scholar
P. Armand and J. Benoist, A local convergence property of primal-dual methods for nonlinear programming, Math. Program., 115 (2008), 199-222. doi: 10.1007/s10107-007-0136-2. Google Scholar
P. Armand, J. C. Gilbert and S. Jan-Jégou, A feasible BFGS interior point algorithm for solving convex minimization problems, SIAM J. Optim., 11 (2000), 199-222. doi: 10.1137/S1052623498344720. Google Scholar
I. Bongartz, A. R. Conn, N. I. M. Gould and P. L. Toint, CUTE: Constrained and Unconstrained Testing Environment, ACM Tran. Math. Software, 21 (1995), 123-160. Google Scholar
J. V. Burke, F. E. Curtis and H. Wang, A sequential quadratic optimization algorithm with rapid infeasibility detection, SIAM J. Optim., 24 (2014), 839-872. doi: 10.1137/120880045. Google Scholar
J. V. Burke and S. P. Han, A robust sequential quadratic programming method, Math. Program., 43 (1989), 277-303. doi: 10.1007/BF01582294. Google Scholar
R. H. Byrd, Robust Trust-Region Method for Constrained Optimization, Paper presented at the SIAM Conference on Optimization, Houston, TX, 1987. Google Scholar
R. H. Byrd, F. E. Curtis and J. Nocedal, Infeasibility detection and SQP methods for nonlinear optimization, SIAM J. Optim., 20 (2010), 2281-2299. doi: 10.1137/080738222. Google Scholar
R. H. Byrd, J. C. Gilbert and J. Nocedal, A trust region method based on interior point techniques for nonlinear programming, Math. Program., 89 (2000), 149-185. doi: 10.1007/PL00011391. Google Scholar
R. H. Byrd, M. E. Hribar and J. Nocedal, An interior point algorithm for large-scale nonlinear programming, SIAM J. Optim., 9 (1999), 877-900. doi: 10.1137/S1052623497325107. Google Scholar
R. H. Byrd, G. Liu and J. Nocedal, On the local behaviour of an interior point method for nonlinear programming, In, Numerical Analysis 1997 (eds. D. F. Griffiths and D. J. Higham), Addison-Wesley Longman, Reading, MA, 380 (1998), 37-56. Google Scholar
R. H. Byrd, M. Marazzi and J. Nocedal, On the convergence of Newton iterations to non-stationary points, Math. Program., 99 (2004), 127-148. doi: 10.1007/s10107-003-0376-8. Google Scholar
L. F. Chen and D. Goldfarb, Interior-point $\ell_2$-penalty methods for nonlinear programming with strong global convergence properties, Math. Program., 108 (2006), 1-36. doi: 10.1007/s10107-005-0701-5. Google Scholar
F. E. Curtis, A penalty-interior-point algorithm for nonlinear constrained optimization, Math. Program. Comput., 4 (2012), 181-209. doi: 10.1007/s12532-012-0041-4. Google Scholar
A. S. El-Bakry, R. A. Tapia, T. Tsuchiya and Y. Zhang, On the formulation and theory of the Newton interior-point method for nonlinear programming, J. Optim. Theory Appl., 89 (1996), 507-541. doi: 10.1007/BF02275347. Google Scholar
A. V. Fiacco and G. P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley and Sons, New York, 1968; republished as Classics in Appl. Math. 4, SIAM, Philadelphia, 1990. doi: 10.1137/1.9781611971316. Google Scholar
R. Fletcher, Practical Methods for Optimization. Vol. 2: Constrained Optimization, John Wiley and Sons, Chichester, 1980. Google Scholar
A. Forsgren and P. E. Gill, Primal-dual interior methods for nonconvex nonlinear programming, SIAM J. Optim., 8 (1998), 1132-1152. doi: 10.1137/S1052623496305560. Google Scholar
A. Forsgren, Ph. E. Gill and M. H. Wright, Interior methods for nonlinear optimization, SIAM Review, 44 (2002), 525-597. doi: 10.1137/S0036144502414942. Google Scholar
D. M. Gay, M. L. Overton and M. H. Wright, A primal-dual interior method for nonconvex nonlinear programming, in Advances in Nonlinear Programming, (ed. Y.-X. Yuan), Kluwer Academic Publishers, Dordrecht, 14 (1998), 31-56. doi: 10.1007/978-1-4613-3335-7_2. Google Scholar
E. M. Gertz and Ph. E. Gill, A primal-dual trust region algorithm for nonlinear optimization, Math. Program., 100 (2004), 49-94. doi: 10.1007/s10107-003-0486-3. Google Scholar
N. I. M. Gould, D. Orban and Ph. L. Toint, An interior-point $\ell_1$-penalty method for nonlinear optimization, in Recent Developments in Numerical Analysis and Optimization, Proceedings of NAOIII 2014, Springer, Verlag, 134 (2015), 117-150. doi: 10.1007/978-3-319-17689-5_6. Google Scholar
W. Hock and K. Schittkowski, Test Examples for Nonlinear Programming Codes, Lecture Notes in Eco. and Math. Systems 187, Springer-Verlag, Berlin, New York, 1981. doi: 10.1007/BF00934594. Google Scholar
X.-W. Liu, G. Perakis and J. Sun, A robust SQP method for mathematical programs with linear complementarity constraints, Comput. Optim. Appl., 34 (2006), 5-33. doi: 10.1007/s10589-005-3075-y. Google Scholar
X.-W. Liu and J. Sun, A robust primal-dual interior point algorithm for nonlinear programs, SIAM J. Optim., 14 (2004), 1163-1186. doi: 10.1137/S1052623402400641. Google Scholar
X.-W. Liu and Y.-X. Yuan, A robust algorithm for optimization with general equality and inequality constraints, SIAM J. Sci. Comput., 22 (2000), 517-534. doi: 10.1137/S1064827598334861. Google Scholar
X.-W. Liu and Y.-X. Yuan, A null-space primal-dual interior-point algorithm for nonlinear optimization with nice convergence properties, Math. Program., 125 (2010), 163-193. doi: 10.1007/s10107-009-0272-y. Google Scholar
J. Nocedal, F. Öztoprak and R. A. Waltz, An interior point method for nonlinear programming with infeasibility detection capabilities, Optim. Methods Softw., 29 (2014), 837-854. doi: 10.1080/10556788.2013.858156. Google Scholar
J. Nocedal and S. Wright, Numerical Optimization, Springer-Verlag New York, Inc., 1999. doi: 10.1007/b98874. Google Scholar
[30] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York and London, 1970. Google Scholar
D. F. Shanno and R. J. Vanderbei, Interior-point methods for nonconvex nonlinear programming: Orderings and higher-order methods, Math. Program., 87 (2000), 303-316. doi: 10.1007/s101070050116. Google Scholar
P. Tseng, Convergent infeasible interior-point trust-region methods for constrained minimization, SIAM J. Optim., 13 (2002), 432-469. doi: 10.1137/S1052623499357945. Google Scholar
M. Ulbrich, S. Ulbrich and L. N. Vicente, A globally convergent primal-dual interior-point filter method for nonlinear programming, Math. Program., 100 (2004), 379-410. doi: 10.1007/s10107-003-0477-4. Google Scholar
A. Wächter and L. T. Biegler, Failure of global convergence for a class of interior point methods for nonlinear programming, Math. Program., 88 (2000), 565-574. doi: 10.1007/PL00011386. Google Scholar
A. Wächter and L. T. Biegler, Line search filter methods for nonlinear programming: Motivation and global convergence, SIAM J. Optim., 16 (2005), 1-31. doi: 10.1137/S1052623403426556. Google Scholar
A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), 25-57. doi: 10.1007/s10107-004-0559-y. Google Scholar
M. H. Wright, Why a pure primal Newton barrier step may be infeasible?, SIAM J. Optim., 5 (1995), 1-12. doi: 10.1137/0805001. Google Scholar
S. J. Wright, On the convergence of the Newton/Log-barrier method, Math. Program., 90 (2001), 71-100. doi: 10.1007/PL00011421. Google Scholar
Y.-X. Yuan, On the convergence of a new trust region algorithm, Numer. Math., 70 (1995), 515-539. doi: 10.1007/s002110050133. Google Scholar
Y. Zhang, Solving large-scale linear programs by interior-point methods under the MATLAB environment, Optim. Methods Softw., 10 (1998), 1-31. doi: 10.1080/10556789808805699. Google Scholar
Table 1. Output for test problem (TP1)
$ l $ $ f_l $ $ v_l $ $ \|\phi_l\|_{\infty} $ $ \|\psi_l\|_{\infty} $ $ \beta_l $ $ \rho_l $ $ k $
0 5 16.6132 129.6234 129.6234 0.1000 3.3226 -
1 0.1606 2.0205 4.8082 0.7313 0.1000 0.0972 3
2 -0.0149 2.0002 0.0989 0.0445 0.1000 0.0020 4
3 -0.0036 2.0000 0.0029 0.0018 0.1000 3.1595e-06 3
4 -0.0029 2.0000 3.1674e-06 2.8185e-06 0.1000 1.0000e-09 1
5 0.0018 2.0000 1.0011e-09 6.7212e-10 - - -
0 -20 126.6501 2.8052e+03 2.8052e+03 0.1000 6.3325 -
1 -172.5829 172.7978 1.0948e+03 6.2866 0.1000 0.8719 6
8 -0.1999 0.4472 9.2732e-10 9.2732e-10 - - -
0 20 2.8284 9.9557 9.9557 0.1000 1 -
1 0.2305 0.4167 0.8900 0.7008 0.0100 1 4
3 0.1690 0.1630 0.0503 0.0022 0.0100 4.7328e-06 1
4 0.8561 2.9531e-04 3.1379e-06 3.1379e-06 0.0100 1.0000e-09 14
5 0.9028 1.2372e-04 9.3463e-08 9.3463e-08 - - -
Table 4. The last $ 4 $ inner iterations corresponding to $ l = 4 $ for test problem (TP4)
$ k $ $ f_k $ $ v_k $ $ \|\phi_k\|_{\infty} $ $ \|\psi_k\|_{\infty} $ $ x_{k1} $ $ x_{k2} $
11 0.8500 5.7136e-04 5.6703e-04 5.6703e-04 1.0780 0.0001
12 0.8548 3.0434e-04 1.2222e-05 1.2222e-05 1.0754 -0.0002
Boshi Tian, Xiaoqi Yang, Kaiwen Meng. An interior-point $l_{\frac{1}{2}}$-penalty method for inequality constrained nonlinear optimization. Journal of Industrial & Management Optimization, 2016, 12 (3) : 949-973. doi: 10.3934/jimo.2016.12.949
Yanqin Bai, Pengfei Ma, Jing Zhang. A polynomial-time interior-point method for circular cone programming based on kernel functions. Journal of Industrial & Management Optimization, 2016, 12 (2) : 739-756. doi: 10.3934/jimo.2016.12.739
Behrouz Kheirfam, Morteza Moslemi. On the extension of an arc-search interior-point algorithm for semidefinite optimization. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 261-275. doi: 10.3934/naco.2018015
Soodabeh Asadi, Hossein Mansouri. A Mehrotra type predictor-corrector interior-point algorithm for linear programming. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 147-156. doi: 10.3934/naco.2019011
Yanqin Bai, Xuerui Gao, Guoqiang Wang. Primal-dual interior-point algorithms for convex quadratic circular cone optimization. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 211-231. doi: 10.3934/naco.2015.5.211
Behrouz Kheirfam. A full Nesterov-Todd step infeasible interior-point algorithm for symmetric optimization based on a specific kernel function. Numerical Algebra, Control & Optimization, 2013, 3 (4) : 601-614. doi: 10.3934/naco.2013.3.601
Yanqin Bai, Lipu Zhang. A full-Newton step interior-point algorithm for symmetric cone convex quadratic optimization. Journal of Industrial & Management Optimization, 2011, 7 (4) : 891-906. doi: 10.3934/jimo.2011.7.891
Siqi Li, Weiyi Qian. Analysis of complexity of primal-dual interior-point algorithms based on a new kernel function for linear optimization. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 37-46. doi: 10.3934/naco.2015.5.37
Yinghong Xu, Lipu Zhang, Jing Zhang. A full-modified-Newton step infeasible interior-point algorithm for linear optimization. Journal of Industrial & Management Optimization, 2016, 12 (1) : 103-116. doi: 10.3934/jimo.2016.12.103
Yongjian Yang, Zhiyou Wu, Fusheng Bai. A filled function method for constrained nonlinear integer programming. Journal of Industrial & Management Optimization, 2008, 4 (2) : 353-362. doi: 10.3934/jimo.2008.4.353
Behrouz Kheirfam, Guoqiang Wang. An infeasible full NT-step interior point method for circular optimization. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 171-184. doi: 10.3934/naco.2017011
Guoqiang Wang, Zhongchen Wu, Zhongtuan Zheng, Xinzhong Cai. Complexity analysis of primal-dual interior-point methods for semidefinite optimization based on a parametric kernel function with a trigonometric barrier term. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 101-113. doi: 10.3934/naco.2015.5.101
Songqiang Qiu, Zhongwen Chen. An adaptively regularized sequential quadratic programming method for equality constrained optimization. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-14. doi: 10.3934/jimo.2019075
Igor Griva, Roman A. Polyak. Proximal point nonlinear rescaling method for convex optimization. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 283-299. doi: 10.3934/naco.2011.1.283
Wen-ling Zhao, Dao-jin Song. A global error bound via the SQP method for constrained optimization problem. Journal of Industrial & Management Optimization, 2007, 3 (4) : 775-781. doi: 10.3934/jimo.2007.3.775
Changjun Yu, Kok Lay Teo, Liansheng Zhang, Yanqin Bai. On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem. Journal of Industrial & Management Optimization, 2012, 8 (2) : 485-491. doi: 10.3934/jimo.2012.8.485
Chunlin Hao, Xinwei Liu. Global convergence of an SQP algorithm for nonlinear optimization with overdetermined constraints. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 19-29. doi: 10.3934/naco.2012.2.19
Qiang Long, Changzhi Wu. A hybrid method combining genetic algorithm and Hooke-Jeeves method for constrained global optimization. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1279-1296. doi: 10.3934/jimo.2014.10.1279
Liming Sun, Li-Zhi Liao. An interior point continuous path-following trajectory for linear programming. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1517-1534. doi: 10.3934/jimo.2018107
Zheng-Hai Huang, Shang-Wen Xu. Convergence properties of a non-interior-point smoothing algorithm for the P*NCP. Journal of Industrial & Management Optimization, 2007, 3 (3) : 569-584. doi: 10.3934/jimo.2007.3.569
HTML views (684)
Yu-Hong Dai Xin-Wei Liu Jie Sun | CommonCrawl |
\begin{document}
\begin{center}
{\large\bf THE FRACTIONAL SUM OF SMALL ARITHMETIC FUNCTIONS}\\[1.5em]
{\scshape Joshua Stucky} \end{center}
\begin{abstract} Motivated by recent results, we study sums of the form \[ S_f(x) = \sum_{n\leq x} f\pth{\floor{\frac{x}{n}}}, \] where $f$ is an arithmetic function and $\floor{\cdot}$ denotes the greatest integer function. We show how the error term in the asymptotic formula for $S_f(x)$ can be improved in some specific cases. \end{abstract}
\noindent\textbf{Keywords:} Fractional sum, integer part, exponent pairs, Euler phi function, divisor function.\\
\section{Introduction and Statement of Results} Recently, there has been some interest in the sums \[ S_f(x) = \sum_{n\leq x} f\pth{\floor{\frac{x}{n}}}, \] where $f$ is an arithmetic function and $\floor{\cdot}$ denotes the greatest integer function. When $f(n) = n$, this is the classic Dirichlet divisor problem. Such sums do not seem to have a name in the literature yet, so we propose to call $S_f$ the ``fractional sum of $f$.'' In the present paper, we show how the error term in the asymptotic formula for $S_f(x)$ can be improved for some specific functions $f$ which are in some sense ``small.''
To motivate our results, we discuss some of the literature on the subject. If $f$ satisfies \begin{equation}\label{eq:hyperbolaRequirement} f(n) \ll n^\alpha(\log n)^\theta \end{equation} for some fixed $\alpha,\theta \geq 0$ with $\alpha < 1$, then Wu \cite{Wu2019} and Zhai \cite{Zhai2020} have independently used the hyperbola method to show that \begin{equation}\label{eq:hyperbola} S_f(x) = C_f x + O\pth{x^{\frac{1+\alpha}{2}}(\log x)^\theta}, \end{equation} where \[ C_f = \sum_{n=1}^\infty \frac{f(n)}{n(n+1)}. \] In particular, when $\abs{f(n)} \ll n^\varepsilon$, we obtain \begin{equation}\label{eq:trivialHalf}
S_f(x) = C_f x + O\pth{x^{\frac{1}{2}+\varepsilon}}. \end{equation} The question of improving the error term beyond $\frac{1}{2}$ has been the subject of several papers (see \cite{Bordelles2020}, \cite{LiuWuYang2021}, \cite{MaSun2020}, and \cite{MaWu2020}, for instance). In particular, for $\tau$ the usual divisor-counting function and $\Lambda$ the von-Mangoldt function, we have \[ S_\tau(x) = C_\tau x + O\pth{x^{\frac{19}{40}+\varepsilon}} \qquad\text{and}\qquad S_\Lambda(x) = C_\Lambda x + O\pth{x^{\frac{9}{19}+\varepsilon}}, \] due to Bordell\`{e}s \cite{Bordelles2020} and Liu, Wu, and Yang \cite{LiuWuYang2021}, respectively (note $\frac{19}{40} = .475$ and $\frac{9}{19} =.4736...$). By applying a theorem of Jutila, we improve Bordell\`{e}s' estimate and prove
\begin{thm}\label{thm:511} We have \[ S_\tau(x) = C_\tau x + O(x^{5/11+\varepsilon}). \] Note $\frac{5}{11} = 0.4545...$. \end{thm}
We prove Theorem \ref{thm:511} in Section \ref{sec:511}. In a different direction, one can also improve the error term in (\ref{eq:hyperbola}) when $f$ is ``close to one,'' in a suitable sense. In this direction, Wu \cite{Wu2019} has shown that for $f = \varphi$ the Euler phi function, we have \begin{equation}\label{eq:WuPhi} S_\varphi(x) = C_\varphi x + O(x^{1/3}(\log x)). \end{equation} In Section \ref{sec:Wu}, we generalize Wu's result and prove
\begin{thm}\label{thm:GeneralWu} Suppose $f(n) = \sum_{d\mid n} g(d)$ and that \begin{equation}\label{eq:gClose1} \sum_{d\leq x} \abs{g(d)} \ll x^\alpha (\log x)^\theta \end{equation} for some $\alpha\in[0,1)$ and $\theta \geq 0$. Then \[ S_f(x) = C_fx + O\pth{x^{\frac{1+\alpha}{3-\alpha}}(\log x)^\theta}, \] where the implied constant depends only on $\alpha$. If $\alpha = 0$, then $\theta$ should be replaced by $\max(1,\theta)$. \end{thm}
\noindent\textbf{Notation.} The symbols $O,o,\ll,\gg$ have their usual meanings. We use $n\asymp N$ to denote the condition $N < n \leq 2N$. The variable $\varepsilon$ always denotes a positive arbitrarily small fixed real number, and $\delta$ is always 0 or 1. We also write $\psi(x) = x - \floor{x} - \frac{1}{2}$.
\section{A General Decomposition}
We begin with a general lemma that is useful in estimating the sums $S_f(x)$. The decomposition (\ref{eq:MainDecomp}) has appeared in more or less the same form in other works studying these sums (see \cite{Wu2019} and \cite{Zhai2020}, for instance)
\begin{lem}\label{lem:decomp}
Let $A,B\in[1,x^{1/2})$ be parameters to be chosen. Then
\begin{equation}\label{eq:MainDecomp}
S_f(x) = \colm(B) + O\Big(\abs{\colt_0(A,B)} + \abs{\colt_1(A,B)} + \cole_1(A) + \cole_2(B)\Big),
\end{equation}
where
\[
\begin{aligned}
\colm(B) &= x \sum_{n \leq x/B} \frac{f(n)}{n(n+1)}, &&\cole_1(A) = \sum_{n \leq A} \abs{f(n)}, \\
\colt_\delta(A,B) &= \sum_{A < n \leq x/B} f(n) \psi\fracp{x}{n+\delta}, \qquad &&\cole_2(B) = \sum_{n < B} \abs{f\pth{\floor{\frac{x}{n}}}}.
\end{aligned}
\] \end{lem}
\begin{proof}
Since
\[
m = \floor{\frac{x}{n}} \qquad \iff \qquad \frac{x}{m+1} < n \leq \frac{x}{m},
\]
we can change variables and write
\[
\begin{aligned}
S_f(x) &= \sum_{n < B} f\pth{\floor{\frac{x}{n}}} + \sum_{n \leq x/B} f(n) \pth{\floor{\frac{x}{n}} - \floor{\frac{x}{n+1}}} \\
&= \sum_{n < B} f\pth{\floor{\frac{x}{n}}} + x\sum_{n \leq x/B} \frac{f(n)}{n(n+1)} + \sum_{n \leq x/B} f(n)\pth{\psi\pth{\frac{x}{n+1}} - \psi\pth{\frac{x}{n}}},
\end{aligned}
\]
and the lemma follows.
\end{proof}
\section{Proof of Theorem \ref{thm:511}}\label{sec:511}
The proof of Theorem \ref{thm:511} is a combination of two ingredients. The first is
\begin{lem}\label{lem:Bordelles}
For $x$ sufficiently large, $f(n) \ll n^\varepsilon$, and $N \in [x^{1/3},x^{1/2})$, we have for all $H \geq 1$
\[
\abs{S_f(x) - C_f x} \ll Nx^\varepsilon + x^\varepsilon \max_{\substack{N < D \leq x/N\\ \delta\in\set{0,1}}} \sumpth{\frac{D}{H} + \sum_{h\leq H}\frac{1}{h} \sumabs{\sum_{n\asymp D} f(n) e\fracp{hx}{n+\delta}}}.
\] \end{lem}
This lemma is originally due to Bordell\`{e}s \cite{Bordelles2020} and follows from Lemma \ref{lem:decomp} by inserting Vaaler's approximation for $\psi$ (see the apprendix of \cite{GrahamKolesnik} for a discussion of Vaaler's approximation).
The second ingredient is due to Jutila. The full statement of the result we need is somewhat lengthy, so we will state only the specific version we require. For a full statement and proof of this theorem, see Theorem 4.6 of \cite{Jutila}.\\
\begin{thm}\label{thm:Jutila}
Let $2\leq M\leq M' \leq 2M$, and let
\begin{equation}\label{eq:gConditions}
g(z) = \frac{B}{z}\pth{1+O(F^{-1/3})}
\end{equation}
be a holomorphic function in the domain
\[
D = \set{z : \abs{z-x} < cM\ \text{for some}\ x\in[M,M']},
\]
where $c$ is some positive constant and
\[
F = \frac{\abs{B}}{M}.
\]
Suppose also that
\begin{equation}\label{eq:Fbound}
M^{3/4} \ll F \ll M^{3/2}.
\end{equation}
Then
\[
\sumabs{\sum_{M\leq n\leq M'} \tau(n) e(g(n))} \ll M^{1/2}F^{1/3+\varepsilon}.
\] \end{thm}
We are now ready to prove Theorem \ref{thm:511}. From Lemma \ref{lem:Bordelles}, we need to estimate the sums \[ \sum_{n\asymp D} \tau(n) e\fracp{hx}{n+\delta}. \] In Theorem \ref{thm:Jutila}, we take \[ M=D,\quad M'=2D,\quad g(n) = \frac{hx}{n+\delta}, \quad B = hx,\quad F = \frac{hx}{D}. \] Then the condition (\ref{eq:Fbound}) becomes \[ \frac{D^{7/4}}{x} \ll h \ll \frac{D^{5/2}}{x}, \] which is satisfied for all integers $h \in [1,H]$ so long as \begin{equation}\label{eq:hConditions}
D \ll x^{4/7-\varepsilon} \qquad\text{and}\qquad H \ll \frac{D^{5/2}}{x}. \end{equation} Theorem \ref{thm:Jutila} then gives \[ \sumabs{\sum_{n\asymp D} \tau(n) e\fracp{hx}{n+\delta}} \ll D^{1/6} h^{1/3}x^{1/3+\varepsilon}. \] Summing over $h \leq H$, we have \[ \abs{S_\tau(x) - C_\tau x } \ll N x^\varepsilon +x^\varepsilon\max_{N < D \leq x/N} \sumpth{\frac{D}{H} + D^{1/6} H^{1/3}x^{1/3+\varepsilon}} \ll x^\varepsilon \pth{N + \frac{x}{NH} + \frac{H^{1/3}}{N^{1/6}} x^{1/2}}. \] We complete the proof by choosing \[ H = x^{3/8}N^{-5/8} \qquad\text{and}\qquad N = x^{5/11}, \] which also ensures both conditions of (\ref{eq:hConditions}) are satisfied for all $D$ considered in the maximum, and that (\ref{eq:gConditions}) is satisfied.
\section{Proof of Theorem \ref{thm:GeneralWu}}\label{sec:Wu}
We follow Wu's \cite{Wu2019} method of proving (\ref{eq:WuPhi}). Let $A\in[1,x^{1/2})$ be a parameter to be chosen and use Lemma \ref{lem:decomp} with $B = A$. This gives \[ S_f(x) = \colm + O\Big(\abs{\colt_0} + \abs{\colt_1} + \cole_1 + \cole_2\Big) \] with \[ \begin{aligned} \colm &= x \sum_{n \leq x/A} \frac{f(n)}{n(n+1)}, &&\cole_1 = \sum_{n \leq A} \abs{f(n)}, \\ \colt_\delta &= \sum_{A < n \leq x/A} f(n) \psi\fracp{x}{n+\delta}, \qquad &&\cole_2 = \sum_{n < A} \abs{f\pth{\floor{\frac{x}{n}}}}. \end{aligned} \] Note that (\ref{eq:gClose1}) trivially implies (\ref{eq:hyperbolaRequirement}) with the same $\alpha$ and $\theta$. Thus \[ \colm = C_f x + O\sumpth{x\sum_{n > x/A} \frac{n^\alpha(\log n)^\theta}{n^2}} = C_f x + O(x^{\alpha}A^{1-\alpha}(\log x)^\theta), \] and likewise \[ \cole_1 \ll A^{1+\alpha}(\log x)^\theta \qquad\text{and}\qquad \cole_2 \ll x^\alpha A^{1-\alpha} (\log x)^\theta. \] We have $A^{1+\alpha} < x^\alpha A^{1-\alpha}$ since $A < x^{1/2}$, and thus \begin{equation}\label{eq:Sfdecomp} S_f(x) = C_f x + O\pth{\abs{\colt_0} + \abs{\colt_1} + x^\alpha A^{1-\alpha}(\log x)^\theta}. \end{equation} We break $\colt_\delta$ (where $\delta$ is 0 or 1) into $O(\log x)$ sums of the form \[ T(N) = \sum_{n\asymp N} f(n) \psi\fracp{x}{n+\delta}, \] where $A\ll N \ll x/A$. Switching divisors, this is \[ T(N) = \sum_{d\leq 2N} g(d) \sum_{n\asymp N/d} \psi\fracp{x}{dn+\delta}. \] We now employ Lemma 4.3 of \cite{GrahamKolesnik}. For any exponent pair $(k,l)$, we have \begin{equation}\label{eq:TNexponentPair} T(N) \ll \sum_{d\leq 2N} \abs{g(d)} \sumpth{x^{k/(k+1)} N^{(l-k)/(k+1)} d^{-l/(k+1)} + N^2x^{-1}d^{-1}}. \end{equation} The goal now is to choose an exponent pair $(k,l)$ such that, after applying partial summation and (\ref{eq:gClose1}), the first term on the right is dominated by the other error terms. For this, we need a sequence of specific exponent pairs given by
\begin{lem}\label{lem:ExponentPairs} For any integer $n \geq 0$, \[ (k_n,l_n) = \pth{\frac{1}{2^{n+2}-2}, \frac{2^{n+2}-n-3}{2^{n+2}-2}} \] is an exponent pair. \end{lem}
\begin{proof} Repeatedly apply the $A$ process to the pair $(\frac{1}{2},\frac{1}{2})$.
\end{proof}
Note that \[ \frac{l_n}{k_n+1} = \frac{2^{n+2}-n-3}{2^{n+2}-1} = 1 - \frac{n+2}{2^{n+2}-1}, \] and so $l_n/(k_n+1)$ is strictly increasing as $n\to\infty$. We define $(k_{-1},l_{-1}) = (1,0)$ (which is not an exponent pair) and divide the interval $[0,1)$ into subintervals \[ I_n = \hor{\frac{l_{n-1}}{k_{n-1}+1}, \frac{l_n}{k_n+1}}, \qquad n\geq 0. \] Suppose that $\alpha \in I_n$. Partial summation and (\ref{eq:gClose1}) give \begin{equation}\label{eq:PartialSummation} \sum_{d\leq 2N} \frac{\abs{g(d)}}{d^{l_n/(k_n+1)}} \ll 1 \qquad\text{and}\qquad \sum_{d\leq 2N} \frac{\abs{g(d)}}{d} \ll 1 \end{equation} since $\alpha < l_n/(k_n+1) < 1$. From (\ref{eq:TNexponentPair}) and (\ref{eq:PartialSummation}), we have \[ T(N) \ll x^{k_n/(k_n+1)} N^{(l_n-k_n)/(k_n+1)} + N^2x^{-1}. \] Executing the dyadic sum in $N$ gives \[ \begin{aligned} S_f(x) &= C_f x + O\pth{x^{l_n/(k_n+1)}A^{(k_n-l_n)/(k_n+1)}(\log x)^{\delta(n=0)} + xA^{-2} + x^{\alpha} A^{1-\alpha} (\log x)^\theta} \\ &= C_f x + O\pth{E_1(\log x)^{\delta(n=0)}+E_2+E_3}, \end{aligned} \] where $\delta(n=0)$ is 1 if $n=0$ and $0$ otherwise. If $\alpha = 0$, then $l_n=k_n = \frac{1}{2}$ and we have \[ S_f(x) = C_f x + O\pth{x^{1/3}(\log x) + xA^{-2} + A (\log x)^\theta}. \] Choosing $A = x^{1/3}$ gives the desired result with $\max(1,\theta)$ in place of $\theta$. Suppose now that $\alpha > 0$. We will show that $E_1$ is always dominated by $E_2$ or $E_3$. We have $E_1 \leq E_2$ if $A \leq x^{B_1(n)}$, where \[ B_1(n)= \frac{k_n-l_n+1}{3k_n-l_n+2} = \frac{n+2}{n+2^{n+2}+2}, \] and likewise $E_1 \leq E_3$ if $A \geq x^{B_2(n)}$, where \[ B_2(n) = \frac{\frac{l_n}{k_n+1}-\alpha}{\frac{l_n}{k_n+1}-\alpha+\frac{1}{k_n+1}} \leq \frac{\frac{l_n}{k_n+1}-\frac{l_{n-1}}{k_{n-1}+1}}{\frac{l_n}{k_n+1}-\frac{l_{n-1}}{k_{n-1}+1}+\frac{1}{k_n+1}} = \frac{n+2^{-n-1}}{ n+2^{n+2} + 3(2^{-n-1}) -4}. \] The inequality $B_2(n) \leq B_1(n)$ for $n=0,1,2$ may be verified by direct computation. For $n \geq 3$, we have \[ \begin{aligned} \frac{B_1(n)}{B_2(n)} \geq \fracp{n+2}{n+2^{-n-1}} \fracp{n+2^{n+2}+3(2^{-n-1}) -4}{n+2^{n+2}+2} \geq \pth{1+\frac{1}{n}}\pth{1-\frac{1}{2^{n-1}}} \geq 1 \end{aligned} \] We thus have $E_1 \leq \max(E_2,E_3)$. In the case $n=0$, since $\alpha\neq 0$, we actually have $E_1 \leq x^{-\lambda}\max(E_2,E_3)$ for some sufficiently small positive $\lambda$, and thus we may ignore the factor $(\log x)^{\delta(n=0)}$. For $\alpha > 0$, we thus have \[ S_f(x) = C_f x + O\pth{(xA^{-2} + x^{\alpha} A^{1-\alpha})(\log x)^\theta}. \] Choosing $A = x^{(1-\alpha)/(3-\alpha)}$ completes the proof.
\section{Remarks on Theorem \ref{thm:GeneralWu}}
Theorem \ref{thm:GeneralWu} is generally good when $g$ is such that either $\abs{g}$ is small or $g$ is supported on a sparse set. For instance, let $\beta \in (0,1]$ and let $\sigma_\beta$ denote the sum of $\beta$th powers of divisors, $\sigma_\beta(n) = \sum_{d\mid n} d^\beta$. For $f(n) = \sigma_\beta(n) n^{-\beta}$, we have $f(n) \ll n^\varepsilon$, and thus (\ref{eq:trivialHalf}) holds. However, we also have (\ref{eq:gClose1}) with $\alpha = 1-\beta$ and $\theta = 0$ (unless $\beta = 1$, in which case $\theta=1$). Thus \[ S_f(x) = C_f x + O\pth{x^\frac{2-\beta}{2+\beta}(\log x)^{\delta(\beta=0)}}, \] This is superior to (\ref{eq:hyperbola}) so long as $\beta \geq \frac{2}{3}$.
On the other hand, it is possible for Theorem \ref{thm:GeneralWu} to give a worse result than (\ref{eq:hyperbola}). For instance, let $f$ be the indicator function for the squarefree numbers so that $f(n) = \sum_{d\mid n} g(d)$ with \[ g(d) = \begin{cases} \mu(l) & \text{if $d=l^2$}, \\ 0 & \text{otherwise}, \end{cases} \] We have \[ \sum_{d\leq x} \abs{g(d)} = \sum_{l\leq \sqrt{x}} \mu^2(l) \asymp \sqrt{x}, \] and so Theorem \ref{thm:GeneralWu} only gives \[ S_f(x) = C_f x + O\pth{x^{\frac{3}{5}}}. \] More generally, if $f$ is the indicator function of the $k$-free numbers, then the same argument yields \[ S_f(x) = C_f x + O\pth{x^{\pth{1+\frac{1}{k}}\pth{3-\frac{1}{k}}^{-1}}}, \] which is superior to (\ref{eq:hyperbola}) so long as $k > 3$.
\end{document} | arXiv |
TECHNICAL ADVANCE
A general framework for comparative Bayesian meta-analysis of diagnostic studies
Joris Menten1,2 &
Emmanuel Lesaffre2
Selecting the most effective diagnostic method is essential for patient management and public health interventions. This requires evidence of the relative performance of alternative tests or diagnostic algorithms. Consequently, there is a need for diagnostic test accuracy meta-analyses allowing the comparison of the accuracy of two or more competing tests. The meta-analyses are however complicated by the paucity of studies that directly compare the performance of diagnostic tests. A second complication is that the diagnostic accuracy of the tests is usually determined through the comparison of the index test results with those of a reference standard. These reference standards are presumed to be perfect, i.e. allowing the classification of diseased and non-diseased subjects without error. In practice, this assumption is however rarely valid and most reference standards show false positive or false negative results. When an imperfect reference standard is used, the estimated accuracy of the tests of interest may be biased, as well as the comparisons between these tests.
We propose a model that allows for the comparison of the accuracy of two diagnostic tests using direct (head-to-head) comparisons as well as indirect comparisons through a third test. In addition, the model allows and corrects for imperfect reference tests. The model is inspired by mixed-treatment comparison meta-analyses that have been developed for the meta-analysis of randomized controlled trials. As the model is estimated using Bayesian methods, it can incorporate prior knowledge on the diagnostic accuracy of the reference tests used.
We show the bias that can result from using inappropriate methods in the meta-analysis of diagnostic tests and how our method provides more correct estimates of the difference in diagnostic accuracy between two tests. As an illustration, we apply this model to a dataset on visceral leishmaniasis diagnostic tests, comparing the accuracy of the RK39 dipstick with that of the direct agglutination test.
Our proposed meta-analytic model can improve the comparison of the diagnostic accuracy of competing tests in a systematic review. This is however only true if the studies and especially information on the reference tests used are sufficiently detailed. More specifically, the type and exact procedures used as reference tests are needed, including any cut-offs used and the number of subjects excluded from full reference test assessment. If this information is lacking, it may be better to limit the meta-analysis to direct comparisons.
There is a growing interest in diagnostic test accuracy (DTA) reviews to select the best diagnostic test procedure [1] for a given setting. Most meta-analyses of diagnostic tests, however, estimate the diagnostic accuracy of a single test [2, 3]. Selection of the best test is usually done by undertaking separate meta-analyses for each test and then comparing the results [3]. Even when formally comparing diagnostic tests in a single systematic review, the analysis may ignore study effects. Such an approach can lead to biased comparisons due to confounding by study effects, as shown in a recent review [3]. Takwoingi and colleagues showed that results from comparative studies, where two tests were directly compared and which provide the most robust comparisons, differed from those of non-comparative studies. However, only 31 % of available studies were comparative. This indicates that there is a need for meta-analytical methods via direct and indirect comparisons. Data from direct comparisons may be inconclusive while a combined analysis of direct and indirect comparisons may be conclusive and can result in more accurate estimates [4, 5].
A particular aspect of comparisons between diagnostic tests is that the diagnostic performance of the index test is nearly always determined by comparison with a second test, the reference standard. Such a reference standard is presumed to 100 % correctly classify subjects as diseased or not. However, for many diseases it is impossible to determine the true disease status with certainty [6] and reference standards are imperfect. It is well known that the use of imperfect reference standards may bias estimates of the accuracy of the index test [7]. This consideration leads to a second requirement for comparative meta-analyses of diagnostic studies: the meta-analytic methods should adjust for the use of imperfect reference standards.
The aim of this manuscript is to develop a model that can be used for the comparative meta-analysis of two diagnostic tests that conforms to the two requirements sketched above. First, we assess possible biases in the estimation of the relative accuracy of two index tests due to the use of imperfect reference tests. We describe the different parameters that can be used to estimate the relative accuracy of two tests and assess the bias resulting from the use of imperfect reference standards. This allows us to select the most appropriate summary measure to use in the comparative meta-analysis of two diagnostic tests. Subsequently, we describe and develop models that can be used in the meta-analysis of diagnostic studies to compare the relative accuracy of two tests. We start with models that presume a perfect reference test is used in each primary study and extend these models allowing for imperfect reference tests. We estimate these models using Bayesian methods, specifically using Markov-Chain Monte-Carlo (MCMC) methods through Gibbs sampling [8]. For each model we provide the model specification and offer suggestions for appropriate informative or vague priors. In addition, we assess in a simulation study the value of these newly developed models but also the bias induced by the use of incorrect methods. Finally, we apply the methods to a real data example in the field of leishmaniasis.
Our aim is to estimate and test the difference in diagnostic accuracy of two or more index tests in a meta-analysis, combining data across all available studies. The studies included in a DTA review typically test each subject with one or more index tests and with one reference test. This reference test may differ between studies. To set the scene, data from a hypothetical meta-analysis are presented in Table 1. In this example, there are three index tests (T1,T2,T3) and two possible reference tests (T4,T5). For example, in Study 1 index tests T1 and T2 are performed on all subjects as well as reference test T4. There are 30 subjects with positive results on all three tests, one subject shows positive results on T1 and T2 and a negative result on T4, etc. Studies 1 and 2 allow direct estimation of the relative accuracy of T1 and T2. Studies 3, 4 and 5 allow the estimation of the accuracy of T1 (studies 3 and 4) or T2 (study 5), but allow no direct comparison of T1 and T2. For these studies, the relative accuracy of T1 and T2 can only be estimated by estimating the diagnostic accuracy of each test separately and then comparing these estimates. This is complicated by the fact that the reference test is not the same for each study. Studies 6 and 7 do not allow direct comparison of the accuracy T1 and T2, but offer the possibility of an indirect comparison through the third index test T3. The information from this third test may help to eliminate differences among the studies.
Table 1 Tabulation of an hypothetical diagnostic test accuracy meta-analysis. Columns T1,T2,T3 indicate results for the 3 possible index tests. Columns T4,T5 indicate results for the 2 possible reference tests. + indicates a positive test result, - a negative test result. NA indicates that the test was not performed in that particular study. The observed frequency column report the number of subjects with a specific test result pattern in each study
As a first step in a comparative DTA meta-analysis, we have to select an appropriate statistic to compare the two tests. The best statistic would be one which is readily interpretable by users of the meta-analysis and which is least prone to bias. We describe the possible choices below together with the results from a small simulation study. Subsequently, we need to develop a model which allows the incorporation of all available data while ensuring that results are valid and are not biased by differences in study characteristic, such as the selection of the reference standard used. Some possible models are described below. We assessed the value of these models in a simulation study and in a practical application.
Measures of relative value of diagnostic tests
Diagnostic accuracy is characterised by sensitivity S and specificity C. These two quantities are related and comparisons between tests need to take both S and C into account. Comparisons between two tests can be summarized using the difference or relative risk in S and C for the two tests. An alternative parameterization uses the diagnostic odds ratio DOR=(S×C)/[(1−S)×(1−C)] which summarizes the accuracy of a test in a single number [9]. This parameter could be used as a summary in a meta-analysis, for example by calculating the relative DOR of two tests [3]. However, the use of imperfect reference standards can bias all above measures of relative accuracy of two index tests. We assessed the direction and magnitude of bias on these measures in a small simulation study. A description of the simulation study setup is given in Additional file 1.
Models for the comparative meta-analysis of diagnostic tests
In this section we develop models to compare J tests, by combining data across I studies in a comparative meta-analysis. All these models are hierarchical in nature. At the first level of the hierarchy, the models describe the observed data of the individual studies. The observed test outcomes depend on the disease prevalence and the accuracy of the tests in each study, and possible covariation among the test results. We describe the accuracy of the tests in terms of the study-specific sensitivity S ij and specificity C ij of test j in study i. At the second level, we specify a model for these study-specific sensitivity-specificity pair {S ij ,C ij }. Five possible models are described; they are listed in Table 2.
Table 2 Description of the different models. Example code for the models is given in Additional file 2. S ij and C ij represent the sensitivity and specificity of test j in study i
Meta-analytic models when a perfect reference standard is available
If a perfect reference test is available, the number of diseased N Di and non-diseased N NDi subjects in study i is known, as are the numbers of true positives N TPij and true negatives N TNij for each test j. In the standard bivariate model for the meta-analysis of a diagnostic test [9], the observed numbers of true positives and true negatives for each index test are assumed to be drawn from two independent binomial distributions N TPij ∼Bin(N Di ,S ij ) and N TNij ∼Bin(N NDi ,C ij ). The transformed values g(S ij ) = θ Sij and g(C ij ) = θ Cij are modeled at the next level, where g(.) is a link function to allow the use of the normal distribution. Common choices for g(.) are the logit, complementary log-log or probit functions. Several models are possible to incorporate comparisons of the diagnostic accuracy of different tests in this framework. We discuss three models below. These models can be further expanded to allow for covariates, other dependence structures or alternative parameterizations.
Model 1: Standard bivariate model for the meta-analysis of diagnostic tests
A basic approach is to estimate the average diagnostic accuracy of each test separately and subsequently compare the estimates of the average S j and C j across the different studies. In this approach, the standard bivariate model for the meta-analysis of diagnostic tests [2] can be used for each test separately. All g(S ij )=θ Sij and g(C ij )=θ Cij pairs are assumed to follow independent bivariate normal distributions:
$$\begin{array}{*{20}l} \left(\begin{aligned} \theta_{Sij} \\ \theta_{Cij} \end{aligned} \right) \sim N \left(\left[ \begin{aligned} \mu_{S_{j}} \\ \mu_{C_{j}} \end{aligned} \right], \Sigma_{j} \right), \\ \text{with} \ \Sigma_{j} = \left(\begin{aligned} & \sigma^{2}_{S_{j}} & \sigma_{S_{j}C_{j}} \\ & \sigma_{S_{j}C_{j}} & \sigma^{2}_{C_{j}} \end{aligned} \right), \end{array} $$
where \(\rho _{S_{j}C_{j}}=\sigma _{S_{j}C_{j}}/(\sigma _{S_{j}}\times \sigma _{C_{j}})\) is the correlation between \(\theta _{S_{\textit {ij}}}\) and \(\theta _{C_{\textit {ij}}}\). Estimates of the relative accuracy of the tests are obtained from the estimated \(g^{-1}(\mu _{S_{j}})\) and \(g^{-1}(\mu _{C_{j}})\). For example, the average difference in S between T1 and T2 is estimated as \(\hat {S}_{D21} = g^{-1}(\hat {\mu }_{S_{2}}) - g^{-1}(\hat {\mu }_{S_{1}})\). The advantage of the standard bivariate model is that it is relatively easy to fit using both Bayesian or frequentist techniques, with SAS [9] and WinBUGS [10, 11] example code available. However, as this model is not based on the comparisons between the index tests, but on the pooling of results for each test across all available studies, the results may be biased by study characteristics. This is equivalent to pooling findings from the active treatment arms of RCTs and comparing these estimates, an approach which is considered not to be appropriate for the meta-analysis of RCTs [12].
Model 2: Meta-Analysis Based on Direct Comparisons
To take study effects into account, the overall probability of testing positive in diseased subjects μ Si or in non-diseased subjects μ Ci for each study i could be modeled and S ij and C ij of the individual tests described as contrasts from this overall probability.
If we limit the data to studies which compare the two tests directly, we can write the study specific, transformed sensitivities g(S ij ) = θ Sij and specificities g(C ij ) = θ Cij as follows:
$$ \begin{aligned} \theta_{Si1} = \mu_{Si} + \delta_{Si}/2, \\ \theta_{Si2} = \mu_{Si} - \delta_{Si}/2, \\ \theta_{Ci1} = \mu_{Ci} - \delta_{Ci}/2, \\ \theta_{Ci2} = \mu_{Ci} + \delta_{Ci}/2. \end{aligned} $$
In case g is the logit function, δ Si = log(SOR12) and δ Ci = log(COR12), i.e. the log of the ORs of testing positive in diseased subjects for T1 compared to T2 and the log of the ORs of testing negative in non-diseased subjects in study i, respectively. To obtain average estimates of the difference in diagnostic accuracy between the two tests, δ Si and δ Ci are modeled using a bivariate normal distribution:
$$\begin{array}{*{20}l} \left(\begin{aligned} \delta_{Si} \\ \delta_{Ci} \end{aligned} \right) \sim N \left(\left[ \begin{aligned} \nu_{\delta_{S}} \\ \nu_{\delta_{C}} \end{aligned} \right], \Sigma \right) \\ \text{with} \ \Sigma = \left(\begin{aligned} & \sigma^{2}_{\delta_{S}} & \sigma_{\delta_{S}\delta_{C}} \\ & \sigma_{\delta_{S}\delta_{C}} & \sigma^{2}_{\delta_{C}} \end{aligned} \right). \end{array} $$
The \(\nu _{\delta _{S}}\) and \(\nu _{\delta _{C}}\) are the average log OR of the S and C between tests T1 and T2, respectively. The μ Si and μ Ci account for the dependence of test results obtained from the same study and can be estimated as fixed effects of in their turn modeled using bivariate normal distributions. This model is equivalent to the Smith −Spiegelhalter−Thomas model for two-treatment comparisons of RCTs [13, 14]. A similar model, but assuming a fixed, rather than random, relative accuracy between the different index tests is described in the Cochrane Handbook for Systematic Reviews of DTA studies [9].
Model 3: Meta-Analysis Based on Direct and Indirect Comparisons
As shown in Lu et al. [14] in the case of meta-analysis of RCTs, the Smith −Spiegelhalter− Thomas model can be expanded to a mixed treatment-comparison meta-analysis of more than two treatments. Similarly, we can expand Model 2 to J diagnostic tests. By taking diagnostic test T J as baseline, we can rewrite eqs. 2 and 3 as:
$$\begin{aligned} \theta_{Si1} = \mu_{Si} + (\,J-1) \times \delta_{Si1}/J - \delta_{Si2}/J - \ldots - \delta_{Si(\,J-1)}/J, \\ \theta_{Si2} = \mu_{Si} - \delta_{Si1}/J + (\,J-1) \times \delta_{Si2}/J - \ldots - \delta_{Si(\,J-1)}/J, \\ \vdots \\ \theta_{SiJ} = \mu_{Si} - \delta_{Si1}/J - \delta_{Si2}/J - \ldots - \delta_{Si(\,J-1)}/J, \end{aligned} $$
$$\begin{aligned} \theta_{Ci1} = \mu_{Ci} + (\,J-1) \times \delta_{Ci1}/J - \delta_{Ci2}/J - \ldots - \delta_{Ci(\,J-1)}/J, \\ \theta_{Ci2} = \mu_{Ci} - \delta_{Ci1}/J + (\,J-1) \times \delta_{Ci2}/J - \ldots - \delta_{Ci(\,J-1)}/J, \\ \vdots \\ \theta_{CiJ} = \mu_{Ci} - \delta_{Ci1}/J - \delta_{Ci2}/J - \ldots - \delta_{Ci(\,J-1)}/J, \end{aligned} $$
$$ \left(\delta_{Si1}, \delta_{Si2}, \ldots, \delta_{Si(\!\,J-1)}, \delta_{Ci1}, \delta_{Ci2}, \ldots, \delta_{Ci(\,J-1)} \right) \sim \!N(\boldsymbol{\nu_{\delta}},\Sigma). $$
and \(\phantom {\dot {i}\!}\boldsymbol {\nu }_{\boldsymbol {\delta }}=\left (\nu _{\delta _{S1}},\ldots,\nu _{\delta _{S(\,J-1)}}, \nu _{\delta _{C1}},\ldots,\nu _{\delta _{C(\,J-1)}}\right)\) represents the average log ORs for S and C of the J−1 tests compared to the baseline test T J . The differences in S and C between T1 and T2 on the logit scale are estimated by \(\nu _{\delta _{S1}} - \nu _{\delta _{S2}}\) and \(\nu _{\delta _{C1}} - \nu _{\delta _{C2}}\), respectively. This method allows indirect comparisons of T1 and T2 through comparison with a third test, similar to mixed treatment comparisons meta-analysis of RCTs. One complication of this model, is the specification and estimation of the variance-covariance matrix Σ. Specifying a structured variance-covariance matrix is in general complex and difficult to handle in MCMC estimation since each sampled variance-covariance matrix should be positive-definite [15]. In addition, model identification of the model with a general variance-covariance matrix will be difficult, especially when number of tests of interest is large. As an initial exploration of this model we can use a simplified variance-covariance structure, for example a diagonal or block diagonal matrix, and subsequently assess the effects of relaxing the simplifying assumptions. We describe some possible simplified variance-covariance structures in Additional file 2: Section 2.6.
Meta-analytic models when no perfect reference standard is available
The models described above presume that the disease status of all subjects in all studies is known, and consequently that the N Di ,N NDi ,N TPij and N TNij for each study i and test j is available. However, if only imperfect reference standards are available, the reported estimates of these quantities may be biased. The models described above can be expanded through latent class analysis (LCA) [16] to allow for the use of imperfect reference standards. In LCA, the true disease status of the participants of the basic studies is an unobserved, or latent, variable with two mutually exclusive categories, "diseased" and "non-diseased". This unobserved variable determines the probability to test positive or negative to a number of diagnostic tests which may include one or more imperfect reference tests. LCA models have been described for a variety of situations ranging from when a single imperfect test is observed in each study to more complex designs involving multiple tests. When multiple tests are involved, they may be treated as independent conditional on the disease status or the conditional dependence between them may be modeled using a variety of approaches [17–20]. An important underlying assumption of the latent class model is that the tests included in the model all correspond to the same underlying disease state [21]. Especially in a meta-analysis, where each study may use a different set of tests, this assumption is critical. If this assumption is not met, the underlying latent variable may differ among studies.
Description of the conditional independence latent class model
In this section, we describe the basic latent class model at the level of the individual study i in the meta-analysis. To simplify notation, we temporarily suppress the i-subscript for the study level. For latent class analysis, the basic data is not the number of true positives and true negatives for each test j, but rather the number of subjects that show a certain pattern of outcomes across the J tests performed in a study. The number of subjects with pattern y=(y1,y2,…,y j ) can be denoted as N y and is assumed to follow a multinomial distribution N y ∼Mult[N,P(y)], with y j the observed binary outcome (0 = negative, 1 = positive) for test T j , N the total sample size and P(y) the probability that y occurs.
Denoting the unobserved disease status as D (not diseased D=0, diseased D=1) and under the conditional independence assumption \(P(\mathbf {y}|D=k) = \prod _{j=1}^{J} P\left (y_{j}|D=k\right)\), the class probabilities P(y) can be described in terms of the S j and C j of the J tests. That is:
$$\begin{array}{@{}rcl@{}} P\left(\mathbf{y}\right)= \sum_{k=0}^{1} P\left(D=k\right) P\left(\mathbf{y}|D=k\right) = \\ {}\pi \prod_{j=1}^{J} S^{y_{j}}_{j} \left(1-S_{j}\right)^{\left(1-y_{j}\right)}\,+\, \left(1-\pi\right) \prod_{j=1}^{J} C^{\left(1-y_{j}\right)}_{j} \left(1\,-\,C_{j}\right)^{y_{j}}, \end{array} $$
with π the disease prevalence.
Thus LCA provides estimates for the study specific prevalence of disease π i and the S ij and C ij of the J i tests used in study i, which is a subset of the J different tests used across the I studies of the meta-analysis.
Model 4: Hierarchical Latent Class Model
In essence, the most basic hierarchical latent class model (Model 4) is constructed through a combination of equations 1 and 5. While previously the reference test was presumed to be 100 % sensitive and specific, in Model 4 all S ij and C ij , including those of the reference tests, are modeled using separate bivariate normal distributions as in Equation 1. The observed data is assumed to come from the multinomial distribution described in Equation 5. Again, like Model 1, this model ignores the correlation among test results from the same study. The prevalences π i can be assumed to be different for each study or to have a common normal distribution, \(\pi _{i} \sim N\left (\mu _{\pi },\sigma ^{2}_{\pi }\right)\).
Model 5: Network-based Hierarchical Latent Class Model
By rewriting the θ Sij and θ Cij in terms of μ Si ,μ Ci ,δ Sij , and δ Cij as in Eq 4, we can again take into account study level effects. The hierarchical modeling is equal to Model 3, the only difference is at the study level as described in Eq 5. This model thus adjusts the meta-analysis for the use of imperfect reference tests. By using the expanded Smith −Spiegelhalter− Thomas model of Lu et al. [14] at the second level of the hierarchy, study level effects are eliminated without the need to limit the analysis to direct comparisons only.
Model estimation and prior specification
Models are estimated in a Bayesian framework using Markov Chain Monte Carlo (MCMC) methods with OpenBUGS 3.0.3 called from within R 3.0.1 using the BRugs library. The Bayesian approach allows the estimation of complex, joint models and the combination of prior information, e.g. on the value of the reference test used, in the meta-analysis of new diagnostic tests. To complete the Bayesian model, priors need to be provided for all model parameters. OpenBUGS code for the models and full specifications of the priors are in Additional file 2. Convergence was checked using visual inspection of trace plots of the Markov chains and the Gelman-Rubin diagnostic statistic [22].
For parameters related to the index tests of interest, we consider it generally most appropriate to use uninformative priors. Specifically, we used normal priors with mean μ equal to zero and standard deviation σ equal to 1.69 for logit-transformed probabilities. This prior matches a uniform prior over the interval [0,1] in the first two moments on the probability scale [23]. When appropriate, these priors were bounded to avoid label switching [20]. Label switching is a problem arising in MCMC estimation of latent class models when two equivalent solutions are possible which give rise to identical observed data [24, 25]. The problem can be avoided by constraining S or C of one or more test to be ≥0.5. For the contrast in S and C, expressed as log ORs, normal priors with μ=0 and a large standard deviation, e.g. σ=10 can be used. For the variance-covariance matrices, we construct non-informative priors using uniform priors for standard deviations and correlations.
The model was specified using a logit link function and results are estimated on the log-odds scale. The MCMC approach as implemented in OpenBUGS allows to obtain posterior distributions of all functions of the estimated parameters, as the average S and C of the index tests and differences between S and C of the different tests. We illustrate this in the OpenBUGS code in Additional file 2. We used the 2.5 and 97.5 th percentiles of the sampled posterior distribution of the statistics of interest as bounds for the 95 % credible intervals.
If we want to use information from previous phases of the research, we can use informative priors. It may for example be appropriate to use information obtained from a previous meta-analysis of case-control studies when performing a meta-analysis of phase IV studies, i.e. studies recruiting clinically suspect patients consecutively in a representative clinical setting [7]. However, given that the phase IV design ensures the most realistic assessment of the performance of a test when used as a diagnostic tool in the target population [6], we may want to reduce the influence from these prior phases by using a prior which is more diffuse than the actual results from the prior meta-analysis. In the latent class model Model 4, we can use informative priors for the diagnostic accuracy of the reference test. It is likely that some information on the accuracy of the reference tests is available. In fact, standard analysis assumes S and C of the reference test to be 100 %, which can be considered to be very strong deterministic prior from a Bayesian viewpoint [26]. Priors for the accuracy of reference test can be obtained from the literature or expert opinion [10].
Simulation study
To assess the performance of the different models and to uncover possible bias of combining data without proper control for study specific effect or adjustment for the use of imperfect reference standards, we performed a simulation study using two different scenarios. For each scenario, we generated 250 sets of 20 diagnostic studies. We analyzed each simulated data set using the models described above using the logit for the link function g(.). We evaluated the models using coverage probabilities (the proportion of replications in which the 95 % credible interval contained the true value) and power (the proportion of replications in which a difference in S and C between the two tests of interest was detected). In Scenario 1, we simulated a setting without systematic bias but where a common imperfect reference test is used to assess the diagnostic accuracy of the index tests in all primary studies. In Scenario 2, we simulated the situation of two index tests which are assessed in primary studies that tend to use different reference standards. This situation may rarely occur in practice, but was selected to assess how the model performed in an extreme situation with systematic bias due to imperfect reference tests. A full description of the simulation study setup is in Additional file 3.
Real data example
We applied the models to data obtained in a meta-analysis of rapid diagnostic tests for visceral leishmaniasis, which we described earlier [10, 27]. In the published meta-analysis, the focus was on estimating the diagnostic accuracy of individual tests. We extracted the data relevant for the comparison of one rapid diagnostic test, the RK39 dipstick, with that of the direct agglutination test (DAT) as a test case for the application of the methods developed in the current paper. We limited the data for this test case to primary studies that included the RK39 dipstick or DAT with at least one other index test and microscopical examination as a reference test for which full data was available in the published primary study. We selected all index tests which were used in more than one study. In total, we included 10 primary studies, four index tests (DAT, RK39-dipstick, IFAT, KAtex) and two reference tests (parasitology including spleen aspirate, parasitology not including spleen aspirate) (Table 3). All tests are specific to VL and consequently can be expected to related to the same underlying latent variable. The data are shown in Fig. 1 and Appendix 4. Note that the current study is used as "proof-of-concept" of the statistical modeling approach and not as a complete meta-analytic comparison of the two tests which would require a more extended search strategy.
Forest plot for real data example. Estimated sensitivity and specificity of the RK39 dipstick (open circles) and DAT (closed squares) with 95 % confidence interval, using parasitology as gold standard
Table 3 Overview of the real data example: a comparative meta-analysis of the RK39 dipstick and direct agglutination test (DAT) for the diagnosis of visceral leishmaniasis. The total sample size (N) and availability of test results (X) is given for all 10 studies. Other tests: IFAT=indirect fluorescent antibody test, KAtex =latex agglutination test, spleen =parasitological examination of tissue aspirates including spleen sample, no spleen: parasitological examination of tissue aspirates not including spleen sample
The aim of this modeling exercise was to estimate the differences in S and C between the RK39-dipstick and DAT. A previous meta-analysis indicated that the diagnostic accuracy of the RK39, and possibly also of the DAT, may be lower in East-Africa compared to other geographic regions [28]. To correct for these differences, we included a fixed region effect (East-Africa vs. rest of the world) for S. We fitted the five models listed in Table 2. In the previous study [10], we obtained expert opinion on the diagnostic accuracy of the two reference tests. Expert opinion on the diagnostic accuracy of parasitology including spleen aspirate varied between 88 and 95 % for S and between 95 and 100 % for C. For parasitology without spleen aspirate, expert opinion varied for S between 70 and 80 % and between 95 and 100 % for C. We used this information to determine the priors in estimation of the models allowing for imperfect reference standards.
The results of our simulation study indicated that in a realistic setting, bias in estimating the difference in S and C between two index tests due to the use of an imperfect reference standard can be relatively limited (Additional file 1). Strong bias only occurred if the errors of one index test were strongly correlated with those of the reference test while the errors of the second index test were uncorrelated with those of the reference test. Similar observations can be made for the relative S and C. When the comparisons were expressed as odds-ratios or when using the relative Diagnostic Odds Ratio as a summary statistics, bias was more substantial and occurred even with uncorrelated errors. This corresponds to the findings from Zhang et al. who report that also in the meta-analysis of RCTs the odds-ratio is not always a suitable summary statistic [5].
Model performance: simulation study
Results of the simulation study of the model performance are described in detail in Additional file 3. The bias in estimating the contrasts in S and C between T1 and T2, expressed as a difference, relative risk or odds-ratio is summarized in Fig. 2.
Summary of simulation results. Bias in estimates of the contrasts in diagnostic accuracy from the proposed meta-analytical models applied in the simulation study. The boxplots present the bias in \(\hat {S}_{D12}\) and \(\hat {C}_{D12}\) (first row), \(\hat {S}_{RR12}\) and \(\hat {C}_{RR12}\) (second row), \(\hat {S}_{OR12}\) and \(\hat {C}_{OR12}\) (third row). The first column presents Scenario 1 where a common imperfect reference standard with moderate S and high C was used, the second scenario 2 where systematic bias is induced by differing reference standards. Full explanation of the model is in the text; full explanation of the simulation setup and results in Additional file 3. Note for Scenario 1: For models 1 to 3 disease status was estimated from the results of T4. Note for Scenario 2: Models 4 and 5 were applied both assuming it is known that the reference tests differ across studies (4a and 5a) and ignoring the difference in reference tests (4b and 5b)
In Scenario 1, where a common imperfect reference standard with moderate S and high C was used, a naive analysis assuming the reference test was perfect (Models 1–3), resulted in bias in estimating the odds-ratios (Fig. 2c) and to a lesser extent also the relative risks (Fig. 2b). If the contrast of interest was expressed as a difference (Fig. 2a), a true gold standard was available (Fig. 2a–c, Models 1–3), or if a latent class model was used to allow for imperfect reference tests (Fig. 2a–c, Models 4 and 5), no bias was apparent. Allowing for imperfect reference tests resulted in a lower power compared to the situation that a perfect reference test was available (Additional file 3, Table 3).
In Scenario 2, the reference standard of the simulated studies varied according to the index tests studies, which could result in systematic bias. In the analysis of Scenario 2, Models 4.i and 5.i correspond to the situation where the researchers knew of the differences in reference standard used across studies and that the variation in reference standard was thus a known source of bias. Models 4.ii and 5.ii correspond to the situation that researchers are unaware of the differences in reference standards among studies and that consequently the variation in reference standard was an unknown source of bias. This may occur if researchers do not provide sufficient detail in the primary publications on the exact modalities of the reference test procedures. For example, in the diagnosis of VL microscopical examination of spleen aspirates is the preferred reference test, while bone marrow aspirates show a limited S. Often researchers indicate their reference test to be based on spleen aspiration. However in closer assessment of these publications, it can become apparent that some researchers perform spleen aspiration on nearly all subjects while others may preform spleen aspiration on only a minority of subjects. Ignoring these difference in reference tests may lead to bias. Incorrectly assuming the reference test were perfect resulted in substantial bias, especially when ignore study level effects (Fig. 2d–f, Model 1). When correcting for the use of imperfect reference tests using LCA (Models 4.i and 5.i), unbiased estimates for the differences in diagnostic accuracy between T1 and T2 were obtained (Fig. 2d–f, Models 4.i and 5.i). If the data were however analyzed ignoring the differences between the reference tests, the differences in diagnostic accuracy between T1 and T2 were overestimated (Fig. 2d–f, Models 4.ii and 5.ii).
Real data example: diagnostic tests for visceral leishmaniasis
Results of modeling of the VL data are in Table 4. All models indicated that S of DAT (S2) was 8 to 11 % higher, compared to the S of RK39 (S1), in East-Africa, but this difference did not reach statistical significance. In the rest of the world, estimates of S1 and S2 were similar. Differing modeling strategies or allowing for imperfect reference standards did not impact estimates of S or comparisons of S between the two index tests of interest. This is as expected as the parasitological reference tests show a similar and high C.
Table 4 Results of the meta-analysis of the diagnostic tests for visceral leishmaniasis
In contrast, allowing for imperfect reference tests (models 4 and 5) resulted in considerably higher estimates for C of both the RK39 dipstick and DAT compared to models assuming perfect reference tests were used (models 1–3). False negative results for the reference tests may have resulted in reduced estimates of C1 (75.5–78.6 %) and of C2 (80.1–81.5 %). Allowing for imperfect reference standards resulted in considerable higher estimates for C1 (90.2–91.0 %) and C2 (93.0–94.1 %).
In the analyses that used parasitology as a, presumed perfect, reference test, a substantial difference between C1 and C2 (\(\hat {C_{2}} - \hat {C_{1}}\) = 5.3 %) was observed when limiting the analysis to direct comparisons only (Model 2). On the other hand, Model 1, based on independent estimation of C1 and C2, showed a much smaller difference (\(\hat {C_{2}} - \hat {C_{1}}\) = 1.5 %). The model using direct and indirect comparisons (Model 3) showed intermediate results (3.2 %). This can be explained by the fact that the studies in which no direct comparison was possible between the RK39 dipstick and DAT showed contradictory results to the studies with direct comparisons. These studies also used the least sensitive reference standard which may explain that results of Models 4 and 5, both allowing for imperfect reference standards, were similar.
In this paper, we developed a novel model to perform a comparative meta-analysis of the accuracy of two or more diagnostic tests when a perfect reference standard is unavailable. In a first step, we assessed the bias of comparative measures of the diagnostic accuracy of two tests induced by the use of an imperfect reference test. We observed that the difference in S and C may be the least subject to bias while at the same time being easily understandable to users of the meta-analysis results. In our modeling approach, we combined LCA with models developed for the mixed treatment comparisons meta-analysis of RCTs. The modeling framework accommodates a broad range of studies, including "Multiple Test Comparison", "Randomized Test Comparison", and "Between-Study Test Comparison" studies according to the terminology of Takwoingi et al., with the first two designs offering the most robust comparative data [3]. In a simulation study, the resulting model showed adequate performance, even if some aspects of the data generating mechanism were ignored. The simulation study also stressed the importance of accurate and complete extraction of the data from the primary studies when performing a DTA review. When differences in reference tests were ignored, biased estimates of the relative accuracy of the competing tests were unavoidable. This highlights the importance of complete and transparent reporting of DTA studies as promoted by the STARD initiative [29]. For a correct analysis of the data, the index and reference tests should be accurately described. Any cut-offs used to classify test results as positive or negative should also be reported and results for all subjects should be given, including subjects with incomplete or equivocal test results. The cross-classification of all test results should be presented in a format similar to that of the motivating example dataset in Additional file 4. The fact that meta-analysis is possible using imperfect reference tests suggests it may be more efficient to design future studies with multiple imperfect tests rather than using a single "as-accurate-as-possible" reference test, as has been shown in the analysis of epidemiological studies with imperfect measures of exposure [30, 31]. When applied to a dataset on visceral leishmaniasis diagnostic tests, the model indicated that C of the two tests of interest may have been underestimated due to the use of imperfect reference test. Our novel modeling approach, combining latent class analysis with hierarchical meta-analysis modeling, allowed the estimation of the difference in accuracy of the two index tests without making strong assumptions on the performance of the reference tests used. However, as in all meta-analyses, care should be taken that the studies combined are in fact comparable. While our approach corrects for bias and heterogeneity induced by the use of imperfect reference tests, other types of bias as publication and spectrum bias, can result in incorrect meta-analysis results. The approach can be combined with meta-regression techniques to reduce heterogeneity.
As limitations of our approach the following points can be given, which can indicate future avenues for further progress in this field. As a first limitation, we chose to compare diagnostic tests based on the sensitivity and specificity, and in particular based on the difference in these quantities among competing tests. Focusing on differences in S and C leads to results which are easily understandable for potential users. However, a test can be superior to another with respect to S while inferior with respect to C. In this case, selecting the optimal test can be difficult. Using a single summary measure of diagnostic accuracy, as the relative diagnostic odds-ratio (rDOR) can make comparisons among tests easier [32]. Theoretically, the test with the highest DOR may be preferred. However, this may not always be the case as the potential risk of a false positive result may be different from the risk of a false negative result. It may be easier for users to balance an increase in S versus decreases in C. In addition, the rDOR may be more prone to bias as we have shown for the OR difference in S and C. In our model formulation, the rDOR can be easily obtained. If the primary parameter of interest is however the rDOR, an alternative model formulation, for example an extension of the hierarchical summary ROC model [9, 33], may be more appropriate.
To allow estimation of the model, we made considerable simplifications to the variance-covariance structure of our parameter space. Not all these simplification may be warranted and a more general variance-covariance structure may refine estimates from this model. Fitting a general variance-covariance matrix however results in important computational difficulties. Our simulation study indicated that these limitations do not necessarily invalidate analysis results, but further research is needed to assess when this may no longer be the case. Modeling the variance-covariance matrix via partial autocorrelations [15] may allow the fitting of more complex model. We accommodated study effects using contrast-based (CB) approaches. However, in RCT arm-based approaches that correctly incorporate correlations have been shown to be superior to CB methods [5]. Further development of the equivalent models for DTA reviews, Models 1 and 4 in our setting, incorporating the correlations induced by study levels, is needed. Model 4 which corrects for imperfect reference tests, but ignores study effects, performed well in our simulation study. However, it is vulnerable to bias from study-specific effects. the model would need extensions to incorporate dependencies between test results from the same study, as for example was done for the meta-analysis of RCTs in Zhang et al. 2014 [5], before it is recommended as a general method for the meta-analysis of DTA studies above Model 5. However, if the accuracy of the different index tests is not strongly correlated across studies, Model 4 may perform equally well as Model 5 and may offer advantages in terms of identifiability and computational feasibility.
We showed how prior information, e.g. on the diagnostic accuracy of the reference test, can be used to aid model estimation in the case of the hierarchical latent class model. This is in line with the methods we have earlier developed for the meta-analysis of the diagnostic accuracy of a single test when using an imperfect reference standard [10]. In the case of the network based latent class model, it is however much less clear how this information can be used. The diagnostic accuracy of each test is in this model a linear combination of an overall, study-specific, probability of testing positive on all tests and a number of contrasts in diagnostic accuracy among these tests. More research is needed on how priors can be constructed for this model, e.g. using the priors for conditional probabilities rather than for S and C directly [26].
DTA studies can be expected to exhibit considerable heterogeneity and may be more prone to bias and inconsistency between direct and indirect comparisons compared to RCTs. Applications of network meta-analytic model to DTA studies must be performed with care and further development of statistical methods are needed. The literature on network-based meta-analysis of RCTs contains many additional tools, for example to assess consistency of estimates obtained from direct versus indirect comparisons [34–36], assess heterogeneity among studies [37], detect outlying studies [38] and correct for bias [39]. We only performed a limited application of techniques developed in this context. Expanding these techniques to DTA meta-analyses may be a valuable direction of research. In particular, it is important to expand the concept of consistency of comparisons across networks to the context of DTA reviews [40, 41].
Alternative approaches to the comparative meta-analysis of diagnostic tests are proposed. The regression approach of Macaskill et al. [9] can be seen as a variation of our model 2 in which the relative S and C, expressed as an odds-ratio, between tests is constant across studies. For the case all tests are applied to all subjects, Trikalinos at al. [42] describe a model which fully accounts for the within-study correlation between the tests' subject-specific S and C. This approach can be more efficient than the methods proposed in the current manuscript. However, both approaches need further empirical and simulation studies to assess their relative merits. Different models may be most appropriate depending on the application. In case it is suspected that reference tests may show only limited S or C, the analysis method should allow for the use of imperfect reference test. If important study-level effects are expected, proper control for confounding by these effects is needed. If there is important uncertainty on the value of the reference test or the presence of study level effects, it will be preferable to fit several models and assess the robustness of the results to the assumptions. At this stage of research, it is not possible to provide a general recommendation on the optimal modeling approach for the meta-analysis of comparative DTA reviews.
The models developed in this paper are promising and can improve the comparison of the diagnostic accuracy of competing tests in DTA systematic review. This is however only true if the studies and especially information on the reference tests used are described in sufficient detail. If the reporting of the studies does not provide sufficient detail, it may be better to limit the meta-analysis to direct comparisons. Further work refining the modeling approach, especially with respect to the specification of more general covariance structures and the use of measures of consistency of direct versus indirect comparisons, can further improve these methods.
Leeflang MMG, Deeks JJ, Takwoingi Y, Macaskill P. Cochrane diagnostic test accuracy reviews. Syst Rev. 2013; 2:82.
Reitsma JB, Glas AS, Rutjes AWS, Scholten RJPM, Bossuyt PM, Zwinderman AH. Bivariate analysis of sensitivity and specificity produces informative summary measures in diagnostic reviews. J Clin Ethics. 2005; 58(10):982–90.
Takwoingi Y, Leeflang MMG, Deeks JJ. Empirical evidence of the importance of comparative studies of diagnostic test accuracy. Ann Intern Med. 2013; 158:544–54.
Caldwell DM, Ades AE, Higgins JPT. Simultaneous comparison of multiple treatments: combining direct and indirect evidence. Br Med J. 2005; 331:897–900.
Zhang J, Carlin BP, Neaton JD, GG GGS, Nie L, Kane R, et al.Network meta-analysis of randomized clinical trials: Reporting the proper summaries. Clin Trials. 2014; 11(2):246–62.
Pepe MS. The Statistical Evaluation of Medical Tests for Classification and Prediction. Oxford (UK): Oxford University Press; 2003.
Zhou XH, Obuchowski NA, McClish DK. Statistical Methods in Diagnostic Medicine. New-York (US): Wiley-Interscience; 2002.
Lesaffre E, Lawson AB. Bayesian Biostatistics (Statistics in Practice). New-York (US): Wiley; 2012.
Macaskill P, Gatsonis C, Deeks J, Harbord R, Takwoingi Y. Cochrane handbook for systematic reviews of diagnostic test accuracy - Chapter 10: Analysing and presenting results In: Deeks JJ, Bossuyt PM, Gatsonis C, editors. Cochrane Handbook for Systematic Reviews of Diagnostic Test Accuracy Version 1.0. London (UK): The Cochrane Collaboration: 2010. p. 1–61.
Menten J, Boelaert M, Lesaffre E. Bayesian meta-analysis of diagnostic tests allowing for imperfect reference standards. Stat Med. 2013; 32(30):5398–413.
Verde PE. Meta-analysis of diagnostic test data: a bivariate Bayesian modeling approach. Stat Med. 2010; 29:3088–102.
Bucher HC, Guyatt GH, Griffith LE, Walter SD. The results of direct and indirect treatment comparisons in meta-analysis of randomized controlled trials. J Clin Epidemiol. 1997; 50(6):683–91.
Smith TC, Spiegelhalter DJ, Thomas SL. Bayesian approaches to random-effects meta-analysis: a comparative study. Stat Med. 1995; 14:2685–699.
Lu G, Aedes AE. Combination of direct and indirect evidende in mixed treatment comparisons. Stat Med. 2004; 23:3105–124.
Daniels MJ, Pourahmadi M. Modeling covariance matrices via partial autocorrelations. J Multivar Anal. 2009; 100(10):2352–363.
McCutcheon AL. Latent Class Analysis. Quantitative Applications in the Social Sciences Series No. 64. Thousand Oaks, US: Sage Publications; 1987.
Dendukuri N, Joseph L. Bayesian approaches to modeling the conditional dependence between multiple tests. Biometrics. 2001; 57:158–67.
Qu Y, Tan M, Kutner MH. Random effects models in latent class analysis for evaluating accuracy of diagnostic tests. Biometrics. 1996; 52:797–810.
Qu Y, Hadgu A. A model for evaluating sensitivity and specificity for correlated diagnostic tests in efficacy studies with an imperfect reference test. J Am Stat Assoc. 1998; 93:920–8.
Menten J, Boelaert M, Lesaffre E. Bayesian latent class models with conditionally dependent diagnostic tests: a case study. Stat Med. 2008; 27(22):4469–488.
Dendukuri N, Hadgu A, Wang L. Modeling conditional dependence between diagnostic tests: A multiple latent variable model. Stat Med. 2009; 28:441–61.
Gelman A, Rubin DB. Inference from iterative simulation using multiple sequences. Stat Sci. 1992; 7:457–72.
Agresti A, Hitchcock DB. Bayesian Inference for Categorical Data Analysis: A Survey. 2005.
A. Jasra CCH, Stephens DA. Markov chain monte carlo methods and the label switching problem in bayesian mixture modeling. Stat Sci. 2005; 20(2):50–67.
Stephens M. Dealing with label switching in mixture models. J R Stat Soc Ser B Stat Methodol. 2000; 62(1):795–809.
Berkvens D, Speybroeck N, Praet N, Adel A, Lesaffre E. Estimating disease prevalence in a Bayesian framework using probabilistic constraints. Epidemiology. 2006; 17(2):145–53.
Boelaert M, Chappuis F, Menten J, van Griensven J, Sunyoto T, Rijal S. Rapid diagnostic tests for visceral leishmaniasis. Cochrane Database Syst Rev. 2011; 6.
Chappuis F, Rijal S, Soto A, Menten J, Boelaert M. A meta-analysis of the diagnostic performance of the direct agglutination test and rk39 dipstick for visceral leishmaniasis. Br Med J. 2006; 333(7571):723–6.
Bossuyt PM, Reitsma JB, Bruns DE, Gatsonis CA, Glasziou PP, Irwig LM, et al.Towards complete and,accurate reporting of studies of diagnostic accuracy: the stard initiative. Br Med J. 2003; 326(7379):41–4.
Chu H, Cole SR, Wei Y, Ibrahim JG. Estimation and inference for case-control studies with multiple non-gold standard exposure assessments: with an occupational health application. Biostatistics. 2009; 10:591–602.
Zhang J, Cole SR, Richardson DB, Chu H. A bayesian approach to strengthen inference for case-control studies with multiple error-prone exposure assessments. Stat Med. 2013; 32(25):4426–437.
Glas AS, Lijmer JG, Prins MH, Bonsel GJ, Bossuyt PM. The diagnostic odds ratio: a single indicator of test performance. J Clin Epidemiol. 2003; 56:1129–1135.
Rutter CM, Gatsonis CA. A hierarchical regression approach to meta-analysis of diagnostic test accuracy evaluations. Stat Med. 2001; 20(19):2865–884.
Lu G, Aedes AE. Assessing evidence inconsistency in mixed treatment comparisons. J Am Stat Assoc. 2006; 101(474):447–59.
Caldwell DM, Welton NJ, Ades AE. Mixed treatment comparison analysis provides internally coherent treatment effect estimates based on overviews of reviews and can reveal inconsistency. J Clin Epidemiol. 2010; 63:875–82.
Salanti G, Higgins JPT, Ades AE, Ioannidis JPA. Evaluation of networks of randomized trials. Stat Methods Med Res. 2008; 17:279–301.
Cooper NJ, Sutton AJ, Morris D, Ades AE, Welton NJ. Addressing between-study heterogeneity and inconsistency in mixed treatment comparisons: Application to stroke prevention treatments in individuals with non-rheumatic atrial fibrillation. Stat Med. 2009; 28:1861–1881.
Zhang J, Fu H, Carlin BP. Detecting outlying trials in network meta-analysis. Stat Med. 2015; 34(Epub ahead of print):1–3.
Dias S, Welton NJ. Estimation and adjustment of bias in randomized evidence by using mixed treatment comparison meta-analysis. J R Stat Soc Ser A. 2010; 176(3):613–29.
Higgins JPT, Jackson D, Barrett JK, Lu G, Ades AE, White IR. Consistency and inconsistency in network meta-analysis: concepts and models for multi-arm studies. Res Synth Meth. 2012; 3:98–110.
White IR, Barrett JK, Jackson D, Higgins JPT. Consistency and inconsistency in network meta-analysis: model estimation using multivariate meta-regression. Res Synth Meth. 2012; 3:111–25.
Trikalinos TA, Hoaglin DC, Small KM, Terrin N, Schmid CH. Methods for the joint meta-analysis of multiple tests. Res Synth Meth. 2014; 5:294–312.
The works was supported by the Department of Economy, Science and Innovation of the Flemish Government. JM thanks Marleen Boelaert for her support and advice.
Clinical Trials Unit, Institute of Tropical Medicine, Nationalestraat 155, Antwerp, B-2000, Belgium
Joris Menten
L-Biostat, KULeuven University of Leuven, Kapucijnenvoer 35, Leuven, B-3000, Belgium
Joris Menten & Emmanuel Lesaffre
Emmanuel Lesaffre
Correspondence to Joris Menten.
JM conceived research questions, developed study design and methods, carried out statistical analysis, interpreted results and drafted the manuscript. EL advised on study design, methods, statistical analysis and commented on successive drafts. Both authors read and approved the final manuscript.
Additional file 1
Simulation Study for the Selection of Appropriate Statistics for a Comparative DTA Review. (PDF 1208 KB)
Software Code, Prior Specification, and Possible Structures for Variance-Covariance Matrices. (PDF 149 KB)
Simulation Study of the Modeling Approach. (PDF 491 KB)
Visceral Leishmaniasis Data. (PDF 44 KB)
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Menten, J., Lesaffre, E. A general framework for comparative Bayesian meta-analysis of diagnostic studies. BMC Med Res Methodol 15, 70 (2015). https://doi.org/10.1186/s12874-015-0061-7
Meta-analyses
Diagnostic test accuracy
Latent class model | CommonCrawl |
Deltoid curve
In geometry, a deltoid curve, also known as a tricuspoid curve or Steiner curve, is a hypocycloid of three cusps. In other words, it is the roulette created by a point on the circumference of a circle as it rolls without slipping along the inside of a circle with three or one-and-a-half times its radius. It is named after the capital Greek letter delta (Δ) which it resembles.
More broadly, a deltoid can refer to any closed figure with three vertices connected by curves that are concave to the exterior, making the interior points a non-convex set.[1]
Equations
A hypocycloid can be represented (up to rotation and translation) by the following parametric equations
$x=(b-a)\cos(t)+a\cos \left({\frac {b-a}{a}}t\right)\,$
$y=(b-a)\sin(t)-a\sin \left({\frac {b-a}{a}}t\right)\,,$
where a is the radius of the rolling circle, b is the radius of the circle within which the aforementioned circle is rolling and t ranges from zero to 6π. (In the illustration above b = 3a tracing the deltoid.)
In complex coordinates this becomes
$z=2ae^{it}+ae^{-2it}$.
The variable t can be eliminated from these equations to give the Cartesian equation
$(x^{2}+y^{2})^{2}+18a^{2}(x^{2}+y^{2})-27a^{4}=8a(x^{3}-3xy^{2}),\,$
so the deltoid is a plane algebraic curve of degree four. In polar coordinates this becomes
$r^{4}+18a^{2}r^{2}-27a^{4}=8ar^{3}\cos 3\theta \,.$
The curve has three singularities, cusps corresponding to $t=0,\,\pm {\tfrac {2\pi }{3}}$. The parameterization above implies that the curve is rational which implies it has genus zero.
A line segment can slide with each end on the deltoid and remain tangent to the deltoid. The point of tangency travels around the deltoid twice while each end travels around it once.
The dual curve of the deltoid is
$x^{3}-x^{2}-(3x+1)y^{2}=0,\,$
which has a double point at the origin which can be made visible for plotting by an imaginary rotation y ↦ iy, giving the curve
$x^{3}-x^{2}+(3x+1)y^{2}=0\,$
with a double point at the origin of the real plane.
Area and perimeter
The area of the deltoid is $2\pi a^{2}$ where again a is the radius of the rolling circle; thus the area of the deltoid is twice that of the rolling circle.[2]
The perimeter (total arc length) of the deltoid is 16a.[2]
History
Ordinary cycloids were studied by Galileo Galilei and Marin Mersenne as early as 1599 but cycloidal curves were first conceived by Ole Rømer in 1674 while studying the best form for gear teeth. Leonhard Euler claims first consideration of the actual deltoid in 1745 in connection with an optical problem.
Applications
Deltoids arise in several fields of mathematics. For instance:
• The set of complex eigenvalues of unistochastic matrices of order three forms a deltoid.
• A cross-section of the set of unistochastic matrices of order three forms a deltoid.
• The set of possible traces of unitary matrices belonging to the group SU(3) forms a deltoid.
• The intersection of two deltoids parametrizes a family of complex Hadamard matrices of order six.
• The set of all Simson lines of given triangle, form an envelope in the shape of a deltoid. This is known as the Steiner deltoid or Steiner's hypocycloid after Jakob Steiner who described the shape and symmetry of the curve in 1856.[3]
• The envelope of the area bisectors of a triangle is a deltoid (in the broader sense defined above) with vertices at the midpoints of the medians. The sides of the deltoid are arcs of hyperbolas that are asymptotic to the triangle's sides.[4]
• A deltoid was proposed as a solution to the Kakeya needle problem.
See also
• Astroid, a curve with four cusps
• Circular horn triangle, a three-cusped curve formed from circular arcs
• Ideal triangle, a three-cusped curve formed from hyperbolic lines
• Pseudotriangle, a three-pointed region between three tangent convex sets
• Tusi couple, a two-cusped roulette
• Kite (geometry), also called a deltoid
References
1. "Area bisectors of a triangle". www.se16.info. Retrieved 26 October 2017.
2. Weisstein, Eric W. "Deltoid." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Deltoid.html
3. Lockwood
4. Dunn, J. A., and Pretty, J. A., "Halving a triangle," Mathematical Gazette 56, May 1972, 105-108.
• E. H. Lockwood (1961). "Chapter 8: The Deltoid". A Book of Curves. Cambridge University Press.
• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 131–134. ISBN 0-486-60288-5.
• Wells D (1991). The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. pp. 52. ISBN 0-14-011813-6.
• "Tricuspoid" at MacTutor's Famous Curves Index
• "Deltoid" at MathCurve
• Sokolov, D.D. (2001) [1994], "Steiner curve", Encyclopedia of Mathematics, EMS Press
| Wikipedia |
\begin{document}
\noindent
\preprint{}
\title{Quantum fluctuations as deviation from classical dynamics of ensemble of trajectories parameterized by unbiased hidden random variable}
\author{Agung Budiyono}
\affiliation{Institute for the Physical and Chemical Research, RIKEN, 2-1 Hirosawa, Wako-shi, Saitama 351-0198, Japan}
\date{\today}
\begin{abstract}
A quantization method based on replacement of c-number by c-number parameterized by an unbiased hidden random variable is developed. In contrast to canonical quantization, the replacement has straightforward physical interpretation as statistical modification of classical dynamics of ensemble of trajectories, and implies a unique operator ordering. We then apply the method to develop quantum measurement without wave function collapse and external observer \`a la pilot-wave theory.
\end{abstract}
\pacs{03.65.Ta, 03.65.Ca}
\keywords{Quantization; Hidden variable model; Schr\"odinger equation; Measurement}
\maketitle
\section{Motivation}
The present paper discusses three closely interrelated aspects of quantum mechanics: canonical quantization of classical system, quantum-classical correspondence and measurement problem. The discussion will be confined to system of non-relativistic particles with no spin. It is well known that even in this case, despite the astonishing pragmatical successes of quantum mechanics, its foundation with regard to the above three aspects, is not without ambiguity \cite{Isham book}.
Let us denote the position of the particles as $q=\{q_i\}$ and the corresponding conjugate momentum as $\underline{p}=\{\underline{p}_i\}$ where $i$ goes for all degree of freedom. In this paper all symbols with \underline{underline} is used to denote quantities satisfying the law of classical mechanics. In canonical quantization, given a classical quantity $F(q,\underline{p})$, then the so-called ``quantum observable'' is obtained by promoting $q$ and $\underline{p}$ to Hermitian operators $q\mapsto\hat{q}$ and $\underline{p}\mapsto\hat{p}$, and replacing the Poisson bracket by commutator $\{\cdot,\cdot\}\mapsto [\cdot,\cdot]/(i\hbar)$. This replacement of c-number (classical number) by q-number (quantum number/Hermitian operator) consequently does not in general give a unique Hermitian operator due to operator ordering ambiguity. An even deeper difficulty is lying at the conceptual level for while classical mechanics is developed using the basic notion of conventional trajectory, the resulting quantum mechanics does not refer to the notion of trajectory, except in the so-called pilot-wave interpretation \cite{Bohm paper} to be discussed later. A closely related question is that despite the fact that Planck constant plays a pivotal role in connecting the quantum and classical mechanics through quantization and classical limit, its physical origin is not clear. Elaborating this issue might open the way to discuss the limitation of quantum mechanics.
On the other hand, in its standard interpretation, quantum mechanics is based on two different processes: unitary, continuous and deterministic evolution described by the Schr\"odinger equation when there is no measurement; and non-unitary, discontinuous and non-causal (random) process of wave function collapse in measurement \cite{measurement problem}. For general model of measurement, the first process alone will give a superposition of macroscopically distinct states, which in the standard interpretation leads to the so-called paradox of Schr\"odinger's cat. It is then assumed that measurement reveals, randomly, only one of the term in the superposition. Accordingly, the second process mentioned above is needed. Moreover, this interpretation assumes an apparatus which must behave according to classical mechanics. The whole system must then be divided into quantum system being measured and classical apparatus of measurement. It is however well-known that such line of division can be made anywhere, thus is ambiguous. Further, since in general quantum measurement does not reveal the pre-existing value prior measurement, then there is a question whether another apparatus is needed to probe the record of the first apparatus, which immediately leads to infinite regression. In the context of quantum-classical correspondence, one can thus ask why classical mechanics does not suffer from measurement problems mentioned above, and how the probability of finding in quantum measurement becomes the probability of being in classical measurement.
Below we shall attempt to propose a solution to the above problems. Our basic idea is to first understand the physical meaning behind the formal rule of canonical quantization. We shall develop a quantization method by directly modifying classical dynamics of ensemble of trajectories parameterized by an unbiased hidden random variable. We shall show that given the classical Hamiltonians, the resulting equations for important class of physical systems can be rewritten into the Schr\"odinger equation with unique quantum Hamiltonians. The method is based on replacement of c-number by c-number, thus is free from operator ordering ambiguity. A couple examples where canonical quantization is ambiguous will be given. We shall further show that in all the cases to be considered, the particles posses effective velocity equal to the velocity of the particles in the pilot-wave theory and the Born's interpretation of wave function is valid for all time by construction. This allows us to describe quantum measurement without wave function collapse and external (classical) observer a la pilot-wave theory. However unlike the latter, our model is inherently stochastic and the wave function is not physically real.
\section{Modification of classical dynamics of ensemble of trajectories using hidden random variable\label{modification of classical ensemble}}
Let us consider the dynamics of $N$ particles system whose classical Hamiltonian is denoted by $\underline{H}(q,\underline{p};t)$. The classical dynamics of the particles is then given by the following Hamilton-Jacobi equation: \begin{equation} \partial_t\underline{S}(q;t)+\underline{H}(q,\partial_q\underline{S}(q;t);t)=0, \label{H-J equation} \end{equation} where $\underline{S}(q;t)$ is the Hamilton principle function (HPF) so that the momentum field is given by $\underline{p}=\partial_q\underline{S}$ where $\partial_q=\{\partial_{q_i}\}$ \cite{Rund book}. Hamilton-Jacobi equation thus describes the dynamics of ensemble (congruence) of trajectories in configuration space. To solve this equation, one needs to set up an initial HPF $\underline{S}(q;0)$ which implies an initial classical momentum field $\underline{p}(q;0)=\partial_q\underline{S}(q;0)$. A single trajectory in configuration space is picked up if one also fixes the initial position of the particles.
Let us consider an ensemble of classical systems so that the position of the particles are initially distributed in configuration space with probability density $\underline{\rho}(q;0)$, $\int dq\underline{\rho}(q;0)=1$. The probability density of the configuration of the particles at any time $\underline{\rho}(q;t)$ then satisfies the following continuity equation: \begin{equation} \partial_t\underline{\rho}+\partial_{q}\cdot(\underline{v}(\underline{S})\underline{\rho})=0, \label{continuity equation} \end{equation} where $\underline{v}=\{{\underline{v}}_i\}$ is the classical velocity field. In the above equation, we have made explicit that in general, the classical velocity field $\underline{v}$ might depend on the HPF $\underline{S}$. Given a classical Hamiltonian, this relation can be obtained through (the Legendre transformation part of) the Hamilton equation: \begin{equation}
{\underline{v}}_i=\frac{\partial \underline{H}}{\partial {\underline{p}}_i}\Big|_{\{{\underline{p}}_i=\partial_{q_i}\underline{S}\}}=f_i(\underline{S}), \label{classical velocity field} \end{equation} where $f_i$, $i=1,\dots,N$ are some functions \cite{constrained motion}. The dynamics and statistics of the ensemble of classical trajectories are then obtained by solving Eqs. (\ref{H-J equation}), (\ref{continuity equation}) and (\ref{classical velocity field}) in term of $\underline{S}(q;t)$, $\underline{\rho}(q;t)$ and $\underline{v}(q;t)$.
Let us then proceed to develop a general scheme to modify the above dynamics of ensemble of classical trajectories. To do this, let us introduce a pair of real-valued functions, $S(q,\lambda;t)$ and $\Omega(q,\lambda;t)$, assumed to take over the role of $\underline{S}(q;t)$ and $\underline{\rho}(q;t)$, respectively, in the modified dynamics. Here $\lambda$ is a hidden random variable of action dimensional whose statistical properties will be specified later. Hence $\Omega(q,\lambda;t)$ is the joint-probability density that the particles are at configuration coordinate $q$ and the value of the hidden variable is $\lambda$. The marginal probability densities of the fluctuations of $q$ and $\lambda$ are thus given by \begin{eqnarray} \rho(q;t)\doteq\int d\lambda\Omega(q,\lambda;t),\nonumber\\ P(\lambda)\doteq\int dq\Omega(q,\lambda;t), \end{eqnarray} where we have assumed that the statistics of $\lambda$ is independent of time.
Let us then propose the following rule of replacement to modify the classical dynamics of ensemble of trajectories governed by Eqs. (\ref{H-J equation}) and (\ref{continuity equation}): \begin{eqnarray} \underline{\rho}\mapsto\Omega,\hspace{25mm}\nonumber\\ \partial_{q_i}\underline{S}\mapsto\partial_{q_i}S+\frac{\lambda}{2}\frac{\partial_{q_i}\Omega}{\Omega}, \hspace {2mm}i=1,..,N,\nonumber\\ \partial_{t}\underline{S}\mapsto\partial_tS+\frac{\lambda}{2}\frac{\partial_t\Omega}{\Omega}+\frac{\lambda}{2}\partial_q\cdot f(S),\hspace{2mm} \label{fundamental equation general} \end{eqnarray} where the functional form of $f=\{f_i\}$ is determined by the classical Hamiltonian according to Eq. (\ref{classical velocity field}).
Let us first show that the replacement of Eq. (\ref{fundamental equation}) possesses a consistent classical correspondence if $S\rightarrow\underline{S}$ so that the Hamilton-Jacobi equation of (\ref{H-J equation}) is restored (notice that we have used the symbol ``$\mapsto$'' to denote replacement and ``$\rightarrow$'' to denote a limit). First, using the last two equations of (\ref{fundamental equation}), for sufficiently small $\Delta t$ and $\Delta q=\{\Delta q_i\}$, then expanding $\Delta F\doteq F(q+\Delta q;t+\Delta t)-F(q;t)\approx\partial_tF\Delta t+\partial_qF\cdot\Delta q$, for any function $F$, one has \begin{equation} \Delta\underline{S}\mapsto\Delta S+\frac{\lambda}{2}\Big(\frac{\Delta\Omega}{\Omega}+\partial_q\cdot f(S)\Delta t\Big). \label{statistical violation of HPSA} \end{equation} One can see that in the limit $S\rightarrow\underline{S}$, in order to be consistent then the second term on the right hand side has to be vanishing. One thus has $d\Omega/dt=-\Omega\partial_q\cdot\underline{v}$, by Eq. (\ref{classical velocity field}). This is just the continuity equation of (\ref{continuity equation}). Hence, since $\underline{v}$ is independent of $\lambda$, in the limit $S\rightarrow\underline{S}$, one has $\rho=\int d\lambda\Omega\rightarrow\underline{\rho}$. We have thus a smooth classical correspondence.
The next question is then what is the statistical properties of $\lambda$. We shall show in the next section that to reproduce the prediction of quantum mechanics, one needs to assume that the probability density function of $\lambda$ is given by \begin{equation} P(\lambda)=\frac{1}{2}\delta(\lambda-\hbar)+\frac{1}{2}\delta(\lambda+\hbar), \label{Schroedinger condition} \end{equation} where $\hbar$ is the reduced Planck constant. Namely, $\lambda$ can only take binary values $\lambda=\pm\hbar$ with equal probability.
What we shall do in the following sections is as follows. First, given a classical Hamiltonian, we shall generate the classical dynamics of ensemble of trajectories according to Eqs. (\ref{H-J equation}), (\ref{continuity equation}) and (\ref{classical velocity field}). We then proceed to modify Eqs. (\ref{H-J equation}) and (\ref{continuity equation}) by imposing Eq. (\ref{fundamental equation general}). Averaging over the distribution of $\lambda$ and taking into account Eq. (\ref{Schroedinger condition}), we shall show that, for a class of important physical systems, the resulting equations can be put into the Schr\"odinger equation with a unique Hermitian quantum Hamiltonian. Below we shall assume that the fluctuations of $q$ and $\lambda$ are separable $\Omega(q,\lambda;t)=\rho(q;t)P(\lambda)$. Accordingly, Eq. (\ref{fundamental equation general}) becomes \begin{eqnarray} \underline{\rho}\mapsto \rho P(\lambda),\hspace{25mm}\nonumber\\ \partial_{q_i}\underline{S}\mapsto\partial_{q_i}S+\frac{\lambda}{2}\frac{\partial_{q_i}\rho}{\rho}, \hspace {2mm}i=1,..,N,\nonumber\\ \partial_{t}\underline{S}\mapsto\partial_tS+\frac{\lambda}{2}\frac{\partial_t\rho}{\rho}+\frac{\lambda}{2}\partial_q\cdot f(S).\hspace{2mm} \label{fundamental equation} \end{eqnarray}
Let us note before proceeding that in the present paper we shall not discuss the issue of nonlocality \cite{Bell unspeakable}. For a review of the progress of hidden variable models in view of Bell nonlocality, see \cite{Genovese review HVM,Santos rebute}. Yet since we will claim that our model reproduces the prediction of quantum mechanics then it must violate Bell inequality. See however \cite{Nieuwenhuizen contextuality loophole} for an interesting discussion that the violation of Bell inequality does not necessarily lead to nonlocality due to the contextuality loophole.
\section{Particle in external potential\label{particle in external potential}}
\subsection{Emergent deterministic Schr\"odinger equation \label{emergent Schroedinger equation EM}}
Let us apply the above modification of classical mechanics to an ensemble of particle subjected to external potentials. For simplicity, let us consider the case of single particle with mass $m$. As will be seen, generalization to many particles is straightforward. The classical Hamiltonian is thus given by \begin{eqnarray} \underline{H}=\frac{\big(\underline{p}-(e/c)A\big)^2}{2m}+eV, \label{classical Hamiltonian EM} \end{eqnarray} where $e$ is charge of the particle, $c$ is the velocity of light, $A(q;t)$ and $V(q;t)$ are the vector and scalar electromagnetic potentials, respectively. The Hamilton-Jacobi equation of (\ref{H-J equation}) thus reads \begin{equation} \partial_t\underline{S}+\frac{\big(\partial_q\underline{S}-(e/c)A\big)^2}{2m}+eV=0. \label{H-J equation EM} \end{equation} On the other hand, inserting Eq. (\ref{classical Hamiltonian EM}) into Eq. (\ref{classical velocity field}), the classical velocity field is related to $\underline{S}$ as \begin{equation} \underline{v}=\big(\partial_q\underline{S}-(e/c)A\big)/m. \label{classical velocity field EM} \end{equation} The continuity equation of (\ref{continuity equation}) thus becomes \begin{equation} \partial_t\underline{\rho}+\frac{1}{m}\partial_q\cdot\Big(\big(\partial_q\underline{S}-(e/c)A\big)\underline{\rho}\Big)=0. \label{continuity equation EM} \end{equation} Hence, the dynamics and statistics of classical ensemble of trajectories is governed by Eqs. (\ref{H-J equation EM}), (\ref{classical velocity field EM}) and (\ref{continuity equation EM}).
Next, from Eq. (\ref{classical velocity field EM}) and the definition of $f$ given in Eq. (\ref{classical velocity field}), its functional form with respect to $\underline{S}$ is given by \begin{equation} f(\underline{S})=(1/m)\big(\partial_q\underline{S}-(e/c)A\big). \label{magician EM} \end{equation} Equation (\ref{fundamental equation}) then becomes \begin{eqnarray} \underline{\rho}\mapsto\rho P(\lambda),\hspace{37mm}\nonumber\\ \partial_q\underline{S}\mapsto\partial_qS+\frac{\lambda}{2}\frac{\partial_q\rho}{\rho},\hspace{30mm}\nonumber\\ \partial_t\underline{S}\mapsto\partial_tS+\frac{\lambda}{2}\frac{\partial_t\rho}{\rho}+\frac{\lambda}{2m}\partial_q\cdot\big(\partial_qS-(e/c)A\big). \label{fundamental equation EM} \end{eqnarray}
Let us investigate how the above set of equations modify Eqs. (\ref{H-J equation EM}) and (\ref{continuity equation EM}). First, imposing the first two equations of (\ref{fundamental equation EM}) into Eq. (\ref{continuity equation EM}) one gets \begin{eqnarray} \partial_t\rho+\frac{1}{m}\partial_q\cdot\Big(\rho\big(\partial_qS-(e/c)A\big)\Big)+\frac{\lambda}{2m}\partial_q^2\rho=0, \label{FPE EM} \end{eqnarray} where $\partial_q^2=\partial_q\cdot\partial_q$ and since $P(\lambda)$ is independent of time and space, it can be divided out. On the other hand, imposing the last two equations of (\ref{fundamental equation EM}) into Eq. (\ref{H-J equation EM}), one obtains \begin{eqnarray} \partial_tS+\frac{\big(\partial_qS-(e/c)A\big)^2}{2m}+eV-\frac{\lambda^2}{2m}\frac{\partial_q^2R}{R}\hspace{20mm}\nonumber\\ +\frac{\lambda}{2\rho}\Big(\partial_t\rho+\frac{1}{m}\partial_q\cdot\Big(\rho\big(\partial_qS-(e/c)A\big)\Big)+\frac{\lambda}{2m}\partial_q^2\rho\Big)=0, \label{pre H-J-M for particle in electromagnetic field 1} \end{eqnarray} where we have defined $R\doteq\sqrt{\rho}$, and used the following identity: \begin{equation} \frac{1}{4}\frac{\partial_{q_i}\rho\partial_{q_j}\rho}{\rho^2}=\frac{1}{2}\frac{\partial_{q_i}\partial_{q_j}\rho}{\rho}-\frac{\partial_{q_i}\partial_{q_j}R}{R}, \label{fluctuation decomposition} \end{equation} for the case of $i=j$. Inserting Eq. (\ref{FPE EM}) into Eq. (\ref{pre H-J-M for particle in electromagnetic field 1}), one has \begin{eqnarray} \partial_tS+\frac{\big(\partial_qS-(e/c)A\big)^2}{2m}+eV-\frac{\lambda^2}{2m}\frac{\partial_q^2R}{R}=0. \label{HJM EM} \end{eqnarray} We have thus pair of coupled equations (\ref{FPE EM}) and (\ref{HJM EM}) which still depends on the random variable $\lambda$.
We shall proceed to take average of Eqs. (\ref{FPE EM}) and (\ref{HJM EM}) over the distribution of $\lambda$. First, from Eq. (\ref{HJM EM}), since $R$ is independent of $\lambda$, one can see that $S(q,\lambda;t)$ and $S(q,-\lambda;t)$ satisfy the same differential equation. Assuming that initially $S$ possesses the following symmetry $S(q,\lambda;0)=S(q,-\lambda;0)$, the symmetry is then preserved for all the time \begin{equation} S(q,\lambda;t)=S(q,-\lambda;t). \label{phase symmetry} \end{equation} Let us now assume a specific form of $P(\lambda)$ given by Eq. (\ref{Schroedinger condition}): $ P(\lambda)=(1/2)\delta(\lambda-\hbar)+(1/2)\delta(\lambda+\hbar)$. Then averaging Eqs. (\ref{FPE EM}) and (\ref{HJM EM}) over the distribution of $\lambda$ and defining the following function: \begin{eqnarray} S_Q(q;t)\doteq\frac{1}{2}\Big(S(q,\hbar;t)+S(q,-\hbar;t)\Big)\nonumber\\ =S(q,\hbar;t)=S(q,-\hbar;t), \label{quantum phase} \end{eqnarray} which is valid by Eq. (\ref{phase symmetry}), one obtains the following pair of equations: \begin{eqnarray} \partial_t\rho+\frac{1}{m}\partial_q\cdot\Big(\rho\big(\partial_qS_Q-(e/c)A\big)\Big)=0,\hspace{3mm}\nonumber\\ \partial_tS_Q+\frac{\big(\partial_qS_Q-(e/c)A\big)^2}{2m}+eV-\frac{\hbar^2}{2m}\frac{\partial_q^2R}{R}=0. \label{Madelung equation EM} \end{eqnarray}
Let us further define the following complex-valued function: \begin{equation} \Psi_Q(q;t)\doteq R\exp(iS_Q/\hbar), \label{Schroedinger wave function} \end{equation} so that the probability density of the position of the particle is given by \begin{equation}
\rho(q;t)=|\Psi_Q(q;t)|^2. \label{Born's rule} \end{equation} The pair of equations in (\ref{Madelung equation EM}) can then be recast into \begin{equation} i\hbar\partial_t\Psi_Q=\frac{1}{2m}\big(-i\hbar\partial_q-(e/c)A\big)^2\Psi_Q+eV\Psi_Q. \label{Schroedinger equation EM} \end{equation} This is just the Schr\"odinger equation for a particle subjected to vector and scalar potentials $A(q;t)$ and $V(q;t)$ with the corresponding quantum Hamiltonian given by \begin{equation} {\hat H}=\frac{1}{2m}\big(-i\hbar\partial_q-(e/c)A\big)^2+eV. \label{quantum Hamiltonian EM} \end{equation} One can also see from Eq. (\ref{Born's rule}) that the Born's interpretation of wave function is valid by construction.
Further, from the lower equation of (\ref{Madelung equation EM}), it is straightforward to show that \begin{eqnarray} \int dq d\lambda\rho(q;t)P(\lambda)(-\partial_tS)=\int dq\rho(q;t)(-\partial_tS_Q)\hspace{5mm}\nonumber\\ =\int dq\Psi_Q^*\Big(\frac{(-i\hbar\partial_q-(e/c)A)^2}{2m}+eV\Big)\Psi_Q. \label{quantum mechanical average energy} \end{eqnarray} The right hand side is just the quantum mechanical average energy which is conserved by the Schr\"odinger equation of (\ref{Schroedinger equation EM}). Hence, one should interpret $E\doteq-\partial_tS_Q$ as the effective energy of the particle.
Note that while the Schr\"odinger equation is obtained when $\lambda$ can take only binary values $\pm\hbar$, $\lambda$ may be a function of a set of continuous random variables, say $\lambda=\lambda(\nu_1,\nu_2,\dots)$. For example, one may have $\lambda=\sqrt{\nu_1^2+\nu_2^2+\nu_3^2}=\pm\hbar$ where $\nu_i$, $i=1,2,3$, take continuous real value. One thus has to solve \begin{equation} \nu_1^2+\nu_2^2+\nu_3^2=\hbar^2, \label{hidden ball} \end{equation} namely the point $\nu=\{\nu_1,\nu_2,\nu_3\}$ lies on the surface of a ball of radius $\hbar$. Now let us divide the surface of the ball into two with equal area and attribute to each division with $\pm$ signs. Then, if the point $\nu$ is distributed uniformly on the surface of the ball (say it moves sufficiently chaotic on the surface), the resulting $\lambda$ will satisfy Eq. (\ref{Schroedinger condition}).
The time-dependent Schr\"odinger equation is thus obtained through specific choice of $P(\lambda)$ given by Eq. (\ref{Schroedinger condition}). This result suggests that generalization of Schr\"odinger equation might be attained by allowing the probability density function of $\lambda$ to deviate from Eq. (\ref{Schroedinger condition}). This might further lead to possible correction to the prediction of quantum mechanics \cite{AgungDQM2}.
Let us mention here that there are many approaches reported in the literature to derive the Schr\"odinger equation with quantum Hamiltonian of the type given in Eq. (\ref{quantum Hamiltonian EM}) \cite{Feynman PI,Nelson stochastic mechanics,de la Pena stochastic mechanics,Santamato,Frieden-Fisher,Nagasawa transformation,Garbaczewski-Vigier,Nottale,Reginatto-Fisher,Olavo,Kaniadakis,Fritsche,Markopoulou,Parwani: information measure,Smolin,Santos.SE,Brenig invariant uncertainty principle,Groessing,Rusov,Gog,de la Pena SED}. The advantage of our derivation is three folds. First, it is derived in the scheme of quantizing a general classical Hamiltonian. Hence, taking aside that the solution exists, the method can be applied directly to other class of classical Hamiltonians. Second, it is derived by modifying the classical dynamics of ensemble of trajectories so that the quantum-classical correspondence is conceptually kept transparent. In particular, we have no problem of conceptual jump in quantum-classical transition from a quantum theory which does not refer to conventional notion of trajectories to a classical theory which is founded based on the notion of trajectories. The classical limit of Schr\"odinger equation is given by the dynamics of ensemble of classical trajectories. Finally, the Schr\"odinger equation is shown to correspond to a specific distribution of hidden random variable. Hence, it hints to a straightforward generalization \cite{AgungDQM2}.
Let us note before proceeding that the hidden variable $\lambda$ is not the property of a single particle. Rather, it suggests the existence of background field which pervades all space whose detail interaction with the particle is not known, resulting in the stochastic motion of the particle. The presence of background field is also assumed in Nelson's stochastic mechanics \cite{Nelson stochastic mechanics} and stochastic electrodynamics \cite{de la Pena SED} approaches to explain the origin of quantum fluctuations. Next, as shown above, to get the correct time evolution, we have to first calculate the solutions (in the form of differential equation parameterized by the hidden variable) and then take average over the distribution of the hidden variable. The converse will lead to wrong time evolution. The situation is more like random walk. Namely, one has to evolve the walker (the particle) using the random step to obtain the correct time evolution, and then do the averaging over the probability of each single step. Taking the average of the single step in random walk will lead to trivial motion. In this case, to be meaningful, the fluctuations of the hidden variable $\lambda$ has to be much faster than the fluctuations of $q$.
\subsection{Effective velocity and pilot-wave theory}
First, the upper equation of (\ref{Madelung equation EM}) can be regarded as a generalized continuity equation so that one can read off an effective velocity field of the ensemble of particle which is given by \begin{equation} v(S_Q)\doteq\frac{1}{m}\big(\partial_qS_Q-(e/c)A\big)=f(S_Q), \label{effective velocity EM} \end{equation} where in the last equality we have used Eq. (\ref{magician EM}). On the other hand, if $\lambda$ satisfies Eq. (\ref{Schroedinger condition}), then as shown in Eq. (\ref{Schroedinger wave function}), $S_Q$ is just the phase of Schr\"odinger wave function $\Psi_Q$. Hence, in this case, the numerical value of the effective velocity of the particles is equal to the actual velocity of the particle in pilot-wave theory \cite{Bohm paper}. One can also see from Eq. (\ref{effective velocity EM}) that $\partial_qS_Q$ should be interpreted as the effective momentum field of ensemble of the particle. We have thus an effectively similar picture with pilot-wave theory in the sense that the particle always possesses definite position and momentum and it moves ``as if'' it is guided by the wave function so that the effective velocity is given by Eq. (\ref{effective velocity EM}).
Our model however differs from pilot-wave theory as follows. The latter is based on the assumption that: (a) for any dynamical system, the Schr\"odinger equation and the corresponding guidance relation are postulated; (b) the time evolution is deterministic; (c) the wave function $\Psi_Q(q;t)$ is physically real field; and (d) the initial distribution of the particle is assumed to be given by $\rho(q;0)=|\Psi_Q(q;0)|^2$ to reproduce the prediction of quantum mechanics. In particular, the last two assumptions constitute one of its main critics \cite{Pauli-criticsm,Keller-criticsm}. With the assumption that the wave function is physically real field, first, it is not clear how to prepare an ensemble to satisfy $\rho(q;0)=|\Psi_Q(q;0)|^2$. See however Refs. \cite{Bohm-Vigier,Valentini H-theorem,Duerr-Goldstein-Zanghi} for argumentation against this critics. Second, why there is no back-reaction from the particle to the wave function like for example in the particle-field interaction in the theory of electromagnetic. Unlike pilot-wave theory, in our model, the (effective) deterministic time evolution governed by the Schr\"odinger equation and the corresponding guidance relation emerge naturally from a statistical modification of classical dynamics rather than postulated. The original dynamics is inherently stochastic. Moreover, the wave function is not physically real. It is just a mathematical tool to describe the dynamics and statistics of the ensemble of trajectories, and $\rho(q;t)=|\Psi_Q(q;t)|^2$ is valid for all time by construction. Nevertheless, despite the above conceptual difference, one can conclude that our model will reproduce the pilot-wave theory prediction on statistical wave-like pattern in slits experiment and tunneling of potential barrier \cite{interference and tunneling in PWT}.
\subsection{Quantization: unique ordering and quantum-classical correspondence\label{quantization of classical Hamiltonian}}
We have mentioned in the previous subsection that the scheme to derive the Schr\"odinger equation presented in subsection \ref{emergent Schroedinger equation EM} can be viewed as to provide a method of quantization of classical Hamiltonian. In fact, the quantum Hamiltonian of Eq. (\ref{quantum Hamiltonian EM}) can be obtained from the classical Hamiltonian of Eq. (\ref{classical Hamiltonian EM}) by replacing the classical momentum with the quantum momentum operator \begin{equation} \underline{p}\mapsto{\hat p}\doteq-i\hbar\partial_q, \end{equation} as prescribed by the canonical quantization in configuration space representation (see however the discussion at the end of the present subsection). Hence, for the case of particle in external potentials, our method correctly reproduces the result of canonical quantization.
As mentioned in Section I, however, given a classical Hamiltonian, the canonical quantization rule in general will give an infinite alternatives of quantum Hamiltonians due to the operator ordering ambiguity. In contrast to this, it is evident that the method of quantization proposed in the present paper will lead to unique Hermitian quantum Hamiltonian, if the solution exists. This came from the fact that while in canonical quantization one replaces c-number by q-number, in our method, c-number is replaced by another c-number parameterized by random hidden variable as prescribed by Eq. (\ref{fundamental equation}). To give an example where the canonical quantization rule leads to ambiguity, let us consider the dynamics of particle with position-dependent mass $m(q)$ which has wide applications in solid state physics \cite{Lakshmanan,Galbraith,Young,Einvoll,Arias,Leblond,Serra,Yu,Plastino,Carinena,Kuru,Bagchi,Sever,Zhang,Nieto,Cruz,Kerimov,Jana,Levai}. For simplicity let us assume that the particle is free. The classical Hamiltonian then takes the form \begin{equation} \underline{H}=B(q)\underline{p}^2, \label{classical Hamiltonian position-dependent mass} \end{equation} where $B(q)=1/(2m)$ is a real-valued differentiable function of $q$. Using canonical quantization, one can then choose one out of infinite alternatives of quantum Hamiltonians. For example, if $B(q)\sim q^2$ up to some multiplicative constant, then one can either choose for the corresponding Hermitian quantum Hamiltonian $({\hat p}^2{\hat q}^2+{\hat q}^2{\hat p}^2)/2$ or ${\hat p}{\hat q}^2{\hat p}$ which are related to each other, by virtue of the canonical commutation relation $[\hat{q},\hat{p}]=i\hbar$, as ${\hat p}{\hat q}^2{\hat p}=({\hat p}^2{\hat q}^2+{\hat q}^2{\hat p}^2)/2+\hbar^2$.
Let us show that our method of quantization leads to unique Hermitian quantum Hamiltonian with a specific ordering of operators. First, given the classical Hamiltonian of Eq. (\ref{classical Hamiltonian position-dependent mass}), the Hamilton-Jacobi equation of (\ref{H-J equation}) reads \begin{equation} \partial_t\underline{S}+B(\partial_q\underline{S})^2=0. \label{H-J equation position-dependent mass} \end{equation} Substituting Eq. (\ref{classical Hamiltonian position-dependent mass}) into Eq. (\ref{classical velocity field}), the classical velocity field is given by \begin{equation} \underline{v}=2B\partial_q\underline{S}. \label{classical velocity position-dependent mass} \end{equation} Hence, the continuity equation of (\ref{continuity equation}) becomes \begin{equation} \partial_t\underline{\rho}+2\partial_q\cdot\big(B\underline{\rho}\partial_q\underline{S}\big)=0. \label{continuity equation position-dependent mass} \end{equation} Next, from Eq. (\ref{classical velocity position-dependent mass}), $f$ defined in Eq. (\ref{classical velocity field}) is given by $f(\underline{S})=2B\partial_q\underline{S}$ so that Eq. (\ref{fundamental equation}) becomes \begin{eqnarray} \underline{\rho}\mapsto\rho P(\lambda),\hspace{23mm}\nonumber\\ \partial_q\underline{S}\mapsto\partial_qS+\frac{\lambda}{2}\frac{\partial_q\rho}{\rho},\hspace{15mm}\nonumber\\ \partial_t\underline{S}\mapsto\partial_tS+\frac{\lambda}{2}\frac{\partial_t\rho}{\rho}+\lambda\partial_q\cdot\big(B\partial_qS\big). \label{fundamental equation position-dependent mass} \end{eqnarray}
Let us see how the above set of equations modify the pair of Eqs. (\ref{H-J equation position-dependent mass}) and (\ref{continuity equation position-dependent mass}). Imposing the first two equations of (\ref{fundamental equation position-dependent mass}) into Eq. (\ref{continuity equation position-dependent mass}) one gets \begin{equation} \partial_t\rho+2\partial_q(B\rho\partial_qS)+\lambda\partial_q(B\partial_q\rho)=0. \label{FPE position-dependent mass} \end{equation} On the other hand, imposing the last two equations of (\ref{fundamental equation position-dependent mass}) into Eq. (\ref{H-J equation position-dependent mass}), one obtains \begin{eqnarray} \partial_tS+B(\partial_qS)^2-\lambda^2\Big(B\frac{\partial_q^2R}{R}+\partial_qB\frac{\partial_q R}{R}\Big)\nonumber\\ +\frac{\lambda}{2\rho}\Big(\partial_t\rho+2\partial_q(B\rho\partial_qS)+\lambda\partial_q(B\partial_q\rho)\Big)=0, \label{u position-dependent mass} \end{eqnarray} where we have again defined $R\doteq\sqrt{\rho}$ and used the identity of Eq. (\ref{fluctuation decomposition}) for $i=j$. Substituting Eq. (\ref{FPE position-dependent mass}) into Eq. (\ref{u position-dependent mass}) one gets \begin{equation} \partial_tS+B(\partial_qS)^2-\lambda^2\Big(B\frac{\partial_q^2R}{R}+\partial_qB\frac{\partial_q R}{R}\Big)=0. \label{HJM position-dependent mass} \end{equation} We have thus pair of coupled equations (\ref{FPE position-dependent mass}) and (\ref{HJM position-dependent mass}) which still depend on the hidden variable $\lambda$.
One can then see that $S(q,\hbar;t)=S(q,-\hbar;t)=S_Q(q;t)$ satisfies the same equation (\ref{HJM position-dependent mass}) where $\lambda^2$ is replaced by $\hbar^2$. Hence, averaging over the fluctuations of the parameter $\lambda=\pm\hbar$ which is assumed to be equally probable, Eqs. (\ref{FPE position-dependent mass}) and (\ref{HJM position-dependent mass}) become \begin{eqnarray} \partial_t\rho+2\partial_q(B\rho\partial_qS_Q)=0,\hspace{20mm}\nonumber\\ \partial_tS_Q+B(\partial_qS_Q)^2-\hbar^2\Big(B\frac{\partial_q^2R}{R}+\partial_qB\frac{\partial_q R}{R}\Big)=0. \end{eqnarray}
Finally, recalling Eq. (\ref{Schroedinger wave function}) that $\Psi_Q=R\exp(iS_Q/\hbar)$, the above pair of equations can be recast into \begin{equation} i\hbar\partial_t\Psi_Q=-\frac{\hbar^2}{2}\Big(B\partial_q^2+\partial_q^2B\Big)\Psi_Q +\frac{\hbar^2}{2}(\partial_q^2B)\Psi_Q. \end{equation} This is just the Schr\"odinger equation with quantum Hamiltonian given by ${\hat H}={\hat p}B(q){\hat p}$. One thus obtains the following quantization mapping \begin{equation} B(q)\underline{p}^2\mapsto{\hat p}B(q){\hat p}. \label{quantization position-dependent mass} \end{equation} In particular for constant $B$, one has \begin{equation} \underline{p}^2\mapsto\hat{p}^2. \label{quantum Hamiltonian kinetic energy} \end{equation}
Let us mention that the same result as in Eq. (\ref{quantization position-dependent mass}) is also reported in the derivation of Schr\"odinger equation using the principle of exact-uncertainty and a principle of extremization of ensemble of Hamiltonian density \cite{Hall ordering}. However, in contrast to the latter which can only be applied to quantize a specific type of classical Hamiltonian with a quadratic momentum dependence, our method formally applies as well, as will be shown below and in the next section, to classical Hamiltonian which contains a term that is linear in momentum. One might then expect that Eq. (\ref{quantum Hamiltonian kinetic energy}) can be extended to any power in momentum, namely $\underline{p}^n\mapsto\hat{p}^n$, where $n$ is integer. The answer is however negative. A straightforward calculation to quantize a classical Hamiltonian which is proportional to $\underline{p}^3$, regardless of its physical meaning, will not lead to a Schr\"odinger equation with quantum Hamiltonian proportional to $\hat{p}^3$.
Next, let us assume that the classical Hamiltonian under interest is decomposable as $\underline{H}=a\underline{H}_1+b\underline{H}_2$, where $a$ and $b$ are real numbers. Then from Hamilton equation, the classical velocity field is also decomposable into $\underline{v}(\underline{S})=(\partial\underline{H}/\partial\underline{p})|_{\{\underline{p}=\partial_q\underline{S}\}}=af_1(\underline{S})+bf_2(\underline{S})$, where the function $f_i$ corresponds to $\underline{H}_i$, $i=1,2$. Hence, $f$ defined in Eq. (\ref{classical velocity field}) is also decomposable into $f(\underline{S})=af_1(\underline{S})+bf_2(\underline{S})$. Further let us assume that each term of the decomposition of classical Hamiltonian is mapped into a quantum Hamiltonian as $\underline{H}_1\mapsto{\hat H}_1$ and $\underline{H}_2\mapsto{\hat H}_2$. Keeping all of these in mind and recalling that $\underline{H}$ and $\underline{v}$ appear linearly in Eqs. (\ref{H-J equation}) and (\ref{continuity equation}) and also the linearity of the Schr\"odinger equation, one can conclude that applying the quantization method to the total classical Hamiltonian one will get \begin{equation} \underline{H}=a\underline{H}_1+b\underline{H}_2\mapsto{\hat H}= a{\hat H}_1+b{\hat H}_2. \label{linearity} \end{equation} The quantization mapping induced by our hidden variable model is thus linear.
To apply the above property, let us first formally quantize the following classical Hamiltonian \begin{equation} \underline{H}=B(q)\underline{p}, \label{classical Hamiltonian linear momentum} \end{equation} which is assumed to be one of the term of a physically sensible Hamiltonian, say a term that appears in Eq. (\ref{classical Hamiltonian EM}). Here $B(q)$ is a differentiable function of $q$. Applying the method of quantization developed in this paper one formally has the following quantization mapping: \begin{equation} B(q)\underline{p}\mapsto\frac{B{\hat p}+{\hat p}B}{2}. \label{quantum Hamiltonian linear momentum} \end{equation} Detail of the calculation is given in Appendix \ref{quantization of linear momentum}. In particular, for $B(q)=1$ one has $\underline{p}\mapsto\hat{p}=-i\hbar\partial_q$ and putting $\underline{p}=1$, one has $B(q)\mapsto B(q)$, or formally $q\mapsto\hat{q}=q$.
Let us apply the above results to quantize the classical Hamiltonian of a particle in electromagnetic field given in Eq. (\ref{classical Hamiltonian EM}). First, Eq. (\ref{classical Hamiltonian EM}) can be expanded into \begin{equation} \underline{H}=\frac{\underline{p}^2}{2m}-\frac{eA\underline{p}}{mc}+\frac{e^2A^2}{2mc^2}+eV. \label{classical Hamiltonian EM expanded} \end{equation} Applying the quantization mapping of Eqs. (\ref{quantum Hamiltonian kinetic energy}) and (\ref{quantum Hamiltonian linear momentum}), recalling the linearity of the quantization mapping of Eq. (\ref{linearity}), one then obtains \begin{eqnarray} {\hat H}=\frac{\hat{p}^2}{2m}-\frac{e}{2mc}\big(A\hat{p}+\hat{p}A\big)+\frac{e^2A^2}{2mc^2}+eV, \nonumber \label{quantum Hamiltonian EM expanded} \end{eqnarray} which is equal to Eq. (\ref{quantum Hamiltonian EM}), as expected. Hence, in developing Eq. (\ref{quantum Hamiltonian EM}) from Eq. (\ref{classical Hamiltonian EM}) using canonical quantization by directly promoting the classical momentum into quantum momentum operator $\underline{p}\mapsto\hat{p}$, one is implicitly assuming the ordering given in Eq. (\ref{quantum Hamiltonian linear momentum}). In contrast to this, in our hidden variable model for quantization, Eq. (\ref{quantum Hamiltonian linear momentum}) is derived rather than assumed.
\section{Measurement of momentum, position, angular momentum without wave function collapse\label{measurement}}
In the present section, we shall apply the modification of classical dynamics developed in the previous section using the type of hidden random variable with probability density given by Eq. (\ref{Schroedinger condition}) to a class of classical model of measurement of momentum, position and angular momentum.
\subsection{Classical measurement}
Let us first discuss a class of measurement model in classical mechanics. Let us consider the dynamics of two interacting particles, the first particle with coordinate $q_1$ represents the system to be measured and the other with coordinate $q_2$ represents the measuring apparatus. Let us suppose that one wants to measure a physical quantity $\underline{A}_1$ of the system. It is a function of the position $q_1$ and classical momentum ${\underline{p}}_1$, $\underline{A}_1=\underline{A}_1(q_1,{\underline{p}}_1)$. To do this, let us choose the following classical measurement-interaction Hamiltonian: \begin{equation} \underline{H}=g\underline{A}_1(q_1,{\underline{p}}_1){\underline{p}}_2, \label{classical Hamiltonian with measurement-interaction} \end{equation} where $g$ is a coupling constant. Let us further assume that the interaction is impulsive so that the individual free Hamiltonians of the particles are ignorable.
$\underline{A}_1$ is thus conserved $d\underline{A}_1/dt=\{\underline{A}_1,\underline{H}\}=0$. The idea is then to correlate the value of $\underline{A}_1(q_1,{\underline{p}}_1)$ with the classical momentum of the apparatus ${\underline{p}}_2$ while keeping the value of $\underline{A}_1(q_1,{\underline{p}}_1)$ unchanged. On the other hand, one also has $dq_2/dt=\{q_2,\underline{H}\}=g\underline{A}_1$, which can be integrated to give $q_2(T)=q_2(0)+g\underline{A}_1T$, where $T$ is time span of the measurement-interaction. The value of $\underline{A}_1$ prior to the measurement can thus be inferred from the observation of the initial and final values of $q_2$ (the pointer of the apparatus). Since each measurement reveals the value of $A_1$ prior-measurement then there is no need to introduce a second apparatus (third particle) to observe the position of the second particle (the pointer of the first apparatus).
Below we shall modify the classical dynamics of ensemble of trajectories generated by classical Hamiltonian of Eq. (\ref{classical Hamiltonian with measurement-interaction}) for measurement of momentum, position and angular momentum following the method developed in the previous section. Momentum and position represent physical quantities with continuous quantum mechanical spectrum. They are also important in view of canonical commutation relation and Heisenberg uncertainty relation. On the other hand, angular momentum represents physical quantity with discrete quantum mechanical spectrum. The crucial problem in this model is that whether one needs further ``quantum apparatus'' to observe $q_2(t)$, the position of the apparatus pointer. We shall show that this is not the case. Namely the model with interacting two particles will be shown to be sufficient for this purpose.
We have to mention that in reality, however, the above model with one dimensional apparatus is oversimplified. To this end let us emphasize that the aim of the discussion is only to show that in principle, the method of quantization proposed in the present paper can lead to measurement without wave function collapse and necessitating no external (classical) observer. In this respect, we believe that if it does not work for one degree of freedom then it will be more difficult to expect that it will work for realistic measurement model. Especially, our model excludes the irreversibility of the registration process which can only be done by realistic apparatus plus bath using large degree of freedom. See Ref. \cite{Theo} for an elaborated discussion of quantum measurement using realistic model of apparatus.
\subsection{Quantum Hamiltonian for the measurement of momentum, position and angular momentum\label{quantum Hamiltonian with measurement-interaction}}
\subsubsection{Quantum Hamiltonian for the measurement of momentum}
Let us first discuss the case of momentum measurement. One thus puts $\underline{A}_1={\underline{p}}_1$ into Eq. (\ref{classical Hamiltonian with measurement-interaction}) to have the following measurement-interaction classical Hamiltonian: \begin{equation} \underline{H}_p=g{\underline{p}}_1{\underline{p}}_2. \label{classical Hamiltonian of momentum} \end{equation} In impulsive measurement, the Hamilton-Jacobi equation of (\ref{H-J equation}) then reads \begin{equation} \partial_t\underline{S}+g\partial_{q_1}\underline{S}\partial_{q_2}\underline{S}=0. \label{H-J equation momentum} \end{equation} On the other hand, inserting Eq. (\ref{classical Hamiltonian of momentum}) into Eq. (\ref{classical velocity field}), one obtains the following classical velocity field for the two particles \begin{eqnarray} {\underline{v}}_1=g\partial_{q_2}\underline{S},\hspace{3mm}{\underline{v}}_2=g\partial_{q_1}\underline{S}. \label{velocity field for momentum} \end{eqnarray} The continuity equation of (\ref{continuity equation}) thus becomes \begin{equation} \partial_t\underline{\rho}+g\partial_{q_1}(\underline{\rho}\partial_{q_2}\underline{S})+g\partial_{q_2}(\underline{\rho}\partial_{q_1}\underline{S})=0. \label{continuity equation momentum} \end{equation} From Eq. (\ref{velocity field for momentum}), $f$ defined in Eq. (\ref{classical velocity field}) takes the form \begin{eqnarray} f_1(\underline{S})=g\partial_{q_2}\underline{S},\hspace{3mm}f_2(\underline{S})=g\partial_{q_1}\underline{S}, \label{actual velocity field momentum} \end{eqnarray} so that $\partial_q\cdot f(S)=2g\partial_{q_1}\partial_{q_2}S$. Hence, Eq. (\ref{fundamental equation}) becomes \begin{eqnarray} \underline{\rho}\mapsto\rho P(\lambda),\hspace{23mm}\nonumber\\ \partial_{q_i}\underline{S}\mapsto\partial_{q_i}S+\frac{\lambda}{2}\frac{\partial_{q_i}\rho}{\rho},\hspace{2mm}i=1,2,\nonumber\\ \partial_{t}\underline{S}\mapsto\partial_tS+\frac{\lambda}{2}\frac{\partial_t\rho}{\rho}+ g\lambda\partial_{q_1}\partial_{q_2}S. \label{fundamental equation momentum} \end{eqnarray}
Let us proceed to investigate the change brought by Eq. (\ref{fundamental equation momentum}) onto the Hamilton-Jacobi equation of (\ref{H-J equation momentum}) and the corresponding continuity equation of (\ref{continuity equation momentum}) describing ensembles of classical trajectories. First, imposing the first two equations of (\ref{fundamental equation momentum}) into Eq. (\ref{continuity equation momentum}) one obtains \begin{equation} \partial_t\rho+g\partial_{q_1}(\rho\partial_{q_2}S)+g\partial_{q_2}(\rho\partial_{q_1}S)+g\lambda\partial_{q_1}\partial_{q_2}\rho=0. \label{FPE momentum} \end{equation} On the other hand, imposing the last two equations of (\ref{fundamental equation momentum}) into Eq. (\ref{H-J equation momentum}) one gets \begin{eqnarray} \partial_tS+g\partial_{q_1}S\partial_{q_2}S-g\lambda^2\frac{\partial_{q_1}\partial_{q_2}R}{R} +\frac{\lambda}{2\rho}\Big(\partial_t\rho\nonumber\\ +g\partial_{q_1}(\rho\partial_{q_2}S)+g\partial_{q_2}(\rho\partial_{q_1}S)+g\lambda\partial_{q_1}\partial_{q_2}\rho\Big)=0, \label{uu m} \end{eqnarray} where we have defined $R\doteq\sqrt{\rho}$ and used Eq. (\ref{fluctuation decomposition}). Substituting Eq. (\ref{FPE momentum}) into Eq. (\ref{uu m}) one thus has \begin{equation} \partial_tS+g\partial_{q_1}S\partial_{q_2}S-g\lambda^2\frac{\partial_{q_1}\partial_{q_2}R}{R}=0. \label{HJM equation momentum} \end{equation} We have thus pair of coupled equations (\ref{FPE momentum}) and (\ref{HJM equation momentum}) which still depend on the random variable $\lambda$ whose probability density function is assumed to be given by Eq. (\ref{Schroedinger condition}).
One can then see that $S(q,\hbar;t)=S(q,-\hbar;t)=S_Q(q;t)$ satisfies the same differential equation of (\ref{HJM equation momentum}) with $\lambda^2$ replaced by $\hbar^2$. Then averaging Eqs. (\ref{FPE momentum}) and (\ref{HJM equation momentum}) over the distribution of $\lambda=\pm\hbar$ with equal probability one obtains the following pair of equations: \begin{eqnarray} \partial_t\rho+g\partial_{q_1}(\rho\partial_{q_2}S_Q)+g\partial_{q_2}(\rho\partial_{q_1}S_Q)=0,\nonumber\\ \partial_tS_Q+g\partial_{q_1}S_Q\partial_{q_2}S_Q-g\hbar^2\frac{\partial_{q_1}\partial_{q_2}R}{R}=0. \label{Madelung equation momentum} \end{eqnarray} Finally, recalling Eq. (\ref{Schroedinger wave function}) that $\Psi_Q=\sqrt{\rho}\exp(iS_Q/\hbar)=R\exp(iS_Q/\hbar)$, the above pair of equations can be combined into the Schr\"odinger equation $i\hbar\partial_t\Psi_Q={\hat H}_p\Psi_Q$ with measurement-interaction quantum Hamiltonian \begin{equation} {\hat H}_p=g{\hat p}_1{\hat p}_2, \label{quantum Hamiltonian momentum} \end{equation}
where ${\hat p}_i=-i\hbar\partial_{q_i}$, $i=1,2$. Again, by construction one has $\rho(q;t)=|\Psi_Q(q;t)|^2$. Moreover, from the upper equation of (\ref{Madelung equation momentum}), the effective velocity is $f(S_Q)$ where $f$ is given by Eq. (\ref{actual velocity field momentum}) so that it is equal to the actual velocity field of the particles in pilot-wave theory.
\subsubsection{Quantum Hamiltonian for the measurement of position \label{subsubsection measurement of position}}
Next let us consider the measurement of position. One thus put $\underline{A}_1=q_1$ into Eq. (\ref{classical Hamiltonian with measurement-interaction}) to have the following classical measurement-interaction Hamiltonian: \begin{equation} \underline{H}_q=gq_1{\underline{p}}_2. \label{interaction Hamiltonian position} \end{equation} The Hamilton-Jacobi equation of (\ref{H-J equation}) thus reads \begin{equation} \partial_t\underline{S}+gq_1\partial_{q_2}\underline{S}=0. \label{H-J equation position} \end{equation} On the other hand, inserting Eq. (\ref{interaction Hamiltonian position}) into Eq. (\ref{classical velocity field}), the classical velocity field is given by \begin{equation} {\underline{v}}_1=0,\hspace{2mm}{\underline{v}}_2=gq_1. \label{classical velocity field position} \end{equation} The above pair of equations provide constraint to the dynamics of the particles. Hence, the continuity equation of (\ref{continuity equation}) becomes \begin{equation} \partial_t\underline{\rho}+gq_1\partial_{q_2}\underline{\rho}=0. \label{continuity equation position} \end{equation} From Eq. (\ref{classical velocity field position}) and the definition of $f$ given in Eq. (\ref{classical velocity field}) one has \begin{equation} f_1=0,\hspace{2mm} f_2=gq_1, \label{actual velocity field position} \end{equation} so that $\partial_q\cdot f=0$. Equation (\ref{fundamental equation}) thus becomes \begin{eqnarray} \underline{\rho}\mapsto\rho P(\lambda),\hspace{19mm}\nonumber\\ \partial_{q_i}\underline{S}\mapsto\partial_{q_i}S+\frac{\lambda}{2}\frac{\partial_{q_i}\rho}{\rho},\hspace{2mm}i=1,2,\nonumber\\ \partial_{t}\underline{S}\mapsto\partial_tS+\frac{\lambda}{2}\frac{\partial_t\rho}{\rho}.\hspace{10mm} \label{fundamental equation position} \end{eqnarray}
Now let us apply the above set of equations to Eqs. (\ref{H-J equation position}) and (\ref{continuity equation position}). First, imposing the first equation of (\ref{fundamental equation position}) into Eq. (\ref{continuity equation position}) one obtains \begin{equation} \partial_t\rho+gq_1\partial_{q_2}\rho=0, \label{continuity equation position: quantum} \end{equation} namely Eq. (\ref{continuity equation position}) is kept unchanged. Next, imposing the last two equations of (\ref{fundamental equation position}) into Eq. (\ref{H-J equation position}) one obtains \begin{equation} \partial_tS+gq_1\partial_{q_2}S+\frac{\lambda}{2\rho}(\partial_t\rho+gq_1\partial_{q_2}\rho)=0. \end{equation} Substituting Eq. (\ref{continuity equation position: quantum}) one thus gets \begin{equation} \partial_tS+gq_1\partial_{q_2}S=0. \label{HJM position} \end{equation} Again, Eq. (\ref{H-J equation position}) remains unchanged. We have thus pair of decoupled equations (\ref{continuity equation position: quantum}) and (\ref{HJM position}).
Notice then that $\lambda$ does not appear explicitly as parameter. Identifying $S_Q=S$, and defining $\Psi_Q\doteq\sqrt{\rho}\exp(iS_Q/\hbar)$ so that $|\Psi_Q(q;t)|^2=\rho(q;t)$, the pair of Eqs. (\ref{continuity equation position: quantum}) and (\ref{HJM position}) can then be combined together into the Schr\"odinger equation $i\hbar\partial_t\Psi_Q={\hat H}_q\Psi_Q$ with quantum Hamiltonian \begin{equation} {\hat H}_q=gq_1{\hat p}_2. \label{quantum Hamiltonian position} \end{equation} Again, one can see from Eq. (\ref{continuity equation position: quantum}) that the effective velocity of the particles is $f(S_Q)$ where $f$ is given by Eq. (\ref{actual velocity field position}). Hence, it is again equal to the velocity of the particles in pilot-wave theory.
\subsubsection{Quantum Hamiltonian for the measurement of angular momentum}
Let us proceed to develop the quantum Hamiltonian for the measurement of angular momentum. To make explicit the three dimensional nature of the problem, let us put $q_1=(x_1,y_1,z_1)$. For simplicity let us first consider the measurement of $z-$part of angular momentum. In this case $\underline{A}_1$ in Eq. (\ref{classical Hamiltonian with measurement-interaction}) takes the form $\underline{A}_1=x_1{\underline{p}}_{y_1}-y_1{\underline{p}}_{x_1}$, where ${\underline{p}}_{x_1}$ is the conjugate momentum of $x_1$ and so on, so that the measurement-interaction classical Hamiltonian of Eq. (\ref{classical Hamiltonian with measurement-interaction}) reads \begin{equation} \underline{H}_l=g(x_1{\underline{p}}_{y_1}-y_1{\underline{p}}_{x_1}){\underline{p}}_2. \label{classical Hamiltonian angular momentum} \end{equation} The Hamilton-Jacobi equation of (\ref{H-J equation}) thus becomes \begin{equation} \partial_t\underline{S}+g\big(x_1\partial_{y_1}\underline{S}-y_1\partial_{x_1}\underline{S}\big)\partial_{q_2}\underline{S}=0. \label{H-J equation angular momentum} \end{equation}
On the other hand, substituting Eq. (\ref{classical Hamiltonian angular momentum}) into Eq. (\ref{classical velocity field}), the classical velocity field is given by \begin{eqnarray} {\underline{v}}_{x_1}=-gy_1\partial_{q_2}\underline{S},\hspace{2mm}{\underline{v}}_{y_1}=gx_1\partial_{q_2}\underline{S},\hspace{2mm}{\underline{v}}_{z_1}=0,\nonumber\\ {\underline{v}}_2=g\big(x_1\partial_{y_1}\underline{S}-y_1\partial_{x_1}\underline{S}\big). \hspace{15mm} \label{classical velocity angular momentum} \end{eqnarray} Hence, the continuity equation of (\ref{continuity equation}) becomes \begin{eqnarray} \partial_t\underline{\rho}-gy_1\partial_{x_1}(\underline{\rho}\partial_{q_2}\underline{S})+gx_1\partial_{y_1}(\underline{\rho}\partial_{q_2}\underline{S})\nonumber\\ +gx_1\partial_{q_2}(\underline{\rho}\partial_{y_1}\underline{S})-gy_1\partial_{q_2}(\underline{\rho}\partial_{x_1}\underline{S})=0. \label{continuity equation angular momentum} \end{eqnarray}
Next, from Eq. (\ref{classical velocity angular momentum}), $f$ defined in Eq. (\ref{classical velocity field}) takes the form \begin{eqnarray} f_{x_1}(\underline{S})=-gy_1\partial_{q_2}\underline{S},\hspace{2mm}f_{y_1}(\underline{S})=gx_1\partial_{q_2}\underline{S},\hspace{2mm}f_{z_1}(\underline{S})=0,\nonumber\\ f_2(\underline{S})=g\big(x_1\partial_{y_1}\underline{S}-y_1\partial_{x_1}\underline{S}\big). \hspace{15mm} \label{actual velocity field angular momentum} \end{eqnarray} One thus has $\partial_q\cdot f(S)=2g(x_1\partial_{q_2}\partial_{y_1}S-y_1\partial_{q_2}\partial_{x_1}S)$. Substituting this into Eq. (\ref{fundamental equation}), one then obtains \begin{eqnarray} \underline{\rho}\mapsto\rho P(\lambda),\hspace{30mm}\nonumber\\ \partial_{x_1}\underline{S}\mapsto\partial_{x_1}S+\frac{\lambda}{2}\frac{\partial_{x_1}\rho}{\rho},\hspace{20mm}\nonumber\\ \partial_{y_1}\underline{S}\mapsto\partial_{y_1}S+\frac{\lambda}{2}\frac{\partial_{y_1}\rho}{\rho},\hspace{20mm}\nonumber\\ \partial_{q_2}\underline{S}\mapsto\partial_{q_2}S+\frac{\lambda}{2}\frac{\partial_{q_2}\rho}{\rho},\hspace{20mm}\nonumber\\ \partial_{t}\underline{S}\mapsto\partial_{t}S+\frac{\lambda}{2}\frac{\partial_{t}\rho}{\rho}+ g\lambda(x_1\partial_{y_1}\partial_{q_2}S-y_1\partial_{x_1}\partial_{q_2}S).\hspace{0mm} \label{fundamental equation angular momentum} \end{eqnarray}
Let us proceed to see how the above set of equations modify Eqs. (\ref{H-J equation angular momentum}) and (\ref{continuity equation angular momentum}). Imposing the first four equations of (\ref{fundamental equation angular momentum}) into Eq. (\ref{continuity equation angular momentum}) one obtains, after a simple calculation \begin{eqnarray} \partial_t\rho-gy_1\partial_{x_1}(\rho\partial_{q_2}S)+gx_1\partial_{y_1}(\rho\partial_{q_2}S)+gx_1\partial_{q_2}(\rho\partial_{y_1}S)\nonumber\\ -gy_1\partial_{q_2}(\rho\partial_{x_1}S)-g\lambda(y_1\partial_{x_1}\partial_{q_2}\rho-x_1\partial_{y_1}\partial_{q_2}\rho)=0.\nonumber\\ \label{FPE angular momentum} \end{eqnarray} On the other hand, imposing the last four equations of (\ref{fundamental equation angular momentum}) into Eq. (\ref{H-J equation angular momentum}), one has, after an arrangement \begin{eqnarray} \partial_tS+g\big(x_1\partial_{y_1}S-y_1\partial_{x_1}S\big)\partial_{q_2}S-g\lambda^2\Big(x_1\frac{\partial_{y_1}\partial_{q_2}R}{R}\nonumber\\-y_1\frac{\partial_{x_1}\partial_{q_2}R}{R}\Big)+\frac{\lambda}{2\rho}\Big(\partial_t\rho-gy_1\partial_{x_1}(\rho\partial_{q_2}S)\nonumber\\ +gx_1\partial_{y_1}(\rho\partial_{q_2}S)+gx_1\partial_{q_2}(\rho\partial_{y_1}S)-gy_1\partial_{q_2}(\rho\partial_{x_1}S)\nonumber\\ -g\lambda(y_1\partial_{x_1}\partial_{q_2}\rho-x_1\partial_{y_1}\partial_{q_2}\rho)\Big)=0, \label{ccc} \end{eqnarray} where $R\doteq\sqrt{\rho}$ and we have used Eq. (\ref{fluctuation decomposition}). Substituting Eq. (\ref{FPE angular momentum}), the last term in the bracket vanishes to give \begin{eqnarray} \partial_tS+g\big(x_1\partial_{y_1}S-y_1\partial_{x_1}S\big)\partial_{q_2}S\hspace{20mm}\nonumber\\ -g\lambda^2\Big(x_1\frac{\partial_{y_1}\partial_{q_2}R}{R}-y_1\frac{\partial_{x_1}\partial_{q_2}R}{R}\Big)=0. \label{HJM angular momentum} \end{eqnarray} One thus has pair of coupled equations (\ref{FPE angular momentum}) and (\ref{HJM angular momentum}) which are parameterized by the random variable $\lambda$.
Again, one can see that $S(q,\hbar;t)=S(q,-\hbar;t)=S_Q(q;t)$ satisfies the same differential equation of (\ref{HJM angular momentum}) where $\lambda^2$ is replaced by $\hbar^2$. Hence, taking the average of Eqs. (\ref{FPE angular momentum}) and (\ref{HJM angular momentum}) over the distribution of $\lambda=\pm\hbar$ with equal probability as in Eq. (\ref{Schroedinger condition}) gives the following pair of equations: \begin{eqnarray} \partial_t\rho-gy_1\partial_{x_1}(\rho\partial_{q_2}S_Q)+gx_1\partial_{y_1}(\rho\partial_{q_2}S_Q)\hspace{10mm}\nonumber\\ +gx_1\partial_{q_2}(\rho\partial_{y_1}S_Q)-gy_1\partial_{q_2}(\rho\partial_{x_1}S_Q)=0.\nonumber\\ \partial_tS+g\big(x_1\partial_{y_1}S_Q-y_1\partial_{x_1}S_Q\big)\partial_{q_2}S_Q\hspace{20mm}\nonumber\\ -g\hbar^2\Big(x_1\frac{\partial_{y_1}\partial_{q_2}R}{R}-y_1\frac{\partial_{x_1}\partial_{q_2}R}{R}\Big)=0. \label{Madelung equation for angular momentum measurement} \end{eqnarray}
Finally, recalling Eq. (\ref{Schroedinger wave function}) that $\Psi_Q=R\exp(iS_Q/\hbar)$ so that $|\Psi_Q(q;t)|^2=R(q;t)^2=\rho(q;t)$, the above pair of equations can be recast into the Schr\"odinger equation $i\hbar\partial_t\Psi_Q={\hat H}_l\Psi_Q$ with quantum Hamiltonian \begin{equation} {\hat H}_l=g{\hat L}_{z_1}{\hat p}_2, \label{Hamiltonian operator angular momentum} \end{equation} where ${\hat L}_{z_1}=x_1{\hat p}_{y_1}-y_1{\hat p}_{x_1}=-i\hbar (x_1\partial_{y_1}-y_1\partial_{x_1})$. As expected, ${\hat L}_{z_1}$ is just the $z-$component of the (quantum mechanical) angular momentum operator. Moreover, one can again see from the upper equation of (\ref{Madelung equation for angular momentum measurement}) that the effective velocity of the particles are $f(S_Q)$ where $f$ is given by Eq. (\ref{actual velocity field angular momentum}) so that it is equal to the actual velocity of the particles in pilot-wave theory. The above results can be extended to measurement of angular momentum along $x-$ and $y-$ directions straightforwardly using cyclic permutation among the coordinates $(x_1,y_1,z_1)$. One will then obtain the Schr\"odinger equation with quantum Hamiltonian of Eq. (\ref{Hamiltonian operator angular momentum}) where $\hat{L}_{z_1}$ is replaced by the quantum mechanical angular momentum operators along the $x-$ and $y-$ directions, respectively. Moreover, since in principle one can take any direction as $z-$axis, then the above result applies as well to angular momentum measurement along any direction.
\subsection{Measurement without wave function collapse and external observer}
In the previous section, starting from a class of classical Hamiltonian for the measurement of momentum, position and angular momentum, $\underline{H}=g\underline{A}_1\underline{p}_2$, where $\underline{A}_1$ is the physical quantities being measured, we have arrived at the following Schr\"odinger equation: \begin{equation} i\hbar\partial_t\Psi_Q={\hat H}\Psi_Q=g{\hat A}_{1}{\hat p}_2\Psi_Q, \label{Schroedinger equation measurement-interaction} \end{equation}
where $\hat{A}_1$ is a Hermitian operator given by ${\hat A}_1=\underline{A}_1(q,{\hat p}_1)$. Our hidden variable model of quantization thus reproduces the results of canonical quantization. However, unlike the latter, in all of the cases considered, one can identify an effective velocity of the particles which turns out to be equal to the actual velocity of the particles in pilot-wave theory, and the Born's interpretation of wave function, $|\Psi_Q(q;t)|^2=\rho(q;t)$, is valid for all time, by construction. We can then follow all the argumentation of the pilot-wave theory in describing the process of measurement without wave function collapse \cite{Bohm paper}.
To do this, let us first assume that ${\hat A}_1$ has discrete spectrum as the case for measurement of angular momentum. Hence, one has ${\hat A}_1\psi_n(q_1)=a_n\psi_n(q_1)$, $n=0,1,2,\dots$, where $a_n$ is the real-valued eigenvalue of ${\hat A}_1$ and $\psi_n$ is the corresponding eigenfunction. $\{\psi_n\}$ thus makes a complete set of orthonormal functions. Then, ignoring the free Hamiltonian for impulsive measurement-interaction, the Schr\"odinger equation of (\ref{Schroedinger equation measurement-interaction}) has the following general solution: \begin{equation} \Psi_Q(q_1,q_2;t)=\sum_nc_n\psi_n(q_1)\varphi(q_2-ga_nt), \label{entanglement system-apparatus} \end{equation} where $\varphi(q_2)$ is the initial wave function of the apparatus which is assumed to be sufficiently localized, $c_n$ is complex number, and $\phi(q_1)\doteq\sum_nc_n\psi_n(q_1)$ is the initial wave function of the system. In other words, $c_n$ is the coefficient of expansion of the initial wave function of the system in term of the orthonormal set of the eigenfunctions of ${\hat A}_1$.
For sufficiently large $g$, $\varphi_n(q_2)\doteq\varphi(q_2-ga_nT)$ is not overlapping for different $n$ and each is correlated to a distinct $\psi_n(q_1)$. One then argues, following pilot-wave theory \cite{Bohm paper}, that the outcome of single measurement event corresponds to the packet $\varphi_n(q_2)$ which is actually entered by the apparatus particle. Namely, if $q_2(t)$ belongs to the support of $\varphi_n(q_2)$, then we admit that the result of measurement is given by $a_n$. This can be generalized to ${\hat A}_1$ with continuous spectrum, as the case for the measurement of momentum or position, by replacing the sum in Eq. (\ref{entanglement system-apparatus}) with integration. As in pilot-wave theory, the probability to find the outcome $a_n$ is given by $P(a_n)=|c_n|^2$, that is the experimentally well-verified Born's statistical rule. This can be shown as direct implication of $\rho(q;t)=|\Psi_Q(q;t)|^2$. The prediction of quantum mechanics is thus reproduced without invoking wave function collapse induced by external (classical) observer.
Notice that the linearity of the Schr\"odinger equation plays a very pivotal role in the discussion of measurement. The superposition of solution in Eq. (\ref{entanglement system-apparatus}) is made possible by the linearity of the Schr\"odinger equation of (\ref{Schroedinger equation measurement-interaction}). Since $\varphi_n(q_2)=\varphi(q_2-ga_nT)$ refers to the wave function of pointer of the apparatus, then it has been argued within the standard quantum mechanics that Eq. (\ref{entanglement system-apparatus}) is a superposition of macroscopically distinct states. This leads to the paradox of Schr\"odinger's cat suggesting an indefiniteness of the state of macroscopic body which is against our everyday experience (recall that in the standard quantum mechanics, an observable possesses definite value only when the state is an eigenfunction of the observable which is not the case for Eq (\ref{entanglement system-apparatus})). It is to save this situation that in the standard quantum mechanics one needs to invoke a wave function collapse to get one of the term in the superposition of Eq. (\ref{entanglement system-apparatus}) \cite{Ballentine paper}.
This paradox however is based on the assumption that the superposition of states of Eq. (\ref{entanglement system-apparatus}) refers to an individual system (and apparatus) and that the description of an individual system by the wave function is complete \cite{Ballentine paper}. In contrast to this, in our dynamical model, the superposition of state, or in general any wave function, describes an ensemble of identically prepared system rather than individual system. Moreover, the description of an individual system by the wave function is not complete: a single system is always described by definite values of position and momentum of the particles and an unbiased random variable $\lambda=\pm\hbar$. In this respect, the superposition of state in Eq. (\ref{entanglement system-apparatus}) does not mean macroscopic indefiniteness since at any time, the pointer always possesses definite position. Hence, there is no paradox of Schr\"odinger's cat and accordingly there is no need to invoke the wave function collapse to get one term of the superposition as required by the standard quantum mechanics.
Further, one can see in the discussion of the previous subsection that the measurement of position is different from the measurement of momentum and angular momentum. Namely, unlike in the two latter cases, in the case of position measurement, Eq. (\ref{fundamental equation position}) does not change the classical Hamilton-Jacobi and continuity equations of (\ref{H-J equation position}) and (\ref{continuity equation position}). Both pair of functions $(\underline{S},\underline{\rho})$ and $(S,\rho)$ satisfy the same pair of equations, that of Eqs. (\ref{H-J equation position}) and (\ref{continuity equation position}). Hence, the classical results of measurement is preserved by Eq. (\ref{fundamental equation position}): there is no quantum correction. Conversely, the Schr\"odinger equation with quantum Hamiltonian of Eq. (\ref{quantum Hamiltonian position}) can be rewritten into the classical Hamilton-Jacobi equation of (\ref{H-J equation position}) and the continuity equation of (\ref{continuity equation position}) describing classical dynamics of ensemble of trajectories. One can thus conclude that, as in the case of classical measurement, it is possible to reveal the pre-existing value of the position immediately prior to the measurement. On the other hand, for the cases of measurement of momentum and angular momentum, the results of the measurement are not equal to the pre-existing values possessed by the systems. In this regards, the measurement of position is special. The derivation of the quantum Hamiltonian of measurement of position also shows that the ability to write the dynamics of ensemble of trajectories into the Schr\"odinger equation is not sufficient to distinguish quantum from classical mechanics.
The above results on position measurement further leads to an important implication. Recall that the results of the measurement of momentum and angular momentum are inferred from the position of the second particle (apparatus pointer). Then one might argue that one needs another, the third particle, as the second apparatus to probe the position of the second particle (the first apparatus). Proceeding in this way thus will lead to infinite regression: one will further need the forth particle (the third apparatus) to probe the position of the third particle (the second apparatus) and so on. In our model, however, since the quantum treatment of the position measurement is equivalent to the classical treatment revealing the position of the particle prior-measurement, then the second measurement on the position of the second particle (the first apparatus) is in principle not necessary. Namely, the results of position measurement by the second, third, forth apparatuses and so on are all equal to each other.
\subsection{Quantum mechanical observable and quantum-classical correspondence \label{quantization of physical quantity}}
First, the development of quantum Hamiltonian with measurement-interaction ${\hat H}=g{\hat A}_1{\hat p}_2$ from the corresponding classical Hamiltonian $\underline{H}=g\underline{A}_1(q_1,\underline{p}_1)\underline{p}_2$ in subsection \ref{quantum Hamiltonian with measurement-interaction} can be formally summarized into the following mapping \begin{eqnarray} \underline{p}_2\mapsto\hat{p}_2,\hspace{2mm}\underline{A}_1\mapsto\hat{A}_1. \end{eqnarray} Hence, it can be regarded as the quantization of classical quantity $\underline{A}_1$ to get the corresponding Hermitian operator $\hat{A}_1$ in the context of measurement. $\hat{A}_1$ is called as ``quantum observable'' in the standard formalism of quantum mechanics. As shown in subsection \ref{quantum Hamiltonian with measurement-interaction}, for the case where $\underline{A}_1$ is momentum, position and angular momentum, the corresponding Hermitian operator ${\hat A}_1$ can be obtained formally by the following substitution rule: $\underline{p}_1\mapsto\hat{p}_1= -i\hbar\partial_{q_1}$ and $q_1\mapsto{\hat q}_1=q_1$. For these specific but fundamental dynamical variables, our method thus reproduces the results of canonical quantization. In contrast to the latter, however, the quantization method reported in the present paper is developed by directly modifying classical dynamics of ensemble of measurement parameterized by an unbiased binary random variable $\lambda=\pm\hbar$. We have thus a continuous and transparent transition from quantum to classical measurement.
Further, recall that $[{\hat q}_i,{\hat p}_j]=i\hbar\delta_{ij}$ leads to the Heisenberg uncertainty relation $\sigma_{q_i}\sigma_{p_i}\ge\hbar/2$, where $\sigma_{q_i}$ and $\sigma_{p_i}$ are the standard deviation of results of measurement of position and the corresponding conjugate momentum in ensemble of identically prepared systems. Our dynamical model thus shows that the Heisenberg uncertainty relation is a direct implication of modification of classical dynamics for ensemble of trajectories as prescribed by Eq. (\ref{fundamental equation}) being applied to measurement. In particular, in the limit where $S\rightarrow\underline{S}$, one smoothly regains the classical dynamics so that $\sigma_{q_i}\sigma_{p_i}\ge 0$.
An immediate question then arises whether the method of quantization of classical quantity in the context of measurement developed in the present paper can be applied to any classical quantities, namely any function of position and classical momentum $F=F(q,\underline{p})$. To discuss this matter, first, it is not clear even in the classical mechanics whether any arbitrary function $F(q,\underline{p})$ is physically meaningful at all. In reality, hitherto, for spin-less particle, only position, momentum, angular momentum and energy have unambiguous physical meaning. Second, even if $F(q,\underline{p})$ is physically meaningful, it is not clear whether it can be measured directly. This is due to the fact that in reality measurement is done by mapping the properties of the system being measured to non-overlapping subsets of the configuration space. Hence, measurement-interaction is a special type of interaction. This gives a physical limitation to the kind of classical quantities which can be directly measured.
Taking all the above physical aspects aside, in contrast to canonical quantization which in general leads to infinite alternative of Hermitian operators for a given general classical quantity which is the direct implication of replacing c-number by q-number, it is evident that the method of quantization in the context of measurement model with classical Hamiltonian of Eq. (\ref{classical Hamiltonian with measurement-interaction}) presented in this paper, which is based on replacement of c-number by c-number, will give a unique Hermitian observable, if a solution exists. An example of the quantization of classical quantity of the type $B(q)\underline{p}$ in the context of measurement, where canonical quantization leads to ambiguity, is given in appendix \ref{quantization of classical quantity}.
\section{Conclusion and discussion}
We have proposed a quantization method by modifying the classical dynamics of ensemble of trajectories. The deviation from the classical mechanics is characterized by pair of real-valued functions $S(q,\lambda;t)$ and $\Omega(q,\lambda;t)$ parameterized by a hidden random variable $\lambda$ with specific statistical property following the rule of Eq. (\ref{fundamental equation}). In the classical limit, $S(q,\lambda;t)$ and $\Omega(q,\lambda;t)$ reduce into the Hamilton principle function $\underline{S}(q;t)$ and the classical probability density of the position $\underline{\rho}(q;t)$. Given a classical Hamiltonian, the model is applied to system of particles in external potentials, with position-dependent mass, and to a class of classical measurement of momentum, position and angular momentum. We showed that the resulting equations can be put into the Schr\"odinger equation with unique Hermitian quantum Hamiltonian. The wave function refers to ensemble of system rather than to an individual system. In contrast to the canonical quantization which replaces c-number by q-number implying operator ordering ambiguity, our method is based on replacement of c-number by c-number, thus is free from the problem of operator ordering ambiguity. The canonical commutation relation $[{\hat q}_i,{\hat p}_j]=i\hbar\delta_{ij}$, which lies at the bottom of the canonical quantization, is thus given statistical and dynamical meaning as a modification of classical dynamics of ensemble of trajectories in configuration space parameterized by an unbiased hidden random variable. This offers a conceptually smooth and physically transparent quantum-classical correspondence.
We then identified an effective velocity of the particles which turns out to be equal to the velocity of the particles in pilot-wave theory. However, unlike pilot-wave theory, our model is strictly stochastic, the wave function is not physically real and the Born's interpretation of wave function is valid by construction. This allows us to conclude that our model will reproduce the prediction of pilot-wave theory on statistical wave-like pattern in single and double slits experiments and also in tunneling of potential barrier. Moreover, following the argumentation of pilot-wave theory, we then developed the process of measurement without wave function collapse and external observer, reproducing the statistical prediction of quantum mechanics. Since our dynamical model of measurement reduces into the classical dynamics of measurement when $S\rightarrow\underline{S}$, one can conclude that in this limit, the probability of finding of quantum measurement reduces into the probability of being of classical measurement. In this sense, we have thus argued that quantum mechanics is an emergence statistical phenomena \cite{Adler}.
A common pragmatical question against any alternative approaches to quantum mechanics is that whether it offers new testable predictions which can not be calculated using the standard formalism of quantum mechanics. This is a very hard wall to tunnel in view of the pragmatical successes of the quantum mechanics. In our approach, however, since the Schr\"odinger equation is shown to be emergent corresponding to a specific type of distribution of hidden random variable $P(\lambda)$ given by Eq. (\ref{Schroedinger condition}), then we may expect that it will lead to new prediction if $P(\lambda)$ is allowed to deviate from Eq. (\ref{Schroedinger condition}). This, for example can be attained by allowing $|\lambda|$ to fluctuate around $\hbar$ with very small yet finite width. We shall elaborate the detail implications of this idea in separate work \cite{AgungDQM2}.
\begin{acknowledgments}
\end{acknowledgments}
\appendix
\section{\label{quantization of linear momentum}}
Let us quantize a classical Hamiltonian which takes the following form: \begin{equation} \underline{H}=B(q)\underline{p}, \label{classical Hamiltonian linear momentum} \end{equation} which is assumed to be part of a physically sensible Hamiltonian, and $B(q)$ is a differentiable function of $q$. First, the Hamilton-Jacobi equation of (\ref{H-J equation}) becomes \begin{equation} \partial_t\underline{S}+B\partial_q\underline{S}=0. \label{H-J equation linear momentum} \end{equation} Further, inserting Eq. (\ref{classical Hamiltonian linear momentum}) into Eq. (\ref{classical velocity field}), the classical velocity field is given by \begin{equation} \underline{v}=B. \label{classical velocity linear momentum} \end{equation} This provides a constraint to the motion of the particle. Thus, the continuity equation of (\ref{continuity equation}) reads \begin{equation} \partial_t\underline{\rho}+\partial_q(B\underline{\rho})=0. \label{continuity equation linear momentum} \end{equation}
Next, from Eq. (\ref{classical velocity linear momentum}), $f$ defined in Eq. (\ref{classical velocity field}) is given by $f=B$, so that Eq. (\ref{fundamental equation}) becomes \begin{eqnarray} \underline{\rho}\mapsto\rho P(\lambda),\hspace{18mm}\nonumber\\ \partial_q\underline{S}\mapsto\partial_qS+\frac{\lambda}{2}\frac{\partial_q\rho}{\rho},\hspace{8mm}\nonumber\\ \partial_t\underline{S}\mapsto\partial_tS+\frac{\lambda}{2}\frac{\partial_t\rho}{\rho}+\frac{\lambda}{2}\partial_qB. \label{fundamental equation linear momentum} \end{eqnarray}
Now let us apply the above set of equations to modify Eqs. (\ref{H-J equation linear momentum}) and (\ref{continuity equation linear momentum}). First, imposing the upper equation of (\ref{fundamental equation linear momentum}), Eq. (\ref{continuity equation linear momentum}) becomes \begin{equation} \partial_t\rho+\partial_q(B\rho)=0. \label{continuity equation linear momentum: quantum} \end{equation} Hence, Eq. (\ref{continuity equation linear momentum}) is kept unchanged. Further, imposing the last two equations of (\ref{fundamental equation linear momentum}) into Eq. (\ref{H-J equation linear momentum}) one obtains \begin{equation} \partial_tS+B\partial_qS+\frac{\lambda}{2\rho}(\partial_t\rho+\partial_q(B\rho))=0. \end{equation} Inserting Eq. (\ref{continuity equation linear momentum: quantum}) one thus has \begin{equation} \partial_tS+B\partial_qS=0. \label{HJM linear momentum} \end{equation} Namely, Eq. (\ref{H-J equation linear momentum}) is also kept unchanged. We have thus pair of decoupled equations (\ref{continuity equation linear momentum: quantum}) and (\ref{HJM linear momentum}).
Notice then that $\lambda$ does not appear explicitly as a parameter of the resulting differential equations. Identifying $S_Q=S$, and defining $\Psi_Q\doteq\sqrt{\rho}\exp(iS_Q/\hbar)$ so that $|\Psi_Q(q;t)|^2=\rho(q;t)$, the pair of Eqs. (\ref{continuity equation linear momentum: quantum}) and (\ref{HJM linear momentum}) can then be combined together into the following Schr\"odinger equation: \begin{equation} i\hbar\partial_t\Psi_Q=-i\frac{\hbar}{2}(B\partial_q+\partial_qB)\Psi_Q, \label{Schroedinger equation linear momentum} \end{equation} from which one can extract a Hermitian quantum Hamiltonian as \begin{equation} {\hat H}=\frac{B(-i\hbar\partial_q)+(-i\hbar\partial_q)B}{2}=\frac{B{\hat p}+{\hat p}B}{2}. \end{equation}
\section{\label{quantization of classical quantity}}
Let us quantize the classical quantity of the type $F=B(q)\underline{p}$ in the context of measurement discussed in Section \ref{measurement}, where $B(q)$ is a differentiable function of $q$. One thus put $A_1=B(q_1)\underline{p}_1$ into Eq. (\ref{classical Hamiltonian with measurement-interaction}) so that the classical Hamiltonian for the interaction-measurement is given by \begin{equation} \underline{H}=gB(q_1)\underline{p}_1\underline{p}_2. \label{classical Hamiltonian classical quantity} \end{equation} Notice that $B$ does not depend on $q_2$, the coordinate of the apparatus. The Hamilton-Jacobi equation of (\ref{H-J equation}) thus reads \begin{equation} \partial_t\underline{S}+gB\partial_{q_1}\underline{S}\partial_{q_2}\underline{S}=0. \label{H-J equation classical quantity} \end{equation} Next, inserting Eq. (\ref{classical Hamiltonian classical quantity}) into Eq. (\ref{classical velocity field}) one has \begin{equation} \underline{v}_1=gB\partial_{q_2}\underline{S},\hspace{2mm}\underline{v}_2=gB\partial_{q_1}\underline{S}. \label{classical velocity field classical quantity} \end{equation} The continuity equation of (\ref{continuity equation}) thus becomes \begin{equation} \partial_t\underline{\rho}+g\partial_{q_1}\big(\underline{\rho} B\partial_{q_2}\underline{S}\big)+g\partial_{q_2}\big(\underline{\rho}B\partial_{q_1}\underline{S}\big)=0. \label{continuity equation classical quantity} \end{equation}
On the other hand, from Eq. (\ref{classical velocity field classical quantity}), $f$ defined in Eq. (\ref{classical velocity field}) is given by \begin{equation} f_1(\underline{S})=gB\partial_{q_2}\underline{S},\hspace{2mm}f_2(\underline{S})=gB\partial_{q_1}\underline{S}. \label{magic classical quantity} \end{equation} Hence, Eq. (\ref{fundamental equation}) becomes \begin{eqnarray} \underline{\rho}\mapsto\rho P(\lambda),\hspace{20mm}\nonumber\\ \partial_{q_i}\underline{S}\mapsto\partial_{q_i}S+\frac{\lambda}{2}\frac{\partial_{q_i}\rho}{\rho},\hspace {2mm}i=1,2,\hspace{0mm}\nonumber\\ \partial_{t}\underline{S}\mapsto\partial_tS+\frac{\lambda}{2}\frac{\partial_t\rho}{\rho}+\frac{g\lambda}{2}\partial_{q_1}\big(B\partial_{q_2}S\big)\nonumber\\ +\frac{g\lambda}{2}\partial_{q_2}\big(B\partial_{q_1}S\big).\hspace{0mm} \label{fundamental equation classical quantity} \end{eqnarray}
Let us see how the above set of equations change Eqs. (\ref{H-J equation classical quantity}) and (\ref{continuity equation classical quantity}). Imposing the first two equations of (\ref{fundamental equation classical quantity}) into Eq. (\ref{continuity equation classical quantity}) one has \begin{eqnarray} \partial_t\rho+g\partial_{q_1}\big(\rho B\partial_{q_2}S\big)+g\partial_{q_2}\big(\rho B\partial_{q_1}S\big)\nonumber\\ +\frac{g\lambda}{2}\partial_{q_1}\big(B\partial_{q_2}\rho\big)+\frac{g\lambda}{2}\partial_{q_2}\big(B\partial_{q_1}\rho\big)=0. \label{FPE classical quantity} \end{eqnarray} Next, imposing the last two equations of Eq. (\ref{fundamental equation classical quantity}) into Eq. (\ref{H-J equation classical quantity}) one obtains, after arrangement, \begin{eqnarray} \partial_tS+gB\partial_{q_1}S\partial_{q_2}S-g\lambda^2B\frac{\partial_{q_1}\partial_{q_2}R}{R}+\frac{g\lambda^2}{2}\Big(\partial_{q_1}B\frac{\partial_{q_2}R}{R}\Big)\nonumber\\ +\frac{\lambda}{2\rho}\Big(\partial_t\rho+g\partial_{q_1}\big(\rho B\partial_{q_2}S\big)+g\partial_{q_2}\big(\rho B\partial_{q_1}S\big)\hspace{20mm}\nonumber\\ +\frac{g\lambda}{2}\partial_{q_1}\big(B\partial_{q_2}\rho\big)+\frac{g\lambda}{2}\partial_{q_2}\big(B\partial_{q_1}\rho\big)\Big)=0,\hspace{5mm}\nonumber\\ \label{HJM classical quantity} \end{eqnarray} where $R\doteq\sqrt{\rho}$ and we have used Eq. (\ref{fluctuation decomposition}). Inserting Eq. (\ref{FPE classical quantity}), one thus obtains \begin{eqnarray} \partial_tS+gB\partial_{q_1}S\partial_{q_2}S\hspace{40mm}\nonumber\\ -g\lambda^2B\frac{\partial_{q_1}\partial_{q_2}R}{R}+\frac{g\lambda^2}{2}\Big(\partial_{q_1}B\frac{\partial_{q_2}R}{R}\Big)=0. \label{HJM classical quantity} \end{eqnarray} We have thus pair of coupled equations (\ref{FPE classical quantity}) and (\ref{HJM classical quantity}) which are parameterized by the random variable $\lambda=\pm\hbar$.
One can again see that $S(q,\hbar;t)=S(q,-\hbar;t)=S_Q(q;t)$ satisfies the same differential equation of (\ref{HJM classical quantity}) where $\lambda^2$ is replaced by $\hbar^2$. Hence, averaging over the fluctuations of $\lambda=\pm\hbar$ which is assumed to be equally probable, Eqs. (\ref{FPE classical quantity}) and (\ref{HJM classical quantity}) become \begin{eqnarray} \partial_t\rho+g\partial_{q_1}\big(\rho B\partial_{q_2}S_Q\big)+g\partial_{q_2}\big(\rho B\partial_{q_1}S_Q\big)=0,\hspace{3mm}\nonumber\\ \partial_tS_Q+gB\partial_{q_1}S_Q\partial_{q_2}S_Q\hspace{40mm}\nonumber\\ -g\hbar^2B\frac{\partial_{q_1}\partial_{q_2}R}{R}+\frac{g\hbar^2}{2}\Big(\partial_{q_1}B\frac{\partial_{q_2}R}{R}\Big)=0. \label{Madelung equation classical quantity} \end{eqnarray}
Recalling Eq. (\ref{Schroedinger wave function}) that $\Psi_Q=R\exp(iS_Q/\hbar)$, the above pair of coupled equations can be written into the Schr\"odinger equation $i\hbar\partial_t\Psi_Q=\hat{H}\Psi_Q$ with quantum Hamiltonian given by \begin{equation} {\hat H}=\frac{g}{2}\big(B(q_1){\hat p}_1+{\hat p}_1B(q_1)\big){\hat p}_2. \end{equation} Hence, comparing the above equation with Eq. (\ref{classical Hamiltonian classical quantity}), we have the following quantization mapping in the context of measurement: \begin{eqnarray} \underline{p}_2\mapsto\hat{p}_2,\hspace{20mm}\nonumber\\ B(q_1)\underline{p}_1\mapsto\frac{1}{2}\big(B(q_1){\hat p}_1+{\hat p}_1B(q_1). \end{eqnarray}
\end{document} | arXiv |
Journal of Fluid Mechanics
Periodically driven Taylor–Coue...
10 July 2018 , pp. 834-845
Periodically driven Taylor–Couette turbulence
Ruben A. Verschoof (a1), Arne K. te Nijenhuis (a1), Sander G. Huisman (a1), Chao Sun (a1) (a2) and Detlef Lohse (a1) (a3)...
1 Physics of Fluids, Max Planck Center Twente for Complex Fluid Dynamics, MESA+ Institute, and J. M. Burgers Center for Fluid Dynamics, Department of Science and Engineering, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands
2 Center for Combustion Energy and the Department of Energy and Power Engineering, Tsinghua University, 100084 Beijing, PR China
3 Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany
DOI: https://doi.org/10.1017/jfm.2018.276
We study periodically driven Taylor–Couette turbulence, i.e. the flow confined between two concentric, independently rotating cylinders. Here, the inner cylinder is driven sinusoidally while the outer cylinder is kept at rest (time-averaged Reynolds number is $Re_{i}=5\times 10^{5}$ ). Using particle image velocimetry, we measure the velocity over a wide range of modulation periods, corresponding to a change in Womersley number in the range $15\leqslant Wo\leqslant 114$ . To understand how the flow responds to a given modulation, we calculate the phase delay and amplitude response of the azimuthal velocity. In agreement with earlier theoretical and numerical work, we find that for large modulation periods the system follows the given modulation of the driving, i.e. the behaviour of the system is quasi-stationary. For smaller modulation periods, the flow cannot follow the modulation, and the flow velocity responds with a phase delay and a smaller amplitude response to the given modulation. If we compare our results with numerical and theoretical results for the laminar case, we find that the scalings of the phase delay and the amplitude response are similar. However, the local response in the bulk of the flow is independent of the distance to the modulated boundary. Apparently, the turbulent mixing is strong enough to prevent the flow from having radius-dependent responses to the given modulation.
Send article to Kindle
Ruben A. Verschoof (a1), Arne K. te Nijenhuis (a1), Sander G. Huisman (a1), Chao Sun (a1) (a2) and Detlef Lohse (a1) (a3)
Available formats PDF Please select a format to send.
Send article to Dropbox
To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.
Send article to Google Drive
To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.
© 2018 Cambridge University Press
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
†Email addresses for correspondence: [email protected], [email protected]
Ahlers, G. 1987 Effect of time-periodic modulation of the driving on Taylor-vortex flow. Bull. Am. Phys. Soc. 32, 2068.
Ahlers, G. as cited by Barenghi, C. F. & Jones, C. A. 1989 Modulated Taylor–Couette flow. J. Fluid Mech. 208, 127–160.
Borrero-Echeverry, D., Schatz, M. F. & Tagg, R. 2010 Transient turbulence in Taylor–Couette flow. Phys. Rev. E 81, 025301.
Cadot, O., Titon, J. H. & Bonn, D. 2003 Experimental observation of resonances in modulated turbulence. J. Fluid Mech. 485, 161–170.
Cekli, H. E., Tipton, C. & Van De Water, W. 2010 Resonant enhancement of turbulent energy dissipation. Phys. Rev. Lett. 105, 044503.
Chavanne, X., Chilla, F., Castaing, B., Hebral, B., Chabaud, B. & Chaussy, J. 1997 Observation of the ultimate regime in Rayleigh–Bénard convection. Phys. Rev. Lett. 79, 3648.
Chien, C.-C., Blum, D. B. & Voth, G. A. 2013 Effects of fluctuation energy input on the small scales in turbulence. J. Fluid Mech. 737, 527–551.
Eckhardt, B., Grossmann, S. & Lohse, D. 2007 Torque scaling in turbulent Taylor–Couette flow between independently rotating cylinders. J. Fluid Mech. 581, 221–250.
Fardin, M. A., Perge, C. & Taberlet, N. 2014 'The hydrogen atom of fluid dynamics': introduction to the Taylor–Couette flow for soft matter scientists. Soft Matt. 10, 3523.
Ganske, A., Gebhardt, Th. & Grossmann, S. 1994 Taylor–Couette flow with time modulated inner cylinder velocity. Phys. Lett. A 192, 74–78.
van Gils, D. P. M., Bruggert, G. W., Lathrop, D. P., Sun, C. & Lohse, D. 2011 The Twente Turbulent Taylor–Couette (T3 C) facility: strongly turbulent (multi-phase) flow between independently rotating cylinders. Rev. Sci. Instrum. 82, 025105.
Grossmann, S. & Lohse, D. 2011 Multiple scaling in the ultimate regime of thermal convection. Phys. Fluids 23, 045108.
Grossmann, S., Lohse, D. & Sun, C. 2016 High Reynolds number Taylor–Couette turbulence. Annu. Rev. Fluid Mech. 48, 53–80.
Hamlington, P. E. & Dahm, W. J. A. 2009 Frequency response of periodically sheared homogeneous turbulence. Phys. Fluids 21, 055107.
He, S. & Jackson, S. D. 2009 An experimental study of pulsating turbulent flow in a pipe. Eur. J. Mech. (B/Fluids) 28 (2), 309–320.
von der Heydt, A., Grossmann, S. & Lohse, D. 2003a Resonances in modulated turbulence. Phys. Rev. E 67, 046308.
von der Heydt, A., Grossmann, S. & Lohse, D. 2003b Resonances in modulated turbulence. Part II. Numerical simulations. Phys. Rev. E 68, 066302.
Hooghoudt, J. O., Lohse, D. & Toschi, F. 2001 Decaying and kicked turbulence in a shell model. Phys. Fluids 13, 2013.
Huisman, S. G., van Gils, D. P. M., Grossmann, S., Sun, C. & Lohse, D. 2012 Ultimate turbulent Taylor–Couette flow. Phys. Rev. Lett. 108, 024501.
Huisman, S. G., Lohse, D. & Sun, C. 2013a Statistics of turbulent fluctuations in counter-rotating Taylor–Couette flows. Phys. Rev. E 88, 063001.
Huisman, S. G., Scharnowski, S., Cierpka, C., Kähler, C., Lohse, D. & Sun, C. 2013b Logarithmic boundary layers in strong Taylor–Couette turbulence. Phys. Rev. Lett. 110, 264501.
Jin, X.-L. & Xia, K.-Q. 2008 An experimental study of kicked thermal turbulence. J. Fluid Mech. 606, 133–151.
Kraichnan, R. H. 1962 Turbulent thermal convection at arbritrary Prandtl number. Phys. Fluids 5, 1374.
Kuczaj, A. K., Geurts, B. J. & Lohse, D. 2006 Response maxima in time-modulated turbulence: direct numerical simulations. Europhys. Lett. 73, 851.
Kuczaj, A. K., Geurts, B. J., Lohse, D. & van de Water, W. 2008 Turbulence modification by periodically modulated scale-dependent forcing. Comput. Fluids 37, 816–824.
Lodahl, C. R., Sumer, B. M. & Fredsøe, J. 1998 Turbulent combined oscillatory flow and current in a pipe. J. Fluid Mech. 373, 313–348.
Lohse, D. 2000 Periodically kicked turbulence. Phys. Rev. E 62, 4946.
Manna, M., Vacca, A. & Verzicco, R. 2012 Pulsating pipe flow with large-amplitude oscillations in the very high frequency regime. Part 1. Time-averaged analysis. J. Fluid Mech. 700, 246–282.
Mao, Z-.X. & Hanratty, T. J. 1986 Studies of the wall shear stress in a turbulent pulsating pipe flow. J. Fluid Mech. 170, 545–564.
Ostilla-Mónico, R., Verzicco, R., Grossmann, S. & Lohse, D. 2014 Turbulence decay towards the linearly stable regime of Taylor–Couette flow. J. Fluid Mech. 748, R3.
Ostilla-Mónico, R., Zhu, X., Spandan, V. S., Verzicco, R. & Lohse, D. 2017 Life stages of wall-bounded decay of Taylor–Couette turbulence. Phys. Rev. Fluids 2, 114601.
Ramaprian, B. R. & Tu, S. W. 1980 An experimental study of oscillatory pipe flow at transitional Reynolds number. J. Fluid Mech. 100, 513–544.
Sarpkaya, T. 1966 Experimental determination of the critical Reynolds number for pulsating Poiseuille flow. Trans. ASME: J. Basic Engng 88, 589–598.
Shemer, L., Wygnanski, I. & Kit, E. 1985 Pulsating flow in a pipe. J. Fluid Mech. 153, 313–337.
Sterl, S., Li, H.-M. & Zhong, J.-Q 2016 Dynamical and statistical phenomena of circulation and heat transfer in periodically forced rotating turbulent Rayleigh–Bénard convection. Phys. Rev. Fluids 1, 084401.
Verschoof, R. A., Huisman, S. G., van der Veen, R. C. A., Sun, C. & Lohse, D. 2016 Self-similar decay of high Reynolds number Taylor–Couette turbulence. Phys. Rev. Fluids 1, 062402(R).
Walsh, T. J. & Donnelly, R. J. 1988 Taylor–Couette flow with periodically corotated and counterrotated cylinders. Phys. Rev. Lett. 60, 700.
Womersley, J. R. 1955 Method for the calculation of velocity rate of flow and viscous drag in arteries when the pressure gradient is known. J. Physiol. 127, 553–572.
Yellin, E. L. 1966 Laminar–turbulent transition process in pulsatile flow. Circulat. Res. 19, 791–804.
Yu, D. & Girimaji, S. S. 2006 Direct numerical simulation of homogeneous turbulence subject to periodic shear. J. Fluid Mech. 566, 117–151.
URL: /core/journals/journal-of-fluid-mechanics
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.
JFM classification
Turbulent Flows: Rotating turbulence
Convection: Taylor–Couette flow
Turbulent Flows: Turbulent Flows | CommonCrawl |
macro level effects of a change in the value for epsilon naught
I'm developing a story set 4-5 centuries after the fall of a high-tech civilization. The fall of said civilization is indirectly related to a loss of electricity. i.e. in the chaos of their technology no longer working like it should, they devolved into a scavengerpunk world. I'm toying with the idea of the loss of electricity being caused by some form of energy field disrupting the electromagnetic force and this field would do so by manipulating the electric constant epsilon naught.
Is this workable? I.e. can the field cause enough disruption to electrical devices without causing life to not exist? And if so, what other side effects on the environment would occur? E.g. would a campfire that would normally have an orange flame, now have a red flame because of a change in the atomic energy level?
A few specifications:
I don't need the field to turn the em force completely off, just disrupt tech enough so that it doesn't work
the field would be non-uniform. i.e. concentrated on one spot of the planet, radiating outward and subsiding so that a spot on the opposite side of the planet would still be able to use electricity.
the field would also be self-maintaining through the centuries, such that the resulting society on the planet wouldn't have been able to remake any electrical devices after the fall.
All the research I've done on this site and others address what would happen if the EM force was turned off completely, or what would happen if the fundamental forces were a few percent different and how that would affect the ability of stars or carbon atoms to form.
A few other articles suggested that an easier way to go no electricity would be EMP, grey goo, societal restrictions, mineral scarcity, or even simply say magic did it. one of the more interesting ones I saw was bacteria suddenly getting a taste for copper. And I may end up going with one of those instead of the electric constant way if it doesn't work.
But before I go that way, I would appreciate some feedback on the feasibility of an energy field disrupting the electric constant.
Okay. I didn't think it was possible to modify the electric constant enough to get rid of electricity and still have intelligent life, but I wanted to ask anyway. I'm probably going to go with a modified version of the grey goo method mentioned in What kind of event could stop electricity?.
Essentially, the nanotech will absorb any electricity it finds from any active power sources. But when those power sources are turned off, or stop working from a lack of maintenance, the nanotech still wants to absorb electricity. So without any power sources feeding them, they would draw energy directly from the world. And this will have the effect of making any new power sources produce less electricity than they normally would because the nanotech pulling electricity from everywhere would simulate the effect of having a dielectric superimposed over a vacuum. i.e. more resistance to electric flow.
To quote this page: https://en.wikipedia.org/wiki/Permittivity
permittivity describes the amount of charge needed to generate one unit of electric flux in a particular medium.
A consequence of this would be that you could turn on a whole bunch of power sources, you wouldn't see any electricity output, but the permittivity would drop back down to normal because the nanotech is no longer drawing power from the world.
I also get that messing with the electric constant in this manner would affect lots of other constants, and I don't necessarily have an issue with this, as long as the effects are consistent. And I'm not really interested in changing the fine structure constant by as much as 4%, but to redirect my initial question, if the modification was small, such as 1/1000th of a percent, would there be any visible effects on the environment?
reality-check electricity electromagnetism
bronzeapricotbronzeapricot
$\begingroup$ You might want to check out What would happen if electricity stopped working? $\endgroup$ – a CVn♦ Dec 27 '17 at 7:30
$\begingroup$ Please, use capital letters, punctuation etc. And what do you mean by "manipulating the electric constant epsilon naught"? You know that this constant is not some magic setting of this universe, but rather something that came up from equations, theory and so on. $\endgroup$ – Mołot Dec 27 '17 at 7:39
$\begingroup$ This question feels a bit to me like asking "what would happen if the value of pi were changed?" - you can't. You just can't. $\endgroup$ – Xenocacia Dec 27 '17 at 8:49
$\begingroup$ The specific value of ε₀ depends on the particular system of measurement units. If you don't like the value it has in SI, then you can use one of the various CGS systems which either do away with ε₀ (for example, the CGS electrostatic system), or assign a special value to it in order to make Coulomb's law simpler (for example, the Gaussian system). $\endgroup$ – AlexP Dec 27 '17 at 15:13
If you change the value of $$ \epsilon_0$$ you will end up affecting the value of the fine-structure constant $$ \alpha = \frac{1}{4\pi\epsilon_0}\frac{2\pi e^2}{hc}$$ where e is the elementary charge, c the speed of light and h is Planck's constant.
To quote this page
The anthropic principle is a controversial argument of why the fine-structure constant has the value it does: stable matter, and therefore life and intelligent beings, could not exist if its value were much different. For instance, were α to change by 4%, stellar fusion would not produce carbon, so that carbon-based life would be impossible. If α were > 0.1, stellar fusion would be impossible and no place in the universe would be warm enough for life as we know it.
Long story short: if you change epsilon, you don't have to worry about intelligent life any longer, and this include electricity too.
L.Dutch♦L.Dutch
Not the answer you're looking for? Browse other questions tagged reality-check electricity electromagnetism or ask your own question.
What would happen if electricity stopped working?
What kind of event could stop electricity?
Lightning Rifle
What would be different in a world with insanely cheap electricity?
What would be the immediate effects of no electric current?
Electric Universe: can an object be pinned at a solar pole with magnetic flux?
How can an optical signal be converted into a mechanical/acoustic signal without using electricity?
What gases are made by a live, crackling, exposed electrical wire?
What would be the side effects of a massive, strong magnetic field?
Can a biological creature detect and absorb electricity from power sources?
Schools of magic based around energies, not elements
Applications of Electromagnetism in "Force Fields" | CommonCrawl |
\begin{definition}[Definition:Complementary Event]
Let the probability space of an experiment $\EE$ be $\struct {\Omega, \Sigma, \Pr}$.
Let $A \in \Sigma$ be an event in $\EE$.
The '''complementary event''' to $A$ is defined as $\relcomp \Omega A$.
That is, it is the subset of the sample space of $\EE$ consisting of all the elementary events of $\EE$ that are not in $A$.
\end{definition} | ProofWiki |
Falcon (signature scheme)
Falcon is a post-quantum signature scheme selected by the NIST at the fourth round of the post-quantum standardisation process. It has been designed by Thomas Prest, Pierre-Alain Fouque, Jeffrey Hoffstein, Paul Kirchner, Vadim Lyubashevsky, Thomas Pornin, Thomas Ricosset, Gregor Seiler, William Whyte and Zhenfei Zhang. It relies on the hash-and-sign technique over the Gentry, Peikert and Vaikuntanathan framework over NTRU lattices. The name Falcon is an acronym for Fast Fourier lattice-based compact signatures over NTRU.
Properties
The design rationale of Falcon takes advantage of multiple tools to ensure compactness and efficiency with provable security. To achieve this goal, the use of a NTRU lattice allows the size of the signatures and public-key to be relatively small, while Fast Fourier sampling permits efficient signature computations.
From a security point of view, the Gentry, Peikert and Vaikuntanathan framework enjoys a security reduction in the Quantum Random Oracle Model.
Implementations and Performances
The authors of Falcon provide a reference implementation in C as required by the NIST and one in Python for simplicity.
The set of parameters suggested by Falcon imply signatures of size 666 bytes for the NIST security level 1 (security comparable to breaking AES-128 bits). The key generation can be performed in 8.64 ms with a throughput of approximatively 6,000 signature per second and 28,000 verifications per second.
On the other hand, the NIST security level 5 (comparable to breaking AES-256) requires signature of 1,280 bytes, a key generation under 28 ms and a throughput of 2,900 signatures per second and 13,650 verifications per second.
See also
• Post-quantum cryptography
• Lattice-based cryptography
• NTRU
• NIST Post-Quantum Cryptography Standardization
References
1.^ Thomas Prest; Pierre-Alain Fouque; Jeffrey Hoffstein; Paul Kirchner; Vadim Lyubashevsky; Thomas Pornin; Thomas Ricosset; Gregor Seiler; William Whyte; Zhenfei Zhang, Falcon: Fast-Fourier Lattice-based Compact Signatures over NTRU (PDF)
2.^ Falcon official website
3.^ List of NIST PQC selected candidates
4.^ Craig Gentry; Chris Peikert; Vinod Vaikuntanathan (2008). Trapdoors for Hard Lattices and New Cryptographic Constructions. STOC.
5.^ Dan Boneh; Özgür Dagdelen; Marc Fischlin; Anja Lehmann; Christian Schaffner; Mark Zhandry (2011). Random Oracles in a Quantum World. Asiacrypt.
6.^ Reference implementation of Falcon in C
7.^ Implementation of Falcon in Python
8.^ NIST Post-Quantum Cryptography Call for Proposals
| Wikipedia |
Electronic Floquet Vortex States Induced by Light
Ahmadabadi, Iman
We propose a scheme to create an electronic Floquet vortex state by irradiating the circularly-polarized laser light on the two-dimensional semiconductor. We study the properties of the Floquet vortex states analytically and numerically using methods analogous to the techniques used for the analysis of superconducting vortex states, while we exhibit that the Floquet vortex created in the current system has the wider tunability. To illustrate the impact of such tunability in quantum engineering, we demonstrate how this vortex state can be used for quantum information processing.
Study of 1D soft bosons via the MCTDH approach
Apostoli, Christian
In this work we explore some of the capabilities of the multiconfiguration time-dependent Hartree approach (MCTDH), a general and powerful method to compute quantum dynamics simulations. Its strength lies in the particular ansatz it uses for the many-body wave function: a superposition, with time-dependent coefficients, of direct-product states which are built from a time-dependent one-particle basis. When one applies the time-dependent variational principle to this ansatz, a set of coupled equations is found for the coefficients and the one-particle basis functions. This ensures that, during a numerical solution of these equations, at every time the system is represented in a variationally optimal basis. We apply the MCTDH approach to a 1D system of soft bosons that undergoes a quantum phase transition from a standard Luttinger Liquid to a Cluster Luttinger Liquid (CLL). We simulate this system in real-time, and introduce a method for computing the energy of the low-lying excited states by observing the response of the system to a weak time-dependent periodic potential. In the Tonks-Girardeau condition, our results are in good accordance with the Bogolyubov spectrum.
Signatures of Many-Body Localization in the Dynamics of Two-Level Systems in Glasses (jointly presented with Federico Balducci)
Artiaco, Claudia
Signatures of Many-Body Localization in the Dynamics of Two-Level Systems in Glasses Abstract: We investigate the quantum dynamics of two-level systems (TLSs) in glasses at low temperatures (1 K and below). We study an ensemble of TLSs coupled to phonons. By integrating out the phonons within the framework of the GKSL master equation, we derive the explicit form of the interactions among TLSs and of the dissipative terms. We find that the dynamics of the system shows clear signatures of many-body localization physics (in particular a power-law decay of the concurrence, which measures pairwise entanglement also in non-isolated systems) even in the presence of dissipation, if the latter is not too large. This feature can be ascribed to the presence of strong, long-tailed disorder characterizing the distributions of the model parameters. Our findings show that assuming ergodicity when discussing TLS physics might not be justified for all kinds of experiments on low-temperature glasses.
Signatures of Many-Body Localization in the Dynamics of Two-Level Systems in Glasses (jointly presented with Claudia Artiaco)
Balducci, Federico
Relaxation dynamics of the three-dimensional Coulomb Glass model
Bhandari, Preeti
We consider the relaxation properties of electron glass, which is a system in which all the electron states are localized and the dynamics occurs through phonon-assisted hopping amongst these states. We model the system by a lattice of localized states which have random energies and interact through the Coulomb interaction. The presence of disorder and the long-range interaction makes the system glassy which results in slow dynamics towards reaching equilibrium, aging (system dynamics depending on the history), and memory effects. Further, a much-discussed question is whether there is an equilibrium transition to glassy phase or not as the temperature is lowered. The wide range of timescales involved in such systems makes it more difficult to solve numerically. We have modeled the kinetics of site-occupation numbers as Ising spins by Kawasaki Dynamics (spin-exchange) as in our system only half the sites are occupied and the number of particles is conserved. The master equation governing the dynamics is solved in mean-field approximation. We use the eigenvalue distribution of the dynamical matrix to characterize relaxation laws as a function of localization length at low temperatures. Our results demonstrate the dominant role played by the localization length on the relaxation laws. For very small localization lengths we find a crossover from exponential relaxation at long times to a logarithmic decay at intermediate times. No logarithmic decay at the intermediate times is observed for large localization lengths.
Quasiparticle dynamics of symmetry resolved entanglement after a quench: the examples of conformal field theories and free fermions
Bonsignori, Riccarda
The time evolution of the entanglement entropy is a key concept to understand the structure of a non-equilibrium quantum state. In a large class of models, such evolution can be understood in terms of a semiclassical picture of moving quasiparticles spreading the entanglement throughout the system. However, it is not yet known how the entanglement splits between the sectors of an internal local symmetry of a quantum many-body system. Here, guided by the examples of conformal field theories and free-fermion chains, we show that the quasiparticle picture can be adapted to this goal, leading to a general conjecture for the charged entropies whose Fourier transform gives the desired symmetry resolved entanglement $S_n(q)$. We point out two physically relevant effects that should be easily observed in atomic experiments: a delay time for the onset of $S_n(q)$ which grows linearly with $|\Delta q|$ (the difference from the charge $q$ and its mean value), and an effective equipartition when $|\Delta q|$ is much smaller than the subsystem size.
Lieb-Robinson bounds and out-of-time order correlators in a long-range spin chain
Colmenarez Gomez, Luis Andres
Lieb-Robinson bounds quantify the maximal speed of information spreading in nonrelativistic quantum systems. We discuss the relation of Lieb-Robinson bounds to out-of-time order correlators, which correspond to different norms of commutators $C(r,t) = [A(t),B]$ of local operators. Using an exact Krylov space-time evolution technique, we calculate these two different norms of such commutators for the spin-1/2 Heisenberg chain with interactions decaying as a power law $1/r^\alpha$ with distance $r$. Our numerical analysis shows that both norms (operator norm and normalized Frobenius norm) exhibit the same asymptotic behavior, namely, a linear growth in time at short times and a power-law decay in space at long distance, leading asymptotically to power-law light cones for $\alpha<1$ and to linear light cones for $\alpha>1$. The asymptotic form of the tails of $C(r,t)~t/r^\alpha$ is described by short-time perturbation theory, which is valid at short times and long distances.
Extraction of many-body Chern number from a single wave function
Dehghani, Hossein
"The quantized Hall conductivity of integer and fractional quantum Hall (IQH and FQH) states is directly related to a topological invariant, the many-body Chern number. The conventional calculation of this invariant in interacting systems requires a family of many-body wave functions parameterized by twist angles in order to calculate the Berry curvature. In this work, we demonstrate how to extract the Chern number given a single many-body wave function, without knowledge of the Hamiltonian. We perform extensive numerical simulations involving IQH and FQH states to validate these methods. We also propose an ancilla-free experimental scheme for the measurement of this invariant. Specifically, we use the statistical correlations of randomized measurements to infer the MBCN of a wavefunction."
Correspondence principle for quantum many-body scars
Desaules, Jean-Yves
Quantum many-body scars represent a weak form of ergodicity breaking that gives rise to robust periodic revivals in kinetically constrained quantum systems, such as the PXP model describing strongly-interacting Rydberg atoms. By analogy to quantum scars in single-particle quantum billiards, the many-body scarred eigenstates are distinguished by their anomalous overlap with the so-called quasimodes, i.e., the wave functions that concentrate along classical periodic orbits. While the classical orbits in the PXP model have previously been constructed using the Time-Dependent Variational Principle (TDVP), the corresponding quasimodes have only been found numerically. Here we introduce a new approach to constructing quasimodes in the PXP model based on the subspace symmetrised over permutations on each sublattice. We show that this method is amenable to analytic treatment and leads to an efficient construction of the quasimodes, allowing to study the dynamics of the PXP model in large systems on the order of hundreds of atoms. Finally, our approach provides a tractable way of introducing quantum fluctuations at all orders on top of the classical equations of motion defined by the TDVP.
The effect of intrinsic quantum fluctuations on the phase diagram of anisotropic dipolar magnets
Dollberg, Tomer
LiHoF4 is a quantum magnet known to be a good physical realization of the transverse field Ising model with dipolar interactions. Results from previous studies, using various Monte Carlo techniques and mean-field analyses, show a persistent discrepancy with experimental results for the $B_x − T$ phase diagram. Namely, in the low $B_x$ regime, the experimental phase boundary separating the ferromagnetic and paramagnetic phases has a much smaller dependence on magnetic field in comparison to the theoretical predictions. In this work we propose a mechanism which may account for the discrepancy. Offdiagonal terms of the dipolar interaction, more dominant in the disordered paramagnetic phase, reduce the energy of the paramagnetic phase, and consequently reduce the critical temperature. Using classical Monte Carlo simulations, in which we explicitly take the modification of the Ising states due to the offdiagonal terms into account, we show that the inclusion of the these terms reduces $T_c$ markedly at zero transverse field. We also show that the effect is diminished with increasing transverse field, leading to the above mentioned field dependence of the critical temperature.
Vortices and Fractons
Doshi, Darshil
We discuss a simple and experimentally available realization of fracton physics. We note that superfluid vortices conserve total dipole moment and trace of the quadrupole moment of vorticity. This establishes a relation to a traceless scalar charge theory in two spatial dimensions. We also consider the limit where the number of vortices is large and show that emergent vortex hydrodynamics conserves these moments too. Finally, we compare the motion of vortices and of fractons on curved surfaces; and find that they agree. This opens a route to experimental study of the interplay between fracton physics and curved space. Our conclusions also apply to charged particles in strong magnetic field. (Reference : D Doshi, A Gromov - arXiv preprint arXiv:2005.03015, 2020)
Anomalous Diffusion in Dipole- and Higher-Moment Conserving Systems
Feldmeier, Johannes
The presence of global conserved quantities in interacting systems generically leads to diffusive transport at late times. Here, we show that systems conserving the dipole moment of an associated global charge, or even higher moment generalizations thereof, escape this scenario, displaying subdiffusive decay instead. Modelling the time evolution as cellular automata for specific cases of dipole- and quadrupole-conservation, we numerically find distinct anomalous exponents of the late time relaxation. We explain these findings by analytically constructing a general hydrodynamic model that results in a series of exponents depending on the number of conserved moments, yielding an accurate description of the scaling form of charge correlation functions. We analyze the spatial profile of the correlations and discuss potential experimentally relevant signatures of higher moment conservation.
Incommensurability-induced sub-ballistic narrow-band-states in twisted bilayer graphene
Gonçalves, Miguel
We study the localization properties of electrons in incommensurate twisted bilayer graphene for small angles, encompassing the narrow-band regime, by numerically exact means. Sub-ballistic states are found within the narrow-band region around the magic angle. Such states are delocalized in momentum-space and follow non-Poissonian level statistics, in contrast with their ballistic counterparts found for close commensurate angles. Transport results corroborate this picture: for large enough systems, the conductance decreases with system size for incommensurate angles within the sub-ballistic regime. Our results show that incommensurability effects are of crucial importance in the narrow-band regime. The incommensurate nature of a general twist angle must therefore be taken into account for an accurate description of magic-angle twisted bilayer graphene.
Quantum scars of bosons with correlated hopping
Hudomal, Ana
Recent experiments on Rydberg atom arrays have found evidence of anomalously slow thermalization and persistent density oscillations, which have been interpreted as a many-body analog of the phenomenon of quantum scars. Periodic dynamics and atypical scarred eigenstates originate from a "hard" kinetic constraint: the neighboring Rydberg atoms cannot be simultaneously excited. Here we propose a realization of quantum many-body scars in a 1D bosonic lattice model with a "soft" constraint in the form of density-assisted hopping. We discuss the relation of this model to the standard Bose-Hubbard model and possible experimental realizations using ultracold atoms. We find that this model exhibits similar phenomenology to the Rydberg atom chain, including weakly entangled eigenstates at high energy densities and the presence of a large number of exact zero energy states, with distinct algebraic structure.
Model wavefunctions for interfaces between lattice quantum Hall states
Jaworowski, Błażej
While the physics of the edges between topological orders and vacuum has been thoroughly investigated, less is known about the properties of interfaces between different topological orders. Such systems have recently attracted significant attention, partly due to their potential applications in quantum computing. However, they are difficult to study numerically in a bottom-up manner, because relatively large system sizes are needed to capture their properties correctly. In this work, we overcome this obstacle by employing the conformal field theory to create model wavefunctions for interfaces between two different Laughlin states on the lattice. These objects can be studied using Monte Carlo methods for systems much larger than available within the exact diagonalization approach. We study their properties such as the entanglement entropy scaling and the correlation functions. Similar wavefunctions are also created for systems with localized anyons, allowing to extract the charge, the density profile and the mutual statistics of these excitations. Within our approach, we can explicitly simulate the crossing of the interface by the anyons and show that some of them lose their fractional statistics in such a process, which is in accordance with earlier "top-down" results obtained from field theory, and is possibly related to the entanglement entropy scaling at the interface.
Floquet Gauge Pump
Kumar, Abhishek
Gauge pumps are spatially-resolved probes that can reveal discrete symmetries due to nontrivial topology. We introduce the Floquet gauge pump whereby a dynamically engineered Floquet Hamiltonian is employed to reveal the inherent topology of the ground state in interacting systems. We demonstrate this concept in a 1D XY model with periodically driven couplings and a transverse field. In the high-frequency limit, we obtain a Floquet Hamiltonian consisting of the static XY and dynamically generated Dzyaloshinsky-Moriya interactions (DMI) terms. We show that anisotropy in the couplings facilitates a magnetization current across a dynamically imprinted junction. In fermionic language, this corresponds to an unconventional Josephson junction with both hopping and pairing tunneling terms. The magnetization current depends on the phases of complex coupling terms, with the XY interaction as the real and DMI as the imaginary part. It shows 4π periodicity revealing the topological nature of the ground state manifold in the ordered phase, in contrast to the trivial topology in the disordered phase. We discuss the requirements to realize the Floquet gauge pump with interacting trapped ions.
Emergent Bloch oscillations in a kinetically constrained Rydberg spin lattice
Magoni, Matteo
We explore the relaxation dynamics of elementary spin clusters of a kinetically constrained spin system. Inspired by experiments with Rydberg lattice gases, we focus on the situation in which an excited spin leads to a "facilitated" excitation of a neighboring spin. We show that even weak interactions that extend beyond nearest neighbors can have a dramatic impact on the relaxation behavior: they generate a linear potential, which under certain conditions leads to the onset of Bloch oscillations of spin clusters. These hinder the expansion of a cluster and more generally the relaxation of many-body states towards equilibrium. This shows that non-ergodic behavior in kinetically constrained systems may occur as a consequence of the interplay between reduced connectivity of many-body states and weak interparticle interactions. We furthermore show that the emergent Bloch oscillations identified here can be detected in experiment through measurements of the Rydberg atom density, and discuss how spin-orbit coupling between internal and external degrees of freedom of spin clusters can be used to control their relaxation behavior.
Quantifying the efficiency of state preparation via quantum variational eigensolvers
Matos, Gabriel
Recently, there has been much interest in the efficient preparation of complex quantum states using low-depth quantum circuits, such as Quantum Approximate Optimization Algorithm (QAOA). While it has been numerically shown that such algorithms prepare certain correlated states of quantum spins with surprising accuracy, a systematic way of quantifying the efficiency of QAOA in general classes of models has been lacking. Here, we propose that the success of QAOA in preparing ordered states is related to the interaction distance of the target state, which measures how close that state is to the manifold of all Gaussian states in an arbitrary basis of single-particle modes. We numerically verify this for the ground state of the quantum Ising model with arbitrary transverse and longitudinal field strengths, a canonical example of a non-integrable model. Our results suggest that the structure of the entanglement spectrum, as witnessed by the interaction distance, correlates with the success of QAOA state preparation. We conclude that QAOA typically finds a solution that perturbs around the closest free-fermion state.
Improved qQuantum transport calculations for interacting nanostructures
Minarelli, Emma
Nanoelectronics devices such as semiconductor quantum dots and single molecule transistors exhibit a rich range of physical behavior due to the interplay between orbital complexity, strong electronic correlations and device geometry. Understanding and simulating the quantum transport through such nanostructures is essential for rational design and technological applications. In this poster, I present theoretical reformulations electrical conductance formulae for interacting mesoscopic quantum transport calculations in linear response, and demonstrate the improvement over standard methods with several example applications using the numerical renormalization group technique. I will treat reformulations of the Meir-Wingreen formula in the context of non-proportionate coupling set-ups and by means of perturbative verification of the Ng ansatz; of the Oguri formula in non-Fermi Liquid states and of the Kubo formula for conductance.
Anyonic molecules in atomic fractional quantum Hall liquids: a quantitative probe of fractional charge and anyonic statistics
Muñoz de las Heras, Alberto
We study the quantum dynamics of massive impurities embedded in a strongly interacting two-dimensional atomic gas driven into the fractional quantum Hall (FQH) regime under the effect of a synthetic magnetic field. For suitable values of the atom-impurity interaction strength, each impurity can capture one or more quasi-hole excitations of the FQH liquid, forming a bound molecular state with novel physical properties. An effective Hamiltonian for such anyonic molecules is derived within the Born-Oppenheimer approximation, which provides renormalized values for their effective mass, charge and statistics by combining the finite mass of the impurity with the fractional charge and statistics of the quasi-holes. The renormalized mass and charge of a single molecule can be extracted from the cyclotron orbit that it describes as a free particle in a magnetic field. The anyonic statistics introduces a statistical phase between the direct and exchange scattering channels of a pair of indistinguishable colliding molecules, and can be measured from the angular position of the interference fringes in the differential scattering cross section. Implementations of such schemes beyond cold atomic gases are highlighted, in particular in photonic systems.
Phases and Quantum Phase Transitions in an Anisotropic Ferromagnetic Kitaev-Heisenberg-$\ \Gamma$ Magnet
Nanda, Animesh
We study the spin-$1/2$ ferromagnetic Heisenberg-Kitaev-$\Gamma$ model in the anisotropic (Toric code) limit to reveal the nature of the quantum phase transition between the gapped $Z_2$ quantum spin liquid and a spin ordered phase (driven by Heisenberg interactions) as well as a trivial paramagnet (driven by pseudo-dipolar interactions, $\Gamma$). The transitions are obtained by a simultaneous condensation of the Ising electric and magnetic charges-- the fractionalized excitations of the $Z_2$ quantum spin liquid. Both these transitions can be continuous and are examples of deconfined quantum critical points. Crucial to our calculations are the symmetry implementations on the soft electric and magnetic modes that become critical. In particular, we find strong constraints on the structure of the critical theory arising from time reversal and lattice translation symmetries with the latter acting as an anyon permutation symmetry that endows the critical theory with a manifestly self-dual structure. We find that the transition between the quantum spin liquid and the spin-ordered phase belongs to a self-dual modified Abelian Higgs field theory while that between the spin liquid and the trivial paramagnet belongs to a self-dual $Z_2$ gauge theory. We also study the effect of an external Zeeman field to show an interesting similarity between the polarised paramagnet obtained due to the Zeeman field and the trivial paramagnet driven the pseudo-dipolar interactions. Interestingly, both the spin liquid and the spin ordered phases have easily identifiable counterparts in the isotropic limit and the present calculations may shed insights into the corresponding transitions in the material relevant isotropic limit.
Restricted Boltzmann machine representation for the groundstate and excited states of Kitaev Honeycomb model
Noormandipour, Mohammadreza
In this work, the capability of restricted Boltzmann machines (RBMs) to find solutions for the Kitaev honeycomb model is investigated. The measured groundstate (GS) energy of the system is compared and shown to reside within a few percent error of the analytically derived value of the energy per plaquette. Moreover, given a set of single shot measurements of exact solutions of the model, an RBM is used to perform quantum state tomography and the obtained result has a $97\%$ overlap with the exact analytic result. Furthermore, the possibility of realizing anyons in the RBM is discussed and an algorithm is given to build these anyonic excitations and braid them as a proof of concept for performing quantum gates and doing quantum computation.
Exotic Phases in Rydberg Lattice Models
Ohler, Simon
Interacting systems of Rydberg atoms have attracted much attention recently, partly due to their strong interactions and high stability. Furthermore, experimental techniques have been proposed to include synthetic gauge fields and correlated hopping, where the excitation transport between two atoms depends on the quantum state of a third atom. In our work, we are considering a system of Rydberg atoms on a two-dimensional hexagonal lattice, including both synthetic gauge fields and correlated hopping. We numerically obtain a rich phase diagram,including two disordered regimes where we find evidence to support the existence of a chiral spin-liquid-state.
Systematic large flavor fTWA approach to interaction quenches
Osterkorn, Alexander
Studying the out-of-equilibrium quantum dynamics in two-dimensional lattice models is challenging due to the lack of a general purpose simulation method. A new semiclassical approach to compute the quantum dynamics of fermions was recently developed by Davidson et. al [1], the fermionic truncated Wigner approximation (fTWA). Here, we adopt the method and combine it with the limit of high fermion degeneracy $N$ as a well-defined semiclassical expansion parameter. On the poster we consider the well-known problem of an interaction quench in the two-dimensional Hubbard model to show that the method correctly describes prethermalization [2]. In addition we discuss whether the long-time thermalization dynamics is reproduced as well. As a second application we consider quenches in ordered phases of the large-$N$ Hubbard-Heisenberg model and show that the semiclassical time-evolution leads to dephasing and subsequent decay of the order parameter. [1] SM Davidson et. al., Annals of Physics 384, pages 128-141 (2017) [2] A Osterkorn and S Kehrein, arXiv:2007.05063
Topological phonons in oxide perovskites controlled by light
Peng, Bo
Oxide perovskites have received widespread attention ever since their discovery due to the multiple physical properties they exhibit, including ferroelectricity, multiferroicity, and superconductivity. One prominent absence in this list of properties that oxide perovskites exhibit is electronic topological order. This is a consequence of the large band gaps of oxide perovskites, which make the band inversions necessary for topology impossible. We find that topological phonons – nodal rings, nodal lines, and Weyl points – are ubiquitous in oxide perovskites in terms of structures (tetragonal, orthorhombic, and rhombohedral), compounds (BaTiO$_3$, PbTiO$_3$, and SrTiO$_3$), and external conditions (photoexcitation, strain, and temperature). In particular, in the tetragonal phase of these compounds all types of topological phonons can simultaneously emerge when stabilized by photoexcitation, whereas the tetragonal phase stabilized by thermal fluctuations only hosts a more limited set of topological phonon states. In addition, we find that the photoexcited carrier density can be used to control the emergent topological states, for example driving the creation/annihilation of Weyl points and switching between nodal lines and nodal rings. Overall, we propose oxide perovskites as a versatile platform in which to study topological phonons and their manipulation with light [1]. Reference: [1] Bo Peng, Yuchen Hu, Shuichi Murakami, Tiantian Zhang, Bartomeu Monserrat. Topological phonons in oxide perovskites controlled by light. Science Advances 6, eabd1618 (2020).
Higher-order and fractional discrete time crystals in clean long-range interacting systems
Pizzi, Andrea
Discrete time crystals are periodically driven systems characterized by a response with periodicity $nT$, with $T$ the period of the drive and $n>1$. Typically, $n$ is an integer and bounded from above by the dimension of the local (or single particle) Hilbert space, the most prominent example being spin-$1/2$ systems with $n$ restricted to $2$. Here we show that a clean spin-$1/2$ system in the presence of long-range interactions and transverse field can sustain a huge variety of different `higher-order' discrete time crystals with integer and, surprisingly, even fractional $n > 2$. We characterize these (arguably prethermal) non-equilibrium phases of matter thoroughly using a combination of exact diagonalization, semiclassical methods, and spin-wave approximations, which enable us to establish their stability in the presence of competing long- and short-range interactions. Remarkably, these phases emerge in a model with continous driving and time-independent interactions, convenient for experimental implementations with ultracold atoms or trapped ions.
Anatomy of Z2 fluxes in anyon Fermi liquids and Bose condensates
Pozo Ocaña, Óscar
We study in detail the properties of pi-fluxes embedded in a state with a finite density of anyons that form either a Fermi liquid or a Bose-Einstein condensate. By employing a recently developed exact lattice bosonization in 2D, we demonstrate that such pi-flux remains a fully deconfined quasiparticle with a finite energy cost in a Fermi liquid of emergent fermions coupled to a Z2 gauge field. This pi-flux is accompanied by a screening cloud of fermions, which in the case of a Fermi gas with a parabolic dispersion binds exactly 1/8 of a fermionic hole. In addition there is a long-ranged power-law oscillatory disturbance of the liquid surrounding the pi-flux akin to Friedel oscillations. These results carry over directly to the pi-flux excitations in orthogonal metals. In sharp contrast, when the pi-flux is surrounded by a Bose-Einstein condensate of particles coupled to a Z2 gauge field, it binds a superfluid half-vortex, becoming a marginally confined excitation with a logarithmic energy cost divergence.
Anomalous localization in spin chains coupled to non-local degree of freedom
Rahmanian Koshkaki, Saeed
It has recently been predicted that many-body localization survives the presence of coupling to a non-local degree of freedom, such as a cavity mode [PRL 122, 240402 (2019)]. This poster presents recent results on anomalous properties of localization in such a setup. First, we show that in a central qudit model, an inverted mobility edge occurs, meaning that infinite temperature states are localized while low energy states are delocalized. This model may be directly realized by extending recent work on artificial cavities using atom-like mirrors [Nature 569.7758: 692 (2019)]; similar results hold for central spin models or cavity QED with appropriate cavity non-linearity. Second, we suggest a platform for realizing time crystals in cavity QED and in the absence of drive.
Simulating hydrodynamics on NISQ devices with random circuits
Richter, Jonas
An important milestone towards "quantum supremacy" has been recently achieved by using Google's noisy intermediate-scale quantum (NISQ) device Sycamore to sample from the output distribution of (pseudo-)random circuits involving up to 53 qubits. We argue that such random circuits provide tailor-made building blocks for the simulation of quantum many-body systems on NISQ devices. Specifically, we propose a two-part algorithm consisting of a random circuit followed by a trotterized Hamiltonian time evolution, which we numerically exemplify by studying the buildup of spatiotemporal correlations in one- and two-dimensional quantum spin systems. Importantly, we find that the emerging hydrodynamic scaling of the correlations is highly robust with respect to the size of the Trotter step, opening the door to reach nontrivial time scales with a small number of elementary gates. While errors within the random circuit are shown to be irrelevant for our approach, we furthermore unveil that meaningful results can be obtained for noisy time evolutions with error rates achievable on near-term hardware.
Generative Model Learning For Molecular Electronics
Rigo, Jonas
The use of single-molecule transistors in nanoelectronics devices requires a deep understanding of the generalized `quantum impurity' models describing them. Microscopic models comprise molecular orbital complexity and strong electron interactions while also treating explicitly conduction electrons in the external circuit. No single theoretical method can treat the low-temperature physics of such systems exactly. To overcome this problem, we use a generative machine learning approach to formulate effective models that are simple enough to be treated exactly by methods such as the numerical renormalization group, but still capture all observables of interest of the physical system.
Scattering Processes via Tensor Network Simulations
Rigobello, Marco
Scattering processes are a crucial ingredient for the investigation of the fundamental interactions. Working in the framework of Hamiltonian lattice quantum field theory, we attack this problem via numerical tensor network simulations. We focus on the theory of quantum electrodynamics in $1+1$ spacetime dimensions but develop a set of tools which are relevant for a broader class of $1+1$ dimensional quantum field theories. Specifically, we identify a matrix product state representation of the initial momentum wave packet and compute its real-time dynamics. The outcome of some scattering simulations is presented.
Long-range Ising chains: eigenstate thermalization and symmetry breaking of excited states
Russomanno, Angelo
We use large-scale exact diagonalization to study the quantum Ising chain in a transverse field with long-range power-law interactions decaying with exponent $\alpha$. Analyzing various eigenstate and eigenvalue properties, we find numerical evidence for ergodic behavior in the thermodynamic limit for $\alpha>0$, i.e. for the slightest breaking of the permutation symmetry at $\alpha=0$. Considering an excited-states fidelity susceptibility, an energy-resolved average level-spacing ratio and the eigenstate expectations of observables, we observe that a behavior consistent with eigenstate thermalization first emerges at high energy densities for finite system sizes, as soon as $\alpha>0$. We argue that ergodicity moves towards lower energy densities for increasing system sizes. While we argue the system to be ergodic for any $\alpha>0$, we also find a peculiar behaviour near $\alpha=2$ suggesting the proximity to a yet unknown integrable point. We further study the symmetry-breaking properties of the eigenstates. We argue that for weak transverse fields the eigenstates break the $\mathbb{Z}_2$ symmetry, and show long-range order, at finite excitation energy densities for all the values of $\alpha$ we can technically address ($\alpha\leq 1.5$). Our contribution settles central theoretical questions on long-range quantum Ising chains and are also interesting for the nonequilibrium dynamics of trapped ions.
Pyrochlore $S=\frac{1}{2}$ Heisenberg antiferromagnet at finite temperature
Schäfer, Robin
We use a combination of three computational methods to investigate the notoriously difficult frustrated three-dimensional pyrochlore S=12 quantum antiferromagnet, at finite temperature T: canonical typicality for a finite cluster of 2×2×2 unit cells (i.e., 32 sites), a finite-T matrix product state method on a larger cluster with 48 sites, and the numerical linked cluster expansion (NLCE) using clusters up to 25 lattice sites, including nontrivial hexagonal and octagonal loops. We calculate thermodynamic properties (energy, specific heat capacity, entropy, susceptibility, magnetization) and the static structure factor. We find a pronounced maximum in the specific heat at $T=0.57J$, which is stable across finite size clusters and converged in the series expansion. At $T\approx 0.25J$ (the limit of convergence of our method), the residual entropy per spin is $0.47k_B\log(2)$, which is relatively large compared to other frustrated models at this temperature. We also observe a nonmonotonic dependence on T of the magnetization at low magnetic fields, reflecting the dominantly nonmagnetic character of the low-energy states. A detailed comparison of our results to measurements for the $S=1$ material $NaCaNi_2F_7$ yields a rough agreement of the functional form of the specific heat maximum, which in turn differs from the sharper maximum of the heat capacity of the spin ice material $Dy_2Ti_2O_7$.
Scattering of mesons in quantum simulators
Surace, Federica Maria
Simulating real-time evolution in theories of fundamental interactions represents one of the central challenges in contemporary theoretical physics. Cold-atom platforms represent promising candidates to realize quantum simulations of non-perturbative phenomena in gauge theories, such as vacuum decay and hadron collisions, in extreme conditions prohibitive for direct experiments. In this work, we demonstrate that present-day quantum simulators can give access to S-matrix measurements of elastic and inelastic meson collisions in Abelian gauge theories, mimicking experiments with linear particle accelerators. Considering for definiteness a $(1 + 1)$-dimensional $\mathbb{Z}_2$-lattice gauge theory realizable with Rydberg-atom arrays, we solve the meson scattering problem exactly in the limit of large fermion mass and for arbitrary coupling strength.
Neural network wave functions and the sign problem
Szabó, Attila
Neural quantum states are a promising approach to study many-body quantum physics. However, they face a major challenge when applied to lattice models: Neural networks struggle to converge to ground states with a nontrivial sign structure. In this talk, I present a neural network architecture with a simple, explicit, and interpretable phase ansatz, which can robustly represent such states and achieve state-of-the-art variational energies for both conventional and frustrated antiferromagnets. In the first case, the neural network correctly recovers the Marshall sign rule without any prior knowledge. For frustrated magnets, our approach uncovers low-energy states that exhibit the Marshall sign rule but does not reach the true ground state, which is expected to have a different sign structure. I discuss the possible origins of this "residual sign problem" as well as strategies for overcoming it, which may allow using neural quantum states for challenging spin liquid problems.
Separation-dependent emission pathways of quantum emitters
Talukdar, Jugal
System-environment interactions have been studied extensively for many decades and recent developments in quantum optics and circuit QED provide intriguing possibilities for realizing non-linear environments. The Bose-Hubbard lattice for photons, e.g., has been realized experimentally using superconducting circuits, thereby providing an exciting platform to study effective interactions between quantum emitters mediated by the engineered photonic environment. We consider a collection of macroscopically separated two-level emitters coupled to a non-linear environment and study the dissipative dynamics. Specifically, we report our theoretical progress on understanding the criteria for the existence of specific emission pathways as a function of the positions of the emitters.
Continuous matrix product operator approach to finite temperature quantum states
Tang, Wei
We present an algorithm for studying quantum systems at finite temperature using continuous matrix product operator representation. The approach handles both short-range and long-range interactions in the thermodynamic limit without incurring any time discretization error. Moreover, the approach provides direct access to physical observables including the specific heat, local susceptibility, and local spectral functions. After verifying the method using the prototypical quantum XXZ chains, we apply it to quantum Ising models with power-law decaying interactions and on the infinite cylinder, respectively. The approach offers predictions that are relevant to experiments in quantum simulators and the nuclear magnetic resonance spin-lattice relaxation rate.
Semiclassical theory of finite temperature dynamics of the sine-Gordon model
Vörös, Dániel
We investigate the finite temperature dynamics of the sine-Gordon model by studying its dynamical correlation functions at low temperatures in the semiclassical approach. Going beyond previous analyses based on perfectly reflective or transmissive collision dynamics of the gapped solitonic excitations, we focus on the generic case when both transmissive and reflective scatterings are present. We argue that the behaviour of the correlation functions is qualitatively different from both special cases, in particular, the autocorrelation function decays in time neither exponentially nor as a power-law, but assumes a squeezed exponential form. Supporting our claim, we perform numerical simulations utilizing the exact S-matrix of the model.
Vortex-Phase in Non-Centrosymmetric Antiferromagnets
Wolba, Benjamin
In this work we consider two-dimensional, non-centrosymmetric antiferromagnets, for which the competition between exchange and Dyzaloshinskii-Moriya interaction leads to the formation of spatially modulated phases of the staggered order parameter. Within the framework of Ginzburg-Landau theory we show that by applying a magnetic field parallel to the c-axis, which thus induces easy-plane anisotropy, one can stabilize a square lattice of vortices close to Neel temperature. Upon decreasing temperature, this vortex phase undergoes spontaneous symmetry breaking into a rectangular phase, which was not anticipated before. We discuss the relevance of our results for the chiral antiferromagnet Ba2CuGe2O7.
The Mott Transition as a Topological Phase Transition
Wong, Patrick
We show that the Mott metal-insulator transition in the standard one-band Hubbard model can be understood as a topological phase transition. Our approach is inspired by the observation that the mid-gap pole in the self-energy of a Mott insulator resembles the spectral pole of the localized surface state in a topological insulator. We use numerical renormalization group--dynamical mean-field theory to solve the infinite-dimensional Hubbard model and represent the resulting local self-energy in terms of the boundary Green's function of an auxiliary tight-binding chain without interactions. The auxiliary system is of generalized Su-Schrieffer-Heeger model type; the Mott transition corresponds to a dissociation of domain walls.
Dipolar dimer liquid
Zhang, Junyi
Motivated by water, we proposed a lattice liquid model of dipolar dimers. We show that on bipartite lattice it can be exactly mapped to annealed Ising models on random graphs. With exactly solved Ising models, we cannot only prove the existence of the liquid-liquid phase transition, but also bound the critical temperature tightly around $k_BT_c = 3.5J$ , which is also confirmed by Monte Carlo simulation.
Random multipolar driving: tunably slow heating through spectral engineering
Zhao, Hongzheng
We study heating in interacting quantum many-body systems driven by random sequences with $n-$multipolar correlations, corresponding to a polynomially suppressed low frequency spectrum. For $n\geq1$, we find a prethermal regime, the lifetime of which grows algebraically with the driving rate, with exponent ${2n+1}$. A simple theory based on Fermi's golden rule accounts for this behaviour. The quasiperiodic Thue-Morse sequence corresponds to the $n\to \infty$ limit, and accordingly exhibits an exponentially long-lived prethermal regime. Despite the absence of periodicity in the drive, and in spite of its eventual heat death, the prethermal regime can host versatile non-equilibrium phases, which we illustrate with a random multipolar discrete time crystal.
Subdiffusive dynamics and critical quantum correlations in a disorder-free localized Kitaev honeycomb model
Zhu, Guo-Yi
Disorder-free localization has recently emerged as a mechanism for ergodicity breaking in homogeneous lattice gauge theories. In this work we show that this mechanism can lead to unconventional states of quantum matter as the absence of thermalization lifts constraints imposed by equilibrium statistical physics. We study a Kitaev honeycomb model in a skew magnetic field subject to a quantum quench from a fully polarized initial product state and observe nonergodic dynamics as a consequence of disorder-free localization. We find that the system exhibits a subballistic entanglement growth and quantum correlation spreading, which is otherwise typically associated with thermalizing systems. In the asymptotic steady state the Kitaev model develops volume-law entanglement and power-law decaying dimer quantum correlations at an energy density where the equilibrium model only displays paramagnetic and noncritical properties. Our work sheds light onto the potential for disorder-free localized lattice gauge theories to realize quantum states in two dimensions with properties beyond what is possible in an equilibrium context. | CommonCrawl |
\begin{definition}[Definition:Unital Associative Commutative Algebra/Definition 1]
Let $R$ be a commutative ring with unity.
A '''unital associative commutative algebra''' over $R$ is an algebra $\left({A, *}\right)$ over $R$ that is unital, associative and commutative and whose underlying module is unitary.
\end{definition} | ProofWiki |
To be a bookworm
I have a complete set of encyclopaedias on my bookshelf, beautifully bound and neatly arranged with volume 1 on the left and volume 32 on the right. They were a talking point when I bought them 30-odd years ago and were regularly used, but as the internet took off so my use of them declined, preferring to use the online resources which were so conveniently at my fingertips.
So it was that the books remained unexamined for a number of years, until I recently moved house, whereupon I discovered an awful fact: an insect larva - specifically, a bookworm (Anobium punctatum, I've been told) - had found its way into the books and had munched its way through them.
The entomologist who identified it for me was curious about it, and took it for study. One question she did ask, though, was just how far the creature had travelled. I looked back at the books and, as I'd previously noted, confirmed that it had started at the first page of volume 1 (evidence suggests that it hatched there, the parent having departed a different way) and had travelled until it had reached the final page of the final volume, which is where I discovered it, so I gave this information to her, along with the measurements of the books, which were all identical:
each cover is 6 mm thick
each book is 40 mm thick in total
So, how far had the bookworm gone?
calculation-puzzle lateral-thinking
ClickRickClickRick
This riddle, like the question about the direction of a bus, is actually a trick riddle – the answer is very simple, but most people won't get it on the first try (and won't understand why their answer is incorrect until you tell them) due to an inadvertent oversimplification on their part. And just like the direction-of-the-bus riddle, this riddle depends on an implied convention.
The intended naïve line of thought is that for the bookworm to have travelled from the first page of volume 1 to the last page of volume 32, that it would have to have chewed clean through all 32 volumes of the encyclopedia, excepting the two covers on the edge of the row of the bookshelf. This gives us a total distance of $40 \times 32 - 6 \times 2 = 1268\ \text{mm}$.
However, this line of reasoning is incorrect, because the front cover of a book is on the right side of its spine. So the bookworm in fact didn't chew through the pages of volumes 1 or 32 at all – just their covers. This means that the actual distance travelled is $40 \times 30 + 6 \times 2 = 1212\ \text{mm}$.
However, even this answer is not necessarily correct. There's no information given about what language the encyclopedia is in. Certain languages like Chinese or Arabic are written from right-to-left, and the books in that language that are written in right-to-left generally have their front covers on the left side of the spine. Supposing the encyclopedia was written in one of these languages, the initial answer of $1268\ \text{mm}$ would in fact be correct.
Of course, this answer is slightly unsatisfactory because volumes of an encyclopedia written in a right-to-left language are generally also organized from right to left, so the fact that they were organized left-to-right on this particular shelf would seem somewhat unusual.
user88user88
$\begingroup$ +1. And for completeness, in the absence of any specific information to the contrary, the English-speaking conventions apply to the question which was written in English. $\endgroup$
– ClickRick
$\begingroup$ I understand that. I'm just pointing out that it's a trick riddle because of that convention. $\endgroup$
Jul 3 '14 at 0:20
$\begingroup$ So if I were to make explicit that it is in English, which would lose nothing from the primary trick, would it become a better riddle? I'm open to suggestions which would turn it around from downvotes to upvotes. $\endgroup$
$\begingroup$ I'm pretty sure the English part isn't the problem (as you said, people will assume it's English if the problem is written in English) - it's more that people on this site don't really seem to like trick riddles much in the first place. $\endgroup$
$\begingroup$ @ClickRick I don't really think this is a trick riddle with a stupid answer, everything is pretty clear in the question and undebatable solvable if the language formulation is added (I would just add it as flavour: A 32 volume british encyclopedia) $\endgroup$
– Falco
Is the answer 12 mm? He says nothing about having all the volumes, just that he arranged volume 1 and 32.
Building on the most upvoted answer, the first page of volume one sits on the right having only volume 1's cover separating it from volume 32 cover, that has its final page sitting on the left side of the book.
So the worm would travel the distance of two covers: $6 * 2 = 12$ mm.
d'alar'cop
Francisco GrossoFrancisco Grosso
$\begingroup$ look up and learn some LaTeX, there you will learn how to use MathJax (to create nice formulas) $\endgroup$
– d'alar'cop
If the bookworm was born before the move and found while on the new home, the answer is obvious. It travelled the distance from the old home to the new home.
Tulio F.Tulio F.
I feel like I'm missing something, but isn't this just $$(6*2)*32+(40-6*2)*32-6-6=1268 \text{ mm} = 1.268 \text{ m}$$ Which can be reduced to $$ 40*32-6-6 = 1268 \text{ mm} $$
The extra two $-6$ are for skipping the first cover of the first book (started on the first page) and the last cover of the last book (ended on the last page).
I don't see the puzzle here.
BobsonBobson
$\begingroup$ Your answer is incorrect. To explain the correct answer would spoil the puzzle, however, because it depends on a specific trick. $\endgroup$
$\begingroup$ @JoeZ. - I knew there had to be a trick that I was missing. I wasn't expecting that one, though. $\endgroup$
– Bobson
Not the answer you're looking for? Browse other questions tagged calculation-puzzle lateral-thinking or ask your own question.
Which way is the bus going?
Directions on an infinite compass rose
The Pie-Maker's Son
An Unfamiliar Day in the Life
This is it. This is the one. Find your wife
How did he count my money so fast?
Ernie and the Mastermind
Weren't we supposed to know more of the story by now...?
A talk with Four Brothers | CommonCrawl |
\begin{definition}[Definition:Limit of Real Function/Left]
Let $\openint a b$ be an open real interval.
Let $f: \openint a b \to \R$ be a real function.
Let $L \in \R$.
Suppose that:
:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R: b - \delta < x < b \implies \size {\map f x - L} < \epsilon$
where $\R_{>0}$ denotes the set of strictly positive real numbers.
That is, for every real strictly positive $\epsilon$ there exists a real strictly positive $\delta$ such that ''every'' real number in the domain of $f$, less than $b$ but within $\delta$ of $b$, has an image within $\epsilon$ of $L$.
:400px
Then $\map f x$ is said to '''tend to the limit $L$ as $x$ tends to $b$ from the left''', and we write:
:$\map f x \to L$ as $x \to b^-$
or
:$\ds \lim_{x \mathop \to b^-} \map f x = L$
This is voiced:
:'''the limit of $\map f x$ as $x$ tends to $b$ from the left'''
and such an $L$ is called:
:'''a limit from the left'''.
\end{definition} | ProofWiki |
# Efficiency in iterative methods: theoretical background
The computational efficiency index (CEI) is a measure of the efficiency of an iterative method. It takes into account the order of convergence, the number of evaluations, and the computational cost per iteration. The CEI is defined as follows:
$$CEI(\mu_0, \mu_1, m) = \frac{\rho_1}{C(\mu_0, \mu_1, m)}$$
where $\rho_1$ is the order of convergence, $C(\mu_0, \mu_1, m)$ is the computational cost per iteration, $\mu_0$ and $\mu_1$ are the ratios between products and evaluations, and $m$ is the number of unknowns.
## Exercise
Calculate the CEI for a given iterative method with an order of convergence $\rho_1 = 2$, $\mu_0 = \mu_1 = 1$, and $m = 10$.
### Solution
$$CEI(1, 1, 10) = \frac{2}{10}$$
# MATLAB: an introduction and its importance in implementing iterative methods
MATLAB is a high-level programming language that is designed for numerical computations. It is widely used in various fields, including engineering, physics, and economics. MATLAB provides a user-friendly interface for creating and manipulating matrices, solving mathematical equations, and visualizing data.
Implementing iterative methods in MATLAB is crucial for solving complex problems efficiently. MATLAB's built-in functions and libraries make it easy to implement various iterative methods, such as Gauss-Seidel, Jacobi, and Successive Over-Relaxation. These methods can be used to solve optimization problems, solve linear systems, and perform least squares.
# Basics of MATLAB programming
MATLAB uses a simple and intuitive syntax for programming. Variables are used to store data, and data types include integers, floating-point numbers, and complex numbers. Arrays are used to store multiple values in a single variable, and they can be vectors, matrices, or higher-dimensional arrays.
Functions are blocks of code that perform a specific task. In MATLAB, functions can be defined using the `function` keyword, and they can take input arguments and return output values. MATLAB also provides built-in functions for various mathematical operations and data manipulation tasks.
## Exercise
Create a function in MATLAB that calculates the sum of two numbers.
### Solution
```matlab
function sum = addNumbers(a, b)
sum = a + b;
end
```
# Implementing iterative methods in MATLAB: convergence analysis
To implement the Gauss-Seidel method in MATLAB, you can use the following code:
```matlab
function x = gaussSeidel(A, b, x0, tol, maxIter)
[n, m] = size(A);
x = x0;
for iter = 1:maxIter
xNew = b / diag(A);
for i = 1:n
xNew(i) = xNew(i) + (b(i) - A(i, :) * x(1:end-1)) / A(i, i);
end
if norm(xNew - x) < tol
break;
end
x = xNew;
end
end
```
Similarly, you can implement the Jacobi and Successive Over-Relaxation methods using MATLAB. The convergence analysis of these methods can be performed using the CEI and other convergence criteria, such as the residual norm or the number of iterations.
## Exercise
Implement the Jacobi method in MATLAB and analyze its convergence properties.
### Solution
```matlab
function x = jacobi(A, b, x0, tol, maxIter)
[n, m] = size(A);
x = x0;
for iter = 1:maxIter
xNew = b / diag(A);
for i = 1:n
xNew(i) = xNew(i) + (b(i) - A(i, :) * x(1:end-1)) / A(i, i);
end
if norm(xNew - x) < tol
break;
end
x = xNew;
end
end
```
# Examples of iterative methods: Gauss-Seidel, Jacobi, and Successive Over-Relaxation
The Gauss-Seidel method is an iterative method that converges to a solution of a linear system with positive definite matrix. It is particularly useful for solving large-scale linear systems.
The Jacobi method is another iterative method that converges to a solution of a linear system. It is less sensitive to the initial guess than the Gauss-Seidel method but may require more iterations.
The Successive Over-Relaxation (SOR) method is a variant of the Gauss-Seidel method that uses a relaxation parameter to improve convergence. It is particularly useful for solving ill-conditioned systems.
## Exercise
Implement the Successive Over-Relaxation method in MATLAB and compare its performance with the Gauss-Seidel and Jacobi methods.
### Solution
```matlab
function x = SOR(A, b, x0, tol, maxIter, omega)
[n, m] = size(A);
x = x0;
for iter = 1:maxIter
xNew = b / diag(A);
for i = 1:n
xNew(i) = xNew(i) + (b(i) - A(i, :) * x(1:end-1)) / A(i, i);
end
x = (1 - omega) * x + omega * xNew;
if norm(xNew - x) < tol
break;
end
end
end
```
# Optimization problems and iterative methods: solving linear systems and least squares
Iterative methods can be used to solve linear systems, which are equations of the form Ax = b, where A is a matrix, x is a vector of unknowns, and b is a vector of constants. Examples of linear systems include systems of equations, linear regression, and eigenvalue problems.
Iterative methods can also be used to solve least squares problems, which are equations of the form ||Ax - b||^2, where A is a matrix, x is a vector of unknowns, and b is a vector of constants. Least squares problems are commonly used in data fitting and curve fitting.
## Exercise
Implement the Gauss-Seidel method in MATLAB to solve a linear system and compare its performance with other iterative methods.
### Solution
```matlab
% Solve a linear system using the Gauss-Seidel method
x = gaussSeidel(A, b, x0, tol, maxIter);
% Compare the performance of the Gauss-Seidel method with other iterative methods
xJacobi = jacobi(A, b, x0, tol, maxIter);
xSOR = SOR(A, b, x0, tol, maxIter, omega);
```
# Strategies for improving the efficiency of iterative methods in MATLAB
Preconditioning is a technique that can be used to improve the efficiency of iterative methods by transforming the original linear system into a more easily solvable system. In MATLAB, preconditioning can be implemented using the `pcg` and `pcg_precond` functions.
Sparse matrices are matrices that have a large number of zero elements. In MATLAB, sparse matrices can be used to save memory and computational resources when solving large linear systems. Sparse matrices can be created using the `sparse` function.
Parallel computing is a technique that involves solving a problem by dividing it into smaller subproblems and solving them simultaneously. In MATLAB, parallel computing can be implemented using the `parfor` and `spmd` functions.
## Exercise
Implement the Gauss-Seidel method in MATLAB to solve a linear system using preconditioning and sparse matrices.
### Solution
```matlab
% Create a sparse matrix A
A = sparse([1, 0, 0; 0, 2, 0; 0, 0, 3]);
% Create a preconditioner for A
P = inv(diag(A));
% Solve the linear system using the Gauss-Seidel method with preconditioning
x = gaussSeidel(A, b, x0, tol, maxIter, P);
```
# Comparing the performance of different iterative methods
The performance of iterative methods can be analyzed using the CEI, the residual norm, and the number of iterations. Different methods may have different convergence rates, computational costs, and scalability properties.
In practice, the choice of an iterative method depends on the specific problem, the desired accuracy, and the available computational resources. It is essential to carefully analyze the performance of different methods and choose the most efficient and accurate method for a given problem.
## Exercise
Compare the performance of the Gauss-Seidel, Jacobi, and Successive Over-Relaxation methods for solving a linear system.
### Solution
```matlab
% Solve the linear system using the Gauss-Seidel method
xGS = gaussSeidel(A, b, x0, tol, maxIter);
% Solve the linear system using the Jacobi method
xJacobi = jacobi(A, b, x0, tol, maxIter);
% Solve the linear system using the Successive Over-Relaxation method
xSOR = SOR(A, b, x0, tol, maxIter, omega);
```
# Applications of efficient iterative methods in MATLAB: image processing, signal processing, and machine learning
Iterative methods can be used to solve optimization problems and perform data analysis in image processing, signal processing, and machine learning. Examples include image denoising, image segmentation, and feature extraction.
In signal processing, iterative methods can be used to solve linear systems and perform filtering, compression, and classification tasks. Examples include the Wiener filter, the Kalman filter, and the K-means clustering algorithm.
In machine learning, iterative methods can be used to train and optimize models, perform dimensionality reduction, and perform regression and classification tasks. Examples include gradient descent, stochastic gradient descent, and the k-nearest neighbors algorithm.
## Exercise
Implement the Gauss-Seidel method in MATLAB to solve a linear system for image denoising.
### Solution
```matlab
% Denoise an image using the Gauss-Seidel method
xDenoised = gaussSeidel(A, b, x0, tol, maxIter);
```
# Conclusion: the future of iterative methods in MATLAB and beyond
In this textbook, we have explored the efficiency of iterative methods in MATLAB and their applications in various fields, including image processing, signal processing, and machine learning.
The future of iterative methods in MATLAB and beyond is promising, as researchers continue to develop new methods and techniques to improve their efficiency and accuracy. This includes the development of hybrid methods that combine the strengths of different iterative methods, the use of advanced optimization algorithms, and the integration of machine learning techniques for better problem solving and decision-making.
As the field of computational science and engineering continues to advance, the importance of efficient iterative methods will only grow. By understanding the theoretical background, implementing iterative methods in MATLAB, and analyzing their performance, we can develop more efficient and accurate solutions to complex problems. | Textbooks |
\begin{document}
\title{Integer and fractional packing of families of graphs} \author{Raphael Yuster \thanks{ e-mail: [email protected] \qquad World Wide Web: http:$\backslash\backslash$research.haifa.ac.il$\backslash$\symbol{126}raphy} \\ Department of Mathematics\\ University of Haifa at Oranim\\ Tivon 36006, Israel}
\date{}
\maketitle \setcounter{page}{1} \begin{abstract} Let ${\cal F}$ be a family of graphs. For a graph $G$, the {\em ${\cal F}$-packing number}, denoted $\nu_{{\cal F}}(G)$, is the maximum number of pairwise edge-disjoint elements of ${\cal F}$ in $G$. A function $\psi$ from the set of elements of ${\cal F}$ in $G$ to $[0,1]$
is a {\em fractional ${\cal F}$-packing} of $G$ if $\sum_{e \in H \in {\cal F}} {\psi(H)} \leq 1$ for each $e \in E(G)$. The {\em fractional ${\cal F}$-packing number}, denoted $\nu^*_{{\cal F}}(G)$, is defined to be the maximum value of $\sum_{H \in {{G} \choose {{\cal F}}}} \psi(H)$ over all fractional ${\cal F}$-packings $\psi$. Our main result is that $\nu^*_{{\cal F}}(G)-\nu_{{\cal F}}(G) = o(|V(G)|^2)$. Furthermore, a set of $\nu_{{\cal F}}(G) -o(|V(G)|^2)$ edge-disjoint elements of ${\cal F}$ in $G$ can be found in randomized polynomial time. For the special case ${\cal F}=\{H_0\}$ we obtain a significantly simpler proof of a recent difficult result of Haxell and R\"odl \cite{HaRo}
that $\nu^*_{H_0}(G)-\nu_{H_0}(G) = o(|V(G)|^2)$. \end{abstract}
\section{Introduction} All graphs considered here are finite and have no loops, multiple edges or isolated vertices. For the standard terminology used the reader is referred to \cite{Bo}. Let ${\cal F}$ be any fixed finite or infinite family of graphs. For a graph $G$, the {\em ${\cal F}$-packing number}, denoted $\nu_{{\cal F}}(G)$, is the maximum number of pairwise edge-disjoint copies of elements of ${\cal F}$ in $G$. A function $\psi$ from the set of copies of elements of ${\cal F}$ in $G$ to $[0,1]$ is a {\em fractional ${\cal F}$-packing} of $G$ if $\sum_{e \in H \in {\cal F}} {\psi(H)} \leq 1$ for each $e \in E(G)$. For a fractional ${\cal F}$-packing $\psi$, let $w(\psi)=\sum_{H \in {{G} \choose {{\cal F}}}} \psi(H)$. The {\em fractional ${\cal F}$-packing number}, denoted $\nu^*_{{\cal F}}(G)$, is defined to be the maximum value of $w(\psi)$ over all fractional packings $\psi$. Notice that, trivially, $\nu^*_{{\cal F}}(G) \geq \nu_{{\cal F}}(G)$. If ${\cal F}$ consists of a single graph $H_0$ we shall denote the parameters above by $\nu_{H_0}(G)$ and $\nu^*_{H_0}(G)$.
Since computing $\nu^*_{{\cal F}}(G)$ amounts to solving a linear program, it can be computed in polynomial time for every finite ${\cal F}$. On the other hand, it was proved by Dor and Tarsi in \cite{DoTa} that computing $\nu_{H_0}(G)$ is NP-Hard for every $H_0$ with a component having at least three edges. Thus, it is interesting to determine when $\nu^*_{{\cal F}}(G)$ and $\nu_{{\cal F}}(G)$ are ``close'', thereby getting a polynomial time approximating algorithm for an NP-Hard problem. The following result was proved by Haxell and R\"odl in \cite{HaRo}. \begin{theo} \label{t0} If $H_0$ is a fixed graph and $G$ is a graph with $n$ vertices, then $\nu^*_{H_0}(G)-\nu_{H_0}(G) = o(n^2)$. \end{theo} The 25 page proof of Theorem \ref{t0} presented in \cite{HaRo} is very difficult. The major difficulty lies in the fact that their method requires proving that there is a fractional packing which is only slightly less than optimal, and which assigns to every copy of $H_0$ either $0$ or a value greater than $\tau$ for some $\tau > 0$ which is only a function of $H_0$.
In this paper we present a significantly simpler proof of Theorem \ref{t0}. Our proof method enables us to generalize Theorem \ref{t0} to the ``family'' case. There does not seem to be an easy way to generalize the proof in \cite{HaRo} to the family case. \begin{theo} \label{t1} If ${\cal F}$ is a fixed family of graphs and $G$ is a graph with $n$ vertices, then $\nu^*_{{\cal F}}(G)-\nu_{{\cal F}}(G) = o(n^2)$. \end{theo} Notice that Theorem \ref{t1} immediately yields a polynomial time algorithm for approximating $\nu_{{\cal F}}(G)$ to within an additive term of $\epsilon n^2$ for every $\epsilon > 0$. Furthermore, if ${\cal F}$ is finite, the degree of the polynomial depends only on ${\cal F}$, and not on $1/\epsilon$. Our proof also supplies a randomized polynomial time algorithm that {\em finds} a set of $\nu_{{\cal F}}(G)-o(n^2)$ edge-disjoint copies of elements of ${\cal F}$ in $G$.
\section{Tools used in the main result}
As in \cite{HaRo}, a central ingredient in our proof of the main result is Szemer\'edi's regularity lemma \cite{Sz}. Let $G=(V,E)$ be a graph, and let $A$ and $B$ be two disjoint subsets of $V(G)$. If $A$ and $B$ are non-empty, let $E(A,B)$ denote set of edges between them, and put $e(A,B)=|E(A,B)|$. The {\em density of edges} between $A$ and $B$ is defined as $$
d(A,B) = \frac{e(A,B)}{|A||B|}. $$ For $\gamma>0$ the pair $(A,B)$ is called {\em $\gamma$-regular} if for every $X \subset A$ and $Y \subset B$ satisfying
$|X|>\gamma |A|$ and $|Y|>\gamma |B|$ we have $$
|d(X,Y)-d(A,B)| < \gamma. $$
An {\em equitable partition} of a set $V$ is a partition of $V$ into pairwise disjoint classes $V_1,\ldots,V_m$ whose sizes are as equal as possible. An equitable partition of the set of vertices $V$ of a graph $G$ into the classes $V_1,\ldots,V_m$ is called {\em $\gamma$-regular} if $|V_i| < \gamma |V|$ for every $i$ and all but at most $\gamma {m \choose 2}$ of the pairs $(V_i,V_j)$ are $\gamma$-regular. The regularity lemma states the following: \begin{lemma} \label{l21} For every $\gamma>0$, there is an integer $M(\gamma)>0$ such that for every graph $G$ of order $n > M$ there is a $\gamma$-regular partition of the vertex set of $G$ into $m$ classes, for some $1/\gamma < m < M$.
\square
\end{lemma}
Let $H_0$ be a fixed graph with the vertices $\{1,\ldots,k\}$, $k \geq 3$. Let $W$ be a $k$-partite graph with vertex classes $V_1,\ldots,V_k$. A subgraph $J$ of $W$ with ordered vertex set $v_1,\ldots,v_k$ is {\em partite-isomorphic} to $H_0$ if $v_i \in V_i$ and the map $v_i \rightarrow i$ is an isomorphism from $J$ to $H_0$.
The following lemma is almost identical to the (2 page) proof of Lemma 15 in \cite{HaRo} and hence the proof is omitted. \begin{lemma} \label{l22} Let $\delta$ and $\zeta$ be positive reals. There exist $\gamma=\gamma(\delta, \zeta, k)$ and $T=T(\delta, \zeta, k)$ such that the following holds. Let $W$ be a $k$-partite graph with vertex classes $V_1,\ldots,V_k$
and $|V_i|=t > T$ for $i=1,\ldots,k$. Furthermore, for each $(i,j) \in E(H_0)$, $(V_i,V_j)$ is a $\gamma$-regular pair with density $d(i,j) \geq \delta$ and for each $(i,j) \notin E(H_0)$,
$E(V_i,V_j)=\emptyset$. Then, there exists a spanning subgraph $W'$ of $W$, consisting of at least $(1-\zeta)|E(W)|$ edges such that the following holds. For an edge $e \in E(W')$, let $c(e)$ denote the number of subgraphs of $W'$ that are partite isomorphic to $H_0$ and that contain $e$. Then, for all $e \in E(W')$, if $e \in E(V_i,V_j)$ then $$
\left|c(e) - t^{k-2} \frac{\prod_{(s,p) \in E(H_0)}d(s,p)}{d(i,j)}\right| < \zeta t^{k-2}. $$
\square
\end{lemma}
Finally, we need to state the seminal result of Frankl and R\"odl \cite{FrRo} on near perfect coverings and matchings of uniform hypergraphs. Recall that if $x,y$ are two vertices of a hypergraph then $deg(x)$ denotes the degree of $x$ and $deg(x,y)$ denotes the number of edges that contain both $x$ and $y$ (their {\em co-degree}). We use the version of the Frankl and R\"odl Theorem due to Pippenger (see, e.g., \cite{Fu}). \begin{lemma} \label{l23} For an integer $r \geq 2$ and a real $\beta > 0$ there exists a real $\mu > 0$ so that: If the $r$-uniform hypergraph $L$ on $q$ vertices has the following properties for some $d$:\\ (i) $(1-\mu)d < deg(x) < (1+\mu)d$ holds for all vertices,\\ (ii) $deg(x,y) < \mu d$ for all distinct $x$ and $y$,\\ then $L$ has a matching of size at least $(q/r)(1-\beta)$.
\square
\end{lemma}
\section{Proof of the main result} Let ${\cal F}$ be a family of graphs, and let $\epsilon > 0$. To avoid the trivial case we assume $K_2 \notin {\cal F}$. We shall prove there exists $N=N({\cal F},\epsilon)$ such that for all $n > N$, if $G$ is an $n$-vertex graph then $\nu_{{\cal F}}^*(G) - \nu_{{\cal F}}(G) < \epsilon n^2$.
Let $k_\infty$ denote the maximal order of a graph in ${\cal F}$. Let $k_0=\min\{k_\infty,\lceil 20/\epsilon \rceil\}$. Let $\delta=\beta=\epsilon/4$. For all $r=2,\ldots,{k_0}^2$, let $\mu_r=\mu(\beta,r)$ be as in Lemma \ref{l23}, and put $\mu=\min_{r=2}^{k_0^2} \{\mu_r\}$. Let $\zeta=\mu \delta^{{k_0}^2}/2$. For $k=3, \ldots, k_0$, let $\gamma_k=\gamma(\delta, \zeta, k)$ and $T_k=T(\delta, \zeta, k)$ be as in Lemma \ref{l22}. Let $\gamma=\min_{k=3}^{k_0} \{\gamma_k\}$. Let $M=M(\gamma\epsilon/(25{k_0}^2))$ be as in Lemma \ref{l21}. Finally, we shall define $N$ to be a sufficiently large constant, depending on the above chosen parameters, and for which various conditions stated in the proof below hold (it will be obvious in the proof that all these conditions indeed hold for $N$ sufficiently large). Thus, indeed, $N=N({\cal F}, \epsilon)$.
Fix an $n$-vertex graph $G$ with $n > N$ vertices. Fix a fractional ${\cal F}$-packing $\psi$ with $w(\psi)=\nu_{{\cal F}}^*(G)$. We may assume that $\psi$ assigns a value to each {\em labeled} copy of an element of ${\cal F}$ simply by dividing the value of $\psi$ on each nonlabeled copy by the size of the automorphism group of that element. If $\nu_{{\cal F}}^*(G) < \epsilon n^2$ we are done. Hence, we assume $\nu_{{\cal F}}^*(G) =\alpha n^2 \geq \epsilon n^2$.
We apply Lemma \ref{l21} to $G$ and obtain a $\gamma'$-regular partition with $m'$ parts, where $\gamma'=\gamma \epsilon/(25{k_0}^2)$ and $1/\gamma' < m' < M(\gamma')$. Denote the parts by $U_1,\ldots,U_{m'}$. Notice that the size of each part is either $\lfloor n/{m'} \rfloor$ or $\lceil n/{m'} \rceil$. For simplicity we may and will assume that $n/{m'}$ is an integer, as this assumption does not affect the asymptotic nature of our result. For the same reason we may and will assume that $n/(25m'{k_0}^2/\epsilon)$ is an integer.
We randomly partition each $U_i$ into $25{k_0}^2/\epsilon$ equal parts of size $n/(25m'{k_0}^2/\epsilon)$ each. All $m'$ partitions are independent. We now have $m=25m'{k_0}^2/\epsilon$ {\em refined} vertex classes, denoted $V_1,\ldots,V_m$. Suppose $V_i \subset U_s$ and $V_j \subset U_t$ where $s \neq t$. We claim that if
$(U_s,U_t)$ is a $\gamma'$-regular pair then $(V_i,V_j)$ is a $\gamma$-regular pair. Indeed, if $X \subset V_i$ and $Y \subset V_j$ have $|X|, |Y| > \gamma n/(25m'{k_0}^2/\epsilon)$ then
$|X|, |Y| > \gamma' n/m'$ and so $|d(X,Y) - d(U_s,U_t)| < \gamma'$. Also
$|d(V_i,V_j) - d(U_s,U_t)| < \gamma'$. Thus, $|d(X,Y) - d(V_i,V_j)| < 2\gamma' < \gamma$.
Let $H$ be a labeled copy of some $H_0 \in {\cal F}$ in $G$. If $H$ has $k$ vertices and $k \leq k_0$ then the expected number of pairs of vertices of $H$ that belong to the same vertex class in the refined partition is clearly at most ${k \choose 2}\epsilon/(25{k_0}^2) < \epsilon/50$. Thus, the probability that $H$ has two vertices in the same vertex class is also at most $\epsilon/50$. We call $H$ {\em good} if it has $k \leq k_0$ vertices and its $k$ vertices belong to $k$ distinct vertex classes of the refined partition. By the definition of $k_0$, if $H$ has $k > k_0$ vertices and $\psi(H) > 0$ then we must have $k > 20/\epsilon$. Since graphs with $k$ vertices have at least $k/2$ edges, the contribution of graphs with $k > k_0$ vertices to $\nu_{{\cal F}}^*(G)$ is at most ${n \choose 2}/(10/\epsilon) < \epsilon n^2/20$. Hence, if $\psi^{**}$ is the restriction of $\psi$ to good copies (the bad copies having $\psi^{**}(H)=0$) then the expectation of $w(\psi^{**})$ is at least $(\alpha-\epsilon/50-\epsilon/20)n^2$. We therefore {\em fix} a partition $V_1,\ldots,V_m$ for which $w(\psi^{**}) \geq (\alpha-0.07\epsilon)n^2$.
Let $G^*$ be the spanning subgraph of $G$ consisting of the edges with endpoints in distinct vertex classes of the refined partition that form a $\gamma$-regular pair with density at least $\delta$ (thus, we discard edges inside classes, between non regular pairs, or between sparse pairs). Let $\psi^*$ be the restriction of $\psi^{**}$ to the labeled copies of elements of ${\cal F}$ in $G^*$. We claim that $\nu_{{\cal F}}^*(G^*) \geq w(\psi^*) > w(\psi^{**})-0.72\delta n^2 \geq (\alpha-0.07\epsilon-0.72\delta)n^2=(\alpha - \delta)n^2$. Indeed, by considering the number of discarded edges we get (using $m' > 1/\gamma'$ and $\delta >> \gamma'$) $$
w(\psi^{**}) - w(\psi^*) \leq |E(G) - E(G^*)| < \gamma' {{m'} \choose 2}\frac{n^2}{{m'}^2} + {{m'} \choose 2}(\delta+\gamma')\frac{n^2}{{m'}^2} + {m'}{{n/{m'}} \choose 2} < 0.72 \delta n^2. $$ Let $R$ denote the $m$-vertex graph whose vertices are $\{1,\ldots,m\}$ and $(i,j) \in E(R)$ if and only if $(V_i,V_j)$ is a $\gamma$-regular pair with density at least $\delta$. We define a (labeled) fractional ${\cal F}$-packing $\psi'$ of $R$ as follows. Let $H$ be a labeled copy of some $H_0 \in {\cal F}$ in $R$ and assume that the vertices of $H$ are $\{u_1,\ldots,u_k\}$ where $u_i$ plays the role of vertex $i$ in $H_0$. We define $\psi'(H)$ to be the sum of the values of $\psi^*$ taken over all subgraphs of $G^*[V_{u_1},\ldots,V_{u_k}]$ which are partite isomorphic to $H_0$, divided by $n^2/m^2$. Notice that by normalizing with $n^2/m^2$ we guarantee that $\psi'$ is a proper fractional ${\cal F}$-packing of $R$ and that $\nu_{{\cal F}}^*(R) \geq w(\psi') =m^2w(\psi^*)/n^2 \geq m^2(\alpha - \delta)$.
We use $\psi'$ to define a random coloring of the edges of $G^*$. Our ``colors'' are the labeled copies of elements of ${\cal F}$ in $R$. Let $d(i,j)$ denote the density of $(V_i,V_j)$ and notice that $|E_{G^*}(V_i,V_j)|=d(i,j)n^2/m^2$. Let $H$ be a labeled copy of some $H_0 \in {\cal F}$ in $R$, and assume that $H$ contains the edge $(i,j)$. Each $e \in E(V_i,V_j)$ is chosen to have the ``color'' $H$ with probability $\psi'(H) /d(i,j)$. The choices made by distinct edges of $G^*$ are independent. Notice that this random coloring is legal (in the sense that the sum of probabilities is at most one) since the sum of $\psi'(H)$ taken over all labeled copies of elements of ${\cal F}$ containing $(i,j)$ is at most $d(i,j) \leq 1$. Notice also that some edges might stay uncolored in our random coloring of the edges of $G^*$.
Let $H$ be a labeled copy of some $H_0 \in {\cal F}$ in $R$, and assume that $\psi'(H) > m^{1-k_0}$. Without loss of generality, assume that the vertices of $H$ are $\{1,\ldots,k\}$ where $i \in V(H)$ plays the role of $i \in V(H_0)$. Let $r$ denote the number of edges of $H$. Notice that $r < k_0^2$. Let $W_H=G^*[V_1,\ldots,V_k]$ (in fact we only consider edges between pairs that correspond to edges of $H_0$). Notice that $W_H$ is a subgraph of $G^*$ which satisfies the conditions in Lemma \ref{l22}, since $t=n/m > N\epsilon/(25{k_0}^2M) > T_k$ (here we assume $N > 25{k_0}^2MT_k/\epsilon$). Let $W'_H$ be the spanning subgraph of $W_H$ whose existence is guaranteed in Lemma \ref{l22}. Let $X_H$ denote the spanning subgraph of $W'_H$ consisting only of the edges whose color is $H$. Notice that $X_H$ is a random subgraph of $W'_H$. For an edge $e \in E(X_H)$, let $C_H(e)$ denote the set of subgraphs of $X_H$ that contain $e$
and that are partite isomorphic to $H_0$. Put $c_H(e)=|C_H(e)|$. A crucial argument is the following: \begin{lemma} With probability at least $1-m^3/n$, for all $e \in E(X_H)$, \label{l31} \begin{equation} \label{e0}
\left|c_H(e) - t^{k-2} \psi'(H)^{r-1} \right| < \mu \psi'(H)^{r-1} t^{k-2}. \end{equation} \end{lemma}
{\bf Proof:}\, Let $C(e)$ denote the set of subgraphs of $W'_H$ that contain $e$ and that are partite isomorphic to $H_0$. Put $c(e)=|C(e)|$. According to Lemma \ref{l22}, if $e \in E(V_i,V_j)$ then \begin{equation} \label{e1}
\left|c(e) - t^{k-2} \frac{\prod_{(s,p) \in E(H_0)}d(s,p)}{d(i,j)}\right| < \zeta t^{k-2}. \end{equation} Fix an edge $e \in E(X_H)$ belonging to $E(V_i,V_j)$. The probability that an element of $C(e)$ also belongs to $C_H(e)$ is precisely $$ \rho=\psi'(H)^{r-1} \cdot \frac{d(i,j)}{\prod_{(s,p) \in E(H_0)}d(s,p)}. $$
We say that two distinct elements $Y,Z \in C(e)$ are {\em dependent} if they share at least one edge other than $e$. Consider the dependency graph $B$ whose vertex set is $C(e)$ and the edges connect dependent pairs. Since two dependent elements share at least three vertices (including the two endpoints of $e$), we have that $\Delta(B) = O(t^{k-3})$. Hence, $\chi(B)=O(t^{k-3})$. Put $s=\chi(B)$. Let $C^1(e), \ldots, C^s(e)$ denote a partition of $C(e)$ to independent sets. Let $C^q_H(e) =C^q(e) \cap C_H(e)$, $c^q(e)=|C^q(e)|$ and
$c^q_H(e)=|C^q_H(e)|$. Clearly, $c^1(e)+ \cdots + c^s(e)=c(e)$ and $c^1_H(e)+ \cdots + c^s_H(e)=c_H(e)$. The expectation of $c^q_H(e)$ is $\rho c^q(e)$. Consider some $C^q(e)$ with $c^q(e) > \sqrt{t}$. According to a large deviation inequality of Chernoff (cf. \cite{AlSp} Appendix A), for every $\eta > 0$, and in particular for $\eta=\mu/8$, if $n$ (and hence $t$ and hence $c^q(e)$) is sufficiently large, $$
\Pr[| c^q_H(e) - \rho c^q(e)| > \eta \rho c^q(e) ] < e^{-\frac{2(\eta\rho c^q(e))^2}{c^q(e)}} = e^{-2\eta^2\rho^2c^q(e)} << t^{-k-1}. $$ It follows that with probability at least $1-st^{-k-1} > 1-t^{-3}$, for all $C^q(e)$ with $c^q(e) > \sqrt{t}$, $ (1-\eta)\rho c^q(e) \leq c^q_H(e) \leq (1+\eta)\rho c^q(e)$ holds. Since the sum of $c^q(e)$ having $c^q(e) \le \sqrt{t}$ is $O(t^{k-2.5})$ and since $c(e)=\Theta(t^{k-2})$ we have that this sum is much less than $\rho\eta c(e)$. Thus, together with (\ref{e1}) and the fact that $\rho < \psi'(H)^{r-1}\delta^{-r}$ we have \begin{equation} \label{e2} c_H(e) = \sum_{q=1}^s c^q_H(e) \leq \rho(1+\eta)(\sum_{q=1}^s c^q(e)) +\rho \eta c(e) =\rho(1+2\eta)c(e) \leq \end{equation} $$
\rho(1+2\eta)t^{k-2}(\zeta+\frac{\prod_{(s,p) \in E(H_0)}d(s,p)}{d(i,j)})= (1+2\eta)t^{k-2}(\psi'(H)^{r-1}+\zeta\rho) \leq $$ $$ t^{k-2}\psi'(H)^{r-1}(1+2\eta)(1+\zeta \delta^{-r})\leq t^{k-2}\psi'(H)^{r-1}(1+\mu/4)(1+\mu/2) \leq (1+\mu)t^{k-2}\psi'(H)^{r-1}. $$ Similarly, \begin{equation} \label{e3} c_H(e) \geq \rho(1-\eta)c(e) - \rho \eta c(e) =\rho(1-2\eta)c(e) \geq \end{equation} $$ \rho(1-2\eta)t^{k-2}(\frac{\prod_{(s,p) \in E(H_0)}d(s,p)}{d(i,j)}-\zeta)= (1-2\eta)t^{k-2}(\psi'(H)^{r-1}-\zeta\rho) \geq $$ $$ t^{k-2}\psi'(H)^{r-1}(1-2\eta)(1-\zeta \delta^{-r})\geq t^{k-2}\psi'(H)^{r-1}(1-\mu/4)(1-\mu/2) \geq (1-\mu)t^{k-2}\psi'(H)^{r-1}. $$ Combining (\ref{e2}) and (\ref{e3}) we have that (\ref{e0}) holds for a fixed $e \in E(X_H)$ with probability at least $1-t^{-3}$. As $E(X_H) < n^2$ we have that (\ref{e0}) holds for all $e \in E(X_H)$ with probability at least $1-n^2/t^3=1-m^3/n$.
\square
We also need the following lemma that gives a lower bound for the number of edges of $X_H$. \begin{lemma} \label{l32} With probability at least $1-1/n$, $$
|E(X_H)| > (1-2\zeta)r\frac{n^2}{m^2}\psi'(H). $$ \end{lemma}
{\bf Proof:}\, We use the notations from Lemma \ref{l31} and the paragraph preceding it. For $(i,j) \in E(H_0)$, the expected number of edges of $E(V_i,V_j)$ that received the color $H$ is precisely $d(i,j)\frac{n^2}{m^2}\frac{\psi'(H)}{d(i,j)}=\frac{n^2}{m^2}\psi'(H)$. Summing over all $r$ edges of $H_0$, the expected number of edges of $W_H$ that received the color $H$ is precisely $r\frac{n^2}{m^2}\psi'(H)$. As at most $\zeta |E(W_H)|$ edges belong to $W_H$ and do not belong to $W'_H$ we have that the expectation of $|E(X_H)|$ is at least $(1-\zeta)r\frac{n^2}{m^2}\psi'(H)$. As $\zeta$, $r$, $m$ are constants and as $\psi'(H)$ is bounded from below by the constant $m^{1-k_0}$, we have, by the common large deviation inequality of Chernoff (cf. \cite{AlSp} Appendix A), that for
$n > N$ sufficiently large, the probability that $|E(X_H)|$ deviates from its mean by more than $\zeta r\frac{n^2}{m^2}\psi'(H)$ is exponentially small in $n$. In particular, the lemma follows.
\square
Since $R$ contains at most $O(m^{k_0})$ labeled copies of elements of ${\cal F}$ with at most $k_0$ vertices, we have that with probability at least $1-m^{k_0}/n - m^{k_0+3}/n > 0$ (here we assume again that $N$ is sufficiently large) {\em all} labeled copies $H$ of elements of ${\cal F}$ in $R$ with $\psi'(H) > m^{1-k_0}$ satisfy the statements of Lemma \ref{l31} and Lemma \ref{l32}. We therefore fix a coloring for which Lemma \ref{l31} and Lemma \ref{l32} hold for all labeled copies $H$ of elements of ${\cal F}$ in $R$ having $\psi'(H) > m^{1-k_0}$.
Let $H$ be a labeled copy of some $H_0 \in {\cal F}$ in $R$ with $\psi'(H) > m^{1-k_0}$, and let $r$ denote the number of edges of $H$. We construct an $r$-uniform hypergraph $L_H$ as follows. The vertices of $L_H$ are the edges of the corresponding $X_H$ from Lemma \ref{l31}. The edges of $L_H$ correspond to the edge sets of the subgraphs of $X_H$ that are partite isomorphic to $H_0$. We claim that our hypergraph satisfies the conditions of Lemma \ref{l23}. Indeed, let $q$ denote he number of vertices of $L_H$. Notice that Lemma \ref{l32} provides a lower bound for $q$. Let $d=t^{k-2} \psi'(H)^{r-1}$. Notice that by Lemma \ref{l31} {\em all} vertices of $L_H$ have their degrees between $(1-\mu)d$ and $(1+\mu)d$. Also notice that the co-degree of any two vertices of $L_H$ is at most $t^{k-3}$ as two edges cannot belong, together, to more than $t^{k-3}$ subgraphs of $X_H$ that are partite isomorphic to $H_0$. In particular, for $N$ sufficiently large, $\mu d > t^{k-3}$. By Lemma \ref{l23} we have at least $(q/r)(1-\beta)$ edge-disjoint copies of $H_0$ in $X_H$. In particular, we have at least $$ (1-\beta)(1-2\zeta)\frac{n^2}{m^2}\psi'(H) > (1-2\beta)\psi'(H)\frac{n^2}{m^2} $$ such copies. Recall that $w(\psi')\geq m^2(\alpha-\delta)$. Since there are at most $O(m^{k_0})$ labeled copies $H$ of elements of ${\cal F}$ in $R$ with $0 < \psi'(H) \leq m^{1-{k_0}}$, their total contribution to $w(\psi')$ is at most $O(m)$. Hence, summing the last inequality over all $H$ with $\psi'(H) > m^{1-{k_0}}$ we have at least $$ (1-2\beta)m^2(\alpha-\delta-O(\frac{1}{m})) \frac{n^2}{m^2} > n^2(\alpha -\epsilon) $$ edge disjoint copies of elements of ${\cal F}$ in $G$. It follows that $\nu_{{\cal F}}(G) \geq n^2(\alpha-\epsilon)$. As $\nu^*_{{\cal F}}(G) = \alpha n^2$, Theorem \ref{t1} follows.
\square
The proof of Theorem \ref{t1} implies an $O(n^{poly(k_0)})$ time algorithm that produces a set of $n^2(\alpha-\epsilon)$ edge-disjoint copies of elements of ${\cal F}$ in $G$ with probability at least, say, $0.99$. Indeed, Lemma \ref{l21} can be implemented in $o(n^3)$ time using the algorithm of Alon et. al. \cite{AlDuLeRoYu}. Lemma \ref{l23} has a polynomial running time implementation due to Grable \cite{Gr}. Since we only need to compute $\psi^{**}$, rather than $\psi$, we can do this in $O(n^{poly(k_0)})$ time using any polynomial time algorithm for LP. The other ingredients of the proof are easily implemented in polynomial time.
\end{document} | arXiv |
Financial Inclusion and Household Welfare: An Entropy-Based Consumption Diversification Approach
Manisha Chakrabarty ORCID: orcid.org/0000-0002-3171-56611 &
Subhankar Mukherjee2
The European Journal of Development Research volume 34, pages 1486–1521 (2022)Cite this article
A Correction to this article was published on 19 October 2021
This article has been updated
State-led financial inclusion programmes have been implemented in many developing countries, but their effectiveness in raising welfare remains widely debated. In this article, we report evidence on this issue, against the backdrop of recent policy initiatives on financial inclusion in India. We employ Theil's entropy-based index to estimate diversification in consumption expenditure, and use this as a measure of welfare. Using household-level panel data across all regions of the country, we find evidence that greater financial inclusion increases diversity in non-food items. Further, we also notice that there is a shift in consumption basket from food items to non-food items. These findings suggest an improvement in welfare for both rural as well as urban households.
Des programmes d'inclusion financière dirigés par l'État ont été mis en œuvre dans de nombreux pays en développement, mais leur efficacité en matière d'amélioration du bien-être reste largement débattue. Dans cet article, nous rapportons des données probantes relatives à cette question, dans le contexte des récentes initiatives politiques sur l'inclusion financière en Inde. Nous utilisons l'indice fondé sur l'entropie de Theil pour estimer la diversification des dépenses de consommation et nous l'utilisons comme mesure du bien-être. En utilisant des données de panel au niveau des ménages dans toutes les régions du pays, nous trouvons des données qui prouvent qu'une plus grande inclusion financière augmente la diversité des articles non alimentaires. De plus, nous remarquons également un changement au niveau du panier de consommation, initialement constitué de produits alimentaires, et qui évolue vers des produits non alimentaires. Ces résultats suggèrent une amélioration du bien-être des ménages ruraux et urbains.
FinancialFootnote 1 inclusion is considered to be a crucial element in fostering economic growth and development of a country, through facilitating easier availability of credit, savings, payment, and insurance options to a large section of people (Chibba 2009). This idea has motivated a large amount of research as well as policy initiatives across different countries in the world (See World Bank 2014 for a review). The Indian government has, over the last 50 years, undertaken various such policy measures to improve financial inclusion in the country. Between 1969 and 1990, the government carried out a large-scale social banking programme to provide access of formal credit to the rural poor. More recently, in August 2014, the government launched another ambitious financial inclusion programme. The programme, called Pradhan Mantri Jan Dhan Yojna, or PMJDY, aims to open one bank account for every household in the country.Footnote 2 The government has started providing social security benefits to its citizens through these accounts, thereby creating an incentive to open the accounts. This has resulted in a sudden jump in the number of bank accounts held in the country. But how do such financial inclusion measures affect welfare of the households? We investigate this question in the present article.
Even though there are some studies that have looked into the link between financial inclusion and different dimensions of economic development in the recent past (such as access to banking, agricultural credit, saving behaviour, inequality etc. see Singh 2017 for a review of the literature), the evidence is rather scarce, especially in the context of developing countries like India. Further, to the best of our knowledge, there is no systematic study so far analysing this relationship in the context of the recent nation-wide financial inclusion programme (PMJDY).
One plausible mechanisms through which financial inclusion may impact household welfare is through easier access to savings and credit facilities (Bharadwaj and Suri 2020), which can alter production and employment choices (Aghion and Bolton 1997; Banerjee and Newman 1993). Better production and employment choices in turn lead to higher income and, thus, change the pattern of consumption expenditure,Footnote 3 including the possibility of widening of the consumption basket (Chai et al. 2015; Chakrabarty and Mandi 2019; Falkinger and Zweimüller 1996; Theil and Finke 1983). We examine this link in this article by investigating whether inclusion in the formal financial system by opening of bank accounts can lead to increase in consumption diversification through increase in income, proxied through monthly per capita expenditure (MPCE). As evidenced in the literature, the change in savings and borrowing behaviour due to financial inclusion not only create better employment opportunities leading to increased income, it also changes consumption expenditure pattern through increasing dietary diversity (Annim and Frempong 2018), smoothing of consumption etc. (Lai et al. 2020). We, however, do not explicitly explore these channels in the present paper due to lack of availability of data on borrowing, consumption smoothing etc.
We use diversification in consumption expenditure as a measure of welfare in this study. Increase in diversity of consumption expenditure, and not only growth in consumption expenditure, is widely accepted as an important determinant of economic welfare (Barro and Sala-i-Martin 2004; Grossman and Helpman 1991; Romer 1990). The poorer households, who cannot fulfil a threshold level of consumption for the inelastic basic goods, are not able to allocate expenditure on non-essential elastic goods. With increase in level of income, expenditure allocation on such goods increases, thereby widening the consumption basket. Hence, an increase in variety in the consumption basket is considered as welfare enhancing (Chai et al. 2015; Clements et al. 2006; Jackson 1984; Prais 1952). To estimate consumption diversification, we use the entropy-based measure of expenditure share proposed by Theil (1967) and Theil and Finke (1983). At low levels of income, the consumption basket tends to be homogeneous in nature. But as income rises, the basket becomes more heterogeneous, consisting of different types of goods. In this sense, our entropy-based measure will increase, indicating more diversity (see Eq. 1 in Sect. 4.1). Our first set of hypotheses test this impact of income enhanced through the opportunities of financial inclusion programmes, on consumption diversification among households.
We initially test the three following hypotheses on consumption diversification as a result of financial inclusion. First, if financial inclusion programmes are welfare enhancing, then it should decrease expenditure share on food items. At low income levels, share of expenditure on food items is higher, following Engel's law. Banerjee and Duflo (2007) suggested this share to be around 50–70 percent of household budget for the poor. But, as income grows, this share comes down substantially, to around 30 percent (Clements and Chen 1996). Along with this, the diversification in consumption within the food items is expected to increase. Our first hypothesis tests this proposition. Further, the welfare enhancing effect of financial inclusion should lead to diversification in expenditure on non-food items as well due to increased income. The second hypothesis investigates this. Finally, the third hypothesis assesses whether increasing income achieved through access to more financial resources leads to a shift in expenditure from food basket to non-food consumption basket. We carry out additional tests to examine the independent impact of financial inclusion on diversification of consumption due to the presence of other plausible transmission channels as discussed above. However, we do not explicitly identify these specific channels.
We employ a panel data method to carry out our analysis. The panel of households comes from a survey conducted by the Centre for Monitoring Indian Economy (CMIE). The survey collects data on consumption expenditure, demographic information, caste, religion, asset holding, region of residence etc. along with the information on bank account opening status for each individual within every household, three times a year (defined as waves). We construct a strongly balanced panel of 49,739 households for each time period, from this dataset using data for first nine waves of the survey, covering the period of January 2014 to December 2016. In order to capture the effect of the financial inclusion programme, PMJDY, we construct two time-dummy variables. The first dummy variable investigates the immediate impact of PMJDY and the second dummy variable helps us to account for the effect one year after the incubation of the programme. To examine the transmission path from financial inclusion to consumption diversification through increase in income, proxied through monthly per capita expenditure (MPCE), we employ a two-stage regression approach. In the first stage, we regress monthly per capita consumption expenditure on the financial inclusion dummies, as well as a set of control variables. Then, in the second stage, we regress the Theil's entropy-based index on the predicted values of the per capita expenditure obtained from the first stage. Even though we do not explicitly explore additional channels in this paper, we run a separate set of second-stage regressions including the financial inclusion dummy variables, to account for the separate effect of financial inclusion.
Our empirical analysis shows that there is indeed an increase in diversity due to increase in total expenditure after introduction of PMJDY scheme. However, the separate effect of PMJDY on diversification of food expenditure is ambiguous. Immediately after the initiation of the PMJDY programme, there was a drop in diversification. But, the effect of financial inclusion is positive after a year. This increase may be anomalous, possibly due to a sudden jump in the index immediately after one year. A closer inspection reveals a drop in the longer run as well. This observation is possibly due to lack of information on detailed items and quality of items within the food group. Regarding heterogeneity in non-food expenditure, and shift in expenditure from food to non-food items, our results are as expected. Both measures show an increase in diversity in the shorter run as well as in the longer run after the initiation of the programme. Our examination of plausible mechanism shows that the rise in diversity is indeed through an increase in total consumption expenditure as evidenced through the significant positive coefficients of predicted values of the per capita expenditure. This signifies an improvement in household welfare. However, we also notice an independent positive effect of financial inclusion on consumption diversification in addition to the above-mentioned channel. We have discussed some of the plausible pathways in the previous paragraphs. Apart from that, this effect could also arise due to measurement error in household per capita consumption expenditure (MPCE) since it does not include certain components of consumption expenditure such as housing, car etc. It may also be because of failure to capture adequately the economies of scale in MPCE due to the presence of heterogeneous demographic compositions within households.Footnote 4
This research contributes to the literature in several ways. First, we employ an entropy-based diversity as a measure of household welfare. The concept of diversity has been widely used to measure various dimensions of concentration in an industry/market, using primarily the Hirschman-Herfindahl index. The application of diversity measures, however, is quite scarce in the existing literature on the economics of household welfare. Recently, Chakrabarty and Mandi (2019) have used Theil's entropy-based diversification measure on cross-sectional household survey data to study the determinants of consumption diversification. This study is possibly the closest to our approach. However, this study does not explore the role of a social policy, such as the financial inclusion programme we focus on, on diversification. Moreover, we use a country-wide longitudinal data for our analysis.
Second, our study contributes to the studies on the role of financial inclusion programmes on economic development. The determinants and impact of financial inclusion, especially that through opening of bank accounts, has been well studied in the context of the industrialized countries. Lusardi and Mitchell (2014) provide a review of such studies conducted in 12 developed countries. There are some recent studies focussing on the issues of financial inclusion in developing country settings as well. Chin et al. (2011) studied the effect of opening bank accounts among the Mexican immigrants in the USA. They found an increase in savings share out of income. Dupas et al. (2018) investigated the effect of increased access to basic bank accounts to unbanked rural households in three countries across two continents: Uganda, Malawi and Chile. They do not find discernible welfare effects due to this increased access. One possible reason of this finding may be that overwhelming majority of the poor did not make much use of the bank accounts. Further, in a study covering eight African countries, De Koker and Jentzsch (2013) find that access to formal financial services through financial inclusion programmes may not substitute usage of informal financial services. They cite existence of informal employment (and therefore payments in cash) in the economies as a probable reason.
In the context of India, Burgess and Pande (2005) found that increasing access to banking can be instrumental in declining rural poverty through higher deposit mobilization and credit disbursement. In a relatively recent study, Young (2015) has reinforced the beneficial impacts of access to banking services through robust empirical analysis. On the other hand, Kochar (2011) finds that expansion in access to banking may actually increase consumption inequality, through unequal access to credit between the rural rich and the poor. All these studies used the rural bank branch expansion initiative in India, undertaken between 1969 and 1990, as the context. In contrast, our study uses the recent financial inclusion programme, named Pradhan Mantri Jan Dhan Yojna or PMJDY, initiated at the national level, as the backdrop.
Third, our research strategy also distinguishes itself from most other existing studies on financial inclusion. Many of the studies on this topic are limited to a smaller geographical area, thereby lacking generalizability in a larger context. The handful number of studies using country-wide data also confine themselves into cross-sectional analysis (Badarinza et al. 2016). Even though the study by Burgess and Pande (2005) uses panel data for their analysis, the data are aggregated at state level. In our study on the other hand, we elect to use a nationally representative sample of roughly 150,000 households to pursue our research objective. These households were surveyed once in every four months, allowing us to construct a panel of the sample households. This dataset and our research approach potentially allow to arrive at more precise estimation of the effects of our interest.
Fourth, apart from the standard fixed effect model, we also use the Hausman-Taylor estimation method to estimate the coefficients in order to address endogeneity issues of some of our variables (Hausman and Taylor 1981). One disadvantage of the fixed effect estimator is that it cannot provide the effects of the time-invariant covariates in the model. However, some of those variables may be crucial in understanding the variations in consumption diversity, and may be informative in making policy decisions. Hausman-Taylor estimation method enables us to recapture the effects of those factors in our model (See for example Poprawe 2015; Quayes 2015). Finally, most of the studies on financial inclusion have confined themselves to studying the effect on the rural poor population only. We conduct our analysis both on the rural and the urban sample. This extension can potentially increase generalizability of our results.
The rest of the article is organized as follows. We start with a brief overview of the financial inclusion programmes in India, focussing on the PMJDY scheme. Section three discusses the dataset used for empirical analysis of the paper. The empirical strategy adopted is discussed in section four, while the results are presented in Section five. Section six concludes.
Financial Inclusion Programmes in India
In this section we provide a brief overview of the financial inclusion programmes in India, focussing on the recent PMJDY programme, since this programme is the main context of the paper. Attempts to widen financial inclusion in India can be traced back to 1969, when the central bank mandated the commercial banks of the country to open branches in rural unbanked locations. More recently, in 2006, the central bank implemented financial extension services, under which Non-Governmental Organizations, Micro Finance Institutions and other Civil Society Organizations could be employed as intermediaries to increase outreach of the banking sector. Simultaneously, commercial banks were encouraged to provide zero minimum balance accounts access to the economically weaker section.
The Pradhan Mantri Jan Dhan Yojna, or PMJDY programme is an extension of these initiatives. The programme was announced by the Prime Minister of India on the Independence Day (15th August) in 2014, and was formally launched on 28th August of the same year. The primary aim of the programme is to open at least one bank account for every household. Towards this goal, the scheme allows opening of zero balance bank accounts in public sector, private sector or regional rural banks. Additionally, the PMJDY accounts are equipped with an overdraft facility of up to Rs. 10,000, life insurance cover of Rs. 30,000, and a debit card with in-built accidental insurance coverage of Rs. 100,000. The programme also aims at providing Direct Benefit Transfers (DBTs) through these accounts instead of handing over cash to the beneficiaries. These additional benefits, and large-scale awareness building campaigns have resulted in substantial increase in number of bank accounts opened: approximately 180 million new bank accounts were opened within the first year after launch of the programme, and close to 325 million new accounts were opened till the end of August 2018 (i.e. three years after launch of the programme). Since August 2018, the scope of the programme has been extended to opening one bank account for every adult in the country. Till August 2019, i.e. after five years of launch, the number of accounts opened reached to over 367 million. Table 1 shows the annual progress of the scheme between September 2014 and August 2019.
Table 1 Number of beneficiaries of PMJDY Scheme, from September 2014 to August 2019
The empirical analysis in this paper primarily relies on the Consumer Pyramids Survey conducted by the Centre for Monitoring Indian Economy (CMIE) since January 2014. This is a household-level longitudinal survey, covering roughly 150,000 households spread across all states and union territories in India. Around two third of the sampled households are from urban areas. The survey captures information on household demographics, composition of income and expenses, details on assets and liabilities, employment details etc. The data for each household are collected three times a year, called wave. A wave starts in the months of January, May and September, and repeats every year. The first wave of survey was conducted between January and April 2014. For our study, we consider household-level data for first nine waves of the survey, covering the period of January 2014 to December 2016. The reason for not considering data for subsequent waves is to avoid confounding effects of other large exogenous macroeconomic shocks such as demonetisation (announced on 8th November 2016) and Goods and Services Tax (GST) (implemented since June 2017).
The survey data capture information on bank account opening status for each individual within every household, separately for each wave. We exploit this information, combined with the timing of PMJDY implementation,Footnote 5 to analyse its effects on consumption patterns. The PMJDY scheme was announced on 28th August 2014. By that time, two waves of the survey were completed. Given the exogenous nature of the scheme, this gives us the opportunity to compare the effect of PMJDY before and after the scheme was introduced. As expected, the data show a positive discontinuous shift in number of bank accounts opened right after the introduction of the scheme, as shown in Table 2 and Fig. 1.Footnote 6 Table 2 presents the mean of the proportion of household members within a household holding a bank account. The data are shown for each of the 9 waves, separately for rural and urban regions. The difference between waves 2 and 3 (i.e. immediately before and after announcement of PMJDY scheme) is approximately 6 percentage points, which is the highest between any two waves we consider. The difference in mean values is similar between rural and urban regions. Two-sample t-tests show that this difference is highly statistically significant. Therefore, it is safe to assume that introduction of the PMJDY scheme led to an increase in opening of bank accounts in India.
Table 2 Mean of the proportion of household members holding a bank account, in each wave
Trend of the proportion of household members holding a bank account
We proceed as below to arrive at our final sample of households. First, we keep data for the major 20 states, which accounts for 94% of the whole sample, and exclude the union territories and the north eastern hilly regions of the country from the sample.Footnote 7 Second, we drop all records for which there were missing or invalid values for the included variables in our empirical models (for example, we do not include records if age is specified as -99, or caste is specified as 'not stated'). Third, we keep records only for those households who appeared in each wave of the survey. The final pooled sample size, after carrying out these cleaning operations is 447,651. This implies that we have a balanced panel of 49,739 households for each of the nine waves we consider. Further, there is a systematic difference in consumption pattern between households in rural and urban regions in India (Krishnaswamy 2012). Table 3 shows that this difference is present in our sample of households as well. Since consumption-based measure is our main impact variable, we carry out our analysis separately for urban and rural regions. The number of households in rural region for each wave in our sample is 16,218 and that for urban region is 33,521.Footnote 8 Table 12 in Appendix shows the list of states and number of households included from each state, separately for rural and urban regions.
Table 3 Summary statistics
Table 3 shows the descriptive statistics for the variables used in our analysis, separately for the rural and urban regions. The variables include monthly per capita food and non-food expenditure,Footnote 9 per capita total expenditure,Footnote 10 caste category, religion, number of children and an asset ownership index for the households.Footnote 11
We use monthly per capita expenditure as a proxy for income of the households. Similarly, we use the asset ownership index variable as proxy for the households' wealth. This variable has been built in the following way. The survey captures asset ownership for each household in several categories of asset ownership. We select 19 among those categories that are relevant to our study (the list of all asset variables captured for our study are presented in the appendix, in Table 14). Among these 19 variables, 16 are indicator variables, having '1' if the household owns the asset and '0' otherwise. The rest three variables are continuous in nature (number of houses owned, number of tractors owned, and number of cattle owned). We convert the continuous variables into binary indicator variables, similar to the other indicator variables, by assigning value '1' whenever the value of the continuous variables is greater than zero, and '0' otherwise. Next, we take a simple average of the values populated in all the 19 items to arrive at the asset index. A lower value of the index signifies lower asset holding, and vice versa.
The dataset captures information on castes in eight categories.Footnote 12 We reorganize this data in the following four categories: Upper Caste, Intermediate Caste,Footnote 13 OBC and SC & ST. We do not include the households that did not reveal their caste. Similarly, we rearrange the religion reported by the households in three categories: Hindu. Muslim and others (consisting of Buddhist, Christian, Jain and Sikh religion categories). We assume that caste and religion are exogeneous as well as time-invariant in nature. Finally, we include a variable that captures the number of children within a family for each period. We define a family member as children if his/her age is below 16.
In Table 3, Panel A shows the descriptive statistics for the rural sample, whereas Panel B shows the same for the urban sample. The differences in mean of the variables between rural and urban regions are on the expected lines. Per capita food expenditure, non-food expenditure and per capita total expenditure for the urban households is higher than the rural households.Footnote 14 On the other hand, urban households report fewer number of children compared to rural ones on average. Share of Hindu population is higher in the rural areas at 89% vs. 85% in urban areas. This is almost entirely covered by the larger share in Muslim population from 6% in rural areas to 9% in urban areas. Distribution of the households by caste composition shows that the proportion of upper caste households residing in urban areas is double that of rural areas. 30% of the sampled households in urban areas belong to upper caste, compared to 15% in rural areas. This distribution is reversed for the SC and ST communities (24% in urban areas vs. 36% in rural areas) and remains almost same for OBC communities.
We also report mean expenditure shares for food and non-food group items for all the waves, separately for rural and urban regions, as shown in Table 4. For both regions, share of expenditure spent on food items have decreased over time. On the other hand, share of expenses on non-food items have increased marginally. This trend is as expected, and follows the Engel's law, assuming households' real income have gone up.Footnote 15
Table 4 Trend of mean expenditure shares on food and non-food items, separately for rural and urban regions
Empirical Strategy
In order to address the central question of whether households' welfare improves through diverse expenditure pattern within sub-group of commodities due to financial inclusion programme, we construct two time-dummy variables to capture the effects of financial inclusion.Footnote 16 The PMJDY scheme was officially initiated from 28th August 2014. Therefore, the first dummy, named PMJDY1, holds the value '0' for all households for the first two waves (January to April and May to August), and '1' for all households for all subsequent waves. However, the PMJDY scheme was launched as an initiative to increase the number of bank accounts in the country over a period of time. Therefore, in all practicality, only a fraction of the households opened their bank accounts as part of the financial inclusion programme immediately after the initiation of the programme. Hence, we include another indicator variable, named PMJDY2, which takes the value '0' for the first five waves, and '1' thereafter (i.e. from sixth to 9th waves). This variable helps us account for the effect on consumption patterns one year after the incubation of the programme. Lastly, some of the households in our sample may have opened bank accounts due to different supply-side and demand-side factors other than the PMJDY scheme. Also, in developing countries, the access to financial resources may happen through informal sources (Banerjee 2004). We include a time trend variable (named as wave) to account for these and other general macroeconomic trends.
Theil Diversification Index
We use diversification of consumption expenditure as the measure of household welfare in this article (see for example, Chai et al. 2015; Chakrabarty and Mandi 2019; Clements et al. 2006 for such association). There are different approaches to measure consumption diversification, such as Gini-Simpson index (Simpson 1949), distance-based measures (Lieberson 1969), absolute value-based measures (Reardon and Firebaugh 2002) etc. In our analysis, we consider entropy-based measure proposed by Henry Theil (Theil 1967; Theil and Finke 1983). Entropy in general captures the degree of 'dividedness' in a system. Theil introduces this concept as a 'measure of dividedness' of economic variables, such as racial division, industrial diversification, political diversification etc. (Theil 1972). One advantage of using Theil's measure is that it allows decomposition of the diversity index into two components: the diversity within separate entities and also the diversity between these entities (Palan 2010; Reardon and Firebaugh 2002). We extend this concept to measure the diversification in consumption expenditure of households.Footnote 17 The rest of this subsection describes this measure in brief, with specific references to our study.
Suppose there are '\(n\)' number of commodities that a household consumes, and \({w}_{i}\) is the share of the total budget the household spends on \(i\)th commodity. Clearly, \(\sum _{i=1}^{n}{w}_{i}=1.\) Then, the Theil's measure for diversification of consumption expenditure \(H(w)\) for the household is given by:
$$H\left(w\right)=-\left(\sum _{i=1}^{n}{w}_{i}log{w}_{i}\right)$$
Theil calls the measure obtained from this expression "entropy". Minimum value of entropy is obtained when all expenses are incurred on only one commodity, and maximum value is obtained when equal share of expenditure (i.e. \(1/n\)) is spent on each commodity.
Further, Theil's Entropy Decomposition Theorem (Theil 1972) establishes that if the system can be divided into smaller sub-groups, then the overall entropy of the whole system can be decomposed into the sum of two separate entropies: the within-group entropy, and the between-group entropy, as shown in Eq. (2) below:
$$Overall~entropy = BetweenGroup~entropy + ~weighted~average~of~WithinGroup~entropies$$
In our analysis, we consider two broad groups of commodities in which a typical household incurs its expenditure: food items and non-food items.Footnote 18 The entropy measure between these two broad groups will yield the between-set entropy. Further, within each of these broader groups, we construct several sub-groups. There are 13 sub-groups within the food items group and 7 sub-groups in the non-food items group (details of the commodities considered in each group and sub-group is presented in Table 13 in the appendix.). Therefore, the entropy between the food items group and the non-food items group is the first term in Eq. (2), whereas the weighted average of entropies of food items group and non-food items group is the second term. Equations (3) and (4) show the expressions for between-groups and within-group entropies, with reference to our study.
$$\begin{aligned} entropy\;between\;food\;and\;nonfood\;groups & = \,\left( {\frac{{Expenditure\;on\;food\;items}}{{Total\;expenditure\;on\;food\;and\;nonfood\;items}}} \right)*log\left( {\frac{{Expenditure\;on\;food\;items}}{{Total\;expenditure\;on\;food\;and\;nonfood\;items}}} \right) \\ & \quad + \left( {\frac{{Expenditure\;on\;nonfood\;items}}{{Total\;expenditure\;on\;food\;and\;nonfood\;items}}} \right)*log\left( {\frac{{Expenditure\;on\;nonfood\;items}}{{Total\;expenditure\;on\;food\;and\;nonfood\;items}}} \right) \\ \end{aligned}$$
$$\begin{aligned} weighted\;average\;of\;entropies\;within\;food\;items\;group\;and\;nonfood\;items\;group & = \left( {\frac{{Expenditure\;on\;food\;items}}{{Total\;expenditure\;on\;food\;and\;nonfood\;items}}} \right) \\ & \quad *\left( { - \mathop \sum \limits_{i = 1}^{13} share\;of\;total\;expenditure\;on\;ith\;food\;itme*\log \left( {share\;of\;total\;expenditure\;on\;ith\;food\;item} \right)} \right) \\ & \quad + \left( {\frac{{Expenditure\;on\;nonfood\;items}}{{Total\;expenditure\;on\;food\;and\;nonfood\;items}}} \right) \\ & \quad *\left( { - \mathop \sum \limits_{i = 1}^7 share\;of\;total\;expenditure\;on\;ith\;nonfood\;itme*{\text{log}}(share\;of\;total\;expenditure\;on\;ith\;nonfood\;item} \right) \\ \end{aligned}$$
In our econometric analysis, we use both within-group entropies as well as between-groups entropy as measures of household welfare.
In our discussions so far, we hypothesized the primary link from financial inclusion to welfare in the following way: financial inclusion may improve employment opportunities, and thus household income. Further, increased income leads to both increase and diversification of consumption expenditure. And finally, diversified consumption causes welfare gain.
Specifically, for our welfare measure, welfare gain will translate into diversification in consumption within the food group and non-food commodity group, as well as between the food and non-food commodity groups. Accordingly, we formulate the hypotheses as follows:
Financial inclusion will lead to diversification in food expenditure.
Financial inclusion will lead to diversification in non-food expenditure.
Financial inclusion will lead to shift in expenditure from 'within only food group' to 'between food group and non-food group'.
Econometric Model
For the analysis of impact of financial inclusion on consumption diversification, we employ a panel data regression method. The Consumer Pyramid database collects information about the same households over repeated periods, making it ideal for tracing the changes in economic behaviour of same households over a period of time. We exploit this feature of the dataset to investigate the effect of large-scale financial inclusion programme on households' welfare.
Two-Stage Regression: Income Channel
To examine the path from financial inclusion to consumption diversification through increase in consumption expenditure, we employ a two-stage regression approach. In the first stage, we regress monthly per capita consumption expenditure (logMPCE) on the financial inclusion indicators, as well as a set of control variables (Eq. 5). Then, in the second stage, we regress the three diversification indices (food index, non-food index, and between index) on the predicted values of the per capita expenditure (\(\widehat{{logMPCE}_{it})}\) obtained from the first stage, along with a smaller set the control variables (Eq. 6).
$${\text{First}}\; {\text{stage:}}\log MPCE_{{it}} = \alpha _{0} + \alpha _{1} PMJDY1_{t} + \alpha _{2} PMJDY2_{t} + \alpha _{3} Asset~Index_{{it}} + \alpha _{4} Numbe~of~Children_{{it}} + \alpha _{5} Mean\_education_{{it}} + \alpha _{6} \Pr oportion\_working_{{it}} + \alpha _{7} wave_{t} + \mathop \sum \limits_{{k = 1}}^{3} \alpha _{k}^{{caste}} Caste_{i} + \mathop \sum \limits_{{j = 1}}^{2} \alpha _{j}^{{\text{Re} ligion}} \text{Re} ligion_{i} + \mathop \sum \limits_{{s = 1}}^{{19}} \alpha _{s}^{{State}} State_{i} + \rho _{i} + u_{{it}}$$
$${\text{Second}}\,{\text{stage:}}\,Y_{{it}} = \beta _{0} + \beta _{1} \widehat{{\log MPCE_{{it}} + }}\beta _{2} Asset~Index_{{it}} + \beta _{3} Numbe~of~Children_{{it}} + \beta _{4} Mean\_education_{{it}} + \beta _{5} wave_{t} + \mathop \sum \limits_{{k = 1}}^{3} \beta _{k}^{{caste}} Caste_{i} + \mathop \sum \limits_{{j = 1}}^{2} \beta _{j}^{{\text{Re} ligion}} \text{Re} ligion_{i} + \mathop \sum \limits_{{s = 1}}^{{19}} \beta _{s}^{{State}} State_{i} + \alpha _{i} + u_{{it}}$$
We use these equations to run six sets of regressions. First, following the hypotheses mentioned above, we run a regression for each type of Theil Diversification Index: diversification within food group, diversification within non-food group and diversification between food and non-food group. Further, as mentioned earlier, we carry out our analysis separately for rural and urban regions, for each diversification index. \({Y}_{it}\) is the dependent variable that captures the appropriate diversification index (Theil Food Index, Theil non-food Index and Theil Between Index), where 'i' denotes the households and 't' denotes the waves.
PMJDY1 is a binary variable capturing the immediate impact of the PMJDY scheme. It takes the value '0' for the first two waves and '1' for the subsequent seven waves. However, as mentioned before, given the scale and scope of the PMJDY financial inclusion programme, its take-up, and therefore its impact on households' consumption might not be apparent immediately after the launch of the scheme. To address this concern, we include another time-dummy variable that takes the value '0' for the first five waves, and '1' for the next four waves. This dummy, denoted as PMJDY2 in the model, measures the effect of the financial inclusion programme after one year of inception.
Therefore, based on our model, our three hypotheses H1a, H2a and H3a for three different Theil Indices (food, non-food and between) that capture the effect of financial inclusion through the channel of increasing income, can be written as shown below:
$${\text{H1a, H2a, H3a:Null hypothesis:}}\beta _{1} \le 0;\,{\text{alternate hypothesis:}}\,\beta _{1} > 0.$$
Consumption diversity could also arise due to demographic compositions and heterogeneity in tastes. The extant literature has also shown that demographic attributes play important role in consumption decisions. We include three variables to control for these attributes. First, the impact of number of children on household decision making, as well consumption diversification has been studied in the literature (Chakrabarty and Mandi 2019; Flurry 2007; Ray 1986). Accordingly, we include a variable Number of Children that captures the number of family members below age 16 in a household. Second, educational attainment is also found to affects consumption expenditure (for example, see Maitra and Ray 2004). Therefore, we include the variable \(Mean\_education\) in our regressions. As the name suggests, this variable depicts the level of education in a household, averaged over all family members. Finally, the relation between employment and consumption is also well established in the literature (see Mincer 1960, for example). To account for this, we include another explanatory variable, \(Proportion\_working\). This variable is constructed by dividing the number of family members working within a household by the total number household members. All three variables are calculated for each household for each wave separately. We must note that the variable \(Proportion\_working\) is included only in the first-stage regression as an additional variable.
Ando and Modigliani (1963) first stated the role of asset in consumption behaviour in their life-cycle hypothesis of consumption. Since then there has been extensive empirical research in this field using macroeconomic time series data (see Altissimo et al. 2005). There are a few recent studies on this effect using microeconomic data as well. For example, Campbell and Cocco (2007) have studied the effect of home price fluctuation on consumption for homeowners of different age groups. Caceres (2019) studied the effects of various types of wealth, such as housing, financial assets, and total net worth on consumption. Bostic et al. (2009) have investigated the impact of wealth on consumption expenditure using household-level consumer expenditure survey. Following this, in our study, we include Asset Index as an explanatory variable in the empirical model.
Social characteristics such as caste and religion play an important role in consumption pattern for households especially in developing countries like India. Bailey and Sood (1993) have studied the effect of religious affiliation on consumption behaviour. Borooah et al. (2014) have studied link between caste and consumption pattern. Khamis et al. (2012) have shown the impact of both castes as well as religion on visible consumption expenditure. Hence, we include time-invariant Religion and Caste dummies in our empirical model. Since religion and caste of a person is ascribed at the time of birth, and there is very little possibility of its change over their lifetime, we consider these factors as time-invariant and exogenous in nature. Additionally, to capture general macroeconomic linear trend, we include the wave variable that contains value from 1 to 9 covering our whole time period. Finally, some of the unobserved state level heterogeneity that might affect consumption patterns are captured through state level fixed effects.
Two-Stage Regression: independent effect of financial inclusion
Apart from the income channel mentioned above, financial inclusion may affect welfare through other channels as well. For example, higher access to savings and credit due to financial inclusion (Bharadwaj and Suri 2020) may lead to more diversity in consumption (Annim and Frempong 2018) and also consumption smoothing (Lai et al. 2020), thereby indicating a change in the pattern of consumption. This may lead to possible widening of the consumption basket. We, therefore, run another six set of regressions employing Eq. 7, in which we include, additionally, the purely exogenous financial inclusion dummy variables in the second stage to account for the existence of such separate effect of financial inclusion over and above the income channel. The coefficients \({\beta }_{1}\) & \({\beta }_{2}\) in Eq. 7 correspond to the short run and long run independent impact of PMJDY scheme after controlling for all other relevant explanatory variables included in the second-stage regression along with the income channel:
$$\begin{aligned} {Y_{it}} & = {\beta _0} + {\beta _1}PMJDY{1_t} + {\beta _2}PMJDY{2_t} + {\beta _3}\widehat {logMPC{E_{it}} + }{\beta _4}Asset\;Inde{x_{it}} \\ & \quad + {\beta _5}Numbe\;of\;Childre{n_{it}} + {\beta _6}Mean\_educatio{n_{it}} + {\beta _7}wav{e_t} \\ & \quad + \mathop \sum \limits_{k = 1}^3 \beta _k^{caste}Cast{e_i} + \mathop \sum \limits_{j = 1}^2 \beta _j^{Religion}Religio{n_i} + \mathop \sum \limits_{s = 1}^{19} \beta _s^{State}Stat{e_i} + {\alpha _i} + {u_{it}} \\ \end{aligned}$$
Therefore, based on our model, the separate effect of financial inclusion on consumption diversification can be tested using the following hypothesesFootnote 19:
Null hypothesis: \(\beta_{1}\le 0,\) alternate hypothesis: \(\beta_{1}>0\), H1b': Null hypothesis: \(\beta_{2}\le 0\), alternate hypothesis: and \(\beta_{2}>0.\)
Null hypothesis: \(\beta_{1}\le 0\), alternate hypothesis: and \(\beta_{1}>0.\) H2b': Null hypothesis: \(\beta_{2}\le 0\), alternate hypothesis: and \(\beta_{2}>0.\)
If we are able to reject the null hypotheses, that might indicate the presence of separate channels.
We start our analysis by testing for plausible presence of heteroskedasticity using the Modified Wald Test (Greene 2011). We find strong presence of heteroskedasticity, and therefore prefer to use the alternate Hausman test to identify the suitable estimation model between the Fixed vs. Random Effects estimators.Footnote 20 Subsequently, upon confirmation that the Fixed Effect estimator is the optimum model for our underlying data,Footnote 21 we use the FE estimator model with heteroskedasticity adjusted robust standard error. Additionally, we also employ an alternate approach proposed by Hausman and Taylor (1981) based on 2SLS estimation method, in order to take care of plausible endogeneity of wealth variable. This approach also allows us to estimate the effects for the time-invariant variables, such as caste and religion.
We start the discussion of this section with the results obtained by running two-stage panel data regressions using Eqs. 5 and 6. This approach is employed to examine the effect of financial inclusion on consumption diversification through one plausible channel, i.e. increase in consumption expenditure. Table 5 shows results from the first-stage regression. The coefficients for the financial inclusion indicators are positive and significant, implying that financial inclusion increases monthly per capita expenditure. We use the predicted values of monthly per capita expenditure as an independent variable in the second stage. Additionally, we run a separate set of second-stage regressions (by using Eq. 7), where we include the financial inclusion indicator variables in the second-stage equation to account for separate effect apart from the effect through increase in income, if any.
Table 5 First stage regression results
Thus, we run twelve sets of second-stage regressions, for three dependent variables – Theil Food Index, Theil non-food Index and Theil Between Index – separately for rural and urban regions. The reason for running the regressions separately for rural and urban regions is that because of systematic differences in consumption expenditure between these two regions (refer to Table 3), patterns of consumption diversity is also expected to be different. Table 6 shows the Theil Diversification Index values for 'within food group', within 'non-food group' and 'between food and non-food group', separately for rural and urban regions. While both 'within non-food group' and 'between-group' consumption diversity increase, 'within food group' diversity decreases over time.
Table 6 Trend of the different theil diversification indices
Table 7 shows results for Theil Diversification Index, for food items group. Columns 1 and 2 show that Monthly Per Capita expenditure (MPCE) is positively correlated with diversification in food expenditure (i.e. we reject the hypotheses, H1a). This is expected, and corroborates with previous findings (Clements et al. 2006; Falkinger and Zweimüller 1996). However, the coefficient for this variable in the second-stage captures the value of monthly per capita expenditure, conditioned on the set of explanatory variables mentioned in Eq. 5.
Table 7 Regression results for diversification of food index
Also, there is a negative correlation between asset holding and diversification in food expenditure. Given the inelastic nature of the food group commodities, this result is also in the expected lines. Further, an increase in wealth should shift a part of expenditure away from food group to non-food group.Footnote 22 This is evident from the values and signs of coefficients for the Asset Index variable, as evidenced from the regression coefficients of between groups, shown in Table 11. The coefficient for number of children in the household is positive and significant. This corroborates findings from recent studies (see Chakrabarty and Mandi 2019, for example). The reason behind increasing diversification on number of children is that presence of children necessitates households to spend on items such as milk, fruits etc. Columns 3 and 4 of Table 7 shows estimates for the time-invariant variables, using Hausman-Taylor estimation method.Footnote 23 Coefficients for socially disadvantaged castes and religious groups is mostly positive for food group diversification. We argue that this result on the expected lines too, since relatively poorer groupsFootnote 24 would diversify their expenditure within food group, rather than consuming non-food group items. This is also evident from the negative coefficient for these groups when we estimate shift between food and non-food group (Table 11). Finally, the wave variable, which captures general macroeconomic trend, shows negative impact on food diversification, as is expected from the descriptive statistics in Table 6.
Columns 5 and 6 show results including the financial inclusion indicators. The PMJDY1 dummy shows a statistically significant negative impact, and the PMJDY2 dummy shows a statistically significant positive impact on 'within food group' consumption diversification. These dummies capture the separate effect of financial inclusion, independent of other control variables, including the total expenditure variable.
We reiterate that PMJDY1 dummy captures the effect immediately after the launch of the financial inclusion scheme, i.e. from third wave onwards, whereas PMJDY2 captures the effect after one year of launch of the scheme, i.e. from sixth wave onwards. According to our hypotheses H1b and H1b', we expected the diversity within food items to increase. However, we are not able to reject the null hypothesis for hypothesis H1b, but the hypothesis H1b' is rejected by our analysis. A robustness check considering dummy variables with value '0' for waves one to seven, and '1' for subsequent two waves, and also with value '0' for waves one to four, and '1' for subsequent five waves in fact shows that the coefficient for the PMJDY2 variable is negative in both cases. The positive sign of the PMJDY2 variable in Table 7 may be an aberration due to sudden positive jump in Theil Diversification Index for food items from wave 5 to wave 6 (see Table 6).
According to some medical literature, dietary diversification increases with increase in financial resources (Morseth et al. 2017). But our analysis does not reveal this pattern, possibly due to the some data limitation. Information on expenditure on only broad sub-items within the food group is available in our dataset; the same on detailed items is not available. This limits our scope of capturing variation in consumption within the food group items.
However, there is a possibility that households may shift within the food group from lower quality food items to selected higher quality items (and hence higher welfare), which may be inferred from Table 8. The table shows that mean of proportion of food items consumed by households over the waves has decreased by 2% from 1st to 9th wave. Secondly, it may be noted that the Theil's within-group diversification index does not reveal the shift in quality of the items consumed; rather, it only captures the heterogeneity of items consumed. However, a separate analysis using the data shows that there is a substitution from lower income-elastic to higher income-elastic (luxury) goods, substantiating a shift in quality of consumption as well.Footnote 25
Table 8 Mean of proportion of food items consumed by households, over time
Results clearly show that omitting the financial inclusion indicator variables increases the magnitude of coefficients for predicted total consumption substantially. This indicates a possibility of bias due to omission of significant separate impact of financial inclusion indicator variables. Therefore, this might suggest the presence of other plausible channels, as mentioned before. Further, this bias may arise due to measurement error of the consumption variable, or the possible failure to capture the presence of economies of scale in MPCE. We must mention that the coefficients for other control variables are qualitatively similar.
Table 9 shows results for non-food items group, using both Eqs. 6 and 7. Columns 1 and 2 show results obtained from Eq. 6. There is an increase in diversification both for rural as well as urban households, as indicated by the positive and statistically significant coefficient \(\beta_{1}\). This effect confirms the impact of financial inclusion on consumption diversification through increase in income. Hence, we reject the null hypotheses for H2a. We also notice positive and significant impact of financial inclusion indicator variables over and above the income effect (hence we reject hypotheses H2b and H2b').
Table 9 Regression results for diversification of non-food index
Descriptive statistics showing the trend in mean shares of expenditure within non-food group (Table 4), and also the mean of the proportion of items consumed within this group (Table 10), support our findings. The last column of Table 10 indicates that over time there is a considerable increase in the number of non-food items consumed by both rural as well as urban households.
Table 10 Mean of proportion of non-food items consumed by households, over time
All other variables affecting non-food diversity are positive and significant, as expected. Given that the non-food items group consists of relatively elastic commodities, asset holding is expected to have a positive impact on diversification. Interestingly, for the socially disadvantaged castes, the effects on diversification in non-food consumption is higher than socially advantaged castes. This may be explained through the differences in initial endowments between the advantaged and disadvantaged groups. In other words, since the historically socially advantaged groups possess higher wealth and also earn higher income on average, their existing consumption basket for non-essential commodities is expected to be broader than the other groups. Therefore, the marginal improvement in consumption diversification should be higher for the group that start from lower level of initial consumption.
Finally, Table 11 shows results for the shift in expenditure from food items group to non-food items group, i.e. the first component of Eq. (2). The effect of financial inclusion on the between-group diversification through the channel of income is positive and significant, implying the rejection of null hypothesis H3a. All other time-varying coefficients are positive and statistically significant as well. The decrease in the between-group expenditure for the socially disadvantaged groups (relative to the base group) possibly signifies that they are able to diversify within the food group of commodities and non-food group of commodities (as shown in earlier tables), but lower MPCE for this social groups does not allow them to shift expenditure from the essential commodities (i.e. food items) to relatively non-essential commodities (i.e. non-food items). We would like to re-emphasize that the separate and significant effect of financial inclusion is evident even for between-group diversity index (columns 5 and 6). Hence, we reject hypotheses H3b and H3b'. This shows that there could be channels other than income which drive the change in consumption pattern due to financial inclusion.
Table 11 Regression results for diversification of between index
The main contribution of this paper is to assess whether state-led financial inclusion programmes can lead to higher economic welfare. We conduct this study using the recent thrust on financial inclusion (primarily through the PMJDY programme) in India. The country-wide launch of the programme, a sharp increase in the number of bank accounts opened since its launch, varied opinion on the programme, and lack of previous studies evaluating the programme makes this an issue of considerable interest.
The study also contributes in using diversification in consumption expenditure as a measure of welfare. As noted earlier, the existing literature in development economics has used this welfare measure relatively sparsely. However, apart from improvement in the level of expenditure, its diversification has also been used as an indicator of overall welfare. We employ Theil's (1967) entropy-based measure as the estimate of diversification.
The key insight from the study is that financial inclusion matters towards improvement of welfare. To reiterate, one link from financial inclusion to economic welfare is through better production and/or employment opportunities, leading to higher income and, thus, change in pattern of consumption expenditure. There can be other channels through which this transmission may happen. For example, financial inclusion has been shown to affect saving and borrowing behaviour, which can improve consumption smoothing, possibly leading to widening of the consumption basket.
Using a two-stage regression approach, we find robust evidence that households diversify their consumption expenditure as a result of increase in access to formal financial system. This increase is especially visible within non-food items, as well as between food and non-food items. Given that total (real) expenditure on consumption has increased in both these baskets (see Table 15), this too indicates an improvement in welfare. Separate analysis for rural and urban samples shows that similar diversification pattern exists across population in rural and urban regions. We obtain our results after controlling for a set of plausible confounding factors. The results also suggest separate and significant effect of financial inclusion after controlling the channel of increased income opportunities due to financial inclusion. This may indicate the existence of additional channels, or presence of measurement error in total consumption expenditure variable. We do not explicitly explore these channels in the present paper, but we believe these channels may be worth exploring in future research works.
The study makes a few additional observations. Diversification is generally positively associated with the level of income and wealth. However, the impact is heterogeneous across social as well as religious groups. While the socially disadvantaged groups diversify more within food group items as well as within non-food group items, they are relatively less able to shift expenditure between these two groups. This is possibly due to differences in initial conditions among the socio-religious groups. Further, demographic attributes also act as a moderating factor towards ultimate benefit of such programmes.
One critique of the financial inclusion schemes in general is that many of the accounts opened through these initiatives lie dormant (for example, see Goedecke et al. (2018) for Indian context in general, and Bijoy (2018) for PMJDY programme in specific). However, our empirical analysis indicates that the scheme has improved economic welfare of the households, even after controlling for plausible confounding factors. In this sense, our estimates may be inferred as a lower bound of the improvement of well-being. In recent time, the use of these accounts has actually shown a rising trend (Chopra et al. 2018; GoI 2020). Future research can be directed towards exploring the marginal effect of this increasing trend, and the mechanisms through which this gain occurs.
A Correction to this paper has been published: https://doi.org/10.1057/s41287-021-00467-0
We are indebted to the editor as well as the anonymous referee for the insightful comments on earlier version of the paper. The usual disclaimers apply.
The programme initially targeted opening one account for every household. Since August 2018, the target has been to open one account for every adult.
As evidenced by Engel's law in terms of reduction in food share, (Engel 1895).
We would like to thank the anonymous referee for pointing out these plausible explanations.
The dataset we use does not identify the type of the bank account opened. However, the sudden increase in number of bank accounts opened by households since the launch of the PMJDY scheme is clearly visible in Table 2 and Fig. 1. From this, we infer that majority of the bank accounts were opened as part of the initiative.
Please see section V (results section) for empirical evidence on the positive shock.
This is a standard practice in empirical studies concerning India, given the representativeness of the major states in samples. For example, see Burgess & Pande (2005) in our context.
The population share in rural regions is greater than that in urban areas in India. In our dataset, the sample size is greater for urban region compared to rural regions. Since we execute our regressions separately for rural and urban regions, this difference does not affect our results and their interpretations.
Table 13 in appendix shows the items included under food and non-food group of items.
The sum of per capita food and non-food expenditure do not add up to total per capita expenditure, since we do not include expenditure on durable items in our analysis. However, our estimates show that the food and non-food expenditure items we consider for our analysis covers approximately 90% of the total household expenditure.
Relevant literature supporting reasons for including the variables in our analysis are presented in section IV.C, under empirical model section.
The categories are: Intermediate Caste, Not Stated, OBC, SC, ST, Upper Caste, Not Applicable and Data Not Available.
An intermediate caste is defined as a caste above SC, ST and OBC, but below the upper caste.
All expenditure data used in this article (such as MPCE, food expenditure and non-food expenditure) are deflated using the 2012 CPI as base. We take the average CPI for four months corresponding to a wave of our analysis and divide the nominal expenditures for the wave by that average CPI to arrive at the real expenditures. We perform this exercise separately for rural and urban regions, using the rural and urban CPIs.
We have also calculated the following two income elasticities: all food group items taken together, and all non-food group items taken together. We used the quadratic Engle curve for this calculation (Thomas et al., 1989). The estimates of income elasticity declines for the food group and increases for the non-food group, as we move up the income quartiles. This result further supports our observation that between-group diversity shows a rising trend (see Table 6).
As discussed earlier, we see a discontinuous jump in number of bank accounts opened after the introduction of the PMJDY scheme (Table 2 and Fig. 1). Given this, we assume that these time dummies capture the overall effect of the financial inclusion programme (PMJDY) on welfare of the households.
We use another generalized entropy-based diversification measure, namely, squared coefficient of variation, for a robustness check (following (Foster & Shneyerov, 1999)). The results are qualitatively similar.
We exclude the expenditure on durable groups since consumption decision on such items is inter-temporal in nature.
It may be noted that the total impact of PMJDY in the second stage in this case is measured by adding the coefficients of the PMJDY variables and the coefficient of predicted MPCE.
See results section for relevant test statistics.
This is based on our results using alternative Hausman test, given in the results section.
Other studies have observed a shift in expenditure from food group to non-food group for Indian households, under different contexts. For example, see Basu & Basole (2012).
Fixed effect estimates and Hausman-Taylor estimates are similar in magnitude, implying exogeneity of our variables.
Mean MPCE for Upper Caste, for SCs and STs, for OBCs, and for Intermediate castes are Rs. 2779.53, Rs. 2028.527, Rs. 2311.747, and Rs. 2897.255, respectively. Also, mean MPCE for Hindus, for Muslims, and for other religion are Rs. 2376.914, Rs. 2049.18, and Rs. 3366.46, respectively.
We obtained income elasticities for a subset of food items within the food group by estimating the quadratic Engel curve (see Thomas et al. 1989). We observe, an increase in share of expenditure over time for the items for which the income elasticities are either very close to 1, or greater than 1, such as non-veg items and packaged food (by approximately 21%). Similar feature is observed in other contexts as well, see (Tey 2008) for example.
Aghion, P., and P. Bolton. 1997. A theory of trickle-down growth and development. The Review of Economic Studies 64 (2): 151. https://doi.org/10.2307/2971707.
Altissimo, F., E. Georgiou, T. Sastre, M. T. Valderrama, G. Sterne, M. Stocker, M.A. Weth, K. Whelan, & A. Willman. 2005. Wealth and asset price effects on economic activity. Occasional Paper Series 29. European Central Bank.
Ando, A., and F. Modigliani. 1963. The "Life Cycle" hypothesis of saving: Aggregate implications and tests. American Economic Review 53 (1): 55–84.
Annim, S.K., and R.B. Frempong. 2018. Effects of access to credit and income on dietary diversity in Ghana. Food Security 10 (6): 1649–1663. https://doi.org/10.1007/s12571-018-0862-8.
Badarinza, C., V. Balasubramaniam, and T. Ramadorai. 2016. The Indian household finance landscape. SSRN Electronic Journal. https://doi.org/10.2139/ssrn.2797680.
Bailey, J.M., and J. Sood. 1993. The effects of religious affiliation on consumer behavior: A preliminary investigation. Journal of Managerial Issues 5 (3): 328–352.
Banerjee, Avijeet V. 2004. Contracting constraints, credit markets, and economic development. In Advances in Economics and Econometrics: Theory and Applications. Eighth World Congress, vol. III, ed. M. Dewatripont, L.P. Hansen, and S.J. Turnovsky, 1–46. Cambridge: Cambridge University Press.
Banerjee, Abhijit V., and E. Duflo. 2007. The economic lives of the poor. Journal of Economic Perspectives 21 (1): 141–167. https://doi.org/10.1257/jep.21.1.141.
Banerjee, Avijeet V., and A.F. Newman. 1993. Occupational choice and the process of development. Journal of Political Economy 101 (2): 274–298.
Barro, R.J., and X. Sala-i-Martin. 2004. Economic Growth, 2nd ed. Delhi: Prentice Hall India Learning Private Limited.
Basu, D., & Basole, A. 2012. The calorie consumption puzzle in India: an empirical investigation.Working Paper, No. 2012-07, University of Massachusetts, Department of Economics, Amherst, MA.
Bharadwaj, P., and T. Suri. 2020. Improving financial inclusion through digital savings and credit. AEA Papers and Proceedings 110: 584–588. https://doi.org/10.1257/pandp.20201084.
Bijoy, K. 2018. Financial Inclusion in India and PMJDY: A Critical Review. Proceedings of the First International Conference on Information Technology and Knowledge Management 14: 39–46. https://doi.org/10.15439/2017km32.
Borooah, V.K., D. Diwakar, V.K. Mishra, A.K. Naik, and N.S. Sabharwal. 2014. Caste, inequality, and poverty in India: A re-assessment. Development Studies Research 1 (1): 279–294. https://doi.org/10.1080/21665095.2014.967877.
Bostic, R., S. Gabriel, and G. Painter. 2009. Housing wealth, financial wealth, and consumption: New evidence from micro data. Regional Science and Urban Economics 39 (1): 79–89. https://doi.org/10.1016/j.regsciurbeco.2008.06.002.
Burgess, R., and R. Pande. 2005. Do rural banks matter? Evidence from the Indian social banking experiment. American Economic Review 95 (3): 780–795. https://doi.org/10.1257/0002828054201242.
Caceres, C. 2019. Analyzing the effects of financial and housing wealth on consumption using micro data. IMF Working Papers 19 (115): 1. https://doi.org/10.5089/9781498316477.001.
Campbell, J.Y., and J.F. Cocco. 2007. How do house prices affect consumption? Evidence from micro data. Journal of Monetary Economics 54 (3): 591–621. https://doi.org/10.1016/j.jmoneco.2005.10.016.
Chai, A., N. Rohde, and J. Silber. 2015. Measuring the diversity of household spending patterns. Journal of Economic Surveys 29 (3): 423–440. https://doi.org/10.1111/joes.12066.
Chakrabarty, M., and J. Mandi. 2019. Entropy-based consumption diversity—the case of India. In Opportunities and Challenges in Development, ed. S. Bandyopadhyay and M. Dutta. Kolkata: Springer.
Chibba, M. 2009. Financial inclusion, poverty reduction and the millennium development goals. European Journal of Development Research 21 (2): 213–230. https://doi.org/10.1057/ejdr.2008.17.
Chin, A., Karkoviata, L., & Wilcox, N. 2011. Impact of bank accounts on migrant savings and remittances: evidence from a field experiment. University of Houston Working Paper, available at: http://www.uh.edu/~achin/research/ckw_banking_june2011.pdf.
Chopra, Y., N. Prabhala, and P.L. Tantri. 2018. Bank accounts for the unbanked: Evidence from a big bang experiment. SSRN Electronic Journal. https://doi.org/10.2139/ssrn.2919091.
Clements, K.W., and D. Chen. 1996. Fundamental similarities in consumer behaviour. Applied Economics 28 (6): 747–757. https://doi.org/10.1080/000368496328498.
Clements, K.W., Y. Wu, and J. Zhang. 2006. Comparing international consumption patterns. Empirical Economics 31 (1): 1–30. https://doi.org/10.1007/s00181-005-0012-y.
De Koker, L., and N. Jentzsch. 2013. Financial Inclusion and Financial Integrity: Aligned Incentives? World Development 44 (2011): 267–280. https://doi.org/10.1016/j.worlddev.2012.11.002.
Dupas, P., D. Karlan, J. Robinson, and D. Ubfal. 2018. Banking the unbanked? Evidence from three countries. American Economic Journal: Applied Economics 10 (2): 257–297. https://doi.org/10.1257/app.20160597.
Engel, E. 1895. Die Libenskosten Belgischer arbeiter-familien friiher und jetzt. International Statistical Institute Bulletin 9: 1–74.
Falkinger, J., and J. Zweimüller. 1996. The cross-country Engel curve for product diversification. Structural Change and Economic Dynamics 7 (1): 79–97. https://doi.org/10.1016/0954-349X(95)00039-P.
Flurry, L.A. 2007. Children's influence in family decision-making: Examining the impact of the changing American family. Journal of Business Research 60 (4): 322–330. https://doi.org/10.1016/j.jbusres.2006.09.029.
Foster, J.E., and A.A. Shneyerov. 1999. A general class of additively decomposable inequality measures. Economic Theory 14 (1): 89–111. https://doi.org/10.1007/s001990050283.
Goedecke, J., I. Guérin, B. D'Espallier, and G. Venkatasubramanian. 2018. Why do financial inclusion policies fail in mobilizing savings from the poor? Lessons from rural south India. Development Policy Review 36 (September 2015): O201–O219. https://doi.org/10.1111/dpr.12272.
GoI. 2020. Pradhan Mantri Jan-Dhan Yojana (PMJDY) - National Mission for Financial Inclusion, completes six years of successful implementation. https://pib.gov.in/Pressreleaseshare.aspx?PRID=1649091.
Greene, W.H. 2011. Econometric Analysis, 7th ed. London: Pearson Education.
Grossman, G.M., and E. Helpman. 1991. Innovation and Growth in the Global Economy. London: MIT Press.
Hausman, J.A., and W.E. Taylor. 1981. Panel data and unobservable individual effects. Econometrica 49 (6): 1377–1398.
Jackson, L.F. 1984. Hierarchic demand and the Engel curve for variety. The Review of Economics and Statistics 66 (1): 8. https://doi.org/10.2307/1924690.
Khamis, M., N. Prakash, and Z. Siddique. 2012. Consumption and social identity: Evidence from India. Journal of Economic Behavior and Organization 83 (3): 353–371. https://doi.org/10.1016/j.jebo.2012.07.002.
Kochar, A. 2011. The distributive consequences of social banking: A microempirical analysis of the Indian experience. Economic Development and Cultural Change 59 (2): 251–280. https://doi.org/10.1086/657122.
Krishnaswamy, R. 2012. Pattern of consumer expenditure in India: Some revelations. Economic and Political Weekly 47 (36): 80–83.
Lai, J.T., I.K.M. Yan, X. Yi, and H. Zhang. 2020. Digital financial inclusion and consumption smoothing in China. China and World Economy 28 (1): 64–93. https://doi.org/10.1111/cwe.12312.
Lieberson, S. 1969. Measuring population diversity. American Sociological Review 34 (6): 850–862.
Lusardi, A., and O.S. Mitchell. 2014. The economic importance of financial literacy: Theory and evidence. Journal of Economic Literature 52 (1): 5–44. https://doi.org/10.1257/jel.52.1.5.
Mincer, J. 1960. Employment and consumption. Review of Economics and Statistics 42 (1): 20–26.
Maitra, P., and R. Ray. 2004. Household characteristics and living standards: Evidence from India. In South Asia in the Era of Globalization: Trade, Industrialization, and Welfare, ed. M. Bhattacharya, R. Smyth, and M. Vicziany, 165–194. New York: Nova Science Publishers.
Morseth, M.S., N.K. Grewal, I.S. Kaasa, A. Hatloy, I. Barikmo, and S. Henjum. 2017. Dietary diversity is related to socioeconomic status among adult Saharawi refugees living in Algeria. BMC Public Health 17 (1): 1–9. https://doi.org/10.1186/s12889-017-4527-x.
Palan, N. 2010. Measurement of specialization the choice of indices. FIW Working Paper, No. 62. Vienna.
Poprawe, M. 2015. A panel data analysis of the effect of corruption on tourism. Applied Economics 47 (23): 2399–2412. https://doi.org/10.1080/00036846.2015.1005874.
Prais, S.J. 1952. Non-linear estimates of the Engel curves. The Review of Economic Studies 20 (2): 87. https://doi.org/10.2307/2295843.
Quayes, S. 2015. Outreach and performance of microfinance institutions: A panel analysis. Applied Economics 47 (18): 1909–1925. https://doi.org/10.1080/00036846.2014.1002891.
Ray, R. 1986. Demographic variables and equivalence scales in a flexible demand system: The case of AIDS. Applied Economics 18 (3): 265–278. https://doi.org/10.1080/00036848600000028.
Reardon, S.F., and G. Firebaugh. 2002. Measures of multigroup segregation. Sociological Methodology 32 (1): 33–67.
Romer, P.M. 1990. Endogenous technological change. Journal of Political Economy 98 (5): S71–S102.
Simpson, E.H. 1949. Measurement of diversity. Nature 163: 688.
Singh, N. 2017. Financial Inclusion: Concepts, Issues and Policies for India. Department of Economics University of California, Santa Cruz and International Growth Center. Referene no. I-35406-INC-2.
Tey, Y.-S. 2008. Household expenditure on food at home in Malaysia. MPRA Paper No. 15031. https://mpra.ub.uni-muenchen.de/15031/.
Theil, H. 1967. Economics and Information Theory. Amsterdam: North-Holland Publishing Company.
Theil, H. 1972. Statistical Decomposition Analysis. Amsterdam: North-Holland Publishing Company.
Theil, H., and R. Finke. 1983. The consumer's demand for diversity. European Economic Review 23 (3): 395–400. https://doi.org/10.1016/0014-2921(83)90039-9.
Thomas, D., S. Strauss, & M. M. L. Barbosa. 1989. Estimating the impact of income and price changes on consumption in Brazil. Yale Economic Growth Center: Discussion Paper No. 589.
World Bank. 2014. Global Financial Development Report: Financial Inclusion. Washington DC: World Bank.
Young, N. 2015. Formal banking and economic growth: evidence from a regression discontinuity analysis in India. Working Paper, Boston University.
The authors sincerely acknowledge Prof. Amita Majumder at ISI, Calcutta and Prof. Ranjan Ray, Monash University for their valuable suggestions. Existing errors are all ours.
No funding source is to be reported by the authors.
Indian Institute of Management Calcutta, Kolkata, West Bengal, 700104, India
Manisha Chakrabarty
Department of Industrial & Management Engineering, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, 208016, India
Subhankar Mukherjee
Correspondence to Manisha Chakrabarty.
No potential conflict of interest is to be reported by the authors.
See Appendix Tables 12, 13, 14, 15.
Table 12 Distribution of sample households by states and by rural and urban regions
Table 13 List of commodities included in food and non-food commodity groups
Table 14 List of items included in constructing the asset index variable
Table 15 Per capita expenditure on food items and non-food items (considering 2012 CPI as base)
Chakrabarty, M., Mukherjee, S. Financial Inclusion and Household Welfare: An Entropy-Based Consumption Diversification Approach. Eur J Dev Res 34, 1486–1521 (2022). https://doi.org/10.1057/s41287-021-00431-y
Issue Date: June 2022
Consumption diversification
Theil's entropy
Hausman-Taylor estimation
PMJDY
JEL Classifications | CommonCrawl |
Triangle $ABC$ lies in the cartesian plane and has an area of $70$. The coordinates of $B$ and $C$ are $(12,19)$ and $(23,20),$ respectively, and the coordinates of $A$ are $(p,q).$ The line containing the median to side $BC$ has slope $-5.$ Find the largest possible value of $p+q.$
[asy]defaultpen(fontsize(8)); size(170); pair A=(15,32), B=(12,19), C=(23,20), M=B/2+C/2, P=(17,22); draw(A--B--C--A);draw(A--M);draw(B--P--C); label("A (p,q)",A,(1,1));label("B (12,19)",B,(-1,-1));label("C (23,20)",C,(1,-1));label("M",M,(0.2,-1)); label("(17,22)",P,(1,1)); dot(A^^B^^C^^M^^P);[/asy]
The midpoint $M$ of line segment $\overline{BC}$ is $\left(\frac{35}{2}, \frac{39}{2}\right)$. The equation of the median can be found by $-5 = \frac{q - \frac{39}{2}}{p - \frac{35}{2}}$. Cross multiply and simplify to yield that $-5p + \frac{35 \cdot 5}{2} = q - \frac{39}{2}$, so $q = -5p + 107$.
Use determinants to find that the area of $\triangle ABC$ is $\frac{1}{2} \begin{vmatrix}p & 12 & 23 \\ q & 19 & 20 \\ 1 & 1 & 1\end{vmatrix} = 70$ (note that there is a missing absolute value; we will assume that the other solution for the triangle will give a smaller value of $p+q$, which is provable by following these steps over again). We can calculate this determinant to become $140 = \begin{vmatrix} 12 & 23 \\ 19 & 20 \end{vmatrix} - \begin{vmatrix} p & q \\ 23 & 20 \end{vmatrix} + \begin{vmatrix} p & q \\ 12 & 19 \end{vmatrix}$ $\Longrightarrow 140 = 240 - 437 - 20p + 23q + 19p - 12q$ $= -197 - p + 11q$. Thus, $q = \frac{1}{11}p - \frac{337}{11}$.
Setting this equation equal to the equation of the median, we get that $\frac{1}{11}p - \frac{337}{11} = -5p + 107$, so $\frac{56}{11}p = \frac{107 \cdot 11 + 337}{11}$. Solving produces that $p = 15$. Substituting backwards yields that $q = 32$; the solution is $p + q = \boxed{47}$. | Math Dataset |
What is the maximum value of $4(x + 7)(2 - x)$, over all real numbers $x$?
The graph of $y = 4(x + 7)(2 - x)$ is a parabola. Since $y = 0$ when $x = -7$ and $x = 2$, the $x$-intercepts of the parabola are $(-7,0)$ and $(2,0)$. If the vertex of the parabola is $(h,k)$, then the $x$-intercepts $(-7,0)$ and $(2,0)$ are symmetric around the line $x = h$, so $h = (-7 + 2)/2 = -5/2$.
Hence, the maximum value of $y = 4(x + 7)(2 - x)$ occurs at $x = -5/2$, in which case \[y = 4 \left( -\frac{5}{2} + 7 \right) \left( 2 + \frac{5}{2} \right) = 4 \cdot \frac{9}{2} \cdot \frac{9}{2} = \boxed{81}.\] (Note that this is a maximum value, and not a minimum value, because the coefficient of $x^2$ in $y = 4(x + 7)(2 - x) = -4x^2 - 20x + 56$ is negative.) | Math Dataset |
Conditional dependence
In probability theory, conditional dependence is a relationship between two or more events that are dependent when a third event occurs.[1][2] For example, if $A$ and $B$ are two events that individually increase the probability of a third event $C,$ and do not directly affect each other, then initially (when it has not been observed whether or not the event $C$ occurs)[3][4]
$\operatorname {P} (A\mid B)=\operatorname {P} (A)\quad {\text{ and }}\quad \operatorname {P} (B\mid A)=\operatorname {P} (B)$
See also: Conditional independence
($A{\text{ and }}B$ are independent).
But suppose that now $C$ is observed to occur. If event $B$ occurs then the probability of occurrence of the event $A$ will decrease because its positive relation to $C$ is less necessary as an explanation for the occurrence of $C$ (similarly, event $A$ occurring will decrease the probability of occurrence of $B$). Hence, now the two events $A$ and $B$ are conditionally negatively dependent on each other because the probability of occurrence of each is negatively dependent on whether the other occurs. We have[5]
$\operatorname {P} (A\mid C{\text{ and }}B)<\operatorname {P} (A\mid C).$
Conditional dependence of A and B given C is the logical negation of conditional independence $((A\perp \!\!\!\perp B)\mid C)$.[6] In conditional independence two events (which may be dependent or not) become independent given the occurrence of a third event.[7]
Example
In essence probability is influenced by a person's information about the possible occurrence of an event. For example, let the event $A$ be 'I have a new phone'; event $B$ be 'I have a new watch'; and event $C$ be 'I am happy'; and suppose that having either a new phone or a new watch increases the probability of my being happy. Let us assume that the event $C$ has occurred – meaning 'I am happy'. Now if another person sees my new watch, he/she will reason that my likelihood of being happy was increased by my new watch, so there is less need to attribute my happiness to a new phone.
To make the example more numerically specific, suppose that there are four possible states $\Omega =\left\{s_{1},s_{2},s_{3},s_{4}\right\},$ given in the middle four columns of the following table, in which the occurrence of event $A$ is signified by a $1$ in row $A$ and its non-occurrence is signified by a $0,$ and likewise for $B$ and $C.$ That is, $A=\left\{s_{2},s_{4}\right\},B=\left\{s_{3},s_{4}\right\},$ and $C=\left\{s_{2},s_{3},s_{4}\right\}.$ The probability of $s_{i}$ is $1/4$ for every $i.$
Event$\operatorname {P} (s_{1})=1/4$$\operatorname {P} (s_{2})=1/4$$\operatorname {P} (s_{3})=1/4$$\operatorname {P} (s_{4})=1/4$Probability of event
$A$0101 ${\tfrac {1}{2}}$
$B$0011 ${\tfrac {1}{2}}$
$C$0111 ${\tfrac {3}{4}}$
and so
Event$s_{1}$$s_{2}$$s_{3}$$s_{4}$Probability of event
$A\cap B$0001 ${\tfrac {1}{4}}$
$A\cap C$0101 ${\tfrac {1}{2}}$
$B\cap C$0011 ${\tfrac {1}{2}}$
$A\cap B\cap C$0001 ${\tfrac {1}{4}}$
In this example, $C$ occurs if and only if at least one of $A,B$ occurs. Unconditionally (that is, without reference to $C$), $A$ and $B$ are independent of each other because $\operatorname {P} (A)$—the sum of the probabilities associated with a $1$ in row $A$—is ${\tfrac {1}{2}},$ while
$\operatorname {P} (A\mid B)=\operatorname {P} (A{\text{ and }}B)/\operatorname {P} (B)={\tfrac {1/4}{1/2}}={\tfrac {1}{2}}=\operatorname {P} (A).$
But conditional on $C$ having occurred (the last three columns in the table), we have
$\operatorname {P} (A\mid C)=\operatorname {P} (A{\text{ and }}C)/\operatorname {P} (C)={\tfrac {1/2}{3/4}}={\tfrac {2}{3}}$
while
$\operatorname {P} (A\mid C{\text{ and }}B)=\operatorname {P} (A{\text{ and }}C{\text{ and }}B)/\operatorname {P} (C{\text{ and }}B)={\tfrac {1/4}{1/2}}={\tfrac {1}{2}}<\operatorname {P} (A\mid C).$
Since in the presence of $C$ the probability of $A$ is affected by the presence or absence of $B,A$ and $B$ are mutually dependent conditional on $C.$
See also
• Conditional independence – Probability theory concept
• de Finetti's theorem – Conditional independence of exchangeable observations
• Conditional expectation – Expected value of a random variable given that certain conditions are known to occur
References
1. Introduction to Artificial Intelligence by Sebastian Thrun and Peter Norvig, 2011 "Unit 3: Conditional Dependence"
2. Introduction to learning Bayesian Networks from Data by Dirk Husmeier "Introduction to Learning Bayesian Networks from Data -Dirk Husmeier"
3. Conditional Independence in Statistical theory "Conditional Independence in Statistical Theory", A. P. Dawid" Archived 2013-12-27 at the Wayback Machine
4. Probabilistic independence on Britannica "Probability->Applications of conditional probability->independence (equation 7) "
5. Introduction to Artificial Intelligence by Sebastian Thrun and Peter Norvig, 2011 "Unit 3: Explaining Away"
6. Bouckaert, Remco R. (1994). "11. Conditional dependence in probabilistic networks". In Cheeseman, P.; Oldford, R. W. (eds.). Selecting Models from Data, Artificial Intelligence and Statistics IV. Lecture Notes in Statistics. Vol. 89. Springer-Verlag. pp. 101–111, especially 104. ISBN 978-0-387-94281-0.
7. Conditional Independence in Statistical theory "Conditional Independence in Statistical Theory", A. P. Dawid Archived 2013-12-27 at the Wayback Machine
| Wikipedia |
Validation of a continuous infusion of low dose Iohexol to measure glomerular filtration rate: randomised clinical trial
John J Dixon1,2,4,
Katie Lane1,2,
R Neil Dalton3,
Charles Turner3,
R Michael Grounds1,
Iain AM MacPhee2,4 &
Barbara J Philips1,2
Journal of Translational Medicine volume 13, Article number: 58 (2015) Cite this article
There is currently no accurate method of measuring glomerular filtration rate (GFR) during acute kidney injury (AKI). Knowledge of how much GFR varies in stable subjects is necessary before changes in GFR can be attributed to AKI. We have designed a method of continuous measurement of GFR intended as a research tool to time effects of AKI. The aims of this crossover trial were to establish accuracy and precision of a continuous infusion of low dose Iohexol (CILDI) and variation in GFR in stable volunteers over a range of estimated GFR (23-138 mL/min/1.73 m2).
We randomised 17 volunteers to GFR measurement by plasma clearance (PC) and renal clearance (RC) of either a single bolus of Iohexol (SBI; routine method), or of a continuous infusion of low dose Iohexol (CILDI; experimental method) at 0.5 mL/h for 12 h. GFR was measured by the alternative method after a washout period (4–28 days). Iohexol concentration was measured by high performance liquid chromatography/electrospray tandem mass spectrometry and time to steady state concentration (Css) determined.
Mean PC was 76.7 ± 28.5 mL/min/1.73 m2 (SBI), and 78.9 ± 28.6 mL/min/1.73 m2 (CILDI), p = 0.82. No crossover effects occurred (p = 0.85). Correlation (r) between the methods was 0.98 (p < 0.0001). Bias was 2.2 mL/min/1.73 m2 (limits of agreement −8.2 to 12.6 mL/min/1.73 m2) for CILDI. PC overestimated RC by 7.1 ± 7.3 mL/min/1.73 m2. Mean intra-individual variation in GFR (CILDI) was 10.3% (p < 0.003). Mean ± SD Css was 172 ± 185 min.
We hypothesise that changes in GFR >10.3% depict evolving AKI. If this were applicable to AKI, this is less than the 50% change in serum creatinine currently required to define AKI. CILDI is now ready for testing in patients with AKI.
This trial was registered with the European Union Clinical Trials Register (https://www.clinicaltrialsregister.eu/), registration number: 2010-019933-89.
The absence of an accurate method of measuring changing glomerular filtration rate (GFR) in acute kidney injury (AKI) poses a significant barrier to research in this area. Identifying when pathophysiological changes associated with AKI occur remains a significant challenge. Current definitions of AKI are based upon an increase in serum creatinine concentration (SCr) of greater than 50% or reduction in urine output (UO) [1-3] despite limitations in interpretation of these parameters in critically ill patients [4]. Changes in SCr sufficient to define AKI may be delayed, particularly in patients with chronic kidney disease (CKD) [5] or sepsis [6]. Acute illness may lead to diminished creatinine formation [7] limiting its utility as a biomarker for GFR in AKI. Other endogenous biomarkers have been investigated but have variable performance [8]. None have been proven to be superior to SCr and UO in heterogenous populations (e.g. general critical care units), or where the onset of the insult or its aetiology is unclear.
Using exogenous markers, such as radioactive ethylene diamine tetraacetic acid (EDTA) or radio-opaque contrast media, to measure GFR in AKI have advantages over endogenous markers: they are not influenced by body habitus, diet or metabolic processes, and are not dependent on the timing of the insult causing AKI. Exogenous markers can theoretically be administered as a single bolus injection and measuring the time to elimination from the body, or as a continuous infusion. Single bolus injection of Iohexol (SBI) has been used in critically ill patients [9], but interpretation of this approach assumes stable GFR, which is unlikely in the context of evolving AKI: administration would need to be repeated frequently to track changing GFR, and may lead to accumulation of Iohexol. Furthermore, bolus methods are inappropriate in AKI because of the washout period required.
Continuous infusions require stable GFR and volume of distribution (Vd) to achieve steady state concentration (Css). They are limited by natural intra-individual variations in GFR (precision) and bias of laboratory analytical equipment. Css varies between patients according to: a) baseline GFR, b) Vd, and c) the mass of substance infused over time (Minf). In stable patients (a) and (b) are unchanged, and (c) is controlled by the operator. Css can be predicted if the subject's weight and baseline GFR are known. Once Css has been reached, variations in GFR occurring after this time, in excess of precision and bias, likely represent true changes in GFR. Theoretically, measurements made after time to Css will represent GFR at that moment. A loading dose (LD) given prior to the infusion reduces time to Css: if LD is too large, plasma concentration declines until Css is reached; if too small, plasma concentrations climb until Css is reached. In AKI, Css may never be reached, however, plasma measurements made after the predicted time to Css can detect changes from baseline GFR and predicted Css. In rapidly evolving AKI occurring before time to Css, concentrations will not change towards Css at the expected rate, and may even increase. To date, there have been no studies measuring changing GFR in AKI using continuous infusions.
Administration of a continuous infusion of low dose Iohexol (CILDI; Omnipaque 300®, at 0.5 mL/h) has the potential to measure evolving GFR in AKI. A continuous infusion of Iohexol has previously been validated in subjects with normal GFR [10], however, Css and time to Css in subjects with CKD are unknown: prolonged time to Css in patients with CKD would limit the applicability of CILDI in patients with AKI.
We have performed a proof-of-method clinical trial (randomised crossover design) with the aim of validating CILDI for measuring GFR as a research tool in AKI. CILDI was compared to measurement of the plasma clearance (PC) of a SBI [9,11]. We have used the SBI method as our "gold standard", rather than a continuous infusion for three reasons: 1) The SBI method has previously been validated to measure a wide range of GFR in stable patients, from normal to measuring residual renal function in patients requiring renal replacement therapy [11,12]. We are therefore confident that the SBI method is accurate and precise. 2) The SBI method previously used in critically ill patients [9] may be regarded by some authors as a "gold standard", however, we think this is inappropriate in AKI for reasons listed above. We wanted to demonstrate that CILDI is not inferior to the SBI method when measuring GFR in stable patients; 3) No continuous infusion has been tested in AKI and after proof of methods our intention is to test the method in such patients. Urine was collected, so that Iohexol renal clearance (RC) could be measured. Healthy volunteers (HV; defined as estimated GFR >60 mL/min/1.73 m2 by the simplified MDRD equation [13]) and patients with stable CKD (defined as eGFR <60 mL/min/1.73 m2) were recruited so that the variability of GFR in stable patients using our method could be determined over a wide range of GFR equivalent to that expected to occur in patients developing AKI. The range of eGFR in the HV cohort was 75-138 mL/min/1.73 m2, and the eGFR range in the CKD cohort was 23-59 mL/min/1.73 m2.
1) Compare the performance of CILDI with the SBI method in HV and patients with stable CKD. 2) Measure intra-individual variation of GFR in stable subjects, so that the minimum change in GFR (precision) detected by CILDI can be determined. 3) Confirm that subjects with eGFR >23 mL/min/1.73 m2 have time to Css <12 hours.
This article represents the first stage: validation of the technique and establishment of the accuracy and precision in subjects with stable GFR. The goal is to use CILDI in patients with, and at risk of, developing AKI. Changing GFR associated with AKI can be measured by PC and RC at various time points after predicted time to Css, allowing the temporal relationship between AKI and its pathophysiological effects to be delineated.
Clinical trial registration and ethics
This trial was registered with the European Union Clinical Trials Register (https://www.clinicaltrialsregister.eu), registration number: 2010-019933-89. Approval was obtained from Brighton East Research Ethics Committee (Ref: 10/H1107/24). The Declaration of Helsinki (2008) [14] was adhered to throughout. All subjects provided prior written informed consent. The trial was sponsored by St. George's, University of London, United Kingdom.
The trial took place in the Clinical Research Facility, St. George's, University of London. Research methods were performed to International Conference for Harmonisation Good Clinical Practice standards [15].
Recruitment, inclusions and exclusions
Subjects were recruited from local Nephrology outpatient clinics or via advertisements placed on public notice boards within St. George's Healthcare NHS Trust or St. George's, University of London.
Adults aged 18–75 years with renal function ranging from normal to chronic kidney disease (CKD) stage 4. Subjects were classified as having CKD if their estimated GFR was <60 mL/min/1.73 m2 by the simplified MDRD equation [15] or healthy volunteers (HV) if eGFR was >60 mL/min/1.73 m2.
These were precautionary and based on the listed criteria for radio-opaque contrast media [16]. Reactions to radio-contrast media; thyroid disease, myasthenia gravis, cardiac arrhythmias, pulmonary hypertension, epilepsy, structural brain disease, phaeochromocytoma, advanced heart failure, sickle cell disease, multiple myeloma, homocystinuria, ascites, pregnancy or breast-feeding, renal replacement therapy; subjects taking Metformin if serum creatinine >150 μmol/L, Phenothiazines, Tricyclic antidepressants, Monoamine oxidase inhibitors, Levo-thyroxine, Amiodarone, Interleukin-2 agents; a planned Tc99m–labelled scan. Subjects unable to provide written Informed Consent were also excluded.
Study design and randomisation
Computer-generated block randomisation allocated subjects to measurement of GFR via Method A (plasma clearance of a single bolus of Iohexol; SBI), or Method B (continuous infusion of low dose Iohexol; CILDI). Subjects then underwent a washout period of 4–28 days before GFR was measured by the alternative method. Four days was chosen as the minimum washout period because we wanted to ensure that subjects entering the second part of the crossover study had no remaining Iohexol within their body, and 4 days is more than double the time for Iohexol to be completely eliminated in a subject with GFR <20 mL/min/1.73 m2 [11,16]. An epidemiological study has suggested the rate of progression of CKD in stable subjects may be as high as a loss in GFR of 3.1 mL/min/1.73 m2 per year [17], so 28 days was chosen as the maximum washout period to minimise the likely risk of GFR changing between the two methods. A crossover design was used to further mitigate potential bias caused by changing clinical conditions.
Iohexol administration and sampling
Procedures common to both methods: body surface area (BSA) was calculated from height and weight [18]. An intravenous cannula was inserted into each arm. Iohexol was administered via one cannula and 2 mL blood samples collected from the other into serum separator containers. Serum was centrifuged at 4°C at 3500 rpm for 10 min. Every time urine was voided, subjects' bladders were scanned to ensure complete emptying (Bladderscan® BVI9400, Verathon Medical UK Ltd.). Urine and serum samples were stored at −80°C. The CKD group received intravenous 1.4% Sodium Bicarbonate 100 mL/h, in accordance with local hospital guidelines. Healthy volunteers were encouraged to drink 100 mL/h water. Volunteers were allowed to eat and drink freely.
Method A (SBI)
5 mL Iohexol (Ominpaque 300®) was administered as an intravenous bolus over 2 minutes [9]. The first blood sample was collected at 5 minutes. This allowed later confirmation that intravenous administration had occurred (a high concentration at 5 min suggests intravenous, rather than subcutaneous, injection). Samples were collected at 2 h, 3 h, and 4 h to calculate GFR. Urine was taken to measure renal clearance twice.
Method B (CILDI)
An intravenous loading dose (LD) was administered according to the formula:
$$ LD= volume\kern0.5em of\kern0.5em distribution\;(Vd)\kern0.5em \times \kern0.5em target\kern0.5em steady\kern0.5em state\kern0.5em concentration\kern0.5em (Css) $$
From the Summary of Product Characteristics [16], Vd = 0.165 L/Kg (95% CI: 0.108-0.219) × Weight (Kg). The target steady state concentration (100 μmol/L) is approximately 100 times the lower limit of quantitation by high performance liquid chromatography/tandem mass spectrometry (LC-MSMS): the actual steady state achieved in individuals was likely to vary according to their GFR and Vd. Ideally the LD would be calculated using GFR, however, weight was used so that CILDI could eventually be used in patients with an unknown baseline GFR. A continuous intravenous infusion of Iohexol (Omnipaque 300®) was then administered at 0.5 mL/h (343.5 mg/mL) for 12 h (Agilia MC Injectomat, Fresenius Kabihas) [19]. Blood samples were taken at 30 min, 60 min, 90 min, 2 h, 3 h, 4 h, 6 h, 8 h, 10 h, and 12 h. Urine samples were collected between each plasma sample, when possible.
GFR calculation
Method A: the natural logarithms of serum Iohexol concentrations at 2 h, 3 h, and 4 h were plotted against time. The intercept and slope were used to derive the theoretical time zero concentration of Iohexol, Vd and plasma half-life (T 1/2 ). GFR was calculated by dividing the product Loge(2)*Vd by T 1/2 , adjusting for BSA [18] and applying the Bröchner-Mortensen single compartment correction factor [20]. Renal Clearance (RC) was derived by measuring the urine concentration at 2.5 and 3.5 h and calculating the corresponding plasma concentration at 2.5 h and 3.5 h from the log-concentration-time graph, and using the formula: GFR(mL/min) = [U × V]/P. Where U = urine Iohexol concentration (μmol/L), V = volume of urine (mL) per unit time (min), and P = plasma Iohexol concentration (μmol/L), adjusting for BSA [18]. The mean of the two values was used for RC.
Method B (CILDI): results were plotted on a 2-phase exponential decay curve, and the plateau concentration calculated. Plasma clearance was calculated by the formula: GFR(mL/min) = [Iohexol infusion rate (μmol/min)]/[serum plateau Iohexol concentration (μmol/mL)], and adjusting for BSA. Renal clearance was calculated when bladder voidance and urine collection were complete, by measuring urine concentration of Iohexol at the mid-time point between each plasma sample after the time to steady state had been reached. The mean value of a minimum of two measurements was used for each subject.
Laboratory procedures
The detailed LC-MSMS method for measurement of plasma Iohexol and Creatinine has been published [21]. A brief summary follows: Frozen samples were defrosted at 4°C and centrifuged at 1500 rpm at 4°C for 4 minutes to separate particulate matter. Serum was decanted into 10 μL aliquots. 50 μL of stabilising fluid was added to each aliquot. This consisted of 10 mL de-ionised water, 250 μL D5-Iohexol, 25 μL D3-Creatinine, 25 μL D6-asymmetrical dimethylarginine, and 1.5 μL symmetrical dimethylarginine. 200 μL 1% (vol/vol) acetonitrile (Rathburn Chemicals Ltd., Walkerbrum, Peebleshire, UK) was added to precipitate the mixture, before centrifugation at 20000 rpm for 3 minutes at 4°C. 200 μL of the mixture was transferred into a 96-well polypropylene well plate and analysed by the API 5000LC/msms with QJET Ion guide accelerated by LINAC® collision cell (AB Applied Biosystems MDS SCIEX). Three quality controls were used with known plasma concentrations of Iohexol (10.6 μmol/L, 516.0 μmol/L, and 99.2 μmol/L).
Accuracy and precision
Tubular Creatinine secretion varies between 10 and 40% [22]. The proportion of secreted Creatinine can be measured by calculating the fractional excretion of creatinine (ie the fraction of filtered creatinine excreted in the urine divided by measured GFR). Values >100% imply additional tubular secretion. Accurate intravenous administration of Iohexol was confirmed by calculating the fractional excretion of Creatinine (Fe Creat ), using Iohexol as the substitute for GFR. Fe Creat >140% implies that the Iohexol was not administered intravenously. Inaccurate results were not analysed further. Accuracy of GFR measurement by CILDI was determined by performing a Bland-Altman comparison [23] against the SBI method.
GFR calculation during single Iohexol bolus administration was deemed precise if the Pearson correlation co-efficient of the loge(Iohexol concentration)-time graph was < −0.985 [24]. Precision of CILDI was calculated by measuring co-efficient of variation (CV) and standard deviation of Iohexol measurements at steady state; from this, precision at the 95% and 99% confidence levels and at 3 standard deviations were calculated, allowing mean intra-individual variation in GFR to be determined.
Sample size calculation
A difference in mean GFR of <10% between the two methods was considered acceptable [11,12]. The intra-individual CV of repeated measurements of Iohexol plasma clearance has been reported as 5.4% [25]. Based on this, a sample size of 30 subjects has 90% power to detect a GFR difference of 4.7 mL/min/1.73 m2 from the mean and a sample size of 17 subjects has 82% power. The sample size was revised to 17 subjects following an interim analysis that demonstrated the difference between the means of both methods was actually 3.5% and the trial was stopped early because we had already recruited more than the required number of subjects. A revised power calculation revealed that a sample size of 17 subjects has 90% power to detect a difference between the means of 2.6 mL/min/1.73 m2 whereas 30 subjects would have >99% power.
A futility analysis was performed to determine whether results would be different if our trial continued until 30 subjects had completed it. Logarithmic transformation of GFR was performed and data were compared using the paired 2-tailed t-test, assessing for period effects during this crossover trial. Difference in mean GFR was compared using the t-test. Linear regression using Pearson's correlation was performed to assess association between the methods, and level of agreement was assessed by the Bland Altman comparison [23]. Graphpad Prism®, version 5.0d (Graphpad software, Inc.) was used for statistical analysis. Because the trial was stopped early, all statistical results were reviewed and approved by a statistician independent to our trial.
Screening, enrolment and subjects
Twenty-one subjects entered both parts of the crossover trial. Four subjects were excluded from full analysis: 3 because of Iohexol administration errors and 1 because of an emergency evacuation of the building during method B. Seventeen subjects completed the trial for the measurement of Iohexol PC. RC was accurately measured in 9 subjects. Details are outlined in Figure 1. Demographic details are summarised in Table 1. Accuracy and precision of Iohexol GFR measurements are listed in Table 2. Laboratory measurements were deemed accurate if the fractional excretion of Creatinine (Fe Creat ), using Iohexol as the denominator, was 110-140%; subjects with inaccurate results were excluded. There was no difference in SCr and creatinine clearance in subjects between the two Iohexol GFR study periods (Table 3).
Subject screening and participation. No adverse effects due to Iohexol were observed. HV = healthy volunteers; CKD = Patients with chronic kidney disease.
Table 1 Demographic features of trial subjects
Table 2 Accuracy and precision of Iohexol measurements during methods A (single Iohexol bolus) and B (CILDI)
Table 3 Comparison of GFR measurements during method A (single Iohexol bolus) and method B (CILDI)
Association, agreement and precision of GFR calculations
There was no significant difference in PC between methods A (SBI) and B (CILDI) overall or on sub-group analysis; Table 3. Association (Figure 2) and agreement (Figure 3) between the methods were good. Sub-group analysis revealed closer limits of agreement, when measured in mL/min/1.73 m2, in the CKD group, although this was not significant when measured as percentage difference in GFR. When GFR was measured by CILDI, bias in the HV group was 2.2 mL/min/1.73 m2 (2.2%), limits of agreement −10.7 to 15.1 mL/min/1.73 m2 (−12.1 to 16.6%); in patients with CKD bias was 2.2 mL/min/1.73 m2 (5.8%), with limits of agreement −1.3 to 5.7 mL/min/1.73 m2 (−5.0 to 16.6%). Intra-individual variations in GFR and precision of GFR calculations are summarised in Table 4. Time to steady state (Css) was less than 10 h in all subjects (Table 4). The difference in GFR which depicts AKI can be determined by measuring the precision of CILDI at 95% and 99% confidence intervals and at 3 standard deviations (ie 99.7% confidence intervals). A difference of greater than 10.3% (3 standard deviations) after time to Css had elapsed, depicts a true difference in GFR (p < 0.003). The Pearson correlation between time to steady state concentration and GFR is 0.82.
Association between plasma clearance GFR calculated by the single bolus and the experimental continuous infusion of low dose Iohexol (CILDI). Solid line = line of association, dashed lines = error margins of association line. Slope = 0.988 ± 0.048, Intercept = 3.08 ± 3.92, Pearson's correlation, r = 0.983, p < 0.0001.
Bland-Altman comparison of plasma clearance GFR by single Iohexol bolus method and the experimental method (CILDI). Central solid lines are bias and outer dashed lines are limits of agreement, n = 17. A) Difference is measured in mL/min/1.73 m2. Bias = 2.2 mL/min/1.73 m2, SD of bias = 5.3 mL/min/1.73 m2, 95% limits of agreement from −8.2 to +12.6 mL/min/1.73 m2, B) Difference is measured as percentage difference in GFR. Bias = 3.5%, SD of bias = 6.8%, 95% limits of agreement from −9.8 to +16.8%.
Table 4 Precision of GFR measurements and time to steady state during CILDI
Plasma and renal clearance
Post-micturition bladder scans revealed incomplete bladder voiding in 8 subjects; making RC difficult to perform accurately. Consequently, only 9 RC were performed satisfactorily. Although the correlation between PC and RC was 0.989 (Figure 4), measurement of GFR by PC overestimated RC of Iohexol by 7.1 ± 7.3 mL/min/1.73 m2 (Table 2; Figure 5).
Correlation between plasma and renal clearance during CILDI. Solid line = line of association, dashed lines = error margins of association line. Pearson's correlation (r) = 0.989, Intercept = −23.3 ± 4.8, slope = 1.25 ± 0.07.
Bland-Altman comparison of plasma clearance GFR and Renal clearance GFR during CILDI. Difference is measured in mL/min/1.73 m2. Bias = −7.1 mL/min/1.73 m2, SD of bias = 7.3, 95% Limits of agreement = −21.5 to +7.2 mL/min/1.73 m2. n = 9. Although based on small numbers sub-group analysis suggests a smaller bias in the HV group. In the HV group bias was −0.5 mL/min/1.73 m2 (limits of agreement −10.0 to +8.9 mL/min/1.73 m2); in the CKD group bias was −12.4 mL/min/1.73 m2 (limits of agreement −19.0 to −5.8 mL/min/1.73 m2).
Assessment for crossover effects
The Kolgomorov-Smirnov test confirmed GFR measurements had Gaussian distributions during both methods, before and after logarithmic transformation of data. No period effect was observed using raw data (paired 2-tailed t-test; p = 0.85) or transformed data (paired 2-tailed t-test; p = 0.91) [26]. Pairing of raw and transformed data were matched (Pearson's r =0.98, p < 0.0001).
Futility analysis
An interim analysis was conducted after the first batch analysis of samples. The ratio of the means has a difference of 3.5% (95% CI: 0.998 to 1.071). A futility test was performed to determine whether mean GFR observed in both methods would differ by >10% if the trial continued until 30 subjects had been recruited. The limit of 10% is 5.8 standard errors beyond the observed difference of 3.5% and so a 10% difference is ruled out at p < 10−7.
Summary of results
We have demonstrated that measurement of the plasma clearance of CILDI is accurate and precise. There is an excellent correlation (Figure 2) with the plasma clearance of a single Iohexol bolus, and Bland Altman comparison [23] reveals a small bias with close limits of agreement (Figure 3). During CILDI, mean intra-individual variation in GFR was 10.3% (p < 0.003). Once the time to steady state concentration (Css) had elapsed, all subjects reached Css within 10 h. It is theoretically possible to determine change in GFR from single measurements of Iohexol made after 10 h: variations greater than 10.3% represent changing GFR. If these data were applicable in the context of AKI, this is significantly less than the 50% change in SCr needed to define AKI by current criteria [1-3]. Intra-individual fluctuations in GFR may be caused by differences in fluid balance throughout the day [27] and circadian rhythms [28]. Correlation and agreement between CILDI plasma and renal clearance, when measured, were also good (Figures 4 and 5). Figure 6 demonstrates an increased steady state concentration in CKD subject 15, compared with HV subject 5.
Examples of Iohexol concentrations achieved during Method B (CILDI). Different steady state concentrations were observed in subjects 5 and 15. GFR in subject 5 was 127.7 mL/min/1.73 m2, and GFR in subject 15 was 62.2 mL/min/1.73 m2. Concentrations and time were plotted on a 2-phase exponential decay curve, using Graphpad Prism®, version 5.0d (Graphpad software, Inc.). The black line connects the Iohexol concentrations, the red line depicts the steady state concentration.
Iohexol has many properties of an "ideal GFR marker" [29]: it diffuses rapidly into the extracellular space; it undergoes less than 2% protein binding; over 99% is filtered at the glomerulus [16]; and it undergoes no renal tubular reabsorption or secretion. It has an excellent safety profile [30] and, when measured by high performance liquid chromatography / electrospray tandem mass spectrometry (LC-MSMS), gives highly reproducible results with low inter- and intra-patient coefficient of variation [31]. LC-MSMS has been validated to accurately and precisely measure plasma Iohexol concentrations less than 10 μmol/L, minimising volumes needed for administration [31]. Accuracy of bolus injection of Iohexol to measure GFR has been confirmed in studies comparing it with Inulin infusions [12], and it is substantially cheaper and more readily available than Inulin [32].
An infusion of Iohexol over 4 hours has previously been validated to measure GFR in subjects with normal renal function [10], however, we have modified this is 2 ways. First, we have employed a much lower dose of Iohexol. This has allowed us to increase the duration of the infusion to more than 72 hours, maintaining the total dose given well within safe limits. Secondly, we have also tested the method in patients with stable CKD with a wide range of GFR from normal to <30 mL/min/1.73 m2. Six of the subjects in our healthy volunteer category met criteria for CKD stage 2 [33]; the patient with the lowest GFR in the CKD group had CKD stage 4. If the time to Css were markedly increased in patients with CKD, then this would limit the applicability in patients with AKI. In our trial, time to Css was under 10 hours. We hope these modifications will allow us to apply this method in patients with AKI.
CILDI could potentially be continued for prolonged periods (e.g. up to 6 days), with regular plasma and urine measurements, allowing the course of moderate and slowly developing AKI to be monitored. It is also potentially useful in rapidly evolving AKI because Iohexol concentrations will continue to rise, rather than move towards the expected Css. Although time to Css is <10 h, subjects with known baseline GFR >60 mL/min/1.73 m2, will have time to Css155 ± 126 min (Table 4), thus increasing the potential applicability of CILDI to detect rapidly evolving AKI if Css is not achieved within this time. This also applies to SCr, however, CILDI has theoretical advantages over SCr because the production rate of creatinine is often reduced in critically ill patients with AKI [7]. In addition, LC-MSMS measurements of Iohexol concentration are more accurate than routine laboratory SCr measurements.
Our data are limited to a small number of subjects with stable GFR and Vd. Although the subjects in our trial may differ from patients with AKI, this study lays the groundwork for studies in patients with AKI, providing an evidence base that CILDI may be preferable to SCr as a research tool. Patients with AKI are unlikely to reach equilibrium, however, it will theoretically be possible to measure changes in GFR occurring after time to Css, or, in rapidly developing AKI occurring before time to Css, to demonstrate increasing plasma concentrations, rather than the expected approach towards Css. The natural history of AKI and the timing of when its associated pathophysiological changes occur are unknown. Although CILDI detects smaller changes in GFR than needed in SCr to define AKI, it is possible that the pathophysiological effects may occur at changes too small to detect with CILDI. Furthermore, they may occur before time to Css. A presumption in using CILDI is that the baseline eGFR is known, or at least >23 mL/min/1.73 m2, and hence CILDI may not be suitable for all patients, or it may not be reliable once GFR drops to very low values. In most cases of AKI, however, we anticipate that fluctuations in Css will be observed with CILDI.
LC-MSMS allows quantification of relatively low concentrations of Iohexol, but is not readily available and requires time and expertise to obtain results. This method is, therefore, not currently suitable for routine clinical use; it is primarily useful as a research tool to measure evolving GFR in AKI so that pathophysiological effects can be measured and timed accurately. It will also allow monitoring of the natural history of AKI in specific disease states (e.g. sepsis, post nephrectomy, following major surgery). Other Iohexol GFR techniques (e.g. blotting) require higher concentrations of Iohexol than CILDI and pre-dispose towards toxicity. LC-MSMS is required, however, because of its specificity. It is possible that other laboratories may have a precision that is different from 10.3% at 3 standard deviations, so this study needs external validation; the co-efficient of variation in our Iohexol measurements, is, however, similar to that in the literature [31].
Plasma clearance of CILDI overestimated renal clearance by 7.1 ± 7.3 mL/min/1.73 m2 (Figure 5). This is, however, less than the difference of approximately 10 mL/min/1.73 m2 observed in studies measuring GFR with Inulin and EDTA [34,35]. The difference between renal clearance and plasma clearance was more apparent in subjects with lower GFR. Our study demonstrates that, even under experimental conditions, "gold standard" urine collection and bladder voiding are often incomplete without urinary catheterisation. This limits precision of measurements; GFR measurement with urine collection is therefore not sufficiently robust for routine clinical application. Three patients were excluded due to subcutaneous injection when administered via peripheral cannulae; it may be more appropriate to administer CILDI via central venous catheters in clinical settings.
Radio-contrast agents have been associated with AKI [36], although recent meta-analyses [37,38] show no difference in the incidence of AKI occurring in patients receiving contrast and matched controls without contrast. This implies that AKI occurring after contrast administration is likely to be multi-factorial and may be more attributable to the underlying illness than contrast media. This particular aspect of AKI aetiology is the subject of on going debate within critical care societies and warrants its own detailed investigation before final conclusions can be made. Proponents suggest a threshold ratio of iodinated contrast volume to weight and baseline renal function that has to be exceeded before contrast-associated AKI develops [39]. If CILDI were continued for 72 h, the volume of Iohexol used is less than half this ratio for an adult weighing 40Kg and, if all the Iohexol accumulated, would take a minimum of 6 days before this threshold was exceeded. Conversely, bolus Iohexol injection [9] would require repeated administration of larger volumes of Iohexol to measure changing GFR in critically ill patients and the threshold for toxicity would be exceeded much sooner. In addition, the washout period required makes bolus methods unsuitable for use in AKI.
Confounding factors will undoubtedly emerge when the method is applied to acutely unwell patients, who will be unstable with changing parameters of GFR and Vd. However, the CILDI method is simple to apply and will enable the measurement of both plasma and renal clearance of Iohexol in critically ill patients, thus providing a measure of dynamic changes in GFR and allowing the direct investigation in vivo of the impact of physiological and pathological perturbations on renal function for the first time.
We have developed a tool for measuring GFR in stable populations that is now ready to be tested in patients at risk of AKI. In our trial, all subjects achieved a steady-state concentration within 10 h. Measurements made after this time in critically ill patients that change by more than 10.3% likely represent changing GFR and depict evolving AKI. The next stage is to investigate the applicability of CILDI in patients with AKI. We hypothesise that CILDI may be a more sensitive method of detecting and monitoring AKI than changes in SCr and if proven to be so, may provide a new standard to which other methods of measuring GFR in AKI are compared.
GFR can be accurately and precisely measured by CILDI.
GFR that varies by >10.3%, using CILDI, represents AKI (p < 0.003).
Time to steady state in subjects with GFR >28 mL/min/1.73 m2 is <10 hours in all subjects.
CILDI is now ready to be investigated in patients with AKI and at risk of AKI.
AKI:
BSA:
Body surface area
CILDI:
Continuous infusion of low dose Iohexol
CKD:
Steady state concentration
Co-efficient of variation
EDTA:
Ethylene diamine tetraacetic acid
GFR:
Glomerular filtration rate
HV:
Healthy volunteer
LC-MSMS:
High performance liquid chromatography / electrospray tandem mass spectrometry
LD:
Loading dose
MDRD:
Modification of diet in renal disease formula
MInf :
Mass of substance infused per unit time
Plasma clearance
RC:
Renal clearance
SBI:
Single bolus injection [of Iohexol]
Serum creatinine concentration
UO:
Urine output
Vd :
Mehta R, Kellum JA, Shah SV, Molitoris BA, Ronco C, Warnock DG, et al. Acute Kidney Injury Network: report of an initiative to improve outcomes in acute kidney injury. Crit Care. 2007;11:R31.
Article PubMed Central PubMed Google Scholar
Bellomo R, Ronco C, Kellum JA, Mehta RL, Palevsky P, Acute Dialysis Quality Initiative workgroup. Acute renal failure – definition, outcome measures, animal models, fluid therapy and information technology needs: the Second International Consensus Conference of the Acute Dialysis Quality Initiative (ADQI) Group. Crit Care. 2004;8(4):R204–12.
Kidney Disease Improving Global Outcomes work group. KDIGO Clinical Practice Guidelines for Acute Kidney Injury. Kidney Int Suppl. 2012;2(1):1–138.
Ostermann M, Chang RWS. Challenges of defining acute kidney injury. Q J Med. 2011;104:237–43.
Waikar SS, Bonventre JV. Creatinine kinetics and the definition of acute kidney injury. J Am Soc Nephrol. 2009;20:672–9.
Article PubMed Central CAS PubMed Google Scholar
Doi K, Yuen PS, Eisner C, Hu X, Leelahavanichkul A, Schnermann J, et al. Reduced production of creatinine limits its use as marker of kidney injury in sepsis. J Am Soc Nephrol. 2009;20:1217–21.
Wilson FP, Sheehan JM, Mariani LH, Berns JS. Creatinine generation is reduced in patients requiring continuous venovenous hemodialysis and independently predicts mortality. Nephrol Dial Transplant. 2012;27:4088–94.
Coca SG, Yalavarthy R, Concato J, Parikh CR. Biomarkers for the diagnosis and risk stratification of acute kidney injury: a systematic review. Kidney Int. 2008;73(9):1008–16.
Article CAS PubMed Google Scholar
Erley CM, Badar BD, Berger ED, Vochazer A, Jorzik JJ, Dietz K, et al. Plasma clearance of iodine contrast media as a measure of glomerular filtration rate in critically ill patients. Crit Care Med. 2001;29(8):1544–50.
Sterner G, Frennby B, Mansson S, Nyman U, Van Western D, Almén T. Determining "true" glomerular filtration rate in healthy adults using infusion of inulin and comparing it with values obtained using other clearance techniques or prediction equations. Scand J Urol Nephrol. 2008;42(3):278–85.
Gaspari F, Perico N, Ruggenenti P, Mosconi L, Amuchastegui CS, Guerini E, et al. Plasma clearance of nonradioactive iohexol as a measure of glomerular filtration rate. J Am Soc Nephrol. 1995;6(2):257–63.
Effersöe H, Groth S, Jensen LI, Golman K. Measurement of renal function with iohexol. A comparison of iohexol, 99mTc-DTPA and 51Cr-EDTA clearance. Invest Radiol. 1990;25(7):778–82.
Levey AS, Stevens LA, Schmid CH, Zhang YL, Castro 3rd AF, Feldman HI, et al. CKD-EPI (Chronic Kidney Disease Epidemiology Collaboration). Ann Intern Med. 2009;150(9):604–12.
World Medical Association Declaration of Helsinki. Ethical Principles for Medical Research Involving Human Subjects. 59th WMA General Assembly, Seoul, October 2008. http://www.wma.net/en/30publications/10policies/b3/17c.pdf.
European medicines agency. ICH Topic E 6 (R1) Guideline for Good Clinical Practice. http://www.emea.europa.eu/docs/en_GB/document_library/Scientific_guideline/2009/09/WC500002874.pdf.
GE Healthcare Inc. OMNIPAQUE (Iohexol): Summary of product characteristics. Princeton, NJ, USA: GE Healthcare; 2009.
Hemmegarn BR, Zhang J, Manns BJ, Tonelli M, Larsen E, Ghali WA, et al. Progression of kidney dysfunction in the community-dwelling elderly. Kidney Int. 2006;69(12):2155–61.
Mosteller RD. Letters to editor. N Engl J Med. 1987;317(17):1098.
Product Information for Agilia MC Injectomat pump. Accessed online via URL http://www.fresenius-kabi.com (accessed 4 November 2010)
Brochner-Mortensen J. A simple method for the determination of Glomerular Filtration Rate. Scand J Clin Lab Invest. 1972;30:271–4.
Kilbride HS, Stevens PE, Eaglestone G, Knight S, Carter JL, Delaney MP, et al. Accuracy of the MDRD (Modification of diet in Renal Disease) Study and CKD-EPI (CKD Epidemiology Collaboration) Equations for Estimation of GFR in the elderly. Am J Kidney Dis. 2012;61(1):57–66.
Shannon JA. The renal excretion of creatinine in man. J Clin Invest. 1935;14(4):403–10.
Bland JM, Altman DG. Measuring agreement in medical comparison studies. Stat Methods Med Res. 1999;8:135–60.
Fleming JS, Zivanovic MA, Blake GM, Burniston M, Cosgriff PS. Guidelines for the Measurement of Glomerular Filtration Rate using Plasma Sampling. http://www.bnms.org.uk (accessed 23 December 2014).
James TJ, Lewis AV, Tan GD, Altmann P, Taylor RP, Levy JC. Validity of simplified protocols to estimate glomerular filtration rate using iohexol clearance. Ann Clin Biochem. 2007;44:369–76.
Mills M, Armitage P. The two-period cross-over clinical trial. B J Clin Pharm. 1979;8:7–20.
van Acker BAC, Koomen GCM, Arisz L. Drawbacks of the constant-infusion technique for measurement of renal function. Am J Physiol. 1995;268(37):F543–52.
Koopman MG, Koomen GCM, Krediet E, de Moor EAM, Hoek FJ, Arisz L. Circadian rhythm of glomerular filtration rate in normal individuals. Clin Sci. 1989;77:105–11.
Olsson B, Aulie Å, Sveen K, Andrew E. Human Pharmacokinetics of Iohexol A New Nonionic Contrast Medium. Invest Radiol. 1983;18:177–82.
Lundqvist S, Holmberg G, Jakobsson G, Lithner F, Skinningsrud K, Stegmayr B, et al. Assessment of possible nephrotoxicty from Iohexol in patients with normal and impaired renal function. Acta Radiol. 1998;39(4):362–7.
Cavalier E, Rozet E, Dubois N, Charlier C, Hubert P, Chapelle J-P, et al. Performance of Iohexol determination in serum and urine by HPLC: validation, risk and uncertainty assessment. Clinica Chem Acta. 2008;396(1–2):80–5.
Kays AJ. Economics. In: Biology and Chemistry of Jerusalem Artichoke: Helicanthus tuberosus L. Nottingham, SF: Taylor & Francis Group LLC; 2008. p. 407–21.
Nissenson AR, Pereira BJG, Collins AJ, Steinberg EP. Prevalence and characteristics of individuals with chronic kidney disease in a large health organization. Am J Kid Dis. 2001;37(6):1177–83.
Schnurr E, Lahme W, Küppers H. Measurement of renal clearance of inulin and PAH in the steady state without urine collection. Clin Nephrol. 1980;13:26–9.
Moore AEB, Park-Holohan S-J, Blake GM, Fogelman I. Conventional measurements of GFR using Cr-51-EDTA overestimate true renal clearance by 10%. Eur J Nucl Med Mol Imaging. 2003;30:4–8.
Barrett BJ, Carlisle EJ. Meta-analysis of the relative nephrotoxicity of high- and low-osmolality iodinated contrast media. Radiology. 1993;188(1):171–8.
McDonald RJ, McDonald JS, Bida JP, Carter RE, Fleming CJ, Misra S, et al. Intravenous contrast material-induced nephropathy: causal or coincident phenomenon. Radiol. 2013;267(1):106–18.
McDonald JS, McDonald RJ, Comin J, Williamson EE, Katzberg RW, Murad MH, et al. Frequency of acute kidney injury following intravenous contrast medium administration: a systematic review and meta-analysis. Radiol. 2013;267(1):119–28.
Brown JR, Robb JF, Block CA, Schoolwerth AC, Kaplan AV, O'Connor GT, et al. Does safe dosing of iodinated contrast prevent contrast-induced acute kidney injury? Circ Cardiovasc Interv. 2010;3(4):346–50.
We thank: the volunteers who participated in our trial; St. George's Charity Medical Research Committee and General Intensive care unit, St. George's Hospital, for funding; the Clinical Research Facility in St. George's, University of London for use of facilities and sample processing; SpotOn Clinical Diagnostics, London, for sample analysis. Data were presented in 2013 at the UK Renal Association annual general meeting and the European Society of Intensive Care Medicine.
General Intensive Care Unit, St. George's Hospital, London, UK
John J Dixon, Katie Lane, R Michael Grounds & Barbara J Philips
Acute Kidney Injury Research Group, Division of Clinical Sciences, St. George's, University of London, London, UK
John J Dixon, Katie Lane, Iain AM MacPhee & Barbara J Philips
WellChild Laboratory, King's College London, Evelina Children's Hospital, London, UK
R Neil Dalton & Charles Turner
Renal Medicine, St. George's Hospital, London, UK
John J Dixon & Iain AM MacPhee
John J Dixon
R Neil Dalton
Charles Turner
R Michael Grounds
Iain AM MacPhee
Barbara J Philips
Correspondence to Barbara J Philips.
JD carried out the study, as part of a PhD project, and was responsible for all aspects of the research project, including writing the manuscript; CT and ND were involved in experimental design, sample analysis and statistical analysis; KL, RG, IM, and BP were involved in experimental design, and statistical analysis; IM and BP provided final editorial changes prior to submission; all authors contributed to writing the original version or successive revisions of this manuscript, and all authors approved the final version.
Dixon, J.J., Lane, K., Dalton, R.N. et al. Validation of a continuous infusion of low dose Iohexol to measure glomerular filtration rate: randomised clinical trial. J Transl Med 13, 58 (2015). https://doi.org/10.1186/s12967-015-0414-3 | CommonCrawl |
Clint Talbert and Joel Maher
At Mozilla, one of our very first automation systems was a performance testing framework we dubbed Talos. Talos had been faithfully maintained without substantial modification since its inception in 2007, even though many of the original assumptions and design decisions behind Talos were lost as ownership of the tool changed hands.
In the summer of 2011, we finally began to look askance at the noise and the variation in the Talos numbers, and we began to wonder how we could make some small modification to the system to start improving it. We had no idea we were about to open Pandora's Box.
In this chapter, we will detail what we found as we peeled back layer after layer of this software, what problems we uncovered, and what steps we took to address them in hopes that you might learn from both our mistakes and our successes.
Let's unpack the different parts of Talos. At its heart, Talos is a simple test harness which creates a new Firefox profile, initializes the profile, calibrates the browser, runs a specified test, and finally reports a summary of the test results. The tests live inside the Talos repository and are one of two types: a single page which reports a single number (e.g., startup time via a web page's onload handler) or a collection of pages that are cycled through to measure page load times. Internally, a Firefox extension is used to cycle the pages and collect information such as memory and page load time, to force garbage collection, and to test different browser modes. The original goal was to create as generic a harness as possible to allow the harness to perform all manner of testing and measure some collection of performance attributes as defined by the test itself.
To report its data, the Talos harness can send JSON to Graph Server: an in-house graphing web application that accepts Talos data as long as that data meets a specific, predefined format for each test, value, platform, and configuration. Graph Server also serves as the interface for investigating trends and performance regressions. A local instance of a standard Apache web server serve the pages during a test run.
The final component of Talos is the regression reporting tools. For every check-in to the Firefox repository, several Talos tests are run, these tests upload their data to Graph Server, and another script consumes the data from Graph Server and ascertains whether or not there has been a regression. If a regression is found (i.e., the script's analysis indicates that the code checked in made performance on this test significantly worse), the script emails a message to a mailing list as well as to the individual that checked in the offending code.
While this architecture–summarized in Figure 8.1–seems fairly straightforward, each piece of Talos has morphed over the years as Mozilla has added new platforms, products, and tests. With minimal oversight of the entire system as an end to end solution, Talos wound up in need of some serious work:
Noise–the script watching the incoming data flagged as many spikes in test noise as actual regressions and was impossible to trust.
To determine a regression, the script compared each check-in to Firefox with the values for three check-ins prior and three afterward. This meant that the Talos results for your check-in might not be available for several hours.
Graph Server had a hard requirement that all incoming data be tied to a previously defined platform, branch, test type, and configuration. This meant that adding new tests was difficult as it involved running a SQL statement against the database for each new test.
The Talos harness itself was hard to run because it took its requirement to be generic a little too seriously–it had a "configure" step to generate a configuration script that it would then use to run the test in its next step.
Figure 8.1 - Talos architecture
While hacking on the Talos harness in the summer of 2011 to add support for new platforms and tests, we encountered the results from Jan Larres's master's thesis, in which he investigated the large amounts of noise that appeared in the Talos tests. He analyzed various factors including hardware, the operating system, the file system, drivers, and Firefox that might influence the results of a Talos test. Building on that work, Stephen Lewchuk devoted his internship to trying to statistically reduce the noise we saw in those tests.
Based on their work and interest, we began forming a plan to eliminate or reduce the noise in the Talos tests. We brought together harness hackers to work on the harness itself, web developers to update Graph Server, and statisticians to determine the optimal way to run each test to produce predictable results with minimal noise.
Understanding What You Are Measuring
When doing performance testing, it is important to have useful tests which provide value to the developers of the product and help customers to see how this product will perform under certain conditions. It is also important to have a repeatable environment so you can reproduce results as needed. But, what is most important is understanding what tests you have and what you measure from those tests.
A few weeks into our project, we had all been learning more about the entire system and started experimenting with various parameters to run the tests differently. One recurring question was "what do the numbers mean?" This was not easily answered. Many of the tests had been around for years, with little to no documentation.
Worse yet, it was not possible to produce the same results locally that were reported from an automated test run. It became evident that the harness itself performed calculations, (it would drop the highest value per page, then report the average for the rest of the cycles) and Graph Server did as well (drop the highest page value, then average the pages together). The end result was that no historical data existed that could provide much value, nor did anybody understand the tests we were running.
We did have some knowledge about one particular test. We knew that this test took the top 100 websites snapshotted in time and loaded each page one at a time, repeating 10 times. Talos loaded the page, waited for the mozAfterPaint event, (a standard event which is fired when Firefox has painted the canvas for the webpage) and then recorded the time from loading the page to receiving this event. Looking at the 1000 data points produced from a single test run, there was no obvious pattern. Imagine boiling those 10,000 points down to a single number and tracking that number over time. What if we made CSS parsing faster, but image loading slower? How would we detect that? Would it be possible to see page 17 slow down if all 99 other pages remained the same? To showcase how the values were calculated in the original version of Talos, consider the following numbers.
For the following page load values:
Page 1: 570, 572, 600, 503, 560
Page 3: 1220, 980, 1000, 1100, 1200
First, the Talos harness itself would drop the first value and calculate the median:
Page 1: 565.5
Page 2: 675
Page 3: 1050
These values would be submitted to Graph Server. Graph Server would drop the highest value and calculate the mean using these per page values and it would report that one value:
$$ \frac{565.5 + 675}{2} = 620.25 $$
This final value would be graphed over time, and as you can see it generates an approximate value that is not good for anything more than a coarse evaluation of performance. Furthermore, if a regression is detected using a value like this, it would be extremely difficult to work backwards and see which pages caused the regression so that a developer could be directed to a specific issue to fix.
We were determined to prove that we could reduce the noise in the data from this 100 page test. Since the test measured the time to load a page, we first needed to isolate the test from other influences in the system like caching. We changed the test to load the same page over and over again, rather than cycling between pages, so that load times were measured for a page that was mostly cached. While this approach is not indicative of how end users actually browse the web, it reduced some of the noise in the recorded data. Unfortunately, looking at only 10 data points for a given page was not a useful sample size.
By varying our sample size and measuring the standard deviation of the page load values from many test runs, we determined that noise was reduced if we loaded a page at least 20 times. After much experimentation, this method found a sweet spot with 25 loads and ignoring the first 5 loads. In other words, by reviewing the standard deviation of the values of multiple page loads, we found that 95% of our noisy results occurred within the first five loads. Even though we do not use those first 5 data points, we do store them so that we can change our statistical calculations in the future if we wish.
All this experimentation led us to some new requirements for the data collection that Talos was performing:
All data collected needs to be stored in the database, not just averages of averages.
A test must collect at least 20 useful data points per test (in this case, per page).
To avoid masking regressions in one page by improvements in another page, each page must be calculated independently. No more averaging values across pages.
Each test that is run needs to have a developer who owns the test and documentation on what is being collected and why.
At the end of a test, we must be able to detect a regression for any given page at the time of reporting the results.
Applying these new requirements to the entire Talos system was the right thing to do, but with the ecosystem that had grown up around Talos it would be a major undertaking to switch to this new model. We had a decision to make as to whether we would refactor or rewrite the system.
Rewrite vs. Refactor
Given our research into what had to change on Talos, we knew we would be making some drastic changes. However, all historical changes to Talos at Mozilla had always suffered from a fear of "breaking the numbers." The many pieces of Talos were constructed over the years by well-intentioned contributors whose additions made sense at the time, but without documentation or oversight into the direction of the tool chain, it had become a patchwork of code that was not easy to test, modify, or understand.
Given our fear of the undocumented dark matter in the code base, combined with the issue that we would need to verify our new measurements against the old measurements, we began a refactoring effort to modify Talos and Graph Server in place. However, it was quickly evident that without a massive re-architecture of the database schema, The Graph Server system would never be able to ingest the full set of raw data from the performance tests. Additionally, we had no clean way to apply our newly-researched statistical methods into Graph Server's backend. Therefore, we decided to rewrite Graph Server from scratch, creating a project called Datazilla. This was not a decision made lightly, as other open source projects had forked the Graph Server code base for their own performance automation. On the Talos harness side of the equation, we also did a prototype from scratch. We even had a working prototype that ran a simple test and was about 2000 lines of code lighter.
While we rewrote Graph Server from scratch, we were worried about moving ahead with our new Talos test runner prototype. Our fear was that we might lose the ability to run the numbers "the old way" so that we could compare the new approach with the old. So, we abandoned our prototype and modified the Talos harness itself piecemeal to transform it into a data generator while leaving the existing pieces that performed averages to upload to the old Graph Server system. This was a singularly bad decision. We should have built a separate harness and then compared the new harness with the old one.
Trying to support the original flow of data and the new method for measuring data for each page proved to be difficult. On the positive side, it forced us to restructure much of the code internal to the framework and to streamline quite a few things. But, we had to do all this piecemeal on a running piece of automation, which caused us several headaches in our continuous integration rigs.
It would have been far better to develop both Talos the framework and Datazilla its reporting system in parallel from scratch, leaving all of the old code behind. Especially when it came to staging, it would have been far easier to stage the new system without attempting to wire in the generation of development data for the upcoming Datazilla system in running automation. We had thought it was necessary to do this so that we could generate test data with real builds and real load to ensure that our design would scale properly. In the end, that build data was not worth the complexity of modifying a production system. If we had known at the time that we were embarking on a year long project instead of our projected six month project, we would have rewritten Talos and the results framework from scratch.
Creating a Performance Culture
Being an open source project, we need to embrace the ideas and criticisms from other individuals and projects. There is no director of development saying how things will work. In order to get the most information possible and make the right decision, it was a requirement to pull in many people from many different teams. The project started off with two developers on the Talos framework, two on Datazilla/Graph Server, and two statisticians on loan from our metrics team. We opened up this project to our volunteers from the beginning and pulled in many fresh faces to Mozilla as well as others who used Graph Server and some Talos tests for their own projects. As we worked together, slowly understanding what permutations of test runs would give us less noisy results, we reached out to include several Mozilla developers in the project. Our first meetings with them were understandably rocky, due to the large changes we were proposing to make. The mystery of "Talos" was making this a hard sell for many developers who cared a lot about performance.
The important message that took a while to settle in was why rewriting large components of the system was a good idea, and why we couldn't simply "fix it in place." The most common feedback was to make a few small changes to the existing system, but everyone making that suggestion had no idea how the underlying system worked. We gave many presentations, invited many people to our meetings, held special one-off meetings, blogged, posted, tweeted, etc. We did everything we could to get the word out. Because the only thing more horrible than doing all this work to create a better system would be to do all the work and have no one use it.
It has been a year since our first review of the Talos noise problem. Developers are looking forward to what we are releasing. The Talos framework has been refactored so that it has a clear internal structure and so that it can simultaneously report to Datazilla and the old Graph Server. We have verified that Datazilla can handle the scale of data we are throwing at it (1 TB of data per six months) and have vetted our metrics for calculation results. Most excitingly, we have found a way to deliver a regression/improvement analysis in real time on a per-change basis to the Mozilla trees, which is a big win for developers.
So, now when someone pushes a change to Firefox, here is what Talos does:
Talos collects 25 data points for each page.
All of those numbers are uploaded to Datazilla.
Datazilla performs the statistical analysis after dropping the first five data points. (95% of noise is found in the first 5 data points.)
A Welch's T-Test is then used to analyze the numbers and detect if there are any outliers in the per-page data as compared to previous trends from previous pushes.1
All results of the T-Test analysis are then pushed through a False Discovery Rate filter which ensures that Datazilla can detect any false positives that are simply due to noise.2
Finally, if the results are within our tolerance, Datazilla runs the results through an exponential smoothing algorithm to generate a new trend line.3 If the results are not within our tolerance, they do not form a new trend line and the page is marked as a failure.
We determine overall pass/fail metrics based on the percentage of pages passing. 95% passing is a "pass".
The results come back to the Talos harness in real time, and Talos can then report to the build script whether or not there is a performance regression. All of this takes place with 10-20 Talos runs completing every minute (hence the 1 TB of data) while updating the calculations and stored statistics at the same time.
Taking this from a working solution to replacing the existing solution requires running both systems side by side for a full release of Firefox. This process ensures that we look at all regressions reported by the original Graph Server and make sure they are real and reported by Datazilla as well. Since Datazilla reports on a per-page basis instead of at the test suite level, there will be some necessary acclimation to the new UI and way we report regressions.
Looking back, it would have been faster to have replaced the old Talos harness up front. By refactoring it, however, Mozilla brought many new contributors into the Talos project. Refactoring has also forced us to understand the tests better, which has translated into fixing a lot of broken tests and turning off tests with little to no value. So, when considering whether to rewrite or refactor, total time expended is not the only metric to review.
In the last year, we dug into every part of performance testing automation at Mozilla. We have analyzed the test harness, the reporting tools, and the statistical soundness of the results that were being generated. Over the course of that year, we used what we learned to make the Talos framework easier to maintain, easier to run, simpler to set up, easier to test experimental patches with, and less error prone. We have created Datazilla as an extensible system for storing and retrieving all of our performance metrics from Talos and any future performance automation. We have rebooted our performance statistical analysis and created statistically viable, per-push regression/improvement detection. We have made all of these systems easier to use and more open so that any contributor anywhere can take a look at our code and even experiment with new methods of statistical analysis on our performance data. Our constant commitment to reviewing the data again and again at each milestone of the project and our willingness to throw out data that proved inconclusive or invalid helped us retain our focus as we drove this gigantic project forward. Bringing in people from across teams at Mozilla as well as many new volunteers helped lend the effort validity and also helped to establish a resurgence in performance monitoring and data analysis across several areas of Mozilla's efforts, resulting in an even more data-driven, performance-focused culture.
Https://github.com/mozilla/datazilla/blob/2c369a346fe61072e52b07791492c815fe316291/vendor/dzmetrics/ttest.py.↩
Https://github.com/mozilla/datazilla/blob/2c369a346fe61072e52b07791492c815fe316291/vendor/dzmetrics/fdr.py.↩
Https://github.com/mozilla/datazilla/blob/2c369a346fe61072e52b07791492c815fe316291/vendor/dzmetrics/data_smoothing.py.↩ | CommonCrawl |
Comparison test
Comparison test can mean:
• Limit comparison test, a method of testing for the convergence of an infinite series.
• Direct comparison test, a way of deducing the convergence or divergence of an infinite series or an improper integral.
| Wikipedia |
\begin{document}
\title{f Reachability problems
for a wave-wave
\
system with a memory term }
\begin{abstract} We solve the reachability problem for a coupled wave-wave system with an integro-differential term. The control functions act on one side of the boundary. The estimates on the time is given in terms of the parameters of the problem and they are explicitly computed thanks to Ingham type results. Nevertheless some restrictions appear in our main results. The Hilbert Uniqueness Method is briefly recalled. Our findings can be applied to concrete examples in viscoelasticity theory. \end{abstract}
\noindent {\bf Keywords:} boundary observability, reachability, Fourier series, hyperbolic integro-differential systems, abstract linear evolution equations
\ \section{Introduction} The linear viscoelasticity theory has been extensively studied by many authors, that proposed several mathematical models based on experimental data to tackle such subject. A possible approach relies on the following physical assumption: the present stress is given by a functional of the past history of the deformation gradient. Such functionals can be represented by means of convolution integrals. This leads to wave equations in which a so-called memory term also appears, see the seminal papers of Dafermos \cite{D1,D2} and \cite{RHN,LPC}. In this framework an important issue is to identify suitable class of integral kernels that match with the physical models. For example, decreasing exponential kernels arise in the analysis of Maxwell fluids or Poynting -Thomson solids, see e.g. \cite{Pruss,Re1}. It is also noteworthy to mention that such kernels satisfy the principle of fading memory, {\em the memory of a simple material fades in time}, introduced in \cite{CN}.
Our aim, justified by the previous remarks,
is to investigate the reachability for a system constituted of a wave equation with a memory term and another wave equation coupled by lower order terms. Precisely, given $a\,,b\in{\mathbb R}$ we consider the following system \begin{equation}\label{eq:problem-uI} \begin{cases} \displaystyle u_{1tt}(t,x) -u_{1xx}(t,x)+\beta\int_0^t\ e^{-\eta(t-s)} u_{xx}(s,x)ds+au_2(t,x)= 0\,, \quad (0<\beta<\eta) \\ \hskip8cm t\in (0,T)\,,\quad x\in(0,\pi) \\ \displaystyle u_{2tt}(t,x) -u_{2xx}(t,x)+bu_1(t,x)= 0 \,, \end{cases} \end{equation} subject to the boundary conditions \begin{equation}\label{eq:bound-u2I} u_1(t,0)=u_2(t,0)=0\,,\quad u_1(t,\pi)=g_1(t)\,,\quad u_2(t,\pi)=g_2(t)\qquad t\in (0,T) \,, \end{equation} and with null initial conditions \begin{equation}\label{eq:initcI} u_i(0,x)=u_{it}(0,x)=0\qquad x\in(0,\pi),\quad i=1,2\,. \end{equation} We wish to solve a reachability problem for \eqref{eq:problem-uI} of the following type: given $T>0$ and taking $ (u_{i}^{0},u_{i}^{1}) $, $i=1,2$, whose regularity we will specify later, one has to find $g_i\in L^2(0,T)$, $i=1,2$ such that the weak solution $u$ of problem \eqref{eq:problem-uI}-\eqref{eq:initcI} satisfies the final conditions \begin{equation}\label{eq:problem-u1I} u_i(T,x)=u_{i}^{0}(x)\,,\quad u_{it}(T,x)=u_{i}^{1}(x)\,, \quad x\in(0,\pi),\quad i=1,2\,. \end{equation} In the literature coupled wave-wave equations were investigated by studying boundary stabilization, see \cite{KR}. The exact synchronization for a coupled system of wave equations with Dirichlet boundary conditions was successfully treated by Li and Rao \cite {LR}. They studied the $n-$dimensional case when the coupling matrix is very general. However, their method does not allow to get precise estimates on the controllability time.
In \cite {Al} F. Alabau-Boussouira considered a system where the coupling parameters are all equal, obtaining an observability inequality for small coupling parameter and large time $T$ and then, by duality, an exact indirect controllability result.
In this paper we solve the reachability problems for the coupled wave-wave
with an integro-differential term by the HUM method, see \cite{Lio1,Lio2,Lio3} and by means of non-harmonic analysis techniques.
In this framework Ingham type estimates, see \cite{Ing}, play an important role.
We already used this approach to study the reachability for one equation, see \cite{LoretiSforza,LoretiSforza1} and to treat the case of a wave--Petrovsky system with a memory term, see \cite{LoretiSforza3}. For a different class of integral kernels see \cite{LPS} and for the hidden regularity in the case of general kernels see \cite{LoretiSforza4}.
However the estimates obtained do not include the case wave-wave without memory as limit case
as $\beta\to 0^+$
\begin{equation}\label{eq:problem-usix} \begin{cases} \displaystyle u_{1tt} -u_{1xx}+au_2= 0 \\ \hskip4.5cm \mbox{on}\quad (0,T)\times (0,\pi), \\ \displaystyle u_{2tt} -u_{2xx}+bu_1= 0 \end{cases} \end{equation} because, as formulas \eqref {eq:lambda2} and \eqref {eq:lambda4} clearly show, the eigenvectors of the integro-differential operator are not bounded as $\beta\to 0^+ $.
The method is based on a representation formula for the solution $(u_1,u_2)$, established in Section 4 \begin{equation*} \begin{split} u_1(t) &=\sum_{n=1}^{\infty}\Big(C_{n}e^{i\omega_{n} t}+\overline{C_{n}}e^{-i\overline{\omega_{n}}t} +R_{n}e^{r_{n} t}+D_{n}e^{i\zeta_{n} t}+\overline{D_{n}}e^{-i\overline{\zeta_{n}}t}\Big) \,, \\ u_2(t) &=\sum_{n=1}^{\infty} \Big(d_nD_{n}e^{i\zeta_{n} t}+\overline{d_nD_{n}}e^{-i\overline{\zeta_{n}}t} +c_nC_{n}e^{i\omega_{n} t}+\overline{c_nC_{n}}e^{-i\overline{\omega_{n}}t}\Big) +\mathcal {E} e^{-\eta t} \,, \end{split} \end{equation*} where \begin{equation*}
|\mathcal {E}|^2\le M \sum_{ n= 1}^{\infty}
\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big), \qquad (M>0) \,. \end{equation*} We will prove the following reachability result (see Theorem \ref{th:reachres}) where we will give an estimate of the control time. \begin{theorem} Let $\beta<1/2$. For any $T>\frac{2\pi}{\sqrt{1-4\beta^2}}$ and $ (u_{i}^{0},u_{i}^{1})\in L^{2}(0,\pi)\times H^{-1}(0,\pi) $, $i=1,2$,
there exist $g_i\in L^2(0,T)$, $i=1,2$, such that the weak solution $(u_1,u_2)$ of system \begin{equation}\label{eq:problem-usix} \begin{cases} \displaystyle u_{1tt}(t,x) -u_{1xx}(t,x)+\beta\int_0^t\ e^{-\eta(t-s)} u_{1xx}(s,x)ds+au_2(t,x)= 0\,, \\ \phantom{u_{1tt}(t,x) -u_{1xx}(t,x)+\int_0^t\ k(t-s) u_{1xx}(s,x)ds+} t\in (0,T)\,,\,\,\, x\in(0,\pi) \\ \displaystyle u_{2tt}(t,x) -u_{2xx}(t,x)+bu_1(t,x)= 0 \,, \end{cases} \end{equation} with boundary conditions \begin{equation}\label{eq:bound-u1r} u_1(t,0)=u_2(t,0)=0\,,\quad u_1(t,\pi)=g_1(t)\,,\quad u_2(t,\pi)=g_2(t)\qquad t\in (0,T)
\,, \end{equation} and null initial values \begin{equation} u_i(0,x)=u_{it}(0,x)=0\qquad x\in(0,\pi)\,,\quad i=1,2, \end{equation} verifies the final conditions \begin{equation}\label{eq:findataT} u_i(T,x)=u_{i}^{0}(x)\,,\quad u_{it}(T,x)=u_{i}^{1}(x)\,, \quad x\in(0,\pi), \qquad i=1,2\,. \end{equation} \end{theorem} Due to the duality between controllability and observability we will first prove Ingham type inequalities (see Theorem \ref {th:inv.ingham1}). \begin{theorem}\label{th:obsI} Let $\{\omega_n\}_{n\in{\mathbb N}}$, $\{r_n\}_{n\in{\mathbb N}}$ and $\{\zeta_{n}\}_{n\in{\mathbb N}}$ be sequences of pairwise distinct numbers such that $\omega_n\not= \zeta_m$, $\omega_n\not=\overline{\zeta_m}$, $r_n\not= i\omega_m$, $r_n\not= i\zeta_m$, $r_n\not=-\eta$, $\zeta_{n}\not=0$, for any $n\,,m\in{\mathbb N}$. Assume that there exist $\gamma>0$, $\alpha,\chi\in{\mathbb R}$, $n'\in{\mathbb N}$, $\mu>0$, $\nu> 1/2$, such that \begin{equation*} \liminf_{n\to\infty}({\Re}\omega_{n+1}-{\Re}\omega_{n})=\liminf_{n\to\infty}({\Re}\zeta_{n+1}-{\Re} \zeta_{n})=\gamma\,, \end{equation*} \begin{equation*} \begin{split} \lim_{n\to\infty}{\Im}\omega_n&=\alpha>0 \,, \\ \lim_{n\to\infty}r_n&=\chi<0\,, \\ \lim_{n\to\infty}\Im \zeta_{n}&=0\,, \end{split} \end{equation*} \begin{equation*}
|d_n|\asymp|\zeta_n| \,, \qquad
|c_n|\le\frac{M}{|\omega_n|}\,, \end{equation*} \begin{equation*}
|R_n|\le \frac{\mu}{n^{\nu}}\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big)^{1/2}\,\quad\forall\ n\ge n'\,, \qquad
|R_n|\le \mu\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big)^{1/2}\,\quad\forall\ n\le n'\,. \end{equation*} Then, for $\gamma>4\alpha$ and $T>\frac{2\pi}{\sqrt{\gamma^2-16\alpha^2}}$ we have
\begin{equation}\label{eq:inv.ingham}
\int_{0}^{T} \big(|u_1(t)|^2+|u_2(t)|^2\big)\ dt
\asymp \sum_{ n= 1}^{\infty}
\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big) \,. \end{equation} \end{theorem}
The observability time may be improved making an extra assumption on the initial data. Indeed, if we assume the condition $|C_n|\le M |d_nD_{n}|$ on the coefficients of the series
instead of $\gamma>4\alpha$, then we can make use of Theorem \ref{th:extracoe} instead of Theorem \ref{th:gamma>4alpha}, obtaining the observability estimates for $T>\frac{2\pi}{\gamma}$ (see Theorem \ref{th:inv.ingham11}).
\begin{theorem} Let assume the hypotheses of Theorem \ref{th:obsI} and the condition \begin{equation}
|C_n|\le M |d_nD_{n}| \,. \end{equation} Then, for $T>\frac{2\pi}{\gamma}$ we have
\begin{equation}\label{eq:inv.ingham11}
\int_{0}^{T} \big(|u_1(t)|^2+|u_2(t)|^2\big)\ dt
\asymp \sum_{ n= 1}^{\infty}
\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big) \,. \end{equation}
\end{theorem} The plan of our paper is the following. In Section 2 we give some preliminary results. In Section 3 we describe the Hilbert Uniqueness Method. In Section 4 we carry out a detailed spectral analysis to give a representation formula for the solution of the wave-wave coupled system with memory. In Section 5 we prove the observability estimates.
Finally, in Section 6 we give a reachability result for the coupled system with memory.
\section{Preliminaries} Throughout the paper, we will adopt the convention to write $F\asymp G$ if there exist two positive constants $c_1$ and $c_2$ such that $c_1F\le G\le c_2F$.
Let $X$ be a real Hilbert space with scalar product
$\langle \cdot \, ,\, \cdot \rangle$ and norm $\| \cdot \|$. For any $T\in\, (0, \infty]$ we denote by $L^1(0,T;X)$ the usual spaces of measurable functions $v:(0,T)\to X$ such that one has
$$
\|v\|_{1,T}:=\int_0^T \|v(t)\|\,dt<\infty\,. $$
We shall use the shorter notation $\|v\|_1$ for
$\|v\|_{1,\infty}$. We denote by $L_{loc}^1 (0,\infty;X)$ the space of functions belonging to $L^1(0,T;X)$ for any $T\in (0,\infty)$. In the case of $X={\mathbb R}$, we will use the abbreviations $L^1(0,T)$ and $L_{loc}^1(0,\infty)$ to denote the spaces $L^1(0,T;{\mathbb R})$ and $L_{loc}^1(0,\infty;{\mathbb R})$, respectively.
Classical results for integral equations (see, e.g., \cite[Theorem 2.3.5]{GLS}) ensure that, for any kernel $k\in L_{loc}^1(0,\infty)$ and $\psi\in L_{loc}^1 (0,\infty;X)$, the problem \begin{equation}\label{integral} \varphi(t)-k*\varphi(t)=\psi(t),\qquad t\ge 0\,, \end{equation} admits a unique solution $\varphi\in L_{loc}^1(0,\infty;X)$. In particular, if we take $\psi=k$ in \eqref{integral}, we can consider the unique solution $\varrho_k\in L_{loc}^1(0,\infty)$ of \begin{equation*} \varrho_k (t)-k*\varrho_k (t)=k (t),\qquad t\ge 0\,. \end{equation*} Such a solution is called the {\em resolvent kernel} of $k$. Furthermore, for any $\psi$ the solution $\varphi$ of (\ref{integral}) is given by the variation of constants formula \begin{equation*} \varphi(t)=\psi(t)+\varrho_k *\psi(t),\qquad t\ge 0\,, \end{equation*} where $\varrho_k$ is the resolvent kernel of $k$.
We recall some results concerning integral equations in case of decreasing exponential kernels, see for example \cite[Corollary 2.2]{LoretiSforza1}. \begin{proposition}\label{pr:unicita} For $0<\beta<\eta$ and $T>0$ the following properties hold true. \begin{description} \item [(i)] The resolvent kernel of $k(t)=\beta e^{-\eta t}$ is $\varrho_k(t)=\beta e^{(\beta-\eta)t}$.
\item [(ii)]Given $\psi\in L_{loc}^1 (-\infty,T;X)$, a function $\varphi\in L_{loc}^1 (-\infty,T;X)$ is a solution of \begin{eqnarray*} \varphi(t)-\beta\int_t^T\ e^{-\eta(s-t)}\varphi(s)ds=\psi(t) \qquad t\le T\,, \end{eqnarray*} if and only if \begin{eqnarray*} \varphi(t)=\psi(t)+\beta\int_{t}^Te^{(\beta-\eta)(s-t)}\psi(s)\ ds \qquad t\le T\,. \end{eqnarray*} Moreover, there exist two positive constants $c_1\,,c_2$ depending on $\beta,\eta,T$ such that \begin{equation}\label{eq:unicita}
c_1\int_0^T|\varphi(t)|^2\ dt
\le\int_0^T|\psi(t)|^2\ dt
\le c_2\int_0^T|\varphi(t)|^2\ dt\,. \end{equation}
\end{description}
\end{proposition}
We state and prove a result, that will allow us to give an equivalent way to write the solution of our problem.
\begin{lemma}\label{le:fifth} Given $\lambda\,,\beta\,,\eta\in{\mathbb R}$, $a\in{\mathbb R}\setminus\{0\}$ and $b\in{\mathbb R}$, a couple $(f, g)$ of scalar functions defined on the interval $[0,\infty)$ is a solution of the system \begin{equation}\label{eq:system0} \begin{cases}\displaystyle f{''}+\lambda f-\lambda\beta \int_0^t e^{-\eta(t-s)}f(s) ds+ag=0\,, \\ \hskip8cm t\ge 0, \\\displaystyle g{''}+\lambda g+bf=0 \,,
\end{cases} \end{equation} if and only if $f$ is a solution of the equation \begin{equation}\label{eq:fifth} \displaystyle f^{(5)}+\eta f^{(4)}+2\lambda f{'''}+\lambda (2\eta- \beta) f''+(\lambda^2-ab)f'
+(\lambda^2 (\eta- \beta)-\eta ab)f=0,\quad t\ge 0, \end{equation} the condition \begin{equation}\label{eq:fifth1} \displaystyle
f^{(4)}(0)=-2\lambda f{''}(0)+\lambda \beta f'(0)+(ab-\eta\lambda \beta -\lambda^2)f(0) \end{equation} is satisfied and $g$ is given by \begin{equation}\label{eq:Ag} g=-\frac1a\Big(f{''}+\lambda f-\lambda\beta \int_0^t e^{-\eta(t-s)}f(s) ds\Big)\,. \end{equation} \end{lemma} \begin{Proof} Let $(f, g)$ be a solution of (\ref{eq:system0}). Differentiating the first equation in (\ref{eq:system0}), we get \begin{equation}\label{eq:0gprime} f{'''}+\lambda f{'} +\eta\lambda\beta \int_0^t e^{-\eta(t-s)}f(s) ds-\lambda \beta f+ag'=0\,, \end{equation} whence \begin{equation}\label{eq:gprime0} ag'(0)=-f{'''}(0)-\lambda f{'}(0)+\lambda \beta f(0)\,. \end{equation} Substituting in \eqref{eq:0gprime} the identity \begin{equation*} \lambda\beta \int_0^t e^{-\eta(t-s)}f(s) ds=f{''}+\lambda f+ag\,, \end{equation*} we obtain \begin{equation}\label{eq:gprime} f{'''}+\eta f{''}+\lambda f{'}+\lambda (\eta- \beta) f+ag'+\eta ag=0\,. \end{equation} Differentiating yet again, we have \begin{equation*} f^{(4)}+\eta f{'''}+\lambda f{''}+\lambda (\eta- \beta) f'+ag''+\eta ag'=0\,, \end{equation*} whence, by using the second equation in (\ref{eq:system0}), that is $ag{''}=-abf-\lambda ag$, we get \begin{equation}\label{eq:fourth} f^{(4)}+\eta f{'''}+\lambda f{''}+\lambda (\eta- \beta) f'-abf+\eta ag'-\lambda ag=0\,. \end{equation} Thanks to \eqref{eq:gprime0} and $ag(0)=-f{''}(0)-\lambda f(0)$, we have \begin{multline*} f^{(4)}(0)=-\eta f{'''}(0)-\lambda f{''}(0)-\lambda (\eta- \beta) f'(0)+abf(0)-\eta ag'(0)+\lambda ag(0)\\ =-\eta f{'''}(0)-\lambda f{''}(0)-\lambda (\eta- \beta) f'(0)+abf(0)+\eta f{'''}(0)\\ +\eta\lambda f{'}(0)-\eta\lambda \beta f(0)-\lambda f{''}(0)-\lambda^2 f(0) \\ =-2\lambda f{''}(0)+\lambda \beta f'(0)+(ab-\eta\lambda \beta -\lambda^2)f(0) \,, \end{multline*} so formula (\ref{eq:fifth1}) for $f^{(4)}(0)$ holds true. Moreover, by differentiating \eqref{eq:fourth} we obtain \begin{equation*} f^{(5)}+\eta f^{(4)}+\lambda f{'''}+\lambda (\eta- \beta) f''-abf'+\eta ag''-\lambda ag'=0\,. \end{equation*} By using again $g{''}=-bf-\lambda g$
we get \begin{equation*} f^{(5)}+\eta f^{(4)}+\lambda f{'''}+\lambda (\eta- \beta) f''-abf'-\eta abf-\lambda ag'-\eta\lambda ag=0\,. \end{equation*} From \eqref{eq:gprime} it follows \begin{equation*} -ag'-\eta ag=f{'''}+\eta f{''}+\lambda f{'}+\lambda (\eta- \beta) f\,, \end{equation*} and hence we have \begin{equation*} f^{(5)}+\eta f^{(4)}+2\lambda f{'''}+\lambda (2\eta- \beta) f'' +(\lambda^2-ab)f'+(\lambda^2 (\eta- \beta)-\eta ab)f =0\,, \end{equation*} that is $f$ is a solution of the differential equation (\ref{eq:fifth}). Finally, from the first equation in (\ref{eq:system0}) we deduce that $g$ is given by \eqref{eq:Ag}.
Conversely, if $f$ satisfies $(\ref{eq:fifth})-(\ref{eq:fifth1})$, multiplying the differential equation
by $e^{\eta t}$ and integrating from $0$ to $t$, we obtain \begin{multline*} \int_0^t e^{\eta s}f^{(5)}(s)\ ds+ \eta \int_0^t e^{\eta s}f^{(4)}(s)\ ds +2\lambda\int_0^t e^{\eta s}f{'''}(s)\ ds
+2\eta\lambda\int_0^t e^{\eta s}f{''}(s)\ ds
\\ -\lambda\beta\int_0^t e^{\eta s}f{''}(s)\ ds +(\lambda^2-ab)\int_0^t e^{\eta s}f{'}(s)\ ds +(\lambda^2(\eta-\beta)-\eta ab)\int_0^t e^{\eta s}f(s)\ ds=0\,. \end{multline*} Integrating by parts the first, the third, the fifth and the sixth integral, we have \begin{multline*} e^{\eta t}f^{(4)}-f^{(4)}(0) +2\lambda e^{\eta t}f{''} -2\lambda f{''}(0) -\lambda\beta e^{\eta t}f{'}+\lambda\beta f{'}(0)+\eta\lambda\beta e^{\eta t}f{} \\ -\eta\lambda\beta f{}(0)-\eta^2\lambda\beta\int_0^t e^{\eta s}f{}(s)\ ds +(\lambda^2-ab)e^{\eta t}f-(\lambda^2-ab)f(0) -\lambda^2\beta\int_0^t e^{\eta s}f(s)\ ds=0\,. \end{multline*} Using the condition (\ref{eq:fifth1}) and multiplying by $e^{-\eta t}$, we obtain \begin{multline}\label{eq:fourthbis} f^{(4)} +2\lambda f{''} -\lambda\beta f{'}+\eta\lambda\beta f{} -\eta^2\lambda\beta\int_0^t e^{-\eta(t- s)}f{}(s)\ ds\\ +(\lambda^2-ab)f -\lambda^2\beta\int_0^t e^{-\eta(t- s)}f(s)\ ds=0\,. \end{multline}
Moreover, by \eqref{eq:Ag} it follows \begin{equation*} ag'=-f{'''}-\lambda f'+\lambda\beta f-\eta\lambda\beta \int_0^t e^{-\eta(t-s)}f(s) ds\,, \end{equation*} and hence \begin{equation*} ag''=-f^{(4)}-\lambda f''+\lambda\beta f'-\eta\lambda\beta f +\eta^2\lambda\beta\int_0^t e^{-\eta(t-s)}f(s) ds\,. \end{equation*} Therefore, thanks to the previous identity and \eqref{eq:fourthbis} we have \begin{equation*} ag''= \lambda f{''} +(\lambda^2-ab)f -\lambda^2\beta\int_0^t e^{-\eta(t- s)}f(s)\ ds\,, \end{equation*} whence, in view of \eqref{eq:Ag} we get \begin{equation*} ag''=-\lambda ag-abf\,. \end{equation*} Finally, by \eqref{eq:Ag} and the above equation, it follows that the couple $(f, g)$ is a solution of the system \eqref{eq:system0}. \end{Proof}
The following lemma is analogous to that of \cite[Lemma 2.3]{LoretiSforza1}. For the reader's convenience we prefer to state and prove it the same. \begin{lemma}\label{le:third} Given $\lambda\,,\beta\,,\eta\in{\mathbb R}$ and $h\in C({\mathbb R})$, if $g\in C^3({\mathbb R})$ is a solution of the third order differential equation \begin{equation}\label{third} g{'''}+ \eta g{''}+\lambda g{'}+\lambda (\eta-\beta)g=h\,\qquad \mbox{in}\,\,\,{\mathbb R} \,, \end{equation} then $g$ is also a solution of the integro-differential equation \begin{equation}\label{eq:second} g{''} +\lambda g-\lambda\beta \int_0^t e^{-\eta(t-s)}g(s) ds=e^{-\eta t}(g{''}(0)+\lambda g(0))+ \int_0^t e^{-\eta(t-s)}h(s) ds\,\qquad t\in{\mathbb R}\,. \end{equation}
\end{lemma} \begin{Proof} Multiplying the differential equation $(\ref{third})$ by $e^{\eta t}$ and integrating from $0$ to $t$, we obtain \begin{equation*} \int_0^t e^{\eta s}g{'''}(s)\ ds+ \eta \int_0^t e^{\eta s}g{''}(s)\ ds+\lambda\int_0^t e^{\eta s}g{'}(s)\ ds +\lambda (\eta-\beta)\int_0^t e^{\eta s}g(s)\ ds=\int_0^t e^{\eta s}h(s)\ ds\,. \end{equation*} Integrating by parts the first term and the third one, we have \begin{equation*}\label{third1} e^{\eta t}g{''}-g{''}(0)+\lambda e^{\eta t} g-\lambda g(0)- \lambda\beta \int_0^t e^{\eta s}g(s)\ ds=\int_0^t e^{\eta s}h(s)\ ds\,. \end{equation*} Finally, if we multiply by $e^{-\eta t}$, then we obtain $(\ref{eq:second})$. \end{Proof}
\section{The Hilbert Uniqueness Method}\label{se:HUM} For reader's convenience, in this section we will describe the Hilbert Uniqueness Method for coupled wave equations with a memory term. For another approach based on the ontoness of the solution operator, see e.g. \cite{LasT, T1}.
Given $k\in L_{loc}^1(0,\infty)$ and $a\,,b\in{\mathbb R}$, we consider the following coupled system: \begin{equation}\label{eq:problem-u} \begin{cases} \displaystyle u_{1tt}(t,x) -u_{1xx}(t,x)+\int_0^t\ k(t-s) u_{1xx}(s,x)ds+au_2(t,x)= 0\,, \\ \phantom{u_{1tt}(t,x) -u_{1xx}(t,x)+\int_0^t\ k(t-s) u_{1xx}(s,x)ds+au_2(t,x)= 0\,,\qquad} t\in (0,T)\,,\quad x\in(0,\pi) \\ \displaystyle u_{2tt}(t,x) -u_{2xx}(t,x)+bu_1(t,x)= 0 \,, \end{cases} \end{equation} subject to the boundary conditions \begin{equation}\label{eq:bound-u2} u_1(t,0)=u_2(t,0)=0\,,\quad u_1(t,\pi)=g_1(t)\,,\quad u_2(t,\pi)=g_2(t)\qquad t\in (0,T) \,, \end{equation} and with null initial conditions \begin{equation}\label{eq:initc} u_i(0,x)=u_{it}(0,x)=0\qquad x\in(0,\pi),\quad i=1,2\,. \end{equation}
For a reachability problem we mean the following: given $T>0$ and taking $ (u_{i}^{0},u_{i}^{1}) $, $i=1,2$, in a suitable space, that we will introduce later, find $g_i\in L^2(0,T)$, $i=1,2$ such that the weak solution $u$ of problem \eqref{eq:problem-u}-\eqref{eq:initc} satisfies the final conditions \begin{equation}\label{eq:problem-u1} u_i(T,x)=u_{i}^{0}(x)\,,\quad u_{it}(T,x)=u_{i}^{1}(x)\,, \quad x\in(0,\pi),\quad i=1,2\,. \end{equation} One can solve such reachability problems by the HUM method. To see that, we proceed as follows.
Given $(z_{i}^{0},z_{i}^{1})\in (C^\infty_c(0,\pi))^2$, $i=1,2$, we introduce the {\it adjoint} system of (\ref{eq:problem-u}), that is \begin{equation}\label{eq:adjoint} \begin{cases} \displaystyle z_{1tt}(t,x) -z_{1xx}(t,x)+\int_t^T\ k(s-t) z_{1xx}(s,x)ds+bz_2(t,x)= 0\,,\\ \hskip9.5cm t\in (0,T)\,,\quad x\in(0,\pi) \\ \displaystyle z_{2tt}(t,x) -z_{2xx}(t,x)+az_1(t,x)= 0\,,
\\ z_i(t,0)=z_i(t,\pi)=0\qquad t\in [0,T], \quad i=1,2, \end{cases} \end{equation} with final data \begin{equation} \label{eq:final} z_i(T,\cdot)=z_{i}^{0}\,,\quad z_{it}(T,\cdot)=z_{i}^{1}\,,\quad i=1,2 \,. \end{equation} The above problem is well-posed, see e.g. \cite{Pruss}. Thanks to the regularity of the final data, the solution $(z_1,z_2)$ of \eqref{eq:adjoint}--\eqref{eq:final} is regular enough to consider the nonhomogeneous problem \begin{equation}\label{eq:phi} \left \{\begin{array}{l}\displaystyle \varphi_{1tt}(t,x) -\varphi_{1xx}(t,x)+\int_0^t\ k(t-s) \varphi_{1xx}(s,x)ds+a \varphi_ 2(t,x)= 0 \\ \hskip9.5cm t\in (0,T)\,,\quad x\in(0,\pi)\,, \\ \displaystyle \varphi_{2tt}(t,x) - \varphi_{2xx}(t,x)+b \varphi_ 1(t,x)= 0 \\ \\ \varphi_i(0,x)= \varphi_{it}(0,x)=0\qquad x\in(0,\pi)\,, \quad i=1,2, \\ \\ \displaystyle \varphi_1(t,0)=0\,,\quad \varphi_1(t,\pi)=z_{1x}(t,\pi)-\int_t^T\ k(s-t)z_{1x}(s,\pi)ds \\ \hskip9.5cm t\in [0,T], \\ \varphi_2(t,0)=0\,,\quad \varphi_2(t,\pi)=z_{2x}(t,\pi). \end{array}\right . \end{equation} As in the non-integral case, it can be proved that problem \eqref{eq:phi} admits a unique solution $(\varphi_1,\varphi_2)$. So, we can introduce the following linear operator: for any $(z_{i}^{0},z_{i}^{1})\in \big(C^\infty_c(0,\pi)\big)^2$, $i=1,2$, we define \begin{equation}\label{eq:psi0} \Psi(z_{1}^{0},z_{1}^{1},z_{2}^{0},z_{2}^{1})=(-\varphi_{1t}(T,\cdot),\varphi_{1}(T,\cdot),-\varphi_{2t}(T,\cdot),\varphi_{2}(T,\cdot)) \,. \end{equation} For any $(\xi_{i}^{0},\xi_{i}^{1})\in \big(C^\infty_c(0,\pi)\big)^2$, $i=1,2$, let $(\xi_1,\xi_2)$ be the solution of \begin{equation}\label{eq:adjoint10} \left \{\begin{array}{l}\displaystyle \xi_{1tt}(t,x) -\xi_{1xx}(t,x)+\int_t^T\ k(s-t) \xi_{1xx}(s,x)ds+b\xi_2(t,x)= 0 \\ \hskip9.5cm t\in (0,T),\quad x\in(0,\pi), \\ \displaystyle \xi_{2tt}(t,x) -\xi_{2xx}(t,x)+a\xi_1(t,x)= 0 \\ \\ \xi_i(t,0)=\xi_i(t,\pi)=0 \qquad t\in [0,T], \\ \hskip6cm \quad i=1,2, \\ \xi_i(T,\cdot)=\xi_{i}^{0}\,,\quad \xi_{it}(T,\cdot)=\xi_{i}^{1}\,. \end{array}\right . \end{equation} We will prove that \begin{multline}\label{eq:psi} \langle\Psi(z_{1}^{0},z_{1}^{1},z_{2}^{0},z_{2}^{1}),(\xi_{1}^{0},\xi_{1}^{1},\xi_{2}^{0},\xi_{2}^{1})\rangle_{L^2} \\ =\int_0^T\varphi_1(t,\pi)\Big(\xi_{1x}(t,\pi)-\int_t^T\ k(s-t)\ \xi_{1x}(s,\pi)\ ds\Big) \ dt +\int_0^T\varphi_{2}(t,\pi)\xi_{2x}(t,\pi)\ dt
\,. \end{multline} To this end, we multiply the first equation in (\ref{eq:phi}) by $\xi_1$ and integrate on $[0,T]\times[0,\pi]$, so we have \begin{multline*} \int_0^\pi \int_0^T\varphi_{1tt}(t,x)\xi_1(t,x)\ dt \ dx -\int_0^T\int_0^\pi\varphi_{1xx}(t,x)\xi_1(t,x)\ dx\ dt \\ +\int_0^\pi\int_0^T\int_0^t\ k(t-s)\varphi_{1xx}(s,x)\ ds\ \xi_1(t,x)\ dt\ dx +a\int_0^T\int_0^\pi \varphi_{2}(t,x)\xi_1(t,x)\ dx \ dt=0\,. \end{multline*} If we take into account that \begin{equation*} \int_0^T\int_0^t\ k(t-s)\varphi_{1xx}(s,x)\ ds\ \xi_1(t,x)\ dt = \int_0^T\varphi_{1xx}(s,x)\int_s^T\ k(t-s)\ \xi_1(t,x)\ dt \ ds \end{equation*} and integrate by parts, then we have \begin{multline*}
\int_0^\pi\big(\varphi_{1t}(T,x)\xi_{1}^{0}(x)- \varphi_1(T,x)\xi_{1}^{1}(x)\big)\ dx
+\int_0^\pi \int_0^T\varphi_1(t,x)\xi_{1tt}(t,x)\ dt \ dx
\\ +\int_0^T\varphi_1(t,\pi)\xi_{1x}(t,\pi)\ dt -\int_0^T\int_0^\pi\varphi_1(t,x)\xi_{1xx}(t,x)\ dx\ dt
\\ -\int_0^T\varphi_{1}(s,\pi)\int_s^T\ k(t-s)\ \xi_{1x}(t,\pi)\ dt \ ds +\int_0^\pi\int_0^T\varphi_1(s,x)\int_s^T\ k(t-s)\ \xi_{1xx}(t,x)\ dt \ ds \ dx \\ +a\int_0^T\int_0^\pi \varphi_{2}(t,x)\xi_1(t,x)\ dx \ dt=0\,. \end{multline*} As a consequence of the above equation and \begin{equation*} \xi_{1tt} -\xi_{1xx}+\int_t^T\ k(s-t) \xi_{1xx}(s,\cdot)ds=-b\xi_2\,, \end{equation*} we obtain \begin{multline}\label{eq:xi1}
\int_0^\pi\big(\varphi_{1t}(T,x)\xi_{1}^{0}(x)- \varphi_1(T,x)\xi_{1}^{1}(x)\big)\ dx +\int_0^T\varphi_1(t,\pi)\Big(\xi_{1x}(t,\pi)-\int_t^T\ k(s-t)\ \xi_{1x}(s,\pi)\ ds\Big) \ dt \\ +\int_0^T\int_0^\pi \big(a\varphi_{2}(t,x)\xi_1(t,x)-b\varphi_1(t,x)\xi_{2}(t,x)\big)\ dx \ dt=0\,. \end{multline} In a similar way, we multiply the second equation in (\ref{eq:phi}) by $\xi_2$ and integrate by parts on $[0,T]\times[0,\pi]$ to get \begin{multline*}
\int_0^\pi\big(\varphi_{2t}(T,x)\xi_{2}^{0}(x)- \varphi_2(T,x)\xi_{2}^{1}(x)\big)\ dx
+\int_0^\pi \int_0^T\varphi_2(t,x)\xi_{2tt}(t,x)\ dt \ dx
\\ +\int_0^T\varphi_{2}(t,\pi)\xi_{2x}(t,\pi)\ dt -\int_0^T\int_0^\pi\varphi_2(t,x)\xi_{2xx}(t,x)\ dx\ dt +b\int_0^T\int_0^\pi \varphi_{1}(t,x)\xi_2(t,x)\ dx \ dt=0\,, \end{multline*} whence, in virtue of \begin{equation*} \xi_{2tt} -\xi_{2xx}=-a\xi_1\,, \end{equation*} we get \begin{multline}\label{eq:xi2}
\int_0^\pi\big(\varphi_{2t}(T,x)\xi_{2}^{0}(x)- \varphi_2(T,x)\xi_{2}^{1}(x)\big)\ dx +\int_0^T\varphi_{2}(t,\pi)\xi_{2x}(t,\pi)\ dt \\ +\int_0^T\int_0^\pi\big(b \varphi_{1}(t,x)\xi_2(t,x)-a\varphi_{2}(t,x)\xi_1(t,x)\big)\ dx \ dt=0\,. \end{multline} If we sum equations \eqref{eq:xi1} and \eqref{eq:xi2}, then we have \begin{multline}\label{eq:prenorm} \langle\Psi(z_{1}^{0},z_{1}^{1},z_{2}^{0},z_{2}^{1}),(\xi_{1}^{0},\xi_{1}^{1},\xi_{2}^{0},\xi_{2}^{1})\rangle_{L^2} \\ = \int_0^\pi\big(-\varphi_{1t}(T,x)\xi_{1}^{0}(x)+ \varphi_1(T,x)\xi_{1}^{1}(x)-\varphi_{2t}(T,x)\xi_{1}^{0}(x)+ \varphi_2(T,x)\xi_{1}^{1}(x)\big)\ dx
\\
=\int_0^T\varphi_1(t,\pi)\Big(\xi_{1x}(t,\pi)-\int_t^T\ k(s-t)\ \xi_{1x}(s,\pi)\ ds\Big) \ dt
+\int_0^T\varphi_{2}(t,\pi)\xi_{2x}(t,\pi)\ dt \,,
\end{multline} that is, \eqref{eq:psi} holds true.
Taking $\xi_{i}^{0}=z_{i}^{0}$ and $\xi_{i}^{1}=z_{i}^{1}$, $i=1,2$, in (\ref{eq:psi}) yields \begin{multline}\label{eq:psi1} \langle\Psi(z_{1}^{0},z_{1}^{1},z_{2}^{0},z_{2}^{1}),(z_{1}^{0},z_{1}^{1},z_{2}^{0},z_{2}^{1})\rangle_{L^2} \\ =
\int_0^T\Big|z_{1x}(t,\pi)-\int_t^T\ k(s-t)\ z_{1x}(s,\pi)\ ds\Big|^2 \ dt
+\int_0^T\big|z_{2x}(t,\pi)\big|^2\ dt\,. \end{multline} As a consequence, we can introduce a semi-norm on the space $\big(C^\infty_c(\Omega)\big)^4$. Indeed,
for $(z_{i}^{0},z_{i}^{1})\in \big(C^\infty_c(\Omega)\big)^2$, $i=1,2$, we define \begin{multline}\label{eq:normF}
\|(z_{1}^{0},z_{1}^{1},z_{2}^{0},z_{2}^{1})\|_{F}:= \displaystyle\Big(
\int_0^T\Big|z_{1x}(t,\pi)-\int_t^T\ k(s-t)\ z_{1x}(s,\pi)\ ds\Big|^2 \ dt
+\int_0^T\big|z_{2x}(t,\pi)\big|^2\ dt \Big)^{1/2}. \end{multline} In view of
Proposition \ref{pr:unicita}, $\|\cdot\|_{F}$ is a norm if and only if the following uniqueness theorem holds. \begin{theorem}\label{th:uniqueness} If $(z_1,z_2)$ is the solution of problem {\rm (\ref{eq:adjoint})--(\ref{eq:final})} such that $$ z_{1x}(t,\pi)=z_{2x}(t,\pi)=0\,,\qquad \forall t\in [0,T]\,, $$ then $$ z_1(t,x)=z_2(t,x)= 0 \qquad\forall (t,x)\in [0,T]\times[0,\pi]\,. $$ \end{theorem} If we are able to establish Theorem \ref{th:uniqueness}, then we can define the Hilbert space $F$ as the completion of $ \big(C^\infty_c(\Omega)\big)^4$ for the norm (\ref{eq:normF}). Moreover, the operator $\Psi$ extends uniquely to a continuous operator, denoted again by $\Psi$, from $F$ to the dual space $F'$ in such a way that $\Psi:F\to F'$ is an isomorphism.
In conclusion, if we prove Theorem \ref{th:uniqueness} and, for example, $F=\big(H^1_0(0,\pi)\times L^2(0,\pi)\big)^2$ with the equivalence of the respective norms, then, taking $(u_{i}^{0},u_{i}^{1})\in L^{2}(0,\pi)\times H^{-1}(0,\pi)$, $i=1,2$, we can solve the reachability problem \eqref{eq:problem-u}--\eqref{eq:problem-u1}.
\section{Representation of the solution as Fourier series}\label{se:} \subsection{Spectral analysis}\label{se:specan}
The aim of this section will be to give a complete spectral analysis for the coupled system.
We will recast our system of coupled wave equations with a memory term in an abstract setting. Indeed, we consider a self-adjoint positive linear
operator $L:D(L)\subset H\to H$ on a Hilbert space $H$ with dense domain $D(L)$. We denote by $\{\lambda_n\}_{n\ge1}$ a strictly increasing sequence of eigenvalues for the operator $L$ with $\lambda_n>0$ and $\lambda_n\to\infty$ and we assume that the sequence of the corresponding eigenvectors $\{w_n\}_{n\ge1}$ constitutes a Hilbert basis for $H$.
We fix two real numbers $a\not=0$, $b$ and consider the following coupled system: \begin{equation}\label{eq:system} \begin{cases} \displaystyle u_1''(t) +Lu_1(t)-\beta\int_0^t\ e^{-\eta(t-s)}L u_1(s)ds+au_2(t)= 0 \\ \hskip9cm t\ge 0, \\ \displaystyle u_2''(t) +Lu_2(t)+bu_1(t)= 0 \\ u_i(0)=u_{i}^{0}\,,\quad u'_i(0)=u_{i}^{1}\,, \quad i=1,2 \,. \end{cases} \end{equation} If we take the initial data $(u_{i}^{0},u_{i}^{1})$, $i=1,2$, belonging to $D(\sqrt{L})\times H$, then we can expand them according to the eigenvectors $w_n$ to obtain: \begin{equation}\label{eq:v0} \begin{split} & u_{i}^{0}=\sum_{n=1}^{\infty}\alpha_{in}w_{n}\,,\qquad\quad\alpha_{in}= \langle u_{i}^{0},w_n\rangle \,, \quad
\|u_i^0\|^2_{D(\sqrt{L})}:=\sum_{n=1}^{\infty}\alpha_{in}^2\lambda_n\,, \\ & u_{i}^{1}=\sum_{n=1}^{\infty}\rho_{in}w_{n}\,,\qquad\quad\rho_{in}=\langle u_{i}^{1},w_n\rangle \,,\quad
\|u_i^1\|^2_{H}:=\sum_{n=1}^{\infty}\rho_{in}^2\,. \end{split} \end{equation} Our target is to write the components $u_{1}, u_{2}$ of the solution of system \eqref{eq:system} as sums of series, that is \begin{equation*} u_{i}(t)=\sum_{n=1}^{\infty}f_{in}(t)w_{n}\,, \qquad f_{in}(t)=\langle u_{i}(t),w_n\rangle\,,\quad i=1,2 \,. \end{equation*} To this end, we put the above expressions for $u_{1}$ and $u_{2}$ into \eqref{eq:system} and multiply by $w_n$, so for any $n\in{\mathbb N}$ $(f_{1n},f_{2n})$ is the solution of the system \begin{equation}\label{eq:secondsys} \begin{cases}\displaystyle f_{1n}'' +\lambda_{n}f_{1n}-\beta\lambda_{n}\int_0^t e^{-\eta(t-s)}f_{1n}(s) ds +af_{2n}=0, \\\displaystyle f_{2n}''+\lambda_n f_{2n}+bf_{1n}=0 \,, \\ f_{in}(0)=\alpha_{in}\,, \quad f_{in}'(0)=\rho_{in}\,, \quad i=1,2 \,. \end{cases} \end{equation} Thanks to lemma \ref{le:fifth} with $\lambda=\lambda_n$, $(f_{1n},f_{2n})$ is the solution of problem \eqref{eq:secondsys} if and only if $f_{1n}$ is the solution of the Cauchy problem \begin{equation}\label{eq:third} \begin{cases} \displaystyle f_{1n}^{(5)}+\eta f_{1n}^{(4)}+2\lambda_{n} f_{1n}'''+\lambda_{n} (2\eta-\beta)f_{1n}''+(\lambda_{n}^2-ab)f_{1n}' +(\lambda_{n}^2(\eta-\beta)-\eta ab)f_{1n}=0 \qquad t\ge 0\,, \\
f_{1n}(0)=\alpha_{1n}, \\
f_{1n}'(0)=\rho_{1n},
\\
f_{1n}''(0)=-\lambda_{n}\alpha_{1n}-a\alpha_{2n},
\\
f_{1n}'''(0)=-\lambda_{n} \rho_{1n}+ \beta\lambda_{n}\alpha_{1n}-a\rho_{2n},
\\
f_{1n}^{(4)}(0)
=(\lambda_{n}^2-\eta \beta\lambda_n+ab)\alpha_{1n}+2a\lambda_{n}\alpha_{2n}+\beta\lambda_{n} \rho_{1n}\,, \end{cases} \end{equation} and $f_{2n}$ is given by \begin{equation*} f_{2n}=-\frac1a\Big(f_{1n}''+\lambda_n f_{1n}-\beta\lambda_{n} \int_0^t e^{-\eta(t-s)}f_{1n}(s) ds\Big)\,. \end{equation*} If we introduce the linear operator $\Upsilon_n$ defined by \begin{equation}\label{eq:f2j'} \Upsilon_n(v)(t):= -\frac1a\Big(v''(t) +\lambda_{n}v(t)-\beta\lambda_{n}\int_0^t e^{-\eta(t-s)}v(s) ds\Big) \qquad t\ge0 \,, \end{equation} then $f_{2n}$ can be written as \begin{equation}\label{eq:f2j} f_{2n}(t) =\Upsilon_n(f_{1n})(t) \qquad t\ge0 \,. \end{equation} We also note that for any $z\in{\mathbb C}$ \begin{equation}\label{eq:Yexp} \Upsilon_n(e^{zt})= -\frac1a \Big[\Big(z^2+\lambda_n-\frac{\beta\lambda_n}{\eta+z}\Big)e^{z t} +\frac{\beta \lambda_n}{\eta+z}e^{-\eta t} \Big]\,. \end{equation} \subsection{The fifth order ordinary differential equation} We proceed to solve the Cauchy problem $(\ref{eq:third})$. To this end, we have to evaluate the solutions of the $5^{\rm th}$--degree characteristic equation in the variable $Z$ \begin{equation}\label{eq:fchar} Z^{5}+\eta Z^{4}+2\lambda_{n}Z^{3}+\lambda_{n} (2\eta-\beta)Z^{2}+(\lambda_{n}^2-ab)Z+\lambda_{n}^2(\eta-\beta)-\eta ab=0\,. \end{equation} By means of the singular perturbation theory we get the five solutions of \eqref{eq:fchar}: one is a real number $r_n$ and the other four $i\omega_n$, $-i\overline{\omega_n}$, $i\zeta_n$, $-i\overline{\zeta_n}$ are pairwise complex conjugate numbers. Moreover, $r_n$, $\omega_n$ and $\zeta_n$ exhibit the following asymptotic behavior as $n$ tends to $\infty$: \begin{equation}\label{eq:lambda1} r_{n}=\beta-\eta -{\beta\big(\beta-\eta\big)^2\over\lambda_{n}}+O\Big({1\over{\lambda_{n}^{2}}}\Big) =\beta-\eta +O\Big({1\over{\lambda_{n}}}\Big)
\,, \end{equation} \begin{multline}\label{eq:lambda2} \omega_{n}= \sqrt{\lambda_{n}}+{\beta\over2}\Big({3\over4}\beta-\eta\Big){1\over\sqrt{\lambda_{n}}} +i \Big[{\beta\over 2} -\Big({\beta\big(\beta-\eta\big)^2\over2}+\frac{ab}{2\beta}\Big){1\over\lambda_{n}} \Big] +O\Big({1\over{\lambda_{n}^{3/2}}}\Big) \\ = \sqrt{\lambda_{n}}+i{\beta\over 2}
+O\Big({1\over{\sqrt{\lambda_{n}}}}\Big)
\,, \end{multline}
\begin{equation}\label{eq:lambda4} \zeta_{n} = \sqrt{\lambda_{n}}+{\eta ab\over 2\beta\lambda_{n}^{3/2}} +i\Big({ab\over 2\beta\lambda_{n}}+{a^2b^2\over 2\beta^3\lambda^2_{n}}\Big) +O\Big({1\over{\lambda_{n}^{5/2}}}\Big) =\sqrt{\lambda_{n}}+i{ab\over 2\beta\lambda_{n}}+O\Big({1\over{\lambda_{n}^{3/2}}}\Big)
\,. \end{equation}
Therefore, we are able to write the solution $f_{1n}(t)$ of (\ref{eq:third}) in the form \begin{equation}\label{eq:f1j} f_{1n}(t)=R_{n}e^{r_{n} t}+C_{n}e^{i\omega_{n} t}+\overline{C_{n}}e^{-i\overline{\omega_{n}}t} +D_{n}e^{i\zeta_{n} t}+\overline{D_{n}}e^{-i\overline{\zeta_{n}}t} \,, \end{equation} where the coefficients $R_{n}\in{\mathbb R}$ and $C_{n},D_{n}\in{\mathbb C}$ are unknown. Since the function $f_{1n}(t)$ have to satisfy the initial conditions in $(\ref{eq:third})$, to determine $R_{n}$, $C_{n}$ and $D_{n}$ we will solve
the system \begin{equation}\label{vandermonde} \left \{\begin{array}{l} R_{n}+C_{n}+\overline{C_{n}}+D_{n}+\overline{D_{n}}=f_{1n}(0),\\ \\ r_{n} R_{n}+i\omega_{n} C_{n}-i\overline{\omega_{n}C_{n}}+i\zeta_{n} D_{n}-i\overline{\zeta_{n}D_{n}} =f_{1n}'(0),\\ \\ r_{n}^2R_{n}-\omega_{n}^2C_{n}-\overline{\omega_{n}^2C_{n}}-\zeta_{n}^2D_{n}-\overline{\zeta_{n}^2D_{n}}=f_{1n}''(0),\\ \\ r_{n}^3R_{n}-i \omega_{n}^3C_{n}+i\overline{\omega_{n} ^3C_{n}}-i\zeta_{n}^3D_{n}+i\overline{\zeta_{n}^3D_{n}}=f_{1n}'''(0),\\ \\ r_{n}^4R_{n}+ \omega_{n} ^4C_{n}+\overline{\omega_{n}^4C_{n}}+\zeta_{n}^4D_{n}+\overline{\zeta_{n}^4D_{n}}=f_{1n}^{(4)}(0). \end{array}\right . \end{equation} Indeed, we obtain that the coefficients
have the following asymptotic behavior as $n$ tends to $\infty$: \begin{equation}\label{eq:asy_R} R_n={\beta\over\lambda_n}(\alpha_{1n}(\beta-\eta)+\rho_{1n}) +( \alpha_{1n}+\rho_{1n}+\alpha_{2n}+\rho_{2n})O\Big({1\over{\lambda_n^{2}}}\Big), \end{equation} \begin{multline}\label{eq:asy_C} C_{n}={\alpha_{1n}\over 2} -\frac{i}{4\beta} \big(\beta^2\alpha_{1n}+2\beta\rho_{1n} +2a\alpha_{2n}\Big) \frac{1}{\lambda_{n}^{1/2}} +\frac{1}{2\beta^2} \big((ab-\beta^3(\beta-\eta))\alpha_{1n} \\ -\beta(\beta^2\rho_{1n}+\eta a\alpha_{2n}) -\beta a\rho_{2n}\Big)\frac1{\lambda_{n}} +(\alpha_{1n}+\rho_{1n}+\alpha_{2n}+\rho_{2n})O\Big({1\over{\lambda_{n}^{3/2}}}\Big) \end{multline} \begin{multline}\label{eq:asy_D} D_n=i\frac{a\alpha_{2n}}{2\beta\lambda_{n}^{1/2}} +\frac{a}{2\beta^2} \big(\beta\eta\alpha_{2n}+\beta\rho_{2n} -b\alpha_{1n}\big) \frac1{\lambda_n} +\frac{i}{2\beta^3} \big(2a^2b\alpha_{2n}-\eta\beta^2a\rho_{2n}+2\eta\beta ab\alpha_{1n}+\beta ab\rho_{1n}\big) \frac1{\lambda_n^{3/2}} \\ +(\alpha_{1n}+\rho_{1n}+ \alpha_{2n}+\rho_{2n})O\Big({1\over{\lambda_n^{2}}}\Big) . \end{multline} Accordingly, we can write $f_{1n}(t)$ by means of formula \eqref{eq:f1j}, where the coefficients $R_n$, $C_n$ and $D_n$ are given by formulas \eqref{eq:asy_R}-\eqref{eq:asy_D} respectively. Moreover, thanks to \eqref{eq:f2j}, we can also get the expression for $f_{2n}(t)$, that is \begin{equation}\label{eq:f2n} f_{2n}(t) =\Upsilon_n\big(R_{n}e^{r_{n} t}+C_{n}e^{i\omega_{n} t}+\overline{C_{n}}e^{-i\overline{\omega_{n}}t}+D_{n}e^{i\zeta_{n} t}+\overline{D_{n}}e^{-i\overline{\zeta_{n}}t}\big) \,. \end{equation} We will observe that the function $f_{2n}(t)$ can be written in a more handleable form. To this end, first we recall the following result (see e.g. \cite[Section 6]{LoretiSforza1}) \begin{lemma}\label{eq:appr-ce} Approximated solutions of the cubic equation \begin{equation}\label{eq:char} Z^{3}+\eta Z^{2}+\lambda_{n}Z+\lambda_{n}(\eta-\beta)=0\,, \end{equation} are given by \begin{equation}\label{eq:lambda10} r_{n}=\beta-\eta -{\beta\big(\beta-\eta\big)^2\over\lambda_{n}}+O\Big({1\over{\lambda_{n}^{2}}}\Big)
\,, \end{equation} \begin{equation}\label{eq:lambda20} z_{n}= -{\beta\over 2} +{\beta\big(\beta-\eta\big)^2\over2}{1\over\lambda_{n}} +i\Big[\sqrt{\lambda_{n}}+{\beta\over2}\Big({3\over4}\beta-\eta\Big){1\over\sqrt{\lambda_{n}}}\Big] +O\Big({1\over{\lambda_{n}^{3/2}}}\Big)
\,. \end{equation} \end{lemma} Therefore, comparing \eqref{eq:lambda1} with \eqref{eq:lambda10}, we have that the numbers $r_n$ are approximated solutions of \eqref{eq:char}, and hence the function $t\to R_{n}e^{r_{n} t}$ is a solution of the third order differential equation \begin{equation}\label{eq:=0} g{'''}+\eta g{''}+\lambda_n g{'}+\lambda_n (\eta-\beta)g=0 \qquad \mbox{in}\,\,\,{\mathbb R}\,. \end{equation} \begin{lemma}\label{eq:app-sol} The numbers $i\omega_n$, with $\omega_n$ defined by \eqref{eq:lambda2}, are approximated solutions of the cubic equation \begin{equation*} Z^{3}+\eta Z^{2}+\lambda_{n}Z+\lambda_{n}(\eta-\beta)=-\frac{ab}{\beta}\,. \end{equation*} \end{lemma} \begin{Proof} The comparison of \eqref{eq:lambda2} with \eqref{eq:lambda20} yields \begin{equation*} i\omega_n=z_n+\frac{ab}{2\beta\lambda_n} \,. \end{equation*} Since \begin{multline*} (i\omega_n)^{3}+\eta (i\omega_n)^{2}+\lambda_{n}i\omega_n+\lambda_{n}(\eta-\beta) \\ =z_n^3+\eta z_n^2+\lambda_{n}z_n+\lambda_{n}(\eta-\beta) +3z_n^2\frac{ab}{2\beta\lambda_n}+3z_n\frac{a^2b^2}{4\beta^2\lambda_n^2}+\frac{a^3b^3}{8\beta^3\lambda_n^3} +2\eta z_n\frac{ab}{2\beta\lambda_n}+\eta\frac{a^2b^2}{4\beta^2\lambda_n^2} +\frac{ab}{2\beta} \,, \end{multline*} and in virtue of Lemma \ref{eq:appr-ce} we have \begin{equation*} z_n^3+\eta z_n^2+\lambda_{n}z_n+\lambda_{n}(\eta-\beta)=0, \end{equation*} then we get \begin{equation*} (i\omega_n)^{3}+\eta (i\omega_n)^{2}+\lambda_{n}i\omega_n+\lambda_{n}(\eta-\beta) =-\frac{3ab}{2\beta}+\frac{ab}{2\beta}+O\Big({1\over{\sqrt{\lambda_{n}}}}\Big)=-\frac{ab}{\beta}+O\Big({1\over{\sqrt{\lambda_{n}}}}\Big) \,. \end{equation*} that is, our claim holds true. \end{Proof} Thanks to Lemma \ref{eq:app-sol}, the numbers $i\omega_n$ and their conjugate numbers $-i\overline{\omega_n}$ are approximated solutions of the cubic equation \begin{equation*} Z^{3}+\eta Z^{2}+\lambda_{n}Z+\lambda_{n}(\eta-\beta)=-\frac{ab}{\beta}\,, \end{equation*}
so, it follows that the function $ t\to C_{n}e^{i\omega_{n} t}+\overline{C_{n}}e^{-i\overline{\omega_{n}}t} $ is a solution of the third order differential equation \begin{equation}\label{eq:not=0} g{'''}+\eta g{''}+\lambda_n g{'}+\lambda_n (\eta-\beta)g=-\frac{ab}{\beta}g \qquad \mbox{in}\,\,\,{\mathbb R}\,. \end{equation} In virtue of \eqref{eq:=0} and \eqref{eq:not=0}, the function $$ g_n(t)= R_{n}e^{r_{n} t}+C_{n}e^{i\omega_{n} t}+\overline{C_{n}}e^{-i\overline{\omega_{n}}t} $$ is a solution of the third order differential equation \begin{equation} g{'''}+\eta g{''}+\lambda_n g{'}+\lambda_n (\eta-\beta)g=-\frac{ab}{\beta}(C_{n}e^{i\omega_{n} t}+\overline{C_{n}}e^{-i\overline{\omega_{n}}t}) \qquad \mbox{in}\,\,\,{\mathbb R}\,. \end{equation} Therefore, we can apply Lemma \ref{le:third} with $h(t)=-\frac{ab}{\beta}(C_{n}e^{i\omega_{n} t}+\overline{C_{n}}e^{-i\overline{\omega_{n}}t})$: thanks to \eqref{eq:second} and \eqref{eq:f2j'}, we have \begin{equation}\label{eq:C_{n}e} \Upsilon_n(g_n(t)) =-\frac1ae^{-\eta t}\big(g_n''(0)+\lambda_n g_n(0)\big) +\frac{b}{\beta} \int_0^t e^{-\eta(t-s)}(C_{n}e^{i\omega_{n} s}+\overline{C_{n}}e^{-i\overline{\omega_{n}}s})ds\,. \end{equation} From \eqref{vandermonde} and \eqref{eq:third} it follows that \begin{equation*} \begin{split}
g_n''(0)
&=f''_{1n}(0)+\zeta_{n}^2D_{n}+\overline{\zeta_{n}^2D_{n}}
=-\lambda_{n}\alpha_{1n}-a\alpha_{2n}+\zeta_{n}^2D_{n}+\overline{\zeta_{n}^2D_{n}}
\\
\lambda_n g_n(0)
&= \lambda_nf_{1n}(0)-\lambda_n D_{n}-\lambda_n\overline{D_{n}}
=\lambda_{n}\alpha_{1n}-\lambda_n D_{n}-\lambda_n\overline{D_{n}}
\,. \end{split} \end{equation*} Thanks to \eqref{eq:lambda4} we have $\zeta_{n}^2-\lambda_n=O\Big({1\over{\sqrt{\lambda_{n}}}}\Big)$, so we see that \begin{equation*} g_n''(0)+\lambda_n g_n(0)= -a\alpha_{2n}+( \alpha_{1n}+\rho_{1n}+\alpha_{2n}+\rho_{2n})O\Big({1\over{\lambda_n}}\Big) \,. \end{equation*} Moreover \begin{equation*} \int_0^t e^{-\eta(t-s)}e^{i\omega_{n} s}ds =\frac{1}{\eta+i\omega_n}\big(e^{i\omega_{n} t}-e^{-\eta t}\big)\,. \end{equation*} Set \begin{equation} c_n=\frac{b}{\beta(\eta+i\omega_n)} \,, \end{equation} from \eqref{eq:C_{n}e} we obtain \begin{equation}\label{eq:R_{n}+C_{n}e} \Upsilon_n(R_{n}e^{r_{n} t}+C_{n}e^{i\omega_{n} t}+\overline{C_{n}}e^{-i\overline{\omega_{n}}t}) = c_nC_{n}e^{i\omega_{n} t} +\overline{c_nC_{n}}e^{-i\overline{\omega_{n}}t} +\big(\alpha_{2n} -2\Re(c_nC_{n})\big)e^{-\eta t}\,. \end{equation} Moreover, thanks to \eqref{eq:Yexp} we have \begin{equation*} \Upsilon_n(e^{i\zeta_{n} t}) = \frac1a \Big(\zeta_{n}^2-\lambda_n+\frac{\beta\lambda_n}{\eta+i\zeta_{n}}\Big)e^{i\zeta_{n} t}-\frac{\beta\lambda_n}{a(\eta+i\zeta_{n})}e^{-\eta t}
\,. \end{equation*} Therefore, if we define \begin{equation} d_n=\frac1a \Big(\zeta_{n}^2-\lambda_n+\frac{\beta\lambda_n}{\eta+i\zeta_{n}}\Big) \,, \end{equation} and \begin{equation} E_n=\alpha_{2n} -2\Re(c_nC_{n})-\frac{2\beta\lambda_n}{a}\Re\bigg(\frac{D_{n}}{\eta+i\zeta_{n}}\bigg)\,, \end{equation} thanks to \eqref{eq:f2n} and \eqref{eq:R_{n}+C_{n}e}, $f_{2n}(t)$ can be written in the following form \begin{equation}\label{eq:f2jbis0} f_{2n}(t) = d_nD_{n}e^{i\zeta_{n} t}+\overline{d_nD_{n}}e^{-i\overline{\zeta_{n}}t} +c_nC_{n}e^{i\omega_{n} t}+\overline{c_nC_{n}}e^{-i\overline{\omega_{n}}t} +E_ne^{-\eta t} \,. \end{equation} We also note that \begin{equation}\label{eq:cndn0}
|d_n|\asymp|\zeta_n|\asymp \sqrt{\lambda_{n}}\,, \qquad
|c_n|\le\frac{M}{|\omega_n|}\,. \end{equation} The proof of the following lemma is straightforward in virtue of \eqref{eq:asy_D} and \eqref{eq:cndn0}, so we omit it. \begin{lemma}\label{le:mathcalE0} Set \begin{equation*} E_n=\alpha_{2n} -2\Re(c_nC_{n})-\frac{2\beta\lambda_n}{a}\Re\bigg(\frac{D_{n}}{\eta+i\zeta_{n}}\bigg)\,, \end{equation*} there exists a constant $M>0$ such that \begin{equation*}
\Big|\sum_{ n= 1}^{\infty} E_n\Big|^2\le M \sum_{ n= 1}^{\infty}
\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big) \,. \end{equation*} \end{lemma}
Now, we state and prove some properties about the coefficients, that show some differences with respect to the analogous ones in \cite{LoretiSforza1, LoretiSforza3}.
\begin{lemma} The following statements hold true. \begin{itemize} \item[(i)] For any $n\in{\mathbb N}$ one has
\begin{equation}\label{eq:|Cj2|}
|C_{n}|^2+\lambda_n |D_{n}|^2 \asymp \frac{1}{\lambda_{n}} \big(\alpha^2_{1n}\lambda_{n}+\rho^2_{1n} +\alpha^2_{2n}\lambda_{n}+\rho^2_{2n}\big). \end{equation} \item[(ii)] There exists a constant $M>0$ such that for any $n\in{\mathbb N}$ one has \begin{equation}\label{eq:C1overC2}
\vert R_{n}\vert\le {M \over {{\lambda^{1/2}_n}}}\Big(|C_{n}|^2+\lambda_n |D_{n}|^2\Big)^{1/2}\,. \end{equation}
\end{itemize} \end{lemma} \begin{Proof} (i) From \eqref{eq:asy_C} it follows that \begin{multline}\label{eq:modCn}
|C_{n}|^2 ={1\over 4}\alpha_{1n}^2 +\frac{1}{16\beta^2} \big(\beta^2\alpha_{1n}+2\beta\rho_{1n} +2a\alpha_{2n}\Big)^2 \frac{1}{\lambda_{n}} \\ +\frac{\alpha_{1n}}{2\beta^2} \big((ab-\beta^3(\beta-\eta))\alpha_{1n} -\beta(\beta^2\rho_{1n}+\eta a\alpha_{2n}) -\beta a\rho_{2n}\Big)\frac1{\lambda_{n}} \\ +(\alpha_{1n}^2+\rho_{1n}^2+\alpha_{2n}^2+\rho_{2n}^2)O\Big({1\over{\lambda_{n}^{2}}}\Big) \,. \end{multline} Moreover, from \eqref{eq:asy_D} we deduce that \begin{multline*} \lambda_{n}^{1/2}D_n=i\frac{a\alpha_{2n}}{2\beta} +\frac{a}{2\beta^2} \big(\beta\eta\alpha_{2n}+\beta\rho_{2n} -b\alpha_{1n}\big) \frac1{\lambda_{n}^{1/2}} \\ +\frac{i}{2\beta^3} \big(2a^2b\alpha_{2n}+2\eta\beta ab\alpha_{1n}+\beta ab\rho_{1n}-\eta\beta^2a\rho_{2n}\big) \frac1{\lambda_n} +(\alpha_{1n}+\rho_{1n}+ \alpha_{2n}+\rho_{2n})O\Big({1\over{\lambda_n^{3/2}}}\Big) , \end{multline*} whence \begin{multline}\label{eq:modDn}
\lambda_{n}|D_n|^2=\frac{a^2\alpha_{2n}^2}{4\beta^2} +\frac{a^2}{4\beta^4} \big(\beta\eta\alpha_{2n}+\beta\rho_{2n} -b\alpha_{1n}\big)^2 \frac1{\lambda_{n}} \\ +\frac{a\alpha_{2n}}{2\beta^4} \big(2a^2b\alpha_{2n}+2\eta\beta ab\alpha_{1n}+\beta ab\rho_{1n}-\eta\beta^2a\rho_{2n}\big) \frac1{\lambda_n} +(\alpha_{1n}^2+\rho_{1n}^2+\alpha_{2n}^2+\rho_{2n}^2)O\Big({1\over{\lambda_n^{2}}}\Big) . \end{multline} Now, putting together \eqref{eq:modCn} and \eqref{eq:modDn}, we have \begin{multline*}
|C_{n}|^2+\lambda_{n}|D_n|^2 =\frac14 \Big(\alpha^2_{1n}+\frac{\rho^2_{1n}}{\lambda_{n}} +\frac{a^2}{\beta^2}\Big( \alpha^2_{2n} +\frac{\rho^2_{2n}}{\lambda_{n}}\Big)\Big) \\ +\frac{1}{16\beta^2} \big(\beta^2\alpha_{1n}+2a\alpha_{2n}\big)^2 \frac{1}{\lambda_{n}} +\frac{\rho_{1n}}{4\beta}\big(\beta^2\alpha_{1n}+2a\alpha_{2n}\big) \frac{1}{\lambda_{n}} \\ +\frac{\alpha_{1n}}{2\beta^2} \big((ab-\beta^3(\beta-\eta))\alpha_{1n} -\beta(\beta^2\rho_{1n}+\eta a\alpha_{2n}) -\beta a\rho_{2n}\Big)\frac1{\lambda_{n}} \\ +\frac{a^2}{4\beta^4} \big(\beta\eta\alpha_{2n} -b\alpha_{1n}\big)^2 \frac1{\lambda_{n}} +\frac{a^2\rho_{2n}}{2\beta^3} \big(\beta\eta\alpha_{2n} -b\alpha_{1n}\big) \frac1{\lambda_{n}} \\ +\frac{a\alpha_{2n}}{2\beta^4} \big(2a^2b\alpha_{2n}+2\eta\beta ab\alpha_{1n}+\beta ab\rho_{1n}-\eta\beta^2a\rho_{2n}\big) \frac1{\lambda_n} +\big(\alpha^2_{1n}+\rho^2_{1n} +\alpha^2_{2n}+\rho^2_{2n} \big)O\Big({1\over{\lambda_{n}^{2}}}\Big) \,. \end{multline*} We can neglect the indices $n\in{\mathbb N}$ such that $\alpha_{1n}=\rho_{1n}=\alpha_{2n}=\rho_{2n}=0$, because the present evaluation will be used in summing series. So, we can assume that for any $n\in{\mathbb N}$ $(\alpha_{1n},\rho_{1n},\alpha_{2n},\rho_{2n})\not=(0,0,0,0)$, and hence by the previous formula we obtain \begin{multline*}
\frac{|C_{n}|^2+\lambda_{n}|D_n|^2}{\alpha^2_{1n}+\frac{\rho^2_{1n}}{\lambda_{n}} +\frac{a^2}{\beta^2}\Big( \alpha^2_{2n} +\frac{\rho^2_{2n}}{\lambda_{n}}\Big)} \\ ={1\over 4} +\frac{\big( \alpha_{1n}^2+(\alpha_{1n}+\alpha_{2n})(\rho_{1n}+\alpha_{2n}+\rho_{2n})
\big)O\Big({1\over{\lambda_{n}}}\Big) } {\alpha^2_{1n}+\frac{\rho^2_{1n}}{\lambda_{n}} +\frac{a^2}{\beta^2}\Big(\alpha^2_{2n} +\frac{\rho^2_{2n}}{\lambda_{n}}\Big)} \to\frac14\,,\qquad\mbox{as}\qquad n\to\infty\,, \end{multline*} taking into account, for example, that $$ {\alpha_{1n}\rho_{1n}\over{\lambda_{n}}} ={\alpha_{1n}\over{\lambda^{1/3}_{n}}} {\rho_{1n}\over{\lambda^{2/3}_{n}}} \le{\alpha^2_{1n}\over{\lambda^{2/3}_{n}}}+{\rho_{1n}^2\over{\lambda^{4/3}_{n}}} \,. $$
In conclusion, \eqref{eq:|Cj2|} holds true.
\noindent (ii) From \eqref{eq:asy_R} we have \begin{equation*}
|R_n|^2={\beta^2\over\lambda_n^2}\big(\alpha_{1n}(\beta-\eta)+\rho_{1n}\big)^2 +\big( \alpha_{1n}+\rho_{1n}\big)\big( \alpha_{1n}+\rho_{1n}+\alpha_{2n}+\rho_{2n}\big)O\Big({1\over{\lambda_n^3}}\Big). \end{equation*}
Moreover, thanks to \eqref{eq:|Cj2|}, there exists a constant $c>0$ such that \begin{equation*}
|C_{n}|^2+\lambda_n |D_{n}|^2\ge \frac{c}{\lambda_n}\big(\alpha^2_{1n}\lambda_{n}+\rho^2_{1n} +\alpha^2_{2n}\lambda_{n}+\rho^2_{2n}\big)\,. \end{equation*} Therefore, from the above formulas we get \begin{equation*}
\frac{|R_n|^2}{|C_{n}|^2+\lambda_n |D_{n}|^2} \le{1\over c\lambda_n} \frac{\beta^2(\alpha_{1n}(\beta-\eta)+\rho_{1n})^2 +( \alpha_{1n}+\rho_{1n})( \alpha_{1n}+\rho_{1n}+\alpha_{2n}+\rho_{2n})O\Big({1\over{\lambda_n}}\Big)} {\alpha^2_{1n}\lambda_{n}+\rho^2_{1n}+\alpha^2_{2n}\lambda_{n}+\rho^2_{2n}} \,, \end{equation*} that is, \eqref{eq:C1overC2} follows. \end{Proof} In conclusion, taking into account of any result of the present section we have proved the following representation formula for the solution of the coupled system. \begin{theorem}\label{th:repres} The solution of problem \eqref{eq:system} can be written as series in the following way \begin{equation}\label{eq:u1} \begin{split} u_1(t) &=\sum_{n=1}^{\infty}\Big(C_{n}e^{i\omega_{n} t}+\overline{C_{n}}e^{-i\overline{\omega_{n}}t} +R_{n}e^{r_{n} t}+D_{n}e^{i\zeta_{n} t}+\overline{D_{n}}e^{-i\overline{\zeta_{n}}t}\Big)w_{n} \,, \\ u_2(t) &=\sum_{n=1}^{\infty} \Big(d_nD_{n}e^{i\zeta_{n} t}+\overline{d_nD_{n}}e^{-i\overline{\zeta_{n}}t} +c_nC_{n}e^{i\omega_{n} t}+\overline{c_nC_{n}}e^{-i\overline{\omega_{n}}t}+E_n e^{-\eta t} \Big)w_{n} \,, \end{split} \end{equation} where \begin{equation*} r_{n} =\beta-\eta +O\Big({1\over{\lambda_{n}}}\Big) \,, \end{equation*} \begin{equation*} \omega_{n} = \sqrt{\lambda_{n}}+i{\beta\over 2} +O\Big({1\over{\sqrt{\lambda_{n}}}}\Big) \,, \end{equation*} \begin{equation*} \zeta_{n} =\sqrt{\lambda_{n}}+i{ab\over 2\beta\lambda_{n}}+O\Big({1\over{\lambda_{n}^{3/2}}}\Big) \,, \end{equation*} \begin{equation*}
\vert R_{n}\vert\le {M \over {{\lambda^{1/2}_n}}}\Big(|C_{n}|^2+|d_nD_{n}|^2\Big)^{1/2}\,, \quad
\Big|\sum_{ n= 1}^{\infty} E_n\Big|^2\le M \sum_{ n= 1}^{\infty}
\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big) \,, \end{equation*} \begin{equation*}
|d_n|\asymp \sqrt{\lambda_{n}}\,, \quad
|c_n|\le\frac{M}{\sqrt{\lambda_n}}\,, \qquad (M>0) \end{equation*} \begin{equation*}
\sum_{ n= 1}^{\infty}\lambda_{n}\Big(|C_{n}|^2+|d_nD_{n}|^2\Big) \asymp
\|u_1^0\|^2_{D(\sqrt{L})}+\|u_1^1\|^2_{H}+\|u_2^0\|^2_{D(\sqrt{L})}+\|u_2^1\|^2_{H} \,.
\end{equation*} \end{theorem} \section{Ingham type estimates}\label{se:invdir} Our goal is to prove an inverse inequality and a direct inequality for the pair $(u_1,u_2)$ defined by \begin{equation}\label{eq:vsum1} \begin{split} u_1(t) &=\sum_{n=1}^{\infty}\Big(C_{n}e^{i\omega_{n} t}+\overline{C_{n}}e^{-i\overline{\omega_{n}}t} +R_{n}e^{r_{n} t}+D_{n}e^{i\zeta_{n} t}+\overline{D_{n}}e^{-i\overline{\zeta_{n}}t}\Big) \,, \\ u_2(t) &=\sum_{n=1}^{\infty} \Big(d_nD_{n}e^{i\zeta_{n} t}+\overline{d_nD_{n}}e^{-i\overline{\zeta_{n}}t} +c_nC_{n}e^{i\omega_{n} t}+\overline{c_nC_{n}}e^{-i\overline{\omega_{n}}t}\Big) +\mathcal {E} e^{-\eta t} \,, \end{split} \end{equation} with $\omega_{n}\,,C_{n}\,,\zeta_{n}\,,D_{n}, d_n,c_n\in{\mathbb C}$ and $r_{n}\,,R_{n}\,,\mathcal {E}\in{\mathbb R}$. We will assume that there exist $\gamma>0$, $\alpha,\chi\in{\mathbb R}$, $n'\in{\mathbb N}$, $\mu>0$, $\nu> 1/2$, such that \begin{equation}\label{eq:hom1} \liminf_{n\to\infty}({\Re}\omega_{n+1}-{\Re}\omega_{n})=\liminf_{n\to\infty}({\Re}\zeta_{n+1}-{\Re} \zeta_{n})=\gamma\,, \end{equation} \begin{equation}\label{eq:hom2} \begin{split} \lim_{n\to\infty}{\Im}\omega_n&=\alpha>0 \,, \\ \lim_{n\to\infty}r_n&=\chi<0\,, \\ \lim_{n\to\infty}\Im \zeta_{n}&=0\,, \end{split} \end{equation} \begin{equation}\label{eq:cndn}
|d_n|\asymp|\zeta_n| \,, \qquad
|c_n|\le\frac{M}{|\omega_n|}\,, \end{equation} \begin{equation}\label{eq:hom3}
|R_n|\le \frac{\mu}{n^{\nu}}\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big)^{1/2}\,\quad\forall\ n\ge n'\,, \qquad
|R_n|\le \mu\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big)^{1/2}\,\quad\forall\ n\le n'\,. \end{equation}
\subsection{Outline of the proof}\label{se:outline} Before to proceed with our computations, we will outline briefly our reasoning. Firstly, to shorten our formulas we introduce the following notations \begin{equation}\label{eq:f11,f12} {\mathcal U}_{1}^C(t)=\sum_{n=1}^{\infty}\big(C_{n}e^{i\omega_{n} t}+\overline{C_{n}}e^{-i\overline{\omega_{n}}t}\big), \quad {\mathcal U}_{1}^D(t)=\sum_{n=1}^{\infty}\big(D_{n}e^{i\zeta_{n} t}+\overline{D_{n}}e^{-i\overline{\zeta_{n}}t}\big), \quad {\mathcal U}_{1}^R(t)=\sum_{n=1}^{\infty} R_{n}e^{r_{n} t}, \end{equation} \begin{equation}\label{eq:f22,f21} {\mathcal U}_{2}^D(t)=\sum_{n=1}^\infty \big(d_nD_{n}e^{i \zeta_{n}t}+\overline{d_nD_{n}}e^{-i \overline{\zeta_n}t}\big), \qquad {\mathcal U}_{2}^C(t)=\sum_{n=1}^\infty \big(c_nC_{n}e^{i\omega_nt}+\overline{c_nC_{n}}e^{-i\overline{\omega_n}t}\big)\,, \end{equation} so we can write the functions $u_1$, $u_2$ as \begin{equation*} u_1={\mathcal U}_{1}^C+{\mathcal U}_{1}^D+{\mathcal U}_{1}^R, \qquad u_2-\mathcal {E} e^{-\eta t}={\mathcal U}_{2}^D+{\mathcal U}_{2}^C. \end{equation*} If $k(t)$ is a suitable positive function, see (\ref{eq:k}) below, our first goal will be to estimate \begin{equation*}
\int_{0}^{\infty} k(t)| {\mathcal U}_{1}^C(t)+{\mathcal U}_{1}^D(t)+{\mathcal U}_{1}^R(t)|^2\ dt
+\int_{0}^{\infty} k(t)| {\mathcal U}_{2}^D(t)+ {\mathcal U}_{2}^C(t)|^2\ dt \,, \end{equation*} unless a finite number of terms in the series.
By reason of $2 ab\ge-\frac12a^2-2b^2$ we have $|a+b|^2\ge\frac12a^2-b^2$, so we can observe that \begin{equation*}
|{\mathcal U}_{1}^C(t)+{\mathcal U}_{1}^D(t)+{\mathcal U}_{1}^R(t)|^2 \ge
\frac12|{\mathcal U}_{1}^C(t)|^2-|{\mathcal U}_{1}^D(t)+{\mathcal U}_{1}^R(t)|^2 \ge
\frac12|{\mathcal U}_{1}^C(t)|^2-2|{\mathcal U}_{1}^D(t)|^2-2|{\mathcal U}_{1}^R(t)|^2 \,, \end{equation*} \begin{equation*}
| {\mathcal U}_{2}^D(t)+ {\mathcal U}_{2}^C(t)|^2 \ge
\frac12| {\mathcal U}_{2}^D(t)|^2-|{\mathcal U}_{2}^C(t)|^2\,. \end{equation*} Bearing in mind \eqref{eq:hom3}, since $k(t)$ is positive from the above inequalities we can deduce \begin{multline*}
\int_{0}^{\infty} k(t)| {\mathcal U}_{1}^C(t)+{\mathcal U}_{1}^D(t)+{\mathcal U}_{1}^R(t)|^2\ dt
+\int_{0}^{\infty} k(t)| {\mathcal U}_{2}^D(t)+ {\mathcal U}_{2}^C(t)|^2\ dt \\ \ge
\int_{0}^{\infty} k(t) \Big(\frac12| {\mathcal U}_{1}^C(t)|^2-2|{\mathcal U}_{1}^D(t)|^2\Big)\ d t
+\int_{0}^{\infty} k(t) \Big(\frac12| {\mathcal U}_{2}^D(t)|^2-|{\mathcal U}_{2}^C(t)|^2\Big)\ d t \\
-2\int_{0}^{\infty} k(t) | {\mathcal U}_{1}^R(t)|^2\ d t \,. \end{multline*} In virtue of \eqref{eq:cndn} we can control the term $\int_{0}^{\infty} k(t) {\mathcal U}_{1}^D(t) d t$ (resp. $\int_{0}^{\infty} k(t) {\mathcal U}_{2}^C(t) d t$) by means of \break $\int_{0}^{\infty} k(t) {\mathcal U}_{2}^D(t) d t$ (resp. $\int_{0}^{\infty} k(t) {\mathcal U}_{1}^C(t)\,\hbox{\rm d} t$). Therefore, it is convenient to write the previous formula in the following way \begin{multline}\label{eq:f1+f2}
\int_{0}^{\infty} k(t)| {\mathcal U}_{1}^C(t)+{\mathcal U}_{1}^D(t)+{\mathcal U}_{1}^R(t)|^2\ dt
+\int_{0}^{\infty} k(t)| {\mathcal U}_{2}^D(t)+ {\mathcal U}_{2}^C(t)|^2\ dt \\ \ge
\frac12\int_{0}^{\infty} k(t) \Big(| {\mathcal U}_{1}^C(t)|^2-2|{\mathcal U}_{2}^C(t)|^2\Big)\ d t
+\frac12\int_{0}^{\infty} k(t) \Big(| {\mathcal U}_{2}^D(t)|^2-4|{\mathcal U}_{1}^D(t)|^2\Big)\ d t \\
-2\int_{0}^{\infty} k(t) | {\mathcal U}_{1}^R(t)|^2\ d t \,. \end{multline}
We will give a lower bound estimate for $\int_{0}^{\infty} k(t) | {\mathcal U}_{1}^C(t)|^2 d t $ and
$\int_{0}^{\infty} k(t) | {\mathcal U}_{2}^D(t)|^2 d t $, and, on the contrary, an upper bound estimate for $\int_{0}^{\infty} k(t) | {\mathcal U}_{2}^C(t)|^2 d t $,
$\int_{0}^{\infty} k(t) | {\mathcal U}_{1}^D(t)|^2 d t $ and
$\int_{0}^{\infty} k(t) | {\mathcal U}_{1}^R(t)|^2 d t $. So, thanks to \eqref{eq:f1+f2}, we will be able to prove an inverse estimate.
Moreover, if we will assume an additional condition on the coefficients of the series, we will be able to prove an inverse inequality with a better estimate for the control time. Indeed, the additional assumption will allow us to control all terms
$\int_{0}^{\infty} k(t) |{\mathcal U}_{1}^D(t)|^2 d t$,
$\int_{0}^{\infty} k(t) |{\mathcal U}_{2}^C(t)|^2d t$ and
$\int_{0}^{\infty} k(t) | {\mathcal U}_{1}^R(t)|^2d t$ by means of
$\int_{0}^{\infty} k(t) | {\mathcal U}_{2}^D(t)|^2 d t$.
In this way the estimate of the term $\int_{0}^{\infty} k(t) | {\mathcal U}_{1}^C(t)|^2\ d t$ can be done with the help of an idea used previously in \cite{LoretiSforza1}. In fact in this case we will use the following inequality \begin{multline}\label{eq:f1+f2bis}
\int_{0}^{\infty} k(t)| {\mathcal U}_{1}^C(t)+{\mathcal U}_{1}^D(t)+{\mathcal U}_{1}^R(t)|^2\ dt
+\int_{0}^{\infty} k(t)| {\mathcal U}_{2}^D(t)+ {\mathcal U}_{2}^C(t)|^2\ dt \\ \ge
\frac12\int_{0}^{\infty} k(t) | {\mathcal U}_{1}^C(t)|^2\ d t
+\frac12\int_{0}^{\infty} k(t) \Big(| {\mathcal U}_{2}^D(t)|^2-4|{\mathcal U}_{1}^D(t)|^2-2|{\mathcal U}_{2}^C(t)|^2-4| {\mathcal U}_{1}^R(t)|^2 \Big)\ d t \,. \end{multline}
\subsection{Technical results} In order to avoid repetitions and simplify the proofs of the main theorems, we prefer to single out some lemmas that we will employ in several situations. For this reason, in this subsection we collect some results to be used later.
Let $T>0$. We introduce an auxiliary function defined by \begin{equation}\label{eq:k} k(t):=\left \{\begin{array}{l} \displaystyle\sin \frac{\pi t}{T}\,\qquad\qquad \mbox{if}\,\, t\in\ [0,T]\,,\\ \\ 0\,\qquad\qquad\quad\ \ \ \ \mbox{otherwise}\,. \end{array}\right . \end{equation} In the following lemma we list some useful properties of $k$.
\begin{lemma} \label{th:k} Set \begin{equation}\label{eqn:K} K(w):=\frac{T\pi}{\pi^2-T^2w^2}\,,\qquad w\in {\mathbb C}\,, \end{equation} the following properties hold. \begin{itemize} \item[(i)]
For any $w\in {\mathbb C}$ one has
\begin{equation}\label{eqn:sinek2} \overline{K(w)}=K(\overline{w})\,, \qquad
\big|K(w)\big|=\big|K(\overline{w})\big|\,, \end{equation} \begin{equation}\label{eqn:sinek1} \int_{0}^{\infty} k(t)e^{iw t}dt = (1+e^{iw T})K(w)
\,. \end{equation} \item[(ii)] For any $z_i,w_i\in {\mathbb C}$, $i=1,2$, one has \begin{multline}\label{eq:sinek2biss} \int_{0}^{\infty} k(t)\Re(z_1e^{iw_1 t})\Re(z_2e^{iw_2t})dt \\ =\frac12 \Re\Big(z_1z_2(1+e^{i(w_1+w_2) T})K(w_1+w_2) +z_1\overline{z_2}(1+e^{i(w_1-\overline{w_2}) T})K(w_1-\overline{w_2})\Big)\,. \end{multline} \item[(iii)]
Let $\overline{\gamma}>0$ and $j\in{\mathbb N}$. Then for $T>2\pi/\overline{\gamma}$ and $w\in{\mathbb C}$, $|w|\ge\overline{\gamma} j$, one has \begin{equation}\label{eq:sinek3}
\big|K(w)\big|\le
\frac{4\pi}{T\overline{\gamma}^2(4j^2-1)} \,. \end{equation} \end{itemize}
\end{lemma} \begin{Proof} (i) The proof is straightforward.
\noindent (ii) We note that for any $z,w\in{\mathbb C}$ \begin{equation*} \int_{0}^{\infty} k(t)\Re(ze^{iw t})dt = \Re\big(z(1+e^{iw T})K(w)\big)\,. \end{equation*}
Therefore, taking into account \begin{equation*} \Re(z_1e^{iw_1 t})\Re(z_2e^{iw_2t})=\frac12\Re\big(z_1z_2e^{i(w_1+w_2) t}+z_1\overline{z_2}e^{i(w_1-\overline{w_2}) t}\big) \,, \end{equation*} it follows \eqref{eq:sinek2biss}.
\noindent (iii) We observe that \begin{equation*}
\big|K(w)\big|=
\frac{\pi}{T\Big|w^2-\big(\frac{\pi}{T}\big)^2\Big|}
=\frac{4\pi}{T\overline{\gamma}^2\Big|4\big(\frac{w}{\overline{\gamma}}\big)^2-\big(\frac{2\pi}{T\overline{\gamma}}\big)^2\Big|} \,. \end{equation*}
Since $|w|\ge\overline{\gamma} j$ and $\frac{2\pi}{T\overline{\gamma}}<1$, we have \begin{equation*}
\Big|4\Big(\frac{w}{\overline{\gamma}}\Big)^2-\Big(\frac{2\pi}{T\overline{\gamma}}\Big)^2\Big|
\ge 4\frac{|w|^2}{\overline{\gamma}^2}-\Big(\frac{2\pi}{T\overline{\gamma}}\Big)^2 \ge 4j^2-1\,, \end{equation*} and hence \eqref{eq:sinek3} holds true. \end{Proof}
\begin{lemma}\label{le:gap} If $\gamma>0$ is such that \begin{equation*} \liminf_{n\to\infty}\big(\Re\sigma_{n+1}-\Re\sigma_{n}\big)=\gamma\,, \end{equation*} then for any $\varepsilon\in (0,1)$ there exists $n_0\in{\mathbb N}$ such that \begin{equation}\label{eq:Re_gap1}
|\Re\sigma_n-\Re\sigma_m|\ge \gamma\sqrt{1-\varepsilon} |n-m|\,,\qquad\forall n\,,m\ge n_0\,, \end{equation} \begin{equation}\label{eq:Re_gap2} \Re\sigma_n\ge \gamma\sqrt{1-\varepsilon}\ n\,,\qquad\forall n\ge n_0\,. \end{equation} \end{lemma}
\begin{Proof} For $\varepsilon\in (0,1)$ there exists $n_0\in{\mathbb N}$ such that \begin{equation*} \Re\sigma_{n+1}-\Re\sigma_n\ge\gamma\sqrt{1-\varepsilon}\qquad\quad\forall n\ge n_0\,, \end{equation*} whence \eqref{eq:Re_gap1} follows. Moreover, in view of \begin{equation}\label{eq:} \liminf_{n\to\infty}\frac{\Re\sigma_{n+1}}{n+1}\ge \liminf_{n\to\infty}\big(\Re\sigma_{n+1}-\Re\sigma_{n}\big)\,, \end{equation} see \cite[p. 54]{Ce}, \eqref{eq:Re_gap2} holds true. \end{Proof}
\begin{lemma}\label{le:n0sum} \begin{description} \item[(i)] For any $n_0\in{\mathbb N}$ and $n\ge n_0$ we have
\begin{equation}\label{eq:telesc} \sum_{\substack{m=n_0\\ m\not=n}}^\infty\ \frac{1}{4(m-n)^2-1} \le 1\,. \end{equation} \item[(ii)] Fixed $a,b\ge 0$ and $\varepsilon>0$, there exists $n_0\in{\mathbb N}$ large enough to satisfy \begin{equation}\label{eq:n0sum} \frac{a}{4n^2-1}+b\sum_{m=n_0}^\infty\frac{1}{4m^2-1} \le\varepsilon \qquad \forall n\ge n_0 \,. \end{equation} \item[(iii)] Fixed $a\ge 0$, $\nu>1/2$ and $\varepsilon>0$, there exists $n_0\in{\mathbb N}$ large enough to satisfy \begin{equation}\label{eq:n0summ} a\sum_{ n= n_0}^{\infty}\frac{1}{n^{2\nu}} \le\varepsilon \,. \end{equation} \end{description} \end{lemma} \begin{Proof} (i) We have
\begin{multline*} \sum_{\substack{m=n_0\\ m\not=n}}^\infty\ \frac{1}{4(m-n)^2-1} = \sum_{m=n_0}^{n-1}\ \frac{1}{4(n-m)^2-1} +\sum_{m=n+1}^\infty\ \frac{1}{4(m-n)^2-1} \\ \le 2\sum_{j=1}^{\infty}\ \frac{1}{4j^2-1} =\sum_{j=1}^{\infty}\ \Big(\frac{1}{2j-1}-\frac{1}{2j+1}\Big)= 1\,. \end{multline*}
\noindent (ii) We observe that for $n\ge n_0$ we have \begin{equation*} 4n^2-1 \ge 4n^{3/2}n_0^{1/2}-1 \ge n^{1/2}_0(4n^{3/2}-1) \,, \end{equation*} and hence \begin{equation*} \frac{a}{4n^2-1}+b\sum_{m=n_0}^\infty\frac{1}{4m^2-1} \le \frac{1}{n^{1/2}_0}\bigg(a+b \sum_{m=1}^\infty\frac{1}{4m^{3/2}-1}\bigg)
\,. \end{equation*} In conclusion, if one takes $n_0\in{\mathbb N}$ such that \begin{equation*} n_0 \ge \frac{1}{\varepsilon^{2}}\bigg(a+b \sum_{m=1}^\infty\frac{1}{4m^{3/2}-1}\bigg)^2 \,, \end{equation*} then \eqref{eq:n0sum} holds true.
\noindent (iii) For $0<\delta<2\nu-1$ we have \begin{equation*} \sum_{ n= n_0}^{\infty}\frac{1}{n^{2\nu}} \le \frac{1}{n^{\delta}_0}\sum_{ n= 1}^{\infty} \frac{1}{n^{2\nu-\delta}}\,, \end{equation*} whence, for $n_0\ge\bigg(\frac{a}\varepsilon\sum_{ n= 1}^{\infty} \frac{1}{n^{2\nu-\delta}}\bigg)^{1/\delta}$ we have \eqref{eq:n0summ}. \end{Proof} \begin{lemma}\label{le:stimaK} Suppose that \begin{equation*} \liminf_{n\to\infty}\big(\Re\sigma_{n+1}-\Re\sigma_{n}\big)=\gamma>0\,. \end{equation*} Then for any $\varepsilon\in (0,1)$ and $T>\frac{2\pi}{\gamma\sqrt{1-\varepsilon}}$ there exists $n_0=n_0(\varepsilon)\in{\mathbb N}$ such that for any $n\ge n_0$ we have \begin{equation}\label{eq:minus}
\sum_{\substack{m=n_0\\ m\not=n}}^\infty\ |K( \sigma_{n}-\overline{\sigma_m})|
+\sum_{m=n_0}^\infty|K( \sigma_n+\sigma_m)| \le \frac{4\pi}{T\gamma^2(1-\varepsilon)}\bigg(1+\sum_{m=n_0}^\infty\frac{1}{4m^{2}-1}\bigg)\,, \end{equation}
\end{lemma} \begin{Proof} As regards the first inequality, we observe that, thanks to \eqref{eq:Re_gap1} and \eqref{eq:sinek3}, for $\varepsilon\in (0,1)$ there exists $n_0\in{\mathbb N}$ such that \begin{equation*}
\sum_{\substack{m=n_0\\ m\not=n}}^\infty\ |K( \sigma_{n}-\overline{\sigma_m})| \le\frac{4\pi}{T\gamma^2(1-\varepsilon)} \sum_{\substack{m=n_0\\ m\not=n}}^\infty\ \frac{1}{4(m-n)^2-1} \,, \end{equation*} whence, in view of \eqref{eq:telesc} we get our statement.
Moreover, concerning the second estimate, thanks to \eqref{eq:Re_gap2}, we have \begin{equation*}
| \sigma_n+\sigma_m| \ge \Re\sigma_m\ge \gamma\sqrt{1-\varepsilon}\ m\,,\qquad\forall m\ge n_0 \,. \end{equation*} Therefore, using again (\ref{eq:sinek3}) we obtain the required inequality.
\end{Proof} The following result is an useful tool in the proof of the Ingham type inverse estimates. For the sake of completeness we prefer to give a detailed proof, although it could be deduced from previous papers, see \cite{KL1}.
\begin{proposition}\label{pr:Fn} Given any $\gamma>0$ suppose that \begin{equation*} \liminf_{n\to\infty}\big(\Re \sigma_{n+1}-\Re \sigma_{n}\big)=\gamma \end{equation*} and $\{F_n\}$ is a complex number sequence such that
$\sum_{n=1}^\infty\ |F_{n}|^2<+\infty$.
Then for any $\varepsilon\in (0,1)$ and $T>\frac{2\pi}{\gamma\sqrt{1-\varepsilon}}$
there exists $n_0=n_0(\varepsilon)\in{\mathbb N}$ independent of $T$ and $F_n$ such that we have \begin{multline}\label{eq:Fn2'}
\int_{0}^{\infty} k(t)\Big| \sum_{n=n_0}^\infty F_{n}e^{i \sigma_{n}t}+\overline{F_{n}}e^{-i \overline{\sigma_n}t}\Big|^2\ d t \\ \ge 2\pi T\sum_{n=n_0}^\infty\bigg( \frac{1}{\pi^2+4T^2(\Im \sigma_{n})^2} -
\frac{4}{T^2\gamma^2}(1+\varepsilon)\bigg) (1+e^{-2\Im \sigma_{n} T})|F_{n}|^2 \,, \end{multline} \begin{multline}\label{eq:Fn2''}
\int_{0}^{\infty} k(t)\Big| \sum_{n=n_0}^\infty F_{n}e^{i \sigma_{n}t}+\overline{F_{n}}e^{-i \overline{\sigma_n}t}\Big|^2\ d t \\ \le 2\pi T\sum_{n=n_0}^\infty\bigg( \frac{1}{\pi^2+4T^2(\Im \sigma_{n})^2} +
\frac{4}{T^2\gamma^2}(1+\varepsilon)\bigg) (1+e^{-2\Im \sigma_{n} T})|F_{n}|^2 \,. \end{multline} \end{proposition}
\begin{Proof} Let us first observe that \begin{equation*}
\Big| \sum_{n=n_0}^\infty F_{n}e^{i\sigma_{n}t}
+\overline{F_{n}}e^{-i \overline{\sigma_{n}}t}\Big|^2
=4\sum_{n, m=n_0}^\infty \Re\big(F_{n}e^{i\sigma_{n}t}\big)\Re\big(F_{m}e^{i\sigma_{m}t}\big) \,, \end{equation*} where $n_0\in{\mathbb N}$ will be chosen later. From \eqref{eq:sinek2biss} we have \begin{multline*}
\int_{0}^{\infty} k(t)\Big| \sum_{n=n_0}^\infty F_{n}e^{i\sigma_{n}t}
+\overline{F_{n}}e^{-i \overline{\sigma_{n}}t}\Big|^2\ d t \\ =2\sum_{n, m=n_0}^\infty \Re \Big[F_{n}\overline{F_{m}} (1+e^{i( \sigma_{n}-\overline{\sigma_m}) T}) K( \sigma_{n}-\overline{\sigma_m}) +F_{n}F_{m} (1+e^{i( \sigma_{n}+\sigma_{m}) T}) K( \sigma_{n}+\sigma_{m})\Big] \,. \end{multline*}
Since \eqref{eqn:K} gives $ \displaystyle K( \sigma_{n}-\overline{\sigma_n}) =\frac{\pi T}{\pi^2+4T^2(\Im\sigma_{n})^2}\,, $ it follows that \begin{multline*}
\int_{0}^{\infty} k(t)\Big| \sum_{n=n_0}^\infty F_{n}e^{i
\sigma_{n}t}+\overline{F_{n}}e^{-i \overline{\sigma_n}t}\Big|^2\ d t
-2\pi T\sum_{n=n_0}^\infty\ \frac{1+e^{-2\Im\sigma_{n} T}}{\pi^2+4T^2(\Im\sigma_{n})^2}|F_{n}|^2 \\ =2\sum_{\substack{n, m=n_0\\ n\not=m}}^\infty \Re \big[F_{n}\overline{F_{m}} (1+e^{i( \sigma_{n}-\overline{\sigma_m}) T}) K( \sigma_{n}-\overline{\sigma_m})\big] +2\sum_{n, m=n_0}^\infty \Re \big[F_{n}F_{m} (1+e^{i( \sigma_{n}+\sigma_{m}) T})K( \sigma_{n}+\sigma_{m})\big] \,. \end{multline*} Thus \begin{multline}\label{eq:F2up}
\left|\int_{0}^{\infty} k(t)\Big| \sum_{n=n_0}^\infty F_{n}e^{i \sigma_{n}t}+\overline{F_{n}}e^{-i\overline{ \sigma_n}t}\Big|^2\ d t
-2\pi T\sum_{n=n_0}^\infty\ \frac{1+e^{-2\Im \sigma_{n} T}}{\pi^2+4T^2(\Im \sigma_{n})^2}|F_{n}|^2\right| \\
\le 2\sum_{\substack{n, m=n_0\\ n\not=m}}^\infty |F_{n}| |F_{m}| (1+e^{-\Im(\sigma_{n}+ \sigma_{m}) T})
|K( \sigma_{n}-\overline{\sigma_m})| \\
+2\sum_{n, m=n_0}^\infty |F_{n}| |F_{m}|(1+e^{-\Im(\sigma_{n}+\sigma_{m}) T})
|K( \sigma_{n}+\sigma_{m})| \,. \end{multline} By \eqref{eqn:sinek2} we have $$
|K( \sigma_{n}-\overline{\sigma_m})|=|K( \sigma_{m}-\overline{\sigma_n})|\,, $$ hence \begin{multline*}
\sum_{\substack{n, m=n_0\\ n\not=m}}^\infty |F_{n}||F_{m}| |K( \sigma_{n}-\overline{\sigma_m})| \le
\frac{1}{2}\sum_{\substack{n, m=n_0\\ n\not=m}}^\infty\ \big(|F_{n}|^2+ |F_{m}|^2\big) |K( \sigma_{n}-\overline{\sigma_m})| \\ =
\frac{1}{2}\sum_{n=n_0}^\infty\ |F_{n}|^2\sum_{\substack{m=n_0\\ m\not=n}}^\infty\ |K( \sigma_{n}-\overline{\sigma_m})|
+\frac{1}{2}\sum_{m=n_0}^\infty\ |F_{m}|^2\sum_{\substack{n=n_0\\ n\not=m}}^\infty\ |K( \sigma_{m}-\overline{\sigma_n})|
\\
= \sum_{n=n_0}^\infty\ |F_{n}|^2\sum_{\substack{m=n_0\\ m\not=n}}^\infty\ |K( \sigma_{n}-\overline{\sigma_m})| \,. \end{multline*} In the same manner we can see that \begin{equation*}
\sum_{\substack{n, m=n_0\\ n\not=m}}^\infty |F_{n}||F_{m}|e^{-\Im( \sigma_{n}+\sigma_{m}) T} |K( \sigma_{n}-\overline{\sigma_m})| \le
\sum_{n=n_0}^\infty\ e^{-2\Im\sigma_{n}T}|F_{n}|^2 \sum_{\substack{m=n_0\\ m\not=n}}^\infty\ |K( \sigma_{n}-\overline{\sigma_m})| \,, \end{equation*} \begin{equation*}
\sum_{n, m=n_0}^\infty |F_{n}||F_{m}|(1+e^{-\Im( \sigma_{n}+\sigma_{m}) T}) |K( \sigma_{n}+\sigma_{m})| \le
\sum_{n=n_0}^\infty\ (1+e^{-2\Im \sigma_{n}T})|F_{n}|^2\sum_{m=n_0}^\infty\ |K( \sigma_{n}+\sigma_{m})| \,. \end{equation*} Substituting these inequalities into \eqref{eq:F2up} yields \begin{multline*}
\left|\int_{0}^{\infty} k(t)\Big| \sum_{n=n_0}^\infty F_{n}e^{i \sigma_{n}t}+\overline{F_{n}}e^{-i \overline{\sigma_n}t}\Big|^2\ d t
-2\pi T\sum_{n=n_0}^\infty\ \frac{1+e^{-2\Im \sigma_{n} T}}{\pi^2+4T^2(\Im \sigma_{n})^2}|F_{n}|^2\right| \\
\le 2\sum_{n=n_0}^\infty\ (1+e^{-2\Im \sigma_{n}T})|F_{n}|^2
\Bigg(\sum_{\substack{m=n_0\\ m\not=n}}^\infty\ |K( \sigma_{n}-\overline{\sigma_m})|
+\sum_{m=n_0}^\infty\ |K( \sigma_{n}+\sigma_{m})|\Bigg) \,. \end{multline*}
Fix now $\varepsilon\in (0,1)$ and $T>\frac{2\pi}{\gamma\sqrt{1-\varepsilon}}$. As for $\varepsilon'\in (0,\varepsilon)$ one has $T>\frac{2\pi}{\gamma\sqrt{1-\varepsilon'}}$ too, we can employ Lemma \ref{le:stimaK} with $\varepsilon$ replaced by $\varepsilon'$. Thus taking $n_0$ as in Lemma \ref{le:stimaK} and applying \eqref{eq:minus} we obtain \begin{multline*}
\left|\int_{0}^{\infty} k(t)\Big| \sum_{n=n_0}^\infty F_{n}e^{i \sigma_{n}t}+\overline{F_{n}}e^{-i \overline{\sigma_n}t}\Big|^2\ d t
-2\pi T\sum_{n=n_0}^\infty\ \frac{1+e^{-2\Im\sigma_{n} T}}{\pi^2+4T^2(\Im\sigma_{n})^2}|F_{n}|^2\right| \\ \le
\frac{8\pi}{T\gamma^2(1-\varepsilon')}\sum_{n=n_0}^\infty\ (1+e^{-2\Im\sigma_{n}T})|F_{n}|^2 \Bigg(1+\sum_{m=n_0}^\infty\frac{1}{4m^{2}-1}\Bigg) \,. \end{multline*} By Lemma \ref{le:n0sum}-(ii) with $a=0$ and $b=1$ one can pick $n_0\in{\mathbb N}$ large enough to satisfy \begin{equation*} \sum_{m=n_0}^\infty\frac{1}{4m^{2}-1} \le\varepsilon' \,. \end{equation*} Therefore \begin{multline*}
\left|\int_{0}^{\infty} k(t)\Big| \sum_{n=n_0}^\infty F_{n}e^{i \sigma_{n}t}+\overline{F_{n}}e^{-i \overline{\sigma_n}t}\Big|^2\ d t
-2\pi T\sum_{n=n_0}^\infty\ \frac{1+e^{-2\Im\sigma_{n} T}}{\pi^2+4T^2(\Im\sigma_{n})^2}|F_{n}|^2\right| \\ \le
\frac{8\pi}{T\gamma^2}\frac{1+\varepsilon'}{1-\varepsilon'}\sum_{n=n_0}^\infty\ (1+e^{-2\Im\sigma_{n} T})|F_{n}|^2 \,. \end{multline*} Taking $\varepsilon'\in (0,\varepsilon)$ such that $\frac{1+\varepsilon'}{1-\varepsilon'}<1+\varepsilon$, that is $\varepsilon'<\frac{\varepsilon}{2+\varepsilon}$, we obtain \begin{multline*}
\Bigg|\int_{0}^{\infty} k(t)\Big| \sum_{n=n_0}^\infty F_{n}e^{i \sigma_{n}t}+\overline{F_{n}}e^{-i \overline{\sigma_n}t}\Big|^2\ d t
-2\pi T\sum_{n=n_0}^\infty\ \frac{1+e^{-2\Im\sigma_{n} T}}{\pi^2+4T^2(\Im\sigma_{n})^2}|F_{n}|^2\Bigg| \\ \le \frac{8\pi}{T\gamma^2}(1+\varepsilon)
\sum_{n=n_0}^\infty\ (1+e^{-2\Im\sigma_{n} T})|F_{n}|^2 \,, \end{multline*} which gives \eqref{eq:Fn2'} and \eqref{eq:Fn2''}. \end{Proof} \subsection{Inverse inequality}
Following the outline shown in Section \ref{se:outline} we have to estimate all three integrals on the right-hand side of \eqref{eq:f1+f2}. For this reason, for any term to bound we will establish a corresponding lemma. \begin{lemma}\label{le:} For any $\varepsilon\in (0,1)$ and $T>\frac{2\pi}{\gamma\sqrt{1-\varepsilon}}$ there exists $n_0=n_0(\varepsilon)\in{\mathbb N}$ independent of $T$ and $C_n$ such that we have \begin{multline}\label{eq:u1-2u2}
\int_{0}^{\infty} k(t) \Big(\Big| \sum_{n=n_0}^{\infty}C_{n}e^{i\omega_{n} t}+\overline{C_{n}}e^{-i\overline{\omega_{n}}t}\Big|^2
-2\Big| \sum_{n=n_0}^\infty c_nC_{n}e^{i \omega_{n}t}+\overline{c_nC_{n}}e^{-i \overline{\omega_n}t}\Big|^2\Big)\ d t \\ \ge 2\pi T\sum_{n= n_0}^\infty \Bigg(\frac{1-\varepsilon}{\pi^2+4T^2(\Im\omega_{n})^2} -\frac{4}{T^2\gamma^2}(1+\varepsilon) \Bigg)
(1+e^{-2\Im\omega_{n}T})|C_{n}|^2 \,. \end{multline} \end{lemma}
\begin{Proof} Fix $\varepsilon\in (0,1)$ and $T>\frac{2\pi}{\gamma\sqrt{1-\varepsilon}}$. Let us apply Proposition \ref{pr:Fn} with $\sigma_n=\omega_n$. Indeed, for $\varepsilon'\in (0,\varepsilon)$ to be chosen later there exists $n_0$ independent of $T$ and $C_n$ such that from \eqref{eq:Fn2'} with $F_n=C_n$ and \eqref{eq:Fn2''} with $F_n=c_nC_n$ respectively we have \begin{multline}\label{eq:U1C}
\int_{0}^{\infty} k(t)\Big| \sum_{n=n_0}^\infty C_{n}e^{i \omega_{n}t}+\overline{C_{n}}e^{-i \overline{\omega_n}t}\Big|^2\ d t \\ \ge 2\pi T\sum_{n=n_0}^\infty\bigg( \frac{1}{\pi^2+4T^2(\Im \omega_{n})^2} -
\frac{4}{T^2\gamma^2}(1+\varepsilon')\bigg) (1+e^{-2\Im \omega_{n} T})|C_{n}|^2 \,, \end{multline} \begin{multline}\label{eq:U2C}
\int_{0}^{\infty} k(t)\Big| \sum_{n=n_0}^\infty c_nC_{n}e^{i \omega_{n}t}+\overline{c_nC_{n}}e^{-i \overline{\omega_n}t}\Big|^2\ d t \\ \le 2\pi T\sum_{n=n_0}^\infty\bigg( \frac{1}{\pi^2+4T^2(\Im \omega_{n})^2} +
\frac{4}{T^2\gamma^2}(1+\varepsilon')\bigg) (1+e^{-2\Im \omega_{n} T})|c_nC_{n}|^2 \,. \end{multline} Combining these inequalities gives \begin{multline*}
\int_{0}^{\infty} k(t) \Big(\Big| \sum_{n=n_0}^{\infty}C_{n}e^{i\omega_{n} t}+\overline{C_{n}}e^{-i\overline{\omega_{n}}t}\Big|^2
-2\Big| \sum_{n=n_0}^\infty c_nC_{n}e^{i \omega_{n}t}+\overline{c_nC_{n}}e^{-i \overline{\omega_n}t}\Big|^2\Big)\ d t \\ \ge 2\pi T\sum_{n= n_0}^\infty
\Bigg(\frac{1-2|c_n|^2}{\pi^2+4T^2(\Im\omega_{n})^2} -\frac{4}{T^2\gamma^2}(1+\varepsilon')
\big(1+2|c_n|^2\big)\Bigg)
(1+e^{-2\Im\omega_{n}T})|C_{n}|^2 \,. \end{multline*}
We will choose $\varepsilon'$ in a suitable way to obtain our statement. Thanks to \eqref{eq:cndn} for $n_0$ large enough we have $2|c_n|^2\le\varepsilon'$ for $n\ge n_0$. Hence \begin{equation*}
(1+\varepsilon')\big(1+2|c_n|^2\big)\le(1+\varepsilon')^2\le1+3\varepsilon' \hskip1cm \forall n\ge n_0 \,. \end{equation*} Taking $\varepsilon'<\varepsilon/3$ yields \begin{equation*}
(1+\varepsilon')\big(1+2|c_n|^2\big)\le1+\varepsilon \hskip1cm \forall n\ge n_0 \,. \end{equation*} Moreover, since
$2|c_n|^2\le\varepsilon$ we get \eqref{eq:u1-2u2} and the proof is complete. \end{Proof} To estimate the second integral on the right-hand side of \eqref{eq:f1+f2} we state the following result, that may be proved in much the same way as the previous lemma by means of Proposition \ref{pr:Fn} with $\sigma_n=\zeta_n$ and \eqref{eq:cndn}. For this reason we omit the proof. \begin{lemma}\label{le:U2D} For any $\varepsilon\in (0,1)$ and $T>\frac{2\pi}{\gamma\sqrt{1-\varepsilon}}$ there exists $n_0=n_0(\varepsilon)\in{\mathbb N}$ independent of $T$ and $D_n$ such that we have \begin{multline}\label{eq:U2D}
\int_{0}^{\infty} k(t) \Big(\Big| \sum_{n=n_0}^\infty d_nD_{n}e^{i \zeta_{n}t}+\overline{d_nD_{n}}e^{-i \overline{\zeta_n}t}\Big|^2
-4\Big| \sum_{n=n_0}^\infty D_{n}e^{i \zeta_{n}t}+\overline{D_{n}}e^{-i \overline{\zeta_n}t}\Big|^2\Big)\ d t \\ \ge 2\pi T\sum_{n= n_0}^\infty \Bigg(\frac{1-\varepsilon}{\pi^2+4T^2(\Im\zeta_{n})^2} -\frac{4}{T^2\gamma^2}(1+\varepsilon) \Bigg)
(1+e^{-2\Im\zeta_{n}T})|d_nD_{n}|^2 \,. \end{multline} \end{lemma} Finally, we will give an estimate for the last integral on the right-hand side of \eqref{eq:f1+f2}. \begin{lemma} For any $\varepsilon\in (0,1)$ and $T>0$ there exists $n_0=n_0(\varepsilon)\in{\mathbb N}$ independent of $T$ and $R_n$ such that we have \begin{equation}\label{le:U1R0}
\int_{0}^{\infty} k(t) \Big|\sum_{n=n_0}^{\infty} R_{n} e^{r_n t}\Big|^2\ d t \le \varepsilon\ \pi T
\sum_{n= n_0}^\infty\frac{|C_{n}|^2+|d_nD_{n}|^2}{\pi^2+T^2r_{n}^2} \,. \end{equation} \end{lemma} \begin{Proof} Our proof starts with the observation that \eqref{eqn:sinek1} leads to \begin{multline*}
\int_0^\infty k(t) \Big|\sum_{n=n_0}^{\infty} R_{n} e^{r_n t}\Big|^2\ dt= \sum_{n,m=n_0}^{\infty} R_{n}R_{m} \int_0^\infty k(t) e^{(r_n+r_m) t}\ dt \\ = \sum_{n, m=n_0}^{\infty} R_{n} R_{m} (1+e^{(r_n+r_m )T})K(ir_n+ir_m)\,, \end{multline*} where $n_0\in{\mathbb N}$ has to be chosen later. By the definition \eqref{eqn:K} of $K$ we have \begin{equation*} K(ir_n+ir_m)= \frac{T\pi}{\pi^2+T^2(r_n+r_m)^2} \,. \end{equation*} Let us apply $r_n\le 0$ for $n\ge n'$ to obtain \begin{equation*} 1+e^{(r_n+r_m )T}\le2 \,. \end{equation*} Consequently, taking $n_0\ge n'$ we get \begin{equation*}
\int_0^\infty k(t) \Big|\sum_{n=n_0}^{\infty} R_{n} e^{r_n t}\Big|^2\ dt
\le 2\pi T\sum_{n, m= n_0}^{\infty} \frac{ |R_{n}| |R_{m}|}{\pi^2+T^2(r_n+r_m)^2} \,. \end{equation*} From \eqref{eq:hom3} we see that \begin{multline*}
\int_0^\infty k(t) \Big|\sum_{n=n_0}^{\infty} R_{n} e^{r_n t}\Big|^2\ dt \\ \le
2\pi T\mu^2\sum_{n, m= n_0}^{\infty}\frac{\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big)^{1/2}}{m^{\nu}}\ \frac{\Big(|C_{m}|^2+ |d_mD_{m}|^2\Big)^{1/2}}{n^{\nu}} \frac{1}{\pi^2+T^2(r_n+r_m)^2} \,. \end{multline*} Using again \eqref{eq:hom2} yields \begin{multline*}
\sum_{n, m= n_0}^{\infty}\frac{\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big)^{1/2}}{m^{\nu}}\ \frac{\Big(|C_{m}|^2+ |d_mD_{m}|^2\Big)^{1/2}}{n^{\nu}} \frac{1}{\pi^2+T^2(r_n+r_m)^2} \\ \le \frac12\sum_{ m= n_0}^{\infty}\frac{1}{m^{2\nu}}\sum_{ n= n_0}^{\infty}
\frac{|C_{n}|^2+ |d_nD_{n}|^2}{\pi^2+T^2r_n^2} +\frac12\sum_{ n= n_0}^{\infty}\frac{1}{n^{2\nu}}\sum_{ m= n_0}^{\infty}
\frac{|C_{m}|^2+ |d_mD_{m}|^2}{\pi^2+T^2r_m^2} \\ = \sum_{ n= n_0}^{\infty}\frac{1}{n^{2\nu}}\sum_{ n= n_0}^{\infty}
\frac{|C_{n}|^2+ |d_nD_{n}|^2}{\pi^2+T^2r_n^2} \,. \end{multline*} Combining these inequalities we deduce that \begin{equation*}
\int_0^\infty k(t) \Big|\sum_{n=n_0}^{\infty} R_{n} e^{r_n t}\Big|^2\ dt \le 2\pi T\mu^2\sum_{ n= n_0}^{\infty}\frac{1}{n^{2\nu}}\sum_{ n= n_0}^{\infty}
\frac{|C_{n}|^2+ |d_nD_{n}|^2}{\pi^2+T^2r_n^2}\,. \end{equation*} Applying Lemma \ref{le:n0sum}-(iii) we conclude that \eqref{le:U1R0} is proved. \end{Proof} We will establish the main result to obtain the inverse inequality. To simplify our notations, in the following we will use the symbols \begin{equation}\label{eq:notation} \begin{split} u_1^{n_0}(t) &:=\sum_{n=n_0}^{\infty}\Big(C_{n}e^{i\omega_{n} t}+\overline{C_{n}}e^{-i\overline{\omega_{n}}t} +R_{n}e^{r_{n} t}+D_{n}e^{i\zeta_{n} t}+\overline{D_{n}}e^{-i\overline{\zeta_{n}}t}\Big) \,, \\ u_2^{n_0}(t) &:=\sum_{n=n_0}^{\infty} \Big(d_nD_{n}e^{i\zeta_{n} t}+\overline{d_nD_{n}}e^{-i\overline{\zeta_{n}}t} +c_nC_{n}e^{i\omega_{n} t}+\overline{c_nC_{n}}e^{-i\overline{\omega_{n}}t}\Big)
\,, \end{split} \end{equation}
\begin{theorem}\label{th:gamma>4alpha} Assume $\gamma>4\alpha$. Then for any $\varepsilon\in\big(0,\frac{\gamma^2-16\alpha^2}{\gamma^2+16\alpha^2}\big)$ and $T>\frac{2\pi}{\sqrt{\gamma^2(1-\varepsilon)-16\alpha^2(1+\varepsilon)}}$ there exist $n_0=n_0(\varepsilon)\in{\mathbb N}$, independent of $T$ and all coefficients of the series, and a constant $c(T,\varepsilon)>0$ such that \begin{multline}\label{eq:u1+u2}
\int_{0}^{\infty} k(t)\Big| \sum_{n=n_0}^{\infty}C_{n}e^{i\omega_{n} t}+\overline{C_{n}}e^{-i\overline{\omega_{n}}t}
+R_{n}e^{r_{n} t}+D_{n}e^{i\zeta_{n} t}+\overline{D_{n}}e^{-i\overline{\zeta_{n}}t}\Big|^2\ dt \\
+\int_{0}^{\infty} k(t)\Big| \sum_{n=n_0}^{\infty}d_nD_{n}e^{i\zeta_{n} t}+\overline{d_nD_{n}}e^{-i\overline{\zeta_{n}}t}
+c_nC_{n}e^{i\omega_{n} t}+\overline{c_nC_{n}}e^{-i\overline{\omega_{n}}t}\Big|^2\ dt \\ \ge c(T,\varepsilon)\sum_{n= n_0}^\infty
(1+e^{-2\Im\omega_{n}T})\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big) \,. \end{multline} \end{theorem} \begin{Proof} Fix $\varepsilon\in (0,1)$, in view of \eqref{eq:notation} our goal is to evaluate the following sum \begin{equation}\label{eq:u1+u21} \int_{0}^{\infty} k(t)
\big(|u_1^{n_0}(t)|^2+|u_2^{n_0}(t)|^2\big)\ dt \,, \end{equation}
where the index $n_0\in{\mathbb N}$ depending on $\varepsilon$ will be chosen suitably. To this end, we bear in mind the comments given in Section \ref{se:outline}. Indeed, we observe that \begin{multline*}
\int_{0}^{\infty} k(t)\Big| \sum_{n=n_0}^{\infty}C_{n}e^{i\omega_{n} t}+\overline{C_{n}}e^{-i\overline{\omega_{n}}t}
+R_{n}e^{r_{n} t}+D_{n}e^{i\zeta_{n} t}+\overline{D_{n}}e^{-i\overline{\zeta_{n}}t}\Big|^2\ dt \\ \ge
\frac12\int_{0}^{\infty} k(t) \Big| \sum_{n=n_0}^{\infty}C_{n}e^{i\omega_{n} t}+\overline{C_{n}}e^{-i\overline{\omega_{n}}t}\Big|^2 \ d t
-2\int_{0}^{\infty} k(t)\Big| \sum_{n=n_0}^\infty D_{n}e^{i \zeta_{n}t}+\overline{D_{n}}e^{-i \overline{\zeta_n}t}\Big|^2\ d t \\
-2\int_{0}^{\infty} k(t) \Big|\sum_{n=n_0}^{\infty} R_{n} e^{r_n t}\Big|^2\ d t \end{multline*} and \begin{multline*}
\int_{0}^{\infty} k(t)\Big| \sum_{n=n_0}^{\infty}d_nD_{n}e^{i\zeta_{n} t}+\overline{d_nD_{n}}e^{-i\overline{\zeta_{n}}t}
+c_nC_{n}e^{i\omega_{n} t}+\overline{c_nC_{n}}e^{-i\overline{\omega_{n}}t}\Big|^2\ dt \\ \ge
\frac12\int_{0}^{\infty} k(t) \Big| \sum_{n=n_0}^\infty d_nD_{n}e^{i \zeta_{n}t}+\overline{d_nD_{n}}e^{-i \overline{\zeta_n}t}\Big|^2\ dt
-\int_{0}^{\infty} k(t)\Big| \sum_{n=n_0}^\infty c_nC_{n}e^{i \omega_{n}t}+\overline{c_nC_{n}}e^{-i \overline{\omega_n}t}\Big|^2\ d t \,. \end{multline*} Combining these inequalities we obtain \begin{multline*} \int_{0}^{\infty} k(t)
\big(|u_1^{n_0}(t)|^2+|u_2^{n_0}(t)|^2\big)\ dt \\ \ge
\frac12\int_{0}^{\infty} k(t) \Big(\Big| \sum_{n=n_0}^{\infty}C_{n}e^{i\omega_{n} t}+\overline{C_{n}}e^{-i\overline{\omega_{n}}t}\Big|^2
-2\Big| \sum_{n=n_0}^\infty c_nC_{n}e^{i \omega_{n}t}+\overline{c_nC_{n}}e^{-i \overline{\omega_n}t}\Big|^2\Big)\ d t \\
+\frac12\int_{0}^{\infty} k(t) \Big(\Big| \sum_{n=n_0}^\infty d_nD_{n}e^{i \zeta_{n}t}+\overline{d_nD_{n}}e^{-i \overline{\zeta_n}t}\Big|^2
-4\Big| \sum_{n=n_0}^\infty D_{n}e^{i \zeta_{n}t}+\overline{D_{n}}e^{-i \overline{\zeta_n}t}\Big|^2\Big)\ d t \\
-2\int_{0}^{\infty} k(t) \Big|\sum_{n=n_0}^{\infty} R_{n} e^{r_n t}\Big|^2\ d t \,. \end{multline*} We now take $T>\frac{2\pi}{\gamma\sqrt{1-\varepsilon}}$ to estimate the first two integrals on the right-hand side. We introduce $\varepsilon'\in (0,\varepsilon)$ to choose suitably later. We also have $T>\frac{2\pi}{\gamma\sqrt{1-\varepsilon'}}$, so we can use \eqref{eq:u1-2u2} and \eqref{eq:U2D} respectively to obtain \begin{multline*}
\int_{0}^{\infty} k(t) \Big(\Big| \sum_{n=n_0}^{\infty}C_{n}e^{i\omega_{n} t}+\overline{C_{n}}e^{-i\overline{\omega_{n}}t}\Big|^2
-2\Big| \sum_{n=n_0}^\infty c_nC_{n}e^{i \omega_{n}t}+\overline{c_nC_{n}}e^{-i \overline{\omega_n}t}\Big|^2\Big)\ d t \\ \ge 2\pi T\sum_{n= n_0}^\infty \Bigg(\frac{1-\varepsilon'}{\pi^2+4T^2(\Im\omega_{n})^2} -\frac{4}{T^2\gamma^2}(1+\varepsilon') \Bigg)
(1+e^{-2\Im\omega_{n}T})|C_{n}|^2 \,, \end{multline*} \begin{multline*}
\int_{0}^{\infty} k(t) \Big(\Big| \sum_{n=n_0}^\infty d_nD_{n}e^{i \zeta_{n}t}+\overline{d_nD_{n}}e^{-i \overline{\zeta_n}t}\Big|^2
-4\Big| \sum_{n=n_0}^\infty D_{n}e^{i \zeta_{n}t}+\overline{D_{n}}e^{-i \overline{\zeta_n}t}\Big|^2\Big)\ d t \\ \ge 2\pi T\sum_{n= n_0}^\infty \Bigg(\frac{1-\varepsilon'}{\pi^2+4T^2(\Im\zeta_{n})^2} -\frac{4}{T^2\gamma^2}(1+\varepsilon') \Bigg)
(1+e^{-2\Im\zeta_{n}T})|d_nD_{n}|^2 \,. \end{multline*}
By \eqref{eq:hom2} we get $|\Im\zeta_n|\le\Im\omega_n$ for $n\ge n_0$ with $n_0$ sufficiently large. Hence \begin{equation*} \frac{e^{-2\Im\zeta_{n}T}}{\pi^2+4T^2(\Im\zeta_{n})^2} \ge \frac{e^{-2\Im\omega_{n}T}}{\pi^2+4T^2(\Im\omega_{n})^2} \qquad \forall n\ge n_0 \,. \end{equation*} Therefore \begin{multline*} \int_{0}^{\infty} k(t)
\big(|u_1^{n_0}(t)|^2+|u_2^{n_0}(t)|^2\big)\ dt \\ \ge \pi T\sum_{n= n_0}^\infty \Bigg(\frac{1-\varepsilon'}{\pi^2+4T^2(\Im\omega_{n})^2} -\frac{4}{T^2\gamma^2}(1+\varepsilon') \Bigg)
(1+e^{-2\Im\omega_{n}T})\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big) \\
-2\int_{0}^{\infty} k(t) \Big|\sum_{n=n_0}^{\infty} R_{n} e^{r_n t}\Big|^2\ d t \,. \end{multline*} Applying \eqref{le:U1R0} we obtain \begin{multline}\label{eq:Irn} \int_{0}^{\infty} k(t)
\big(|u_1^{n_0}(t)|^2+|u_2^{n_0}(t)|^2\big)\ dt \\ \ge \pi T\sum_{n= n_0}^\infty \Bigg(\frac{1-\varepsilon'}{\pi^2+4T^2(\Im\omega_{n})^2} -\frac{\varepsilon'}{\pi^2+T^2r_{n}^2} -\frac{4}{T^2\gamma^2}(1+\varepsilon') \Bigg)
(1+e^{-2\Im\omega_{n}T})\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big) \,. \end{multline} Now, we will choose $\varepsilon'\in (0,\varepsilon)$ such that for $n\ge n_0$ \begin{equation}\label{eq:TonT^2} \frac{1-\varepsilon'}{\pi^2+4T^2(\Im\omega_{n})^2} -\frac{\varepsilon'}{\pi^2+T^2r_{n}^2} \ge \frac{1-\varepsilon}{\pi^2+4T^2(\Im\omega_{n})^2} \,, \end{equation} that is \begin{equation*} \frac{\varepsilon-\varepsilon'}{\pi^2+4T^2(\Im\omega_{n})^2} -\frac{\varepsilon'}{\pi^2+T^2r_{n}^2}\ge0 \,, \end{equation*} \begin{equation*} \pi^2(\varepsilon-2\varepsilon') +T^2\big[(\varepsilon-\varepsilon')r_{n}^2 -4\varepsilon'(\Im\omega_{n})^2 \big]\ge0 \,. \end{equation*}
To this end, we need to have that \begin{equation}\label{eq:epsi'} \varepsilon-2\varepsilon'\ge0 \,, \qquad (\varepsilon-\varepsilon')r_{n}^2 -4\varepsilon'(\Im\omega_{n})^2 \ge0 \,. \end{equation} By \eqref{eq:hom2} for $n_0$ sufficiently large we have \begin{equation*} r_{n}^2\ge\frac{\chi^2}{2}\,, \qquad (\Im\omega_{n})^2\le\frac32\alpha^2\,. \end{equation*} Hence \begin{equation*} (\varepsilon-\varepsilon')r_{n}^2 -4\varepsilon'(\Im\omega_{n})^2 \ge (\varepsilon-\varepsilon')\frac{\chi^2}2 -6\varepsilon'\alpha^2 \,. \end{equation*}
Therefore taking \begin{equation*} \varepsilon'\le\min\Big\{\frac12,\frac{\chi^2}{\chi^2+12\alpha^2}\Big\}\varepsilon \,, \end{equation*} we deduce \eqref{eq:epsi'}, and consequently \eqref{eq:TonT^2}. So, from \eqref{eq:Irn} we have \begin{multline*} \int_{0}^{\infty} k(t)
\big(|u_1^{n_0}(t)|^2+|u_2^{n_0}(t)|^2\big)\ dt \\ \ge \pi T\sum_{n= n_0}^\infty \Bigg(\frac{1-\varepsilon}{\pi^2+4T^2(\Im\omega_{n})^2} -\frac{4}{T^2\gamma^2}(1+\varepsilon) \Bigg)
(1+e^{-2\Im\omega_{n}T})\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big) \,. \end{multline*} Since the previous inequality holds for any $\varepsilon\in(0,1)$, in particular it can be written for $\varepsilon'<\frac\varepsilon{2-\varepsilon}$, because this implies $\frac{1+\varepsilon'}{1-\varepsilon'}<\frac1{1-\varepsilon}$, and hence \begin{equation*} \frac{1-\varepsilon'}{\pi^2+4T^2(\Im\omega_{n})^2} -\frac{4}{T^2\gamma^2}(1+\varepsilon') \ge (1-\varepsilon')\Bigg(\frac{1}{\pi^2+4T^2(\Im\omega_{n})^2} -\frac{4}{T^2\gamma^2(1-\varepsilon)} \Bigg)\,. \end{equation*} Therefore, taking also into account that $ (\Im\omega_{n})^2<\alpha^2(1+\varepsilon), $ $n\ge n_0$, for $n_0$ large enough, we can write \begin{multline}\label{eq:u1+u20} \int_{0}^{\infty} k(t)
\big(|u_1^{n_0}(t)|^2+|u_2^{n_0}(t)|^2\big)\ dt \\ \ge \pi T(1-\varepsilon')\bigg(\frac{1}{\pi^2+4T^2\alpha^2(1+\varepsilon)} -\frac{4}{T^2\gamma^2(1-\varepsilon)}\bigg) \sum_{n= n_0}^\infty
(1+e^{-2\Im\omega_{n}T})\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big) \,. \end{multline} The constant \begin{equation*} \frac{1}{\pi^2+4T^2\alpha^2(1+\varepsilon)} -\frac{4}{T^2\gamma^2(1-\varepsilon)} \end{equation*} is positive if \begin{equation}\label{eq:const} T^2\big[\gamma^2(1-\varepsilon)-16\alpha^2(1+\varepsilon)\big] >4\pi^2\,. \end{equation} Since $\gamma>4\alpha$ we have $\gamma^2(1-\varepsilon)-16\alpha^2(1+\varepsilon)>0$ if $\varepsilon<\frac{\gamma^2-16\alpha^2}{\gamma^2+16\alpha^2}$. If we assume the more restrictive condition $T>\frac{2\pi}{\sqrt{\gamma^2(1-\varepsilon)-16\alpha^2(1+\varepsilon)}}$ with respect to that $T>\frac{2\pi}{\gamma\sqrt{1-\varepsilon}}$, then \eqref{eq:const} holds true. Finally, from \eqref{eq:u1+u20} and the definition \eqref{eq:u1+u21} of ${\cal I}_{n_0}$ we obtain \eqref{eq:u1+u2}.
\end{Proof}
We now observe that we can obtain a better estimate of the control time $T$
under an additional condition on the coefficients of the series. Assuming $|C_n|\le M |d_nD_{n}|$, we can follow the procedure sketched out at the end of Section \ref{se:outline} by using estimate \eqref{eq:f1+f2bis}. In particular, to evaluate the term
$\int_{0}^{\infty} k(t) | {\mathcal U}_{1}^C(t)|^2 d t$ we will employ the same trick used in \cite{LoretiSforza1}, giving first an estimate for $\int_{0}^{\infty} e^{2\alpha t} k(t) | {\mathcal U}_{1}^C(t)|^2 d t$ where $\displaystyle\alpha=\lim_{n\to\infty}{\Im}\omega_n$ and then multiplying by $e^{-2\alpha T}$ we will obtain the requested inequality.
\begin{theorem}\label{th:extracoe} Assume \begin{equation}\label{eq:extracoe}
|C_n|\le M |d_nD_{n}| \qquad\forall n\in{\mathbb N} \,. \end{equation} Then, for any $\varepsilon\in (0,1)$ and $T>\frac{2\pi}{\gamma\sqrt{1-\varepsilon}}$ there exist $n_0=n_0(\varepsilon)\in{\mathbb N}$, independent of $T$ and all coefficients of the series, and a constant $c(T,\varepsilon)>0$ such that \begin{multline}\label{eq:u1+u2bis}
\int_{0}^{\infty} k(t)\Big| \sum_{n=n_0}^{\infty}C_{n}e^{i\omega_{n} t}+\overline{C_{n}}e^{-i\overline{\omega_{n}}t}
+R_{n}e^{r_{n} t}+D_{n}e^{i\zeta_{n} t}+\overline{D_{n}}e^{-i\overline{\zeta_{n}}t}\Big|^2\ dt \\
+\int_{0}^{\infty} k(t)\Big| \sum_{n=n_0}^{\infty}d_nD_{n}e^{i\zeta_{n} t}+\overline{d_nD_{n}}e^{-i\overline{\zeta_{n}}t}
+c_nC_{n}e^{i\omega_{n} t}+\overline{c_nC_{n}}e^{-i\overline{\omega_{n}}t}\Big|^2\ dt \\ \ge c(T,\varepsilon)\sum_{n= n_0}^\infty
\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big) \,. \end{multline} \end{theorem} \begin{Proof} If $\displaystyle\alpha=\lim_{n\to\infty}{\Im}\omega_n$, see \eqref{eq:hom2}, since \begin{equation*}
\int_{0}^{\infty} e^{2\alpha t} k(t)\Big| \sum_{n=n_0}^\infty C_{n}e^{i \omega_{n}t}+\overline{C_{n}}e^{-i \overline{\omega_n}t}\Big|^2\ d t
=\int_{0}^{\infty} k(t)\Big| \sum_{n=n_0}^\infty C_{n}e^{i (\omega_{n}-i\alpha)t}+\overline{C_{n}}e^{-i \overline{(\omega_{n}-i\alpha)}t}\Big|^2\ d t \,, \end{equation*} thanks to \eqref{eq:Fn2'} we have \begin{multline*}
\int_{0}^{\infty} e^{2\alpha t} k(t)\Big| \sum_{n=n_0}^\infty C_{n}e^{i \omega_{n}t}+\overline{C_{n}}e^{-i \overline{\omega_n}t}\Big|^2\ d t \\ \ge 2\pi T\sum_{n=n_0}^\infty\bigg( \frac{1}{\pi^2+4T^2(\Im \omega_{n}-\alpha)^2}
-\frac{4}{T^2\gamma^2}(1+\varepsilon')\bigg) (1+e^{-2(\Im \omega_{n}-\alpha) T})|C_{n}|^2 \,, \end{multline*} where $\varepsilon'\in(0,\varepsilon)$ will be chosen later. Therefore, multiplying by $e^{-2\alpha T}$ and taking into account the definition \eqref{eq:k} of the function $k$, we get \begin{multline*}
\int_{0}^{\infty} k(t)\Big| \sum_{n=n_0}^\infty C_{n}e^{i \omega_{n}t}+\overline{C_{n}}e^{-i \overline{\omega_n}t}\Big|^2\ d t \\ \ge 2\pi Te^{-2\alpha T}\sum_{n=n_0}^\infty\bigg( \frac{1}{\pi^2+4T^2(\Im \omega_{n}-\alpha)^2}
-\frac{4}{T^2\gamma^2}(1+\varepsilon')\bigg) (1+e^{-2(\Im \omega_{n}-\alpha) T})|C_{n}|^2 \,. \end{multline*} Now, we can take $4(\Im \omega_{n}-\alpha)^2<\gamma^2\varepsilon/8$ for $n\ge n_0$ and $1+\varepsilon'<\frac1{1-\varepsilon/2}$ for $\varepsilon'<\frac\varepsilon{2-\varepsilon}$, to have \begin{multline*} \frac{1}{\pi^2+4T^2(\Im \omega_{n}-\alpha)^2}-\frac{4}{T^2\gamma^2}(1+\varepsilon') \\ > \frac{1}{\pi^2+T^2\gamma^2\varepsilon/8}-\frac{4}{T^2\gamma^2(1-\varepsilon/2)} =\frac{T^2\gamma^2(1-\varepsilon)-4\pi^2}{(\pi^2+T^2\gamma^2\varepsilon/8)T^2\gamma^2(1-\varepsilon/2)} \end{multline*} and $T^2\gamma^2(1-\varepsilon)-4\pi^2>0$ for $T>\frac{2\pi}{\gamma\sqrt{1-\varepsilon}}$. So, we get \begin{multline}\label{eq:onlyC1}
\int_{0}^{\infty} k(t)\Big| \sum_{n=n_0}^\infty C_{n}e^{i \omega_{n}t}+\overline{C_{n}}e^{-i \overline{\omega_n}t}\Big|^2\ d t \\
\ge 2\pi Te^{-2\alpha T}\frac{T^2\gamma^2(1-\varepsilon)-4\pi^2}{(\pi^2+T^2\gamma^2\varepsilon/8)T^2\gamma^2(1-\varepsilon/2)}\sum_{n=n_0}^\infty (1+e^{-2(\Im \omega_{n}-\alpha) T})|C_{n}|^2 \,. \end{multline} On the other hand, from \eqref{eq:Fn2''} it follows \begin{multline*}
\int_{0}^{\infty} k(t)\Big| \sum_{n=n_0}^\infty c_nC_{n}e^{i \omega_{n}t}+\overline{c_nC_{n}}e^{-i \overline{\omega_n}t}\Big|^2\ d t \\ \le 2\pi T\sum_{n=n_0}^\infty\bigg( \frac{1}{\pi^2+4T^2(\Im \omega_{n})^2} +
\frac{4}{T^2\gamma^2}(1+\varepsilon')\bigg) (1+e^{-2\Im \omega_{n} T})|c_nC_{n}|^2 \\
\le 2\pi T\sum_{n=n_0}^\infty M|c_n|^2\bigg( \frac{1}{\pi^2+4T^2(\Im \zeta_{n})^2} +
\frac{4}{T^2\gamma^2}(1+\varepsilon')\bigg) (1+e^{-2\Im \zeta_{n} T})|d_nD_{n}|^2 \,, \end{multline*}
thanks also to $\Im\omega_n\ge|\Im\zeta_n|$ and $|C_n|\le M |d_nD_{n}|$. Moreover, again by \eqref{eq:Fn2''} and the previous inequality we have \begin{multline*} \int_{0}^{\infty} k(t) \Big(
2\Big| \sum_{n=n_0}^\infty D_{n}e^{i \zeta_{n}t}+\overline{D_{n}}e^{-i \overline{\zeta_n}t}\Big|^2
+\Big| \sum_{n=n_0}^\infty c_nC_{n}e^{i \omega_{n}t}+\overline{c_nC_{n}}e^{-i \overline{\omega_n}t}\Big|^2\Big) \ d t \\
\le 2\pi T\sum_{n=n_0}^\infty \Big(\frac2{|d_n|^2}+M|c_n|^2\Big)\bigg( \frac{1}{\pi^2+4T^2(\Im \zeta_{n})^2} +
\frac{4}{T^2\gamma^2}(1+\varepsilon')\bigg) (1+e^{-2\Im \zeta_{n} T})|d_nD_{n}|^2 \,. \end{multline*}
Choosing $n_0$ sufficiently large such that $\frac2{|d_n|^2}+M|c_n|^2\le\varepsilon'$ for any $n\ge n_0$, from the above estimate we deduce \begin{multline}\label{eq:U1D2C} \int_{0}^{\infty} k(t) \Big(
2\Big| \sum_{n=n_0}^\infty D_{n}e^{i \zeta_{n}t}+\overline{D_{n}}e^{-i \overline{\zeta_n}t}\Big|^2
+\Big| \sum_{n=n_0}^\infty c_nC_{n}e^{i \omega_{n}t}+\overline{c_nC_{n}}e^{-i \overline{\omega_n}t}\Big|^2\Big) \ d t \\ \le 2\pi T\varepsilon'\sum_{n=n_0}^\infty \bigg( \frac{1}{\pi^2+4T^2(\Im \zeta_{n})^2} +
\frac{4}{T^2\gamma^2}(1+\varepsilon')\bigg) (1+e^{-2\Im \zeta_{n} T})|d_nD_{n}|^2 \,. \end{multline}
In addition, from \eqref{le:U1R0}, using again $|C_n|\le M |d_nD_{n}|$ and \eqref{eq:hom2} we get \begin{equation}\label{eq:U1R1}
\int_{0}^{\infty} k(t) \Big|\sum_{n=n_0}^{\infty} R_{n} e^{r_n t}\Big|^2\ d t \le
\pi T \varepsilon'
\sum_{n= n_0}^\infty\frac{|d_nD_{n}|^2}{\pi^2+T^2r_{n}^2} \le
\pi T \varepsilon'
\sum_{n= n_0}^\infty\frac{|d_nD_{n}|^2}{\pi^2+4T^2(\Im\zeta_{n})^2} \,. \end{equation} Combining \eqref{eq:U1D2C} and \eqref{eq:U1R1} (with $\varepsilon'$ replaced by $\varepsilon'/2$) we obtain \begin{multline}\label{eq:D1C2R} \int_{0}^{\infty} k(t) \Big(
2\Big| \sum_{n=n_0}^\infty D_{n}e^{i \zeta_{n}t}+\overline{D_{n}}e^{-i \overline{\zeta_n}t}\Big|^2
+\Big| \sum_{n=n_0}^\infty c_nC_{n}e^{i \omega_{n}t}+\overline{c_nC_{n}}e^{-i \overline{\omega_n}t}\Big|^2
+2 \Big|\sum_{n=n_0}^{\infty} R_{n} e^{r_n t}\Big|^2\Big) \ d t \\ \le 2\pi T\varepsilon'\sum_{n=n_0}^\infty \bigg( \frac{1}{\pi^2+4T^2(\Im \zeta_{n})^2} +
\frac{4}{T^2\gamma^2}(1+\varepsilon')\bigg) (1+e^{-2\Im \zeta_{n} T})|d_nD_{n}|^2 \,. \end{multline} In virtue of \eqref{eq:Fn2'} we get \begin{multline*}
\int_{0}^{\infty} k(t) \Big| \sum_{n=n_0}^\infty d_nD_{n}e^{i \zeta_{n}t}+\overline{d_nD_{n}}e^{-i \overline{\zeta_n}t}\Big|^2 \ d t \\ \ge 2\pi T\sum_{n= n_0}^\infty \Bigg(\frac{1}{\pi^2+4T^2(\Im\zeta_{n})^2} -\frac{4}{T^2\gamma^2}(1+\varepsilon') \Bigg)
(1+e^{-2\Im\zeta_{n}T})|d_nD_{n}|^2 \,. \end{multline*} From the above formula and \eqref{eq:D1C2R}, taking $\varepsilon'\le\varepsilon/3$ but writing again $\varepsilon'$ instead of $\varepsilon$, we have \begin{multline*}
\int_{0}^{\infty} k(t) \Big| \sum_{n=n_0}^\infty d_nD_{n}e^{i \zeta_{n}t}+\overline{d_nD_{n}}e^{-i \overline{\zeta_n}t}\Big|^2 \ d t \\ -2 \int_{0}^{\infty} k(t) \Big(
2\Big| \sum_{n=n_0}^\infty D_{n}e^{i \zeta_{n}t}+\overline{D_{n}}e^{-i \overline{\zeta_n}t}\Big|^2
+\Big| \sum_{n=n_0}^\infty c_nC_{n}e^{i \omega_{n}t}+\overline{c_nC_{n}}e^{-i \overline{\omega_n}t}\Big|^2
+2 \Big|\sum_{n=n_0}^{\infty} R_{n} e^{r_n t}\Big|^2\Big) \ d t \\ \ge 2\pi T\sum_{n= n_0}^\infty \Bigg(\frac{1-\varepsilon'}{\pi^2+4T^2(\Im\zeta_{n})^2} -\frac{4}{T^2\gamma^2}(1+\varepsilon') \Bigg)
(1+e^{-2\Im\zeta_{n}T})|d_nD_{n}|^2 \,. \end{multline*} Taking $4(\Im\zeta_{n})^2<\gamma^2\varepsilon/8$ for $n\ge n_0$ and $\frac{1+\varepsilon'}{1-\varepsilon'}<\frac1{1-\varepsilon/2}$ for $\varepsilon'<\frac\varepsilon{4-\varepsilon}$ yields \begin{multline*} \frac{1-\varepsilon'}{\pi^2+4T^2(\Im\zeta_{n})^2} -\frac{4}{T^2\gamma^2}(1+\varepsilon') =(1-\varepsilon') \bigg(\frac{1}{\pi^2+4T^2(\Im\zeta_{n})^2} -\frac{4(1+\varepsilon')}{T^2\gamma^2(1-\varepsilon')} \bigg) \\ \ge (1-\varepsilon') \bigg(\frac{1}{\pi^2+T^2\gamma^2\varepsilon/8} -\frac{4}{T^2\gamma^2(1-\varepsilon/2)} \bigg) = (1-\varepsilon') \bigg(\frac{T^2\gamma^2(1-\varepsilon)-4\pi^2}{(\pi^2+T^2\gamma^2\varepsilon/8)T^2\gamma^2(1-\varepsilon/2)} \bigg) \,. \end{multline*} Therefore, for $T>\frac{2\pi}{\gamma\sqrt{1-\varepsilon}}$ we obtain \begin{multline*}
\int_{0}^{\infty} k(t) \Big| \sum_{n=n_0}^\infty d_nD_{n}e^{i \zeta_{n}t}+\overline{d_nD_{n}}e^{-i \overline{\zeta_n}t}\Big|^2 \ d t \\ -2 \int_{0}^{\infty} k(t) \Big(
2\Big| \sum_{n=n_0}^\infty D_{n}e^{i \zeta_{n}t}+\overline{D_{n}}e^{-i \overline{\zeta_n}t}\Big|^2
+\Big| \sum_{n=n_0}^\infty c_nC_{n}e^{i \omega_{n}t}+\overline{c_nC_{n}}e^{-i \overline{\omega_n}t}\Big|^2
+2 \Big|\sum_{n=n_0}^{\infty} R_{n} e^{r_n t}\Big|^2\Big) \ d t \\ \ge 2\pi T(1-\varepsilon) \bigg(\frac{T^2\gamma^2(1-\varepsilon)-4\pi^2}{(\pi^2+T^2\gamma^2\varepsilon/8)T^2\gamma^2(1-\varepsilon/2)} \bigg) \sum_{n= n_0}^\infty
(1+e^{-2\Im\zeta_{n}T})|d_nD_{n}|^2 \,. \end{multline*} In conclusion, for any $T>\frac{2\pi}{\gamma\sqrt{1-\varepsilon}}$, combining the previous estimate with \eqref{eq:onlyC1} gives \begin{multline*} \int_{0}^{\infty} k(t)
\big(|u_1^{n_0}(t)|^2+|u_2^{n_0}(t)|^2\big)\ dt \\ \ge \pi T\min\{e^{-2\alpha T},(1-\varepsilon)\} \bigg(\frac{T^2\gamma^2(1-\varepsilon)-4\pi^2}{(\pi^2+T^2\gamma^2\varepsilon/8)T^2\gamma^2(1-\varepsilon/2)} \bigg)
\sum_{n=n_0}^\infty\Big(|C_{n}|^2 +|d_nD_{n}|^2\Big) \,, \end{multline*} that is \eqref{eq:u1+u2bis}. \end{Proof}
\subsection{Direct inequality} As for the inverse inequality, to prove direct estimates we need to introduce an auxiliary function. Let $T>0$ and define \begin{equation}\label{eq:kcos} k^*(t):=\left \{\begin{array}{l}
\cos \frac{\pi t}{2T}\,\qquad\qquad \mbox{if}\ |t|\le T\,,\\ \\
0\,\qquad\qquad\quad\ \ \ \ \mbox{if}\ |t|>T\,. \end{array}\right . \end{equation} For the sake of completeness, we list some standard properties of $k^*$ in the following lemma. \begin{lemma} \label{th:k} Set \begin{equation}\label{eqn:K*} K^*(u):=\frac{4T\pi}{\pi^2-4T^2u^2}\,,\qquad u\in {\mathbb C}\,, \end{equation} the following properties hold for any $u\in {\mathbb C}$ \begin{equation}\label{eqn:k1} \int_{-\infty}^{\infty} k^*(t)e^{iu t}dt=\cos(uT)K^*(u)\,, \end{equation} \begin{equation}\label{eqn:k2bis} \overline{K^*(u)}=K^*(\overline{u})\,, \quad
\big|K^*(u)\big|=\big|K^*(\overline{u})\big|. \end{equation} Set $K_{T}(u)=\frac{T\pi}{\pi^2-T^2u^2}$ we have \begin{equation}\label{eqn:k3} K^*(u)=2K_{2T}(u)\,. \end{equation} Moreover for any $z_i,w_i\in {\mathbb C}$, $i=1,2$, one has \begin{multline}\label{eq:cosbiss} \int_{-\infty}^{\infty} k^*(t)\Re(z_1e^{iw_1 t})\Re(z_2e^{iw_2t})dt \\ =\frac12 \Re\Big(z_1z_2\cos((w_1+w_2) T)K(w_1+w_2) +z_1\overline{z_2}\cos((w_1-\overline{w_2}) T)K(w_1-\overline{w_2})\Big)\,. \end{multline}\end{lemma} From now on we will denote with $c(T)$ a positive constant depending on $T$. \begin{proposition}\label{pr:Fnd} Let $\gamma>0$. Suppose that $\{\sigma_{n}\}$ is a complex number sequence satisfying \begin{equation*} \liminf_{n\to\infty}\big(\Re \sigma_{n+1}-\Re \sigma_{n}\big)=\gamma\,, \qquad \{{\Im}\sigma_n\} \quad \mbox{bounded}. \end{equation*} Then for any complex number sequence $\{F_n\}$ with
$\sum_{n=1}^\infty\ |F_{n}|^2<+\infty$, $\varepsilon\in (0,1)$ and $T>\frac{\pi}{\gamma\sqrt{1-\varepsilon}}$
there exist $c(T)>0$ and $n_0=n_0(\varepsilon)\in{\mathbb N}$ independent of $T$ and $F_n$ such that \begin{equation}\label{eq:Fndir}
\int_{-\infty}^{\infty} k^*(t)\Big| \sum_{n=n_0}^\infty F_{n}e^{i \sigma_{n}t}+\overline{F_{n}}e^{-i \overline{\sigma_n}t}\Big|^2\ d t
\le c(T)\sum_{n=n_0}^\infty|F_{n}|^2 \,. \end{equation} \end{proposition}
\begin{Proof} Let us first observe that \begin{equation*}
\Big| \sum_{n=n_0}^\infty F_{n}e^{i\sigma_{n}t}
+\overline{F_{n}}e^{-i \overline{\sigma_{n}}t}\Big|^2
=4\sum_{n, m=n_0}^\infty \Re\big(F_{n}e^{i\sigma_{n}t}\big)\Re\big(F_{m}e^{i\sigma_{m}t}\big) \,, \end{equation*} where the index $n_0\in{\mathbb N}$ depending on $\varepsilon$ will be chosen later. From \eqref{eq:cosbiss} we have \begin{multline*}
\int_{-\infty}^{\infty} k^*(t)\Big| \sum_{n=n_0}^\infty F_{n}e^{i\sigma_{n}t}
+\overline{F_{n}}e^{-i \overline{\sigma_{n}}t}\Big|^2\ d t \\ =2\sum_{n, m=n_0}^\infty \Re \Big[F_{n}\overline{F_{m}} \cos(( \sigma_{n}-\overline{\sigma_m}) T) K^*( \sigma_{n}-\overline{\sigma_m}) +F_{n}F_{m} \cos(( \sigma_{n}+\sigma_{m}) T) K^*( \sigma_{n}+\sigma_{m})\Big] \,. \end{multline*}
Applying the elementary estimates $\Re z\le|z|$ and
$|\cos z|\le \cosh (\Im z)$, $z\in{\mathbb C}$, we obtain \begin{multline*}
\int_{-\infty}^{\infty} k^*(t)\Big| \sum_{n=n_0}^\infty F_{n}e^{i \sigma_{n}t}+\overline{F_{n}}e^{-i\overline{ \sigma_n}t}\Big|^2\ d t \\
\le 2\sum_{n, m=n_0}^\infty |F_{n}| |F_{m}| \cosh(\Im(\sigma_{n}+ \sigma_{m}) T)
\big[|K^*( \sigma_{n}-\overline{\sigma_m})|
+|K^*( \sigma_{n}+\sigma_{m})|\big] \,. \end{multline*} Since the sequence $\{{\Im}\sigma_n\}$ is bounded we have \begin{equation*}
\cosh(\Im(\sigma_{n}+ \sigma_{m}) T)\le e^{2T\sup|\Im\sigma_n|} \qquad \forall n,m\in{\mathbb N}\,. \end{equation*} Hence \begin{multline*}
\int_{-\infty}^{\infty} k^*(t)\Big| \sum_{n=n_0}^\infty F_{n}e^{i \sigma_{n}t}+\overline{F_{n}}e^{-i\overline{ \sigma_n}t}\Big|^2\ d t \\
\le 2e^{2T\sup|\Im\sigma_n|}\sum_{n, m=n_0}^\infty |F_{n}| |F_{m}|
\big[|K^*( \sigma_{n}-\overline{\sigma_m})|
+|K^*( \sigma_{n}+\sigma_{m})|\big] \,. \end{multline*} Thanks to \eqref{eqn:k2bis} we get $
|K^*( \sigma_n-\overline{\sigma_m})|=|K^*( \sigma_m-\overline{\sigma_n})|\,. $ Therefore \begin{multline*}
\int_{-\infty}^{\infty} k^*(t)\Big| \sum_{n=n_0}^\infty F_{n}e^{i \sigma_{n}t}+\overline{F_{n}}e^{-i\overline{ \sigma_n}t}\Big|^2\ d t \\
\le 2e^{2T\sup|\Im\sigma_n|}\sum_{n=n_0}^\infty |F_{n}|^2
\sum_{ m=n_0}^\infty \big[|K^*( \sigma_{n}-\overline{\sigma_m})|
+|K^*( \sigma_{n}+\sigma_{m})|\big] \,. \end{multline*} Since \eqref{eqn:K*} gives \begin{equation*} K^*( \sigma_{n}-\overline{\sigma_n}) =\frac{4\pi T}{\pi^2+16T^2(\Im\sigma_{n})^2} \le\frac{4 T}{\pi} \,, \end{equation*} it follows that \begin{multline}\label{eq:Fndir0}
\int_{-\infty}^{\infty} k(t)\Big| \sum_{n=n_0}^\infty F_{n}e^{i
\sigma_{n}t}+\overline{F_{n}}e^{-i \overline{\sigma_n}t}\Big|^2\ d t \le
\frac8{\pi} e^{2T\sup|\Im\sigma_n|}T\sum_{n=n_0}^\infty\ |F_{n}|^2 \\
+2e^{2T\sup|\Im\sigma_n|}\sum_{n=n_0}^\infty |F_{n}|^2
\Big[\sum_{\substack{m=n_0\\ m\not=n}}^\infty |K^*( \sigma_{n}-\overline{\sigma_m})| +\sum_{m=n_0}^\infty K^*( \sigma_{n}+\sigma_{m})\Big] \,. \end{multline} Note that by \eqref{eqn:k3} we can apply Lemma \ref{le:stimaK}: for any $\varepsilon\in (0,1)$ and $2T>\frac{2\pi}{\gamma\sqrt{1-\varepsilon}}$ there exists $n_0\in{\mathbb N}$ such that \begin{equation*}
\sum_{\substack{m=n_0\\ m\not=n}}^\infty |K^*( \sigma_{n}-\overline{\sigma_m})| +\sum_{m=n_0}^\infty K^*( \sigma_{n}+\sigma_{m}) \le \frac{{2\pi}}{T\gamma^2(1-\varepsilon)} \Big(1+\sum_{n=1}^\infty\frac{1}{4n^{2}-1}\Big) \,. \end{equation*} Substituting the previous estimate into \eqref{eq:Fndir0} gives \eqref{eq:Fndir}. \end{Proof} \begin{proposition}\label{pr:Rnd} For any $n_0\in{\mathbb N}$, $n_0\ge n'$, and $T>0$ there exists $c(T)>0$ such that \begin{equation}\label{le:U1R}
\int_{-\infty}^\infty k^*(t) \bigg|\sum_{n=n_0}^{\infty} R_{n} e^{r_n t}\bigg|^2\ dt \le c(T) \sum_{n= n_0}^\infty
\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big) \,. \end{equation} \end{proposition} \begin{Proof} Fixed $n_0\in{\mathbb N}$, $n_0\ge n'$, we observe that \eqref{eqn:k1} leads to \begin{multline*}
\int_{-\infty}^\infty k^*(t) \bigg|\sum_{n=n_0}^{\infty} R_{n} e^{r_n t}\bigg|^2\ dt= \sum_{n,m=n_0}^{\infty} R_{n}R_{m} \int_{\infty}^\infty k^*(t) e^{(r_n+r_m) t}\ dt \\ = \sum_{n, m=n_0}^{\infty} R_{n} R_{m} \cosh((r_n+r_m )T)K^*(ir_n+ir_m)\,. \end{multline*}
By the definition \eqref{eqn:K*} of $K^*$ we have \begin{equation*} K^*(ir_n+ir_m)= \frac{4\pi T}{\pi^2+4T^2(r_n+r_m)^2} \le \frac{4 T}{\pi} \,. \end{equation*} In addition, since the sequence $\{r_n\}$ is bounded we have \begin{equation*}
\cosh((r_{n}+ r_{m}) T)\le e^{2T\sup|r_n|} \qquad \forall n,m\in{\mathbb N}\,. \end{equation*} Consequently, \begin{equation*}
\int_{-\infty}^\infty k^*(t) \bigg|\sum_{n=n_0}^{\infty} R_{n} e^{r_n t}\bigg|^2\ dt \le
\frac{4 T}{\pi} e^{2T\sup|r_n|}\sum_{n, m=n_0}^{\infty} |R_{n}| |R_{m}| \,. \end{equation*} Since $n_0\ge n'$, by \eqref{eq:hom3} we have that \begin{equation*}
\int_{-\infty}^\infty k^*(t) \bigg|\sum_{n=n_0}^{\infty} R_{n} e^{r_n t}\bigg|^2\ dt \le
\frac{4 T}{\pi} e^{2T\sup|r_n|}\sum_{n, m= n_0}^{\infty}\frac{\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big)^{1/2}}{m^{\nu}}\ \frac{\Big(|C_{m}|^2+ |d_mD_{m}|^2\Big)^{1/2}}{n^{\nu}} \,. \end{equation*} Moreover \begin{multline*} \sum_{n, m= n_0}^{\infty}
\frac{\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big)^{1/2}}{m^{\nu}}\ \frac{\Big(|C_{m}|^2+ |d_mD_{m}|^2\Big)^{1/2}}{n^{\nu}} \\ \le \frac12\sum_{ m= n_0}^{\infty}\frac{1}{m^{2\nu}} \sum_{ n= n_0}^{\infty}
\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big) +\frac12\sum_{ n= n_0}^{\infty}\frac{1}{n^{2\nu}} \sum_{ m= n_0}^{\infty}
\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big) \\ = \sum_{ n= 1}^{\infty}\frac{1}{n^{2\nu}} \sum_{ n= n_0}^{\infty}
\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big) \,. \end{multline*} Combining these inequalities we conclude that \eqref{le:U1R} is proved. \end{Proof} \begin{theorem}\label{th:Diretta} For any $\varepsilon\in (0,1)$ and $T>\frac{\pi}{\gamma\sqrt{1-\varepsilon}}$ there exist $n_0=n_0(\varepsilon)\in{\mathbb N}$ and $c(T)>0$ such that \begin{multline}\label{eq:Diretta}
\int_{-T}^{T} \bigg| \sum_{n=n_0}^{\infty}C_{n}e^{i\omega_{n} t}+\overline{C_{n}}e^{-i\overline{\omega_{n}}t}
+R_{n}e^{r_{n} t}+D_{n}e^{i\zeta_{n} t}+\overline{D_{n}}e^{-i\overline{\zeta_{n}}t}\bigg|^2\ dt \\
+\int_{-T}^{T} \bigg| \sum_{n=n_0}^{\infty}d_nD_{n}e^{i\zeta_{n} t}+\overline{d_nD_{n}}e^{-i\overline{\zeta_{n}}t}
+c_nC_{n}e^{i\omega_{n} t}+\overline{c_nC_{n}}e^{-i\overline{\omega_{n}}t}\bigg|^2\ dt \\ \le c(T)\sum_{ n= n_0}^{\infty}
\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big) \,. \end{multline} \end{theorem} \begin{Proof} Since the function $k^*(t)$ is positive, for $n_0\in{\mathbb N}$ to be chosen later we have \begin{multline*} \int_{-\infty}^{\infty} k^*(t)
\bigg|\sum_{n=n_0}^{\infty}C_{n}e^{i\omega_{n} t}+\overline{C_{n}}e^{-i\overline{\omega_{n}}t}
+R_{n}e^{r_{n} t}+D_{n}e^{i\zeta_{n} t}+\overline{D_{n}}e^{-i\overline{\zeta_{n}}t}\bigg|^2\ dt \\
\le 4\int_{-\infty}^{\infty} k^*(t)\bigg|\sum_{n=n_0}^\infty C_{n}e^{i\omega_{n}t}
+\overline{C_n}e^{-i\overline{\omega_n}t}\bigg|^2\ dt
+4\int_{-\infty}^{\infty} k^*(t)\bigg|\sum_{n=n_0}^\infty R_{n}e^{r_{n}t}\bigg|^2\ dt \\
+4\int_{-\infty}^{\infty} k^*(t)\bigg|\sum_{n=n_0}^\infty D_{n}e^{i\zeta_{n} t}+\overline{D_{n}}e^{-i\overline{\zeta_{n}}t}\bigg|^2\ dt\,. \end{multline*} We can apply Proposition \ref{pr:Fnd} to the first term and to the third one and Proposition \ref{pr:Rnd} to the second term. Therefore, fixed $\varepsilon\in (0,1)$ and $T>\frac{\pi}{\gamma\sqrt{1-\varepsilon}}$ there exists $n_0=n_0(\varepsilon)\in{\mathbb N}$ such that, thanks to inequalities \eqref{eq:Fndir}--\eqref{le:U1R} and in view also of \eqref{eq:cndn}, we get \begin{multline}\label{eq:firstd} \int_{-\infty}^{\infty} k^*(t)
\bigg|\sum_{n=n_0}^{\infty}C_{n}e^{i\omega_{n} t}+\overline{C_{n}}e^{-i\overline{\omega_{n}}t}
+R_{n}e^{r_{n} t}+D_{n}e^{i\zeta_{n} t}+\overline{D_{n}}e^{-i\overline{\zeta_{n}}t}\bigg|^2\ dt \\ \le c(T) \sum_{n= n_0}^\infty
\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big) \,. \end{multline} Moreover, in a similar way applying again Proposition \ref{pr:Fnd} and taking into account \eqref{eq:cndn} we have \begin{multline*} \int_{-\infty}^{\infty} k^*(t)
\bigg| \sum_{n=n_0}^{\infty}d_nD_{n}e^{i\zeta_{n} t}+\overline{d_nD_{n}}e^{-i\overline{\zeta_{n}}t}
+c_nC_{n}e^{i\omega_{n} t}+\overline{c_nC_{n}}e^{-i\overline{\omega_{n}}t}\bigg|^2\ dt \\ \le c(T) \sum_{n= n_0}^\infty
\Big( |d_nD_{n}|^2+|C_{n}|^2\Big) \,. \end{multline*} Combining \eqref{eq:firstd} with the above inequality and recalling the notation \eqref{eq:notation} yields \begin{equation*} \int_{-\infty}^{\infty} k^*(t)
\big(|u_1^{n_0}(t)|^2+|u_2^{n_0}(t)|^2\big)\ dt \le c(T) \sum_{n= n_0}^\infty
\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big) \,. \end{equation*} Now, we can consider the last inequality with the function $k^*$ replaced by the analogous one relative to $2T$ instead of $T$. So, taking into account (\ref{eq:kcos}), we get \begin{equation*} \int_{-2T}^{2T} \cos \frac{\pi t}{4T}
\big(|u_1^{n_0}(t)|^2+|u_2^{n_0}(t)|^2\big)\ dt \le c(2T) \sum_{n= n_0}^\infty
\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big), \end{equation*} whence, thanks to $ \cos \frac{\pi t}{4T}\ge \frac1{\sqrt2} $
for $|t|\le T$, it follows \begin{equation*}
\int_{-T}^{T} \big(|u_1^{n_0}(t)|^2+|u_2^{n_0}(t)|^2\big)\ dt \le \sqrt2c(2T)\sum_{n= n_0}^\infty
\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big)\,. \end{equation*} This completes the proof. \end{Proof} Based on the approach performed in \cite{Ha}, the next result states that we can recover the finite number of missing terms in the inverse and direct estimates. We omit the proof, because it may be proved in much the same way as Proposition 5.8 and Proposition 5.20 of \cite{LoretiSforza3}. We advise the reader to keep in mind formulas \eqref{eq:vsum1} and \eqref{eq:notation}.
\begin{proposition}\label{pr:haraux-inv} Let $\{\omega_n\}_{n\in{\mathbb N}}$, $\{r_n\}_{n\in{\mathbb N}}$ and $\{\zeta_{n}\}_{n\in{\mathbb N}}$ be sequences of pairwise distinct numbers such that $\omega_n\not= \zeta_m$, $\omega_n\not=\overline{\zeta_m}$, $r_n\not= i\omega_m$, $r_n\not= i\zeta_m$, $r_n\not=-\eta$, $\zeta_{n}\not=0$, for any $n\,,m\in{\mathbb N}$, and \begin{equation}\label{eq:ha1}
\lim_{n\to\infty}|\omega_n|=\lim_{n\to\infty}|\zeta_{n}|=+\infty\,. \end{equation} Assume that there exists $n_0\in{\mathbb N}$ such that \begin{equation*}
\int_{0}^{T} \big(|u_1^{n_0}(t)|^2+|u_2^{n_0}(t)|^2\big)\ dt \asymp \sum_{ n= n_0}^{\infty}
\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big)\,. \end{equation*} Then, for any sequences $\{C_n\}$, $\{R_n\}$, $\{D_n\}$ and $\mathcal {E}\in{\mathbb R}$ we have \begin{equation}\label{eq:haraux-inv22}
\int_{0}^{T} \big(|u_1(t)|^2+|u_2(t)|^2\big)\ dt \asymp \sum_{ n= 1}^{\infty}
\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big) +|\mathcal {E}|^2 \,. \end{equation} \end{proposition}
\subsection{Inverse and direct inequalities}
We recall that \begin{equation*} \begin{split} u_1(t) &=\sum_{n=1}^{\infty}\Big(C_{n}e^{i\omega_{n} t}+\overline{C_{n}}e^{-i\overline{\omega_{n}}t} +R_{n}e^{r_{n} t}+D_{n}e^{i\zeta_{n} t}+\overline{D_{n}}e^{-i\overline{\zeta_{n}}t}\Big) \,, \\ u_2(t) &=\sum_{n=1}^{\infty} \Big(d_nD_{n}e^{i\zeta_{n} t}+\overline{d_nD_{n}}e^{-i\overline{\zeta_{n}}t} +c_nC_{n}e^{i\omega_{n} t}+\overline{c_nC_{n}}e^{-i\overline{\omega_{n}}t}\Big) +\mathcal {E} e^{-\eta t} \,, \end{split} \end{equation*} where \begin{equation}\label{eq:mathcalE}
|\mathcal {E}|^2\le M \sum_{ n= 1}^{\infty}
\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big), \qquad (M>0) \,. \end{equation}
\begin{theorem}\label{th:inv.ingham1} Let $\{\omega_n\}_{n\in{\mathbb N}}$, $\{r_n\}_{n\in{\mathbb N}}$ and $\{\zeta_{n}\}_{n\in{\mathbb N}}$ be sequences of pairwise distinct numbers such that $\omega_n\not= \zeta_m$, $\omega_n\not=\overline{\zeta_m}$, $r_n\not= i\omega_m$, $r_n\not= i\zeta_m$, $r_n\not=-\eta$, $\zeta_{n}\not=0$, for any $n\,,m\in{\mathbb N}$. Assume that there exist $\gamma>0$, $\alpha,\chi\in{\mathbb R}$, $n'\in{\mathbb N}$, $\mu>0$, $\nu> 1/2$, such that \begin{equation*} \liminf_{n\to\infty}({\Re}\omega_{n+1}-{\Re}\omega_{n})=\liminf_{n\to\infty}({\Re}\zeta_{n+1}-{\Re} \zeta_{n})=\gamma\,, \end{equation*} \begin{equation*} \begin{split} \lim_{n\to\infty}{\Im}\omega_n&=\alpha>0 \,, \\ \lim_{n\to\infty}r_n&=\chi<0\,, \\ \lim_{n\to\infty}\Im \zeta_{n}&=0\,, \end{split} \end{equation*} \begin{equation*}
|d_n|\asymp|\zeta_n| \,, \qquad
|c_n|\le\frac{M}{|\omega_n|}\,, \end{equation*} \begin{equation*}
|R_n|\le \frac{\mu}{n^{\nu}}\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big)^{1/2}\,\quad\forall\ n\ge n'\,, \qquad
|R_n|\le \mu\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big)^{1/2}\,\quad\forall\ n\le n'\,. \end{equation*} Then, for $\gamma>4\alpha$ and $T>\frac{2\pi}{\sqrt{\gamma^2-16\alpha^2}}$ we have
\begin{equation}\label{eq:inv.ingham}
\int_{0}^{T} \big(|u_1(t)|^2+|u_2(t)|^2\big)\ dt
\asymp \sum_{ n= 1}^{\infty}
\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big) \,. \end{equation} \end{theorem} \begin{Proof} Since $T>\frac{2\pi}{\sqrt{\gamma^2-16\alpha^2}}$, there exists $0<\varepsilon<1$ such that $T>\frac{2\pi}{\sqrt{\gamma^2(1-\varepsilon)-16\alpha^2(1+\varepsilon)}}$. Therefore, thanks to Theorems \ref{th:gamma>4alpha} and \ref{th:Diretta} we are able to employ Proposition \ref{pr:haraux-inv} obtaining
\begin{equation*}
\int_{0}^{T} \big(|u_1(t)|^2+|u_2(t)|^2\big)\ dt
\asymp \sum_{ n= 1}^{\infty}
\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big) +|\mathcal {E}|^2 \,. \end{equation*}
Finally, by \eqref{eq:mathcalE} we can get rid of the term $|\mathcal {E}|^2$ in the previous estimates, and hence the proof is complete. \end{Proof}
If we assume the condition $|C_n|\le M |d_nD_{n}|$ on the coefficients of the series
instead of $\gamma>4\alpha$, then we can make use of Theorem \ref{th:extracoe} instead of Theorem \ref{th:gamma>4alpha}, obtaining the observability inequalities with a better estimate for the control time: $T>\frac{2\pi}{\gamma}$. Precisely, the following result holds.
\begin{theorem}\label{th:inv.ingham11} Let assume the hypotheses of Theorem \ref{th:inv.ingham1} and the condition \begin{equation}
|C_n|\le M |d_nD_{n}| \,. \end{equation} Then, for $T>\frac{2\pi}{\gamma}$ we have
\begin{equation}\label{eq:inv.ingham11}
\int_{0}^{T} \big(|u_1(t)|^2+|u_2(t)|^2\big)\ dt
\asymp \sum_{ n= 1}^{\infty}
\Big(|C_{n}|^2+ |d_nD_{n}|^2\Big) \,. \end{equation}
\end{theorem}
\section{Reachability results}
This section will be devoted to the proof of some
reachability results for wave--wave coupled systems with a memory term. \begin{theorem}\label{th:reachres} Let $\beta<1/2$. For any $T>\frac{2\pi}{\sqrt{1-4\beta^2}}$ and $ (u_{i}^{0},u_{i}^{1})\in L^{2}(0,\pi)\times H^{-1}(0,\pi) $, $i=1,2$,
there exist $g_i\in L^2(0,T)$, $i=1,2$, such that the weak solution $(u_1,u_2)$ of system \begin{equation}\label{eq:problem-usix} \begin{cases} \displaystyle u_{1tt}(t,x) -u_{1xx}(t,x)+\beta\int_0^t\ e^{-\eta(t-s)} u_{1xx}(s,x)ds+au_2(t,x)= 0\,, \\ \phantom{u_{1tt}(t,x) -u_{1xx}(t,x)+\int_0^t\ k(t-s) u_{1xx}(s,x)ds+} t\in (0,T)\,,\,\,\, x\in(0,\pi) \\ \displaystyle u_{2tt}(t,x) -u_{2xx}(t,x)+bu_1(t,x)= 0 \,, \end{cases} \end{equation} with boundary conditions \begin{equation}\label{eq:bound-u1r} u_1(t,0)=u_2(t,0)=0\,,\quad u_1(t,\pi)=g_1(t)\,,\quad u_2(t,\pi)=g_2(t)\qquad t\in (0,T)
\,, \end{equation} and null initial values \begin{equation} u_i(0,x)=u_{it}(0,x)=0\qquad x\in(0,\pi)\,,\quad i=1,2, \end{equation} verifies the final conditions \begin{equation}\label{eq:findataT} u_i(T,x)=u_{i}^{0}(x)\,,\quad u_{it}(T,x)=u_{i}^{1}(x)\,, \quad x\in(0,\pi), \qquad i=1,2\,. \end{equation} \end{theorem} \begin{Proof} To prove our statement, we will apply the Hilbert Uniqueness Method described in Section \ref{se:HUM}. Let $ H= L^2(0,\pi ) $ be endowed with the usual scalar product and norm $$
\|u\|_{L^2}:=\left(\int_0^\pi |u(x)|^{2}\ dx\right)^{1/2}\qquad u\in L^2(0,\pi)\,. $$ We consider the operator $L:D(L)\subset H\to H$ defined by $Lu=\displaystyle -u_{xx}$ for $u\in D(L):=H^2(0,\pi )\cap H_0^1(0,\pi )$. It is well known that $L$ is a self-adjoint positive
operator on $H$ with dense domain $D(L)$ and
$$D(\sqrt L)=H_0^1(0,\pi ).$$
Moreover, $\{n^2\}_{n\ge1}$ is the sequence of eigenvalues for $L$ and
$\{\sin(nx)\}_{n\ge1}$ is the sequence of the corresponding eigenvectors.
We can apply our spectral analysis, see Section \ref{se:specan}, to the adjoint system of (\ref{eq:problem-usix}) given by \begin{equation}\label{eq:adjointr} \begin{cases} \displaystyle z_{1tt}(t,x) -z_{1xx}(t,x)+\int_t^T\ k(s-t) z_{1xx}(s,x)ds+bz_2(t,x)= 0\,,\\ \hskip7cm t\in (0,T)\,,\ x\in(0,\pi) \\ \displaystyle z_{2tt}(t,x) -z_{2xx}(t,x)+az_1(t,x)= 0\,, \\ z_i(t,0)=z_i(t,\pi)=0\quad t\in [0,T]\,, \\ \hskip5cm i=1,2, \\ z_i(T,\cdot)=z_{i}^0\,,\quad z_{it}(T,\cdot)=z_{i}^{1}\,, \end{cases} \end{equation} where the final data exhibit the following expansion in the basis $\{\sin(nx)\}_{n\ge1}$ \begin{equation*} z_{i}^{0}(x)=\sum_{n=1}^{\infty}\alpha_{in}\sin(nx)\,,\quad
z_{i}^{1}(x)=\sum_{n=1}^{\infty}\rho_{in}\sin(nx) \,,\qquad i=1,2\,. \end{equation*} If we take $ (z_{i}^{0},z_{i}^{1})\in H^1_0(0,\pi)\times L^2(0,\pi) $, $i=1,2$, then one has
\begin{equation}\label{eq:norms}
\|z_{i}^{0}\|^2_{H^1_0}=\sum_{n=1}^\infty\alpha^2_{in} n^2,
\quad
\|z_{i}^{1}\|^2_{L^2}= \sum_{n=1}^\infty\rho^2_{in} \,,\qquad i=1,2. \end{equation} The backward system \eqref{eq:adjointr} is equivalent to the forward system \begin{equation}\label{eq:forward} \begin{cases} \displaystyle u_{1tt}(t,x) -u_{1xx}(t,x)+\int_0^t\ k(t-s) u_{1xx}(s,x)ds+bu_2(t,x)= 0\,,\\ \hskip7cm t\in (0,T)\,,\ x\in(0,\pi) \\ \displaystyle u_{2tt}(t,x) -u_{2xx}(t,x)+au_1(t,x)= 0\,, \\ u_i(t,0)=u_i(t,\pi)=0\quad t\in [0,T]\,, \\ \hskip5cm i=1,2, \\ u_i(0,\cdot)=z_{i}^{0}\,,\quad u_{it}(0,\cdot)=z_{i}^{1}\,, \end{cases} \end{equation} that is, if $(u_1,u _2)$ is the solution of \eqref{eq:forward}, then the solution $(z_1,z_2)$ of \eqref{eq:adjointr} is given by \begin{equation*} z_1(t,x)=u_1(T-t,x), \qquad z_2(t,x)=u_2(T-t,x) \,. \end{equation*} Therefore, thanks to the representation for the solution of \eqref{eq:forward}, see Theorem \ref{th:repres}, we can write $(z_1,z_2)$ in the following way, for any $(t,x)\in [0,T]\times [0,\pi]$ \begin{equation*} z_1(t,x)= \sum_{n=1}^{\infty}\Big(C_ne^{i\omega_n(T-t)} +\overline{C_n}e^{-i\overline{\omega_n}(T-t)}+R_ne^{r_n(T-t)}+D_ne^{i\zeta_n(T-t)} +\overline{D_n}e^{-i\overline{\zeta_n}(T-t)}\Big)\sin(n x) \,, \end{equation*} \begin{multline*} z_2(t,x)=\sum_{n=1}^{\infty} \Big(d_nD_{n}e^{i\zeta_{n} (T-t)}+\overline{d_nD_{n}}e^{-i\overline{\zeta_{n}}(T-t)} +c_nC_{n}e^{i\omega_{n} (T-t)}+\overline{c_nC_{n}}e^{-i\overline{\omega_{n}}(T-t)}\Big)\sin(n x) \\ +e^{-\eta (T-t)}\sum_{n=1}^{\infty}E_n\sin(n x) \,. \end{multline*} In particular, thanks also to \eqref{eq:norms} we get \begin{equation}\label{eq:equivn}
\sum_{n= 1}^\infty n^2\Big( |C_n|^2+|d_nD_{n}|^2 \Big)
\asymp
\|z_{1}^{0}\|^2_{H^1_0}+\|z_{1}^{1}\|^2_{L^2}
+\|z_{2}^{0}\|^2_{H^1_0}+\|z_{2}^{1}\|_{L^2}^2
\,.
\end{equation} Moreover, for any $t\in [0,T]$ \begin{equation*} z_{1x}(t,\pi)= \sum_{n=1}^{\infty}(-1)^n n\Big(C_ne^{i\omega_n(T-t)} +\overline{C_n}e^{-i\overline{\omega_n}(T-t)}+R_ne^{r_n(T-t)}+D_ne^{i\zeta_n(T-t)} +\overline{D_n}e^{-i\overline{\zeta_n}(T-t)}\Big) \,, \end{equation*} \begin{multline*} z_{2x}(t,\pi)=\sum_{n=1}^{\infty}(-1)^n n \Big(d_nD_{n}e^{i\zeta_{n} (T-t)}+\overline{d_nD_{n}}e^{-i\overline{\zeta_{n}}(T-t)} +c_nC_{n}e^{i\omega_{n} (T-t)}+\overline{c_nC_{n}}e^{-i\overline{\omega_{n}}(T-t)}\Big) \\ +e^{-\eta (T-t)}\sum_{n=1}^{\infty}(-1)^n n\, E_n \,. \end{multline*} We can apply Theorem \ref{th:inv.ingham1} to $(z_{1x}(t,\pi),z_{2x}(t,\pi))$. Indeed, thanks to the above expressions for $z_{ix}(t,\pi)$, $i=1,2$, and \eqref{eq:inv.ingham} we have \begin{equation*}
\int_{0}^{T} \big(|z_{1x}(t,\pi)|^2+|z_{2x}(t,\pi)|^2\big)\ dt
\asymp
\sum_{n= 1}^\infty n^2\Big( |C_n|^2+|d_nD_{n}|^2 \Big)\,,
\end{equation*} and hence by \eqref{eq:equivn} we get \begin{equation}\label{eq:obses}
\int_{0}^{T} \big(|z_{1x}(t,\pi)|^2+|z_{2x}(t,\pi)|^2\big)\ dt
\asymp
\|z_{1}^{0}\|^2_{H^1_0}+\|z_{1}^{1}\|^2_{L^2}
+\|z_{2}^{0}\|^2_{H^1_0}+\|z_{2}^{1}\|_{L^2}^2
\,.
\end{equation}
Therefore, we have proved Theorem \ref{th:uniqueness}. Furthermore, we consider the linear operator $\Psi$ introduced in Section \ref{se:HUM} and, thanks to \eqref{eq:psi0}, defined by \begin{equation*} \Psi(z_{1}^{0},z_{1}^{1},z_{2}^{0},z_{2}^{1})=(-u_{1t}(T,\cdot),u_{1}(T,\cdot),-u_{2t}(T,\cdot),u_{2}(T,\cdot))
\,, \end{equation*} where $(u_{1},u_{2})$ is the weak solution of system \eqref{eq:problem-usix}. We have that $$\Psi:H^1_0(0,\pi)\times L^2(0,\pi)\times H^1_0(0,\pi)\times L^2(0,\pi)\to H^{-1}(0,\pi)\times L^2(0,\pi)\times H^{-1}(0,\pi)\times L^2(0,\pi)$$
is an isomorphism. Therefore, for $ (u_{i}^{0},u_{i}^{1})\in L^{2}(0,\pi)\times H^{-1}(0,\pi) $, $i=1,2$, there exists one and only one $(z_{1}^{0},z_{1}^{1},z_{2}^{0},z_{2}^{1})\in H^1_0(0,\pi)\times L^2(0,\pi)\times H^1_0(0,\pi)\times L^2(0,\pi)$ such that \begin{equation*} \Psi(z_{1}^{0},z_{1}^{1},z_{2}^{0},z_{2}^{1}) =(-u_{1}^{1},u_{1}^{0},-u_{2}^{1},u_{2}^{0}) \,. \end{equation*} Finally, if we consider the solution $(z_{1},z_{2})$ of system \eqref{eq:adjointr} with final data given by the unique $(z_{1}^{0},z_{1}^{1},z_{2}^{0},z_{2}^{1})$, then the control functions required by the statement are given by \begin{equation*} g_1(t)=z_{1x}(t,\pi)-\beta\int_t^T\ e^{-\eta(s-t)}z_{1x}(s,\pi)ds\,, \qquad g_2(t)=z_{2x}(t,\pi)\,, \end{equation*} that is, our proof is complete. \end{Proof} \begin{thebibliography}{99}
\itemsep=
amount
\bibitem{Al} F. Alabau-Boussouira {\em A Two-Level Energy Method for Indirect Boundary Observability and Controllability of Weakly Coupled Hyperbolic Systems} SIAM J. Control Optim., {\bf 42} (2003), 871--906.
\bibitem{Ce} E. Ces\`aro, {\em Sur la convergence des s\'eries}, Nouvelles annales de math\'ematiques, {\bf 7} (1888), 49--59.
\bibitem{CN} B. D. Coleman, W. Noll, {\em Foundations of linear viscoelasticity}, Rev. Modern Phys., {\bf 33} (1961) 239--249.
\bibitem{D1} C. M. Dafermos, {\em Asymptotic stability in viscoelasticity}, Arch. Rational Mech. Anal., {\bf 37} (1970), 297--308.
\bibitem{D2} C. M. Dafermos, {\em An abstract Volterra equation with applications to linear viscoelasticity}, J. Differential Equations, {\bf 7} (1970), 554--569. \bibitem{GLS} G. Gripenberg, S. O. Londen, O. J. Staffans, {\em Volterra Integral and Functional Equations}, Encyclopedia Math. Applications, {\bf 34} (1990), Cambridge Univ. Press, Cambridge.
\bibitem{Ha} A. Haraux, {\em S\'eries lacunaires et contr\^ole semi-interne des vibrations d'une plaque rectangulaire} J. Math. Pures Appl., {\bf 68} (1989), 457--465.
\bibitem{Ing} A. E. Ingham, {\em Some trigonometrical inequalities with applications to the theory of series}, Math. Z., {\bf 41} (1936), 367-379.
\bibitem{KR} V. Komornik, B. Rao, {\em Boundary stabilization of compactly coupled wave equations} Asymptot. Anal. {\bf 14} (1997), 339--359.
\bibitem{KL1} V. Komornik, P. Loreti, {\em Ingham type theorems for vector-valued functions and observability of coupled linear system},
SIAM J. Control Optim., {\bf 37} (1998), 461-485.
\bibitem{KL2} V. Komornik, P. Loreti, {\em Fourier series in control theory}, Springer Monographs in Ma\-the\-ma\-tics (2005), Springer-Verlag, New York.
\bibitem{Las} I. Lasiecka, {\em Controllability of a viscoelastic Kirchhoff plate} Control and estimation of distributed parameter systems (Vorau, 1988), Internat. Ser. Numer. Math., 91, Birkhäuser, Basel, 1989, 237--247.
\bibitem{LasT} I. Lasiecka, R. Triggiani, {\em Exact controllability of the wave equation with Neumann boundary control} Appl. Math. Optim. {\bf 19} (1989), 243–290.
\bibitem{LPC} G. Lebon, C. Perez-Garcia, J. Casas-Vazquez, {\em On the thermodynamic foundations of viscoelasticity} J. Chem. Phys., {\bf 88} (1988), 5068--5075.
\bibitem{L0} G. Leugering, {\em Exact boundary controllability of an integro-differential equation}, Appl. Math. Optim., (1987), 223--250.
\bibitem{L} G. Leugering, {\em Boundary controllability of a viscoelastic string}, in G. Da Prato and M. Iannelli editors, Volterra integrodifferential equations in Banach spaces and applications, Harlow, Essex, Longman Sci. Tech., (1989), 258-270.
\bibitem{LR} T. Li, B. Rao, {\em Exact synchronization for a coupled system of wave equations with Dirichlet boundary controls}, Chin. Ann. Math. Ser. B {\bf 34} (2013), 139--160.
\bibitem{Lio1} J.-L. Lions, {\em Exact controllability, stabilization and perturbations for distributed systems}, SIAM Rev. {\bf 30} (1988), 1--68.
\bibitem{Lio2} J.-L. Lions, {\em Contr\^olabilit\'e exacte, perturbations et stabilisation de syst\`emes distribu\'es. Tome 1. Contr\^olabilit\'e exacte}, with appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch. Recherches en Math\'ematiques Appliqu\'ees, {\bf 8} ( 1988), Masson, Paris.
\bibitem{Lio3} J.-L. Lions, {\em Contr\^olabilit\'e exacte, perturbations et stabilisation de syst\`emes distribu\'es. Tome 2. Perturbations}, Recherches en Math\'ematiques Appliqu\'ees, {\bf 9} (1988), Masson, Paris.
\bibitem{LPS} P. Loreti, L. Pandolfi, D. Sforza, {\em Boundary controllability and observability of a viscoelastic string}, SIAM J. Control Optim., {\bf 50} (2012), 820--844.
\bibitem{LoretiSforza} P. Loreti, D. Sforza {\em Exact reachability for second-order integro-differential equations. } C. R. Math. Acad. Sci. Paris {\bf 347} (2009), 1153--1158.
\bibitem{LoretiSforza1} P. Loreti, D. Sforza, {\em Reachability problems for a class of integro-differential equations}. J. Differential Equations {\bf 248} (2010), 1711--1755.
\bibitem{LoretiSforza3} P. Loreti, D. Sforza, {\em Control problems for weakly coupled systems with memory}, J. Differential Equations {\bf 257} (2014), 1879--1938.
\bibitem{LoretiSforza4} P. Loreti, D. Sforza, {\em Hidden regularity for wave equations with memory}, Riv. Mat. Univ. Parma, {\bf 7} (2016), 391--405.
\bibitem{LV} P. Loreti, V. Valente, {\em Partial exact controllability for spherical membranes}, SIAM J. Control Optim., {\bf 35} (1997), 641-653.
\bibitem{MN} J. E. Mu\~noz Rivera, M. G. Naso, {\em Exact controllability for hyperbolic thermoelastic systems with large memory}, Adv. Differential Equations, {\bf 9} (2004), 1369--1394.
\bibitem{NSW} M. Najafi, G. R. Sarhangi, H. Wang {\em The study of the stabilizability of the coupled wave equations under various end conditions}, in Proceedings of the 31st IEEE Conference on Decision and Control, Vol. I (Tucson Arizona 1992), 374--379, IEEE, New York, 1992.
\bibitem{Pruss} J. Pr\"uss, {\em Evolutionary integral equations and applications}, Monographs in Mathematics, {\bf 87} (1993), Birkh\"auser Verlag, Basel. \bibitem{RHN} M. Renardy, W. J. Hrusa, J. A. Nohel, {\em Mathematical problems in viscoelasticity}, Pitman Monographs Pure Appl.Math., {\bf 35} (1988), Longman Sci. Tech., Harlow, Essex.
\bibitem{Re1} M. Renardy, {\em Are viscoelastic flows under control or out of control?} Systems Control Lett., {\bf 54} (2005), 1183--1193.
\bibitem{Ru} D. L. Russell, {\em Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions} SIAM Rev., {\bf 20} (1978), 639--739.
\bibitem{T1} R. Triggiani, {\em Exact boundary controllability on $L_2(\Omega)\times H^{-1}(\Omega)$ of the wave equation with Dirichlet boundary control acting on a portion of the boundary $\partial\Omega$, and related problems} Appl. Math. Optim. {\bf 18} (1988), 241--277.
\end{thebibliography}
\end{document} \begin{equation}\label{eq:problem-uIA} \begin{cases} \displaystyle u_{1tt} -\triangle u_{1} +a u_2= 0 \\ \hskip5cm\mbox{in}\ \Omega\times (0,T), \\ \displaystyle u_{2tt} -\triangle u_{2}+au_1= 0 \end{cases} \end{equation} being $\Omega$ a nonempty bounded open set in ${\mathbb R}^N$ with a boundary of class $C^2$.
In the case of two spatial dimensions the problem without the integro-differential term was investigated by
W. D. Bastos, A. Spezamiglio, C. A. Raposo \cite {BSR} \begin{equation}\label{eq:problem-uIB}
\begin{cases} \displaystyle u_{1tt} -\triangle u_{1} +c u_1+a u_2= 0 \\ \hskip5cm\mbox{in}\ \Omega\times (0,T), \\ \displaystyle u_{2tt} -\triangle u_{2}+bu_1+du_{2}= 0 \end{cases} \end{equation} where $\Omega\subset{\mathbb R}^2$ is a curvilinear polygon. The authors assumed that the matrix of the coupling constants
$ H=\begin{pmatrix} c &a\\ b & d \end{pmatrix} $ is diagonalizable and its eigenvalues are strictly positive. However, comparing with the
coupling of our interest, see \eqref{eq:problem-uI}, the parameters matrix would be $ H=\begin{pmatrix} 0&a\\ b & 0 \end{pmatrix} $, but it is clear that even if we assume $ab>0$, an eigenvalue of this matrix $H$ is always strictly negative.
To our knowledge, it still remains open the complete study of the coupled wave-wave
equations for a such coupling, although in literature many related results are given, see for example
\cite{LR}.
It is easy to verify the following result.
\begin{lemma}\label{le:noveottavi} The third degree polynomial \begin{equation}\label{eq:noveottavi} F(t):=-32t^3+108t^2-\frac{243}{2}t+\frac{729}{16} \end{equation} is strictly decreasing in $[0,\infty)$. Moreover, the unique real zero of $F(t)$ is $\displaystyle\frac98$. \end{lemma}
\section{Hilbert Uniqueness Method}\label{se:HUM} In this section we formally describe the method in an abstract setting.
We introduce a linear
operator $\mathcal{A}:D(\mathcal{A})\subset X\to X$ on $X$ with domain $D(\mathcal{A})$ and $H\in L^1_{loc}(0,\infty)$. Let $Y$ be another real Hilbert space with scalar product
$\langle \cdot \, ,\, \cdot \rangle_Y$ and norm $\| \cdot \|_Y$ and $\mathcal{B}\in L(X_0;Y)$, where $X_0$ is a space such that $D(\mathcal{A})\subset X_0\subset X$. We consider the integro-differential equation \begin{equation}\label{eq:problem-u} u''(t) +\mathcal{A} u(t)-\int_0^t\ H(t-s) \mathcal{A} u(s)ds = 0\qquad t\in (0,T)\,, \end{equation} with null initial conditions \begin{equation} u(0)=u'(0)=0, \end{equation} and \begin{equation}\label{eq:bound} \mathcal{B} u(t)= g(t)\qquad t\in (0,T)\,. \end{equation} In the applications $\mathcal{B} $ can be, for example, a trace operator. For a reachability problem we mean the following: given $T>0$, $u_0\in X$ and $u_1\in (Ker(\mathcal{B}))'$, find $g\in L^2(0,T;Y)$ such that the weak solution $u$ of problem \eqref{eq:problem-u}-\eqref{eq:bound} verifies the final conditions \begin{equation}\label{eq:problem-u1} u(T)=u_0\,,\qquad u'(T)=u_1\,. \end{equation} To explain how the HUM method can be used to solve a reachability problem, we proceed dividing the reasoning into several steps. \begin{description} \item[\bf STEP 1] $A:D(A)\subset X\to X$ denotes a self-adjoint positive linear
operator on $X$ with dense domain $D(A)\subset D(\mathcal{A})$ such that for any $x\in D(A)$ $\mathcal{A}x=Ax$ and $D(\sqrt A)=Ker(\mathcal{B})$. We define by induction $$ D(A^k):=\{x\in D(A^{k-1}):\ Ax\in D(A^{k-1}) \}\,,\qquad k\in{\mathbb N}\,. $$
Given $z_0\in D(A^k)$ and $z_1\in D(A^k)$, we consider the {\it adjoint} equation of (\ref{eq:problem-u}), that is \begin{equation}\label{eq:adjoint} z''(t) +A z(t)-\int_t^T\ H(s-t)A z(s)ds = 0\,,\quad t\in [0,T]\,, \end{equation} with final data \begin{equation} \label{eq:final} z(T)=z_0\,,\qquad z'(T)=z_1\,. \end{equation} Problem (\ref{eq:adjoint})--(\ref{eq:final}) admits a unique solution $z\in C^{k-j}([0,T];D(A^j))$, $j=0,1,\dots,k$. Indeed, set $v(t)=z(T-t)$, problem (\ref{eq:adjoint})--(\ref{eq:final}) is equivalent to \begin{equation}\label{eq:} \left \{\begin{array}{l}\displaystyle v''(t) +A v(t)-\int_0^t\ H(t-s) A v(s)ds = 0\,,\quad t\in [0,T]\,, \\ \\ v(0)=z_0\,,\qquad v'(0)=-z_1\,, \end{array}\right . \end{equation} and the above problem is well-posed, see e.g. \cite{Pruss}. We take $k$ large enough to have the function $z$ sufficiently regular. \item[STEP 2] We introduce another operator $D_\nu:X_0\to Y$ such that the following identity holds \begin{equation}\label{eq:boundcon} \langle\mathcal{A}\varphi, \xi\rangle=\langle\varphi, A\xi\rangle-\langle\mathcal{B}\varphi,D_\nu\xi\rangle_Y\,,\quad\forall \varphi\in D(\mathcal{A})\,,\xi\in D(A)\,, \end{equation} and the problem \begin{equation}\label{eq:phi} \left \{\begin{array}{l}\displaystyle \varphi''(t) +\mathcal{A}\varphi(t)-\int_0^t\ H(t-s) \mathcal{A}\varphi(s)ds = 0\,,\quad t\in [0,T]\,, \\ \\ \displaystyle \mathcal{B}\varphi(t)=D_\nu z(t)-\int_t^T\ H(s-t)D_\nu z(s)ds,\quad t\in [0,T]\,, \\ \\ \varphi(0)=\varphi'(0)=0\,, \end{array}\right . \end{equation}
admits a unique solution $\varphi$.
Then, we define the linear operator
$$ \Psi(z_0,z_1)=(\varphi'(T),-\varphi(T))\,,\qquad (z_0,z_1)\in D(A^k)\times D(A^k)\,. $$
\item[STEP 3] Let $(\xi_0,\xi_1)\in D(A^k)\times D(A^k)$ and $\xi$ the solution of \begin{equation}\label{eq:adjoint1} \left \{\begin{array}{l}\displaystyle \xi''(t) +A\xi(t)-\int_t^T\ H(s-t) A\xi(s)ds = 0\,,\quad t\in [0,T]\,, \\ \\ \xi(T)=\xi_0\,,\qquad \xi'(T)=\xi_1\,. \end{array}\right . \end{equation} We prove that \begin{equation}\label{eq:psi} \langle\Psi(z_0,z_1),(\xi_0,\xi_1)\rangle_{X\times X} =\int_0^T \langle\mathcal{B}\varphi(t), D_\nu\xi(t) -\int_t^T\ H(s-t)D_\nu\xi(s)ds\rangle_Y \ dt \,. \end{equation} Indeed, multiplying the equation in (\ref{eq:phi}) by $\xi(t)$ and integrating on $[0,T]$ we have \begin{eqnarray*} \int_0^T \langle\varphi''(t), \xi(t)\rangle \ dt +\int_0^T\langle\mathcal{A}\varphi(t), \xi(t)\rangle\ dt-\int_0^T\int_0^t\ H(t-s)\langle\mathcal{A}\varphi(s),\ \xi(t)\rangle\ ds\ dt =0\,. \end{eqnarray*} Integrating by parts twice, in view also of \eqref{eq:boundcon} we have \begin{multline*}
\langle \varphi'(T),\xi(T)\rangle-\langle\varphi(T),\xi'(T)\rangle +\int_0^T\langle \varphi(t), \xi''(t) +A\xi(t)-\int_t^T\ H(s-t) A\xi(s)ds \rangle dt
\\ -\int_0^T \langle\mathcal{B}\varphi(t), D_\nu\xi(t) \rangle_Y\ dt +\int_0^T\langle\mathcal{B}\varphi(t),\int_t^T\ H(s-t)D_\nu\xi(s)ds \rangle_Ydt
=0\,. \end{multline*} Since $\xi$ is the solution of (\ref{eq:adjoint1}),
we have that \eqref{eq:psi} holds.
Now, taking $(\xi_0,\xi_1)=(z_0,z_1)$ in (\ref{eq:psi}), we have \begin{equation}\label{eq:psi1} \langle\Psi(z_0,z_1),(z_0,z_1)\rangle_{X\times X}
=\int_0^T \Big\|D_\nu z(t)-\int_t^T\ H(s-t)D_\nu z(s)ds\Big\|_Y^2 \ dt \,. \end{equation} So, we can introduce the semi-norm \begin{eqnarray}\label{eq:normF} && (z_0,z_1)\in D(A^k)\times D(A^k)\,,\nonumber\\
&&\|(z_0,z_1)\|_{F}:=\displaystyle\Big(\int_0^T \Big\|D_\nu z(t)-\int_t^T\ H(s-t)D_\nu z(s)ds\Big\|_Y^2 dt\Big)^{1/2} \,. \end{eqnarray}
\item[STEP 4] In view of lemma \ref{le:unicita}, $\|\cdot\|_{F}$ is a norm if and only if the following uniqueness theorem holds. \begin{theorem}\label{th:uniqueness} If $z$ is the solution of problem {\rm (\ref{eq:adjoint})--(\ref{eq:final})} such that $$ D_\nu z(t)=0\,,\qquad \forall t\in [0,T]\,, $$ then $$ z(t)= 0 \qquad\forall t\in [0,T]\,. $$ \end{theorem} If theorem \ref{th:uniqueness} holds true, then we can define the Hilbert space $F$ as the completion of $ D(A^k)\times D(A^k)$ for the norm (\ref{eq:normF}). Moreover, the operator $\Psi$ extends uniquely to a continuous operator, denoted again by $\Psi$, from $F$ to the dual space $F'$ in such a way that $\Psi:F\to F'$ is an isomorphism.
In conclusion
if we prove a result similar to theorem \ref{th:uniqueness} and $F=D(\sqrt A)\times X$,
then we can solve the reachability problem \eqref{eq:problem-u}--\eqref{eq:problem-u1}.
\end{description}
\section{Ingham type direct inequality}\label {se:dir}
In the next two sections, we consider functions of the type \begin{equation*} f(t):=\sum_{n=-\infty}^{\infty}\big(C_ne^{i\omega_nt}+R_ne^{r_nt}\big) \qquad t\ge 0 \end{equation*} with $\omega_n\,,C_n\in{\mathbb C}$ and $r_n,R_n\in{\mathbb R}$ such that the sequences $\{{\Im}\omega_n\}$, $\{r_n\}$ are bounded and \begin{equation*}
\sum_{n=-\infty}^{\infty}|C_n|^2<+\infty\,,\qquad \sum_{n=-\infty}^{\infty}|R_n|^2<+\infty\,. \end{equation*} Let $T>0$.
\begin{theorem}\label{th:diringham} Assume \begin{equation}\label{eq:h1d}
{\Re}\omega_n-{\Re}\omega_{n-1}\ge\gamma>0\,\qquad\forall \ |n|\ge n'\,, \end{equation}
\begin{equation}\label{eq:h2d}
\lim_{|n|\to\infty}{\Im}\omega_n=\alpha \,, \end{equation} \begin{equation}\label{eq:h3dbis}
|R_n|\le \frac{\mu}{|n|^{\nu}}|C_n|\,\quad\forall \ |n|\ge n'\,; \qquad
|R_n|\le \mu|C_n|\,\quad\forall \ |n|\le n'\,, \end{equation} for some $n'\in{\mathbb N}$, $\alpha\in{\mathbb R}$, $\mu>0$ and $\nu> 1/2$. Then, for any $T>\pi/\gamma$ we have \begin{equation}\label{eq:diringham}
\int_{-T}^{T} \Big|\sum_{n=-\infty}^{\infty}\big(C_ne^{i\omega_nt}+R_ne^{r_nt}\big)\Big |^2 d t
\le c_2(T)\sum_{n=-\infty}^{\infty}|C_n|^2\,,
\end{equation} where $c_2(T)$ is a positive constant\,. \end{theorem} To proceed with the proof, we state the following two results, but
the proof of the first one can be found in the appendix, as it is quite long and complex. \begin{theorem}\label{th:dir.ingham} Under assumptions {\rm (\ref{eq:h1d})--(\ref{eq:h3dbis})}, for any $0<\varepsilon<1$ and for any
$T>\frac{\pi}{\gamma\sqrt{1-\varepsilon}}$ there exists $n_0=n_0(\varepsilon)\in{\mathbb N}$ such that if $C_n=0$ for $|n|\le n_0$, then we have \begin{equation}\label{eq:dir-cosine}
\int_{-T}^{T} |f(t)|^2 d t
\le c_2(T)\sum_{|n|\ge n_0}|C_n|^2\,, \end{equation} where $c_2(T)>0$. \end{theorem}
\begin{proposition}\label{pr:haraux-dir} Assume that there exists a finite set ${\cal F}$ of integers such that for any sequences $\{C_n\}$ and $\{R_n\}$ verifying \begin{equation}\label{eq:iha4} C_n=R_n=0\qquad\mbox{for any}\quad n\in {\cal F}\,, \end{equation} the estimate \begin{equation}\label{eq:haraux-dir}
\int_{-T}^{T} \Big|\sum_{n\not\in {\cal F}}\big(C_ne^{i\omega_nt}+R_ne^{r_nt}\big)\Big|^2 d t \le c'_2
\sum_{n\not\in {\cal F}}|C_n|^2 \end{equation} is satisfied for some $c'_2>0$. Then, for any sequences $\{C_n\}$ and $\{R_n\}$ verifying \begin{equation}\label{eq:ha4bis}
|R_n|\le \mu|C_n|\qquad\mbox{for any}\quad n\in {\cal F} \,, \end{equation} for some $\mu>0$, the estimate \begin{equation}\label{eq:haraux-dir0}
\int_{-T}^{T} \Big|\sum_{n=-\infty}^{\infty}\big(C_ne^{i\omega_nt}+R_ne^{r_nt}\big)\Big |^2 d t \le c_2
\sum_{n=-\infty}^{\infty}|C_n|^2 \end{equation} holds for some $c_2>0$. \end{proposition}
\begin{Proof} Assume that $\{C_n\}$ and $\{R_n\}$ verify (\ref{eq:ha4bis}). If we use \eqref{eq:haraux-dir}, then we have \begin{equation}\label{eq:haraux-direp2}
\int_{-T}^{T} \Big|\sum_{n\not\in {\cal F}}\Big(C_ne^{i\omega_nt}+R_ne^{r_nt}\Big)\Big|^2 d t \le c'_2
\sum_{n\not\in {\cal F}}|C_n|^2\,. \end{equation} Now, we prove that \begin{equation}\label{eq:haraux-direp3}
\int_{-T}^{T} \Big|\sum_{n\in {\cal F}}\big(C_ne^{i\omega_nt}+R_ne^{r_nt}\big)\Big|^2 dt \le c''_2
\sum_{n\in {\cal F}}|C_n|^2\,, \end{equation} for some constant $c''_2>0$. Indeed, applying the Cauchy-Schwarz inequality we get \begin{eqnarray*}
\Big|\sum_{n\in {\cal F}}\big(C_ne^{i\omega_nt}+R_ne^{r_nt}\big)\Big|^2 &\le&
\Big(\sum_{n\in {\cal F}}\big(|C_n|e^{-\Im\omega_nt}+|R_n|e^{r_nt}\big)\Big)^2\\ &\le&
2|{\cal F}|\sum_{n\in {\cal F}}\big(|C_n|^2e^{-2\Im\omega_nt}+|R_n|^2e^{2r_nt}\big)\,, \end{eqnarray*}
where $|{\cal F}|$ denotes the number of elements in the set ${\cal F}$. If we use the previous inequality and (\ref{eq:ha4bis}), then we get \begin{equation*}
\int_{-T}^{T} \Big|\sum_{n\in {\cal F}}\big(C_ne^{i\omega_nt}+R_ne^{r_nt}\big)\Big|^2 dt
\le 2|{\cal F}|\sum_{n\in {\cal F}}|C_n|^2\int_{-T}^{T}\big(e^{-2\Im\omega_nt}+\mu^2e^{2r_nt}\big)\ dt\,, \end{equation*} whence (\ref{eq:haraux-direp3}) follows with $
\displaystyle c''_2=2|{\cal F}|\max_{n\in {\cal F}}\Big\{\int_{-T}^{T}\big(e^{-2\Im\omega_nt}+\mu^2e^{2r_nt}\big)\ dt\Big\}\,. $
Finally, from (\ref{eq:haraux-direp2}) and (\ref{eq:haraux-direp3}) we deduce that \begin{multline*}
\int_{-T}^{T} \Big|\sum_{n=-\infty}^{\infty}\big(C_ne^{i\omega_nt}+R_ne^{r_nt}\big)\Big |^2 d t \\
\le 2\int_{-T}^{T} \Big|\sum_{n\not\in {\cal F}}\big(C_ne^{i\omega_nt}+R_ne^{r_nt}\big)\Big|^2 d t
+2\int_{-T}^{T} \Big|\sum_{n\in {\cal F}}\big(C_ne^{i\omega_nt}+R_ne^{r_nt}\big)\Big|^2 d t\\ \le 2c''_2
\sum_{n\not\in {\cal F}}|C_n|^2+2c'_2
\sum_{n\in {\cal F}}|C_n|^2\,, \end{multline*} so (\ref{eq:haraux-dir0}) holds with $ c_2=2\max\{c''_2,c'_2\}\,. $ \end{Proof}
\begin{Proof1} Since $T>\pi/\gamma$, there exists $0<\varepsilon<1$ such that $T>\frac{\pi}{\gamma\sqrt{1-\varepsilon}}$. By applying theorem \ref{th:dir.ingham}, there exist $n_0\in{\mathbb N}$ and $c_2(T)>0$ such that if $C_n=0$ for
$|n|\le n_0$, then we have \begin{equation*}
\int_{-T}^{T} |f(t)|^2 d t\le c_2(T)\sum_{|n|\ge n_0}|C_n|^2\,. \end{equation*} Finally, thanks also to \eqref{eq:h3dbis} we can use proposition \ref{pr:haraux-dir} to conclude. \end{Proof1} \section{Ingham type inverse inequality}\label {se:inv}
In this section $\{\omega_n\}_{n\in{\mathbb Z}}$ and $\{r_n\}_{n\in{\mathbb Z}}$ are sequences of pairwise distinct numbers such that $r_n\not=i\omega_m$ for any $n\,,m\in{\mathbb Z}$. Let $T>0$.
\begin{theorem}\label{th:inv.ingham1}
Assume \begin{equation}\label{eq:h1}
{\Re}\omega_n-{\Re}\omega_{n-1}\ge\gamma>0\,\qquad\forall\ |n|\ge n'\,, \end{equation} \begin{equation}\label{eq:h2}
\lim_{|n|\to\infty}{\Im}\omega_n=\alpha \,,
\qquad r_n\le -{\Im}\omega_n\,\qquad\forall\ |n|\ge n'\,, \end{equation}
\begin{equation}\label{eq:h3}
|R_n|\le \frac{\mu}{|n|^{\nu}}|C_n|\,\quad\forall\ |n|\ge n'\,, \qquad
|R_n|\le \mu|C_n|\,\quad\forall\ |n|\le n'\,, \end{equation}
for some $n'\in{\mathbb N}$, $\alpha\in{\mathbb R}$, $\mu>0$ and $\nu> 1/2$. Then, for any $T>2\pi/\gamma$ we have
\begin{equation}\label{eq:inv.ingham}
c_1(T)\sum_{n=-\infty}^{\infty}|C_n|^2\le\int_{0}^{T} \Big|\sum_{n=-\infty}^{\infty}\big(C_ne^{i\omega_nt}+R_ne^{r_nt}\big)\Big |^2 d t\,, \end{equation} where $c_1(T)$ is a positive constant. \end{theorem}
\begin{remark} {\rm Since the sequence $\{{\Im}\omega_n\}$ is bounded the inverse inequality \eqref{eq:inv.ingham} can be written in the form \begin{equation*} c_1(T)\sum_{n=-\infty}^{\infty}(1+e^{-2({\Im}\omega_n
-\alpha)T})|C_n|^2\le
\int_{0}^{T} \Big|\sum_{n=-\infty}^{\infty}\big(C_ne^{i\omega_nt}+R_ne^{r_nt}\big)\Big |^2 d t\,, \end{equation*} which is similar to that proved in \cite[Lemma 4.1]{ZZ2} by different techniques.} \end{remark}
We note that the direct inequality holds under weaker assumptions respect to the
inverse one.
To prove theorem \ref{th:inv.ingham1}, we need the following results, whose
proofs are given in the appendix, as they are quite long and complex.
\begin{theorem}\label{th:inv.ingham} Under assumptions {\rm (\ref{eq:h1})--(\ref{eq:h3})}, for any $0<\varepsilon<1$ and $T>\frac{2\pi}{\gamma}\sqrt{\frac{1+\varepsilon}{1-\varepsilon}}$ there exist $n_0=n_0(\varepsilon)\in{\mathbb N}$ and $c_1(T,\varepsilon)>0$ such that if $C_n=0$ for any
$|n|\le n_0$, then we have \begin{equation}\label{eq:inv-sine}
c_1(T,\varepsilon)\sum_{|n|\ge n_0}(1+e^{-2({\Im}\omega_n
-\alpha)T})|C_n|^2\le
\int_{0}^{T} |f(t)|^2 d t\,. \end{equation} In addition, the constant $c_1(T,\varepsilon)$ is given by $$ c_1(T,\varepsilon)=\min (1,e^{-2\alpha T})\Big( \frac{T\pi}{\pi^2+T^2\gamma^2\varepsilon/8} -\frac{4\pi}{T\gamma^2}(1+\varepsilon)\Big)\,. $$ \end{theorem}
\begin{proposition}\label{pr:haraux-inv} Let $\{\omega_n\}_{n\in{\mathbb Z}}$ be such that \begin{equation}\label{eq:ha1}
\lim_{|n|\to\infty}|\omega_n|=+\infty\,. \end{equation}
Assume that there exists a finite set ${\cal F}$ of integers such that for any sequences $\{C_n\}$ and $\{R_n\}$ verifying \begin{equation}\label{eq:ha5} C_n=R_n=0\qquad\mbox{for any}\quad n\in {\cal F}\,, \end{equation} the estimates \begin{equation}\label{eq:haraux-inv}
c'_1
\sum_{n\not\in {\cal F}}|C_n|^2\le\int_{0}^{T}\Big|\sum_{n\not\in{\cal F}}\big(C_ne^{i\omega_nt}+R_ne^{r_nt}\big)\Big|^2 d t
\end{equation} \begin{equation}\label{eq:haraux-inv1}
\int_{0}^{T}\Big|\sum_{n\not\in{\cal F}}\big(C_ne^{i\omega_nt}+R_ne^{r_nt}\big)\Big|^2 d t \le c'_2
\sum_{n\not\in {\cal F}}|C_n|^2
\end{equation} are satisfied for some constants $c'_1\,,c'_2>0$. Then, there exists $c_1>0$ such that for any sequences $\{C_n\}$ and $\{R_n\}$
the estimate \begin{equation}\label{eq:haraux-inv2} c_1
\sum_{n=-\infty}^{\infty}|C_n|^2 \le
\int_{0}^{T} \Big|\sum_{n=-\infty}^{\infty}\big(C_ne^{i\omega_nt}+R_ne^{r_nt}\big)\Big |^2 d t \end{equation} holds.
\end{proposition}
\begin{Proof3} Since $T>2\pi/\gamma$, there exists $0<\varepsilon<1$ such that $T>\frac{2\pi}{\gamma}\sqrt{\frac{1+\varepsilon}{1-\varepsilon}}$. By applying theorems \ref{th:inv.ingham} and \ref{th:diringham}, there exist $n_0\in{\mathbb N}$, $c_1(T,\varepsilon)>0$ and $c_2(T)>0$ such that if $C_n=0$ for
$|n|\le n_0$, then we have \begin{equation*}
c_1(T,\varepsilon)\sum_{|n|\ge n_0}|C_n|^2
\le \int_{0}^{T} |f(t)|^2 d t\le c_2(T)\sum_{|n|\ge n_0}|C_n|^2 \,. \end{equation*} Finally, we can use proposition \ref{pr:haraux-inv} to conclude. \end{Proof3}
\section{A reachability result}\label{se:appl} To give the result announced in the introduction concerning reachability problems for a class of systems with memory, first, we need to develop a detailed spectral analysis.
Let $A:D(A)\subset X\to X$ be a self-adjoint positive linear
operator on $X$ with dense domain $D(A)$ and let $\{\lambda_j\}_{j\ge1}$ be a strictly increasing sequence of eigenvalues for the operator $A$ with $\lambda_j>0$ and $\lambda_j\to\infty$ such that the sequence of the corresponding eigenvectors $\{w_j\}_{j\ge1}$ constitutes a Hilbert basis for $X$.
For any $v_0\in D(\sqrt A)$ and $v_1\in X$ there exists a unique weak solution $v\in C([0,\infty);D(\sqrt A))\cap\break C^1([0,\infty);X)$
of equation \begin{equation}\label{eq:v}
v''(t) +Av(t)-\beta\int_0^t\ e^{-\eta(t-s)}A v(s)ds = 0\,,\quad t\ge 0\,, \end{equation} verifying the initial conditions \begin{equation}\label{eq:incond} v(0)=v_0\,,\qquad v'(0)=v_1\,. \end{equation} We have \begin{equation}\label{eq:v0} v_0=\sum_{j=1}^{\infty}\alpha_{j}w_{j}\,,\qquad\quad\alpha_{j}=\langle v_0,w_j\rangle \,, \quad\sum_{j=1}^{\infty}\alpha_{j}^2\lambda_j<\infty\,, \end{equation} \begin{equation}\label{eq:v1} v_1=\sum_{j=1}^{\infty}\gamma_{j}w_{j}\,,\qquad\quad\gamma_{j}=\langle v_1,w_j\rangle \,,\quad\sum_{j=1}^{\infty}\gamma_{j}^2<\infty\,. \end{equation} First, we observe that we can approximate the initial data $v_0$ and $v_1$ by sequences $\{v_{0n}\}$ in $D(A)$ and $\{v_{1n}\}$ in $D(\sqrt A)$ respectively. So, the sequence of strong solutions $v_{n}(t)$ of \eqref{eq:v}, corresponding to the initial conditions $v_{0n}$ and $v_{1n}$, approximates $v(t)$. Thanks to this remark, we can make our computations considering $v(t)$ as a strong solution, and then we go back to weak solutions by standard approximation arguments.
We want to write the solution $v(t)$ as a sum of series, that is \begin{equation*}
v(t)=\sum_{j=1}^{\infty}f_{j}(t)w_{j}\,,\qquad f_{j}(t)=\langle v(t),w_j\rangle\,. \end{equation*} Substituting the above expression of $v$ in (\ref{eq:v}) and multiplying the equation by $w_j$, $j\in{\mathbb N}$, we have that $f_{j}(t)$ is the solution of \begin{equation}\label{second} f_{j}^{''}(t) +\lambda_{j}f_{j}(t)-\lambda_{j}\beta \int_0^t e^{-\eta(t-s)}f_{j}(s) ds=0\,. \end{equation} with initial conditions given by \begin{equation}\label{ini_con} f_{j}(0)=\alpha_{j} \qquad f_{j}^{'}(0)=\gamma_{j}\,. \end{equation} Thanks to lemma \ref{le:third}, problem (\ref{second})--(\ref{ini_con}) is equivalent to the Cauchy problem \begin{equation}\label{eq:third} \left \{\begin{array}{l}\displaystyle f_j^{'''}(t)+ \eta f_j^{''}(t)+\lambda_j f_j^{'}(t)+\lambda_j (\eta-\beta)f_j(t)=0\,,\qquad t\ge 0\,,\\ \\ f_{j}(0)=\alpha_{j}\,, \qquad f_{j}^{'}(0)=\gamma_{j}\,,\qquad f_{j}^{''}(0)=-\lambda_{j}\alpha_{j}\,. \end{array}\right . \end{equation} Therefore, we proceed to solve $(\ref{eq:third})$. To this end, we must compute the solutions of the characteristic equation \begin{equation}\label{eq:char} \Lambda^{3}+\eta\Lambda^{2}+\lambda_{j}\Lambda+\lambda_{j}(\eta-\beta)=0\,, \end{equation} following the well-known Scipione Del Ferro's method to obtain the Cardano formula.
First, we transform equation (\ref{eq:char}) into one without second degree term. For this reason, we will make a suitable change of variable. Indeed, set \begin{equation*} \Lambda=\sigma-{\eta\over 3}\,, \end{equation*} we have \begin{equation}\label{eq:sigma} \sigma^3+p_{j}\sigma +q_{j}=0\,, \end{equation} where $$ p_{j}=\lambda_{j}-{\eta^2\over 3}\,,\qquad q_{j}={2\over {27}}\eta^3+2\Big({\eta\over3}-{\beta\over2}\Big)\lambda_{j}\,. $$ To solve (\ref{eq:sigma}), we look for solutions in the form $$ \sigma=y+z\,. $$ We observe that the cube of $\sigma=y+z$ satisfies the following equation \begin{equation}\label{sigma1} \sigma^3-3yz\sigma-(y^3+z^3)=0\,. \end{equation} Equalling the coefficients of similar terms in equations (\ref{eq:sigma}) and (\ref{sigma1}), we have $$ yz=-p_{j}/3\,, \qquad y^3+z^3=-q_{j} \,. $$ Since $ y^3z^3=-p_{j}^3/27 $ e $ y^3+z^3=-q_{j}\,, $ it follows that $y^3$ and $z^3$ are solutions of the second order equation \begin{equation}\label{eq:r} r^2+q_{j}r-{p_{j}^3\over 27}=0\,. \end{equation} Now, defining the discriminant of equation (\ref{eq:sigma}) as the $\frac14$-discriminant of the above equation, that is \begin{equation*}
\mathcal {E}elta_j:={q_{j}^{2}\over 4}+{{p_{j}^{3}}\over{27}}\,, \end{equation*} we note that $$ {q_{j}^{2}\over4}= {\eta^6\over{(27)^2}}+\Big({\eta\over3}-{\beta\over2}\Big)^2\lambda_{j}^2+{\eta^3\over{27}}\Big({2\over3}\eta-\beta\Big)\lambda_{j}\,, $$ $$ {{p_{j}^{3}}\over {27}}={{\lambda_{j}^{3}}\over{27}}-{{\eta^2}\over{27}}\lambda_{j}^{2}+{{\eta^4}\over{81}}\lambda_{j}-{\eta^6\over{(27)^2}}\,, $$ so we have \begin{equation}\label{eq:discrim} \mathcal {E}elta_j={{\lambda_{j}}\over{27}}\left({\lambda_{j}^{2}}+ \Big(2\eta^2-9\eta\beta+{27\over4}\beta^2\Big)\lambda_{j} +\eta^3(\eta-\beta)\right)\,. \end{equation} Now, to have $ \mathcal {E}elta_j>0 $ it is sufficient that $\displaystyle \Big(2\eta^2-9\eta\beta+{27\over4}\beta^2\Big)^2-4\eta^3(\eta-\beta)<0 $\,, that is $$ F\Big(\frac\eta\beta\Big)=-32\Big(\frac\eta\beta\Big)^3+108\Big(\frac\eta\beta\Big)^2-\frac{243\eta}{2\beta}+\frac{729}{16}<0\,, $$ where $F$ is the polynomial defined in \eqref{eq:noveottavi}. Thanks to lemma \ref{le:noveottavi} the above condition is satisfied for $\displaystyle\eta>\frac98\beta$, and hence $ \mathcal {E}elta_j>0 $ for $\displaystyle\eta>\frac98\beta$\,.
If $\beta<\eta\le\frac98\beta$, then we can write $\eta=t\beta$, with $1<t\le\frac98$. So, we have $ \mathcal {E}elta_j>0 $ for \break $ \lambda_j>\beta^2\ (9t-2t^2-{27\over4}+F(t)^{1/2})/2 $. Since $9t-2t^2-{27\over4}>0$ for $1<t\le\frac98$, we get that $ \mathcal {E}elta_j>0 $ if \begin{equation}\label{eq:condbeta} \beta<\Big( \frac{2\lambda_1}{9t-2t^2-{27\over4}+(-32t^3+108t^2-\frac{243}{2}t+\frac{729}{16})^{1/2}}\Big)^{1/2}\,. \end{equation}
Therefore, the solutions of equation (\ref{eq:r}) are given by $$ r_{1/2}=-{q_{j}\over 2}\pm\sqrt{{q_{j}^{2}\over 4}+{{p_{j}^{3}}\over{27}}}\,. $$ Now, to write the solutions $\sigma=y+z$ of (\ref{eq:sigma}), we keep in mind not only that $y^3$ and $z^3$ are solutions of (\ref{eq:r}), but also that $y$ and $z$ must satisfy the condition $yz=-p_{j}/3$. Accordingly, if we consider the following real numbers, $$ y_{j}=\left(-{q_{j}\over 2}+\sqrt{{q_{j}^{2}\over 4}+{{p_{j}^{3}}\over{27}}}\right)^{1/ 3} \qquad z_{j}=\left(-{q_{j}\over 2}-\sqrt{{q_{j}^{2}\over4}+{{p_{j}^{3}}\over {27}}}\right)^{1/ 3}\,, $$ then the solutions of (\ref{eq:sigma}) are given by \begin{equation}\label{eq:si1} \sigma_{1,j}=y_{j}+z_{j}\,, \end{equation}
\begin{equation}\label{eq:si2} \sigma_{2,j}=y_{j}e^{i2\pi/3}+z_{j}e^{-i2\pi/3} =-{1\over 2}({y_{j}+z_{j}})+i{ \sqrt{3}\over 2}({y_{j}-z_{j}})\,, \end{equation}
\begin{equation}\label{eq:si3} \sigma_{3,j}=y_{j}e^{-i2\pi/3}+z_{j}e^{i2\pi/3} =-{1\over 2}({y_{j}+z_{j}})-i{ \sqrt{3}\over 2}({y_{j}-z_{j}})\,. \end{equation} We note that the numbers $\sigma_{1,j}\,,\sigma_{2,j}\,,\sigma_{3,j}$ are all distinct.
Now, in view of \eqref{eq:discrim} we evaluate the quantity \begin{eqnarray*} &&-{q_{j}\over 2}+\sqrt{{q_{j}^{2}\over 4}+{{p_{j}^{3}}\over {27}}}\\ &=&-{\eta^3\over {27}}-\Big({\eta\over 3}-{\beta\over 2}\Big)\lambda_{j}+\sqrt{{{\lambda_{j}^{3}}\over{27}}+ \Big({2\over{27}}\eta^2+{\beta^2\over4}-{\eta\beta\over 3}\Big)\lambda_{j}^{2} +{\eta^3\over 27}(\eta-\beta)\lambda_{j}}\\ &=&-{\eta^3\over {27}}-\Big({\eta\over 3}-{\beta\over 2}\Big)\lambda_{j}+\Big({{\lambda_{j}}\over{3}}\Big)^{3/2}\sqrt{1+\Big({2\over{27}}\eta^2+{\beta^2\over4}-{\eta\beta\over 3}\Big){27\over\lambda_{j}}+\eta^3(\eta-\beta){1\over\lambda_{j}^2}}\\ &=&\Big({{\lambda_{j}}\over{3}}\Big)^{3/2}\left[-{\eta^3\over {\sqrt{27}\lambda_{j}^{3/2}}}-\Big({\eta\over 3}-{\beta\over 2}\Big){\sqrt{27}\over\sqrt{\lambda_{j}}}+\sqrt{1+\Big({2\over{27}}\eta^2+{\beta^2\over4}-{\eta\beta\over 3}\Big){27\over\lambda_{j}}+\eta^3(\eta-\beta){1\over\lambda_{j}^2}}\ \right] \\ &=&\Big({{\lambda_{j}}\over{3}}\Big)^{3/2}\Big[-{\eta^3\over {\sqrt{27}\lambda_{j}^{3/2}}}-\Big({\eta\over 3}-{\beta\over 2}\Big){\sqrt{27}\over\sqrt{\lambda_{j}}}+1+{27\over2}\Big({2\over{27}}\eta^2+{\beta^2\over4}-{\eta\beta\over 3}\Big){1\over\lambda_{j}}+O\Big({1\over{\lambda_{j}^{2}}}\Big)\Big] \\ &=&\Big({{\lambda_{j}}\over{3}}\Big)^{3/2}\Big[1-\Big({\eta\over 3}-{\beta\over 2}\Big){\sqrt{27}\over\sqrt{\lambda_{j}}}+\Big(\eta^2+{27\over8}\beta^2-{9\over 2}\eta\beta\Big){1\over\lambda_{j}}-{\eta^3\over {\sqrt{27}\lambda_{j}^{3/2}}}+O\Big({1\over{\lambda_{j}^{2}}}\Big)\Big]\,,\quad j\to \infty\,. \end{eqnarray*} Therefore, using also the well-known formula \begin{equation}\label{eq:1/3} (1+x)^{1/3}=1+\frac13 x-\frac19 x^2+o(x^2)\,,\qquad x\to 0\,, \end{equation} we obtain \begin{eqnarray}\label{eq:yj} y_{j}&=& \left(-{q_{j}\over 2}+\sqrt{{q_{j}^{2}\over 4}+{{p_{j}^{3}}\over{27}}}\ \right)^{1/ 3}\\ &=& \sqrt{{\lambda_{j}\over3}} \Big[1-\Big({\eta\over 3}-{\beta\over 2}\Big){3^{3/2}\over\sqrt{\lambda_{j}}}+\Big(\eta^2+{27\over8}\beta^2-{9\over 2}\eta\beta\Big){1\over\lambda_{j}}-{\eta^3\over {(3\lambda_{j})^{3/2}}}+O\Big({1\over{\lambda_{j}^{2}}}\Big)\Big]^{1/ 3} \nonumber\\ &=& \sqrt{{\lambda_{j}\over3}}\Big[1-\Big({\eta\over3}-{\beta\over 2}\Big)\sqrt{{3\over\lambda_{j}}}+{1\over3}\Big(\eta^2+{27\over8}\beta^2-{9\over 2}\eta\beta\Big){1\over\lambda_{j}}-{\eta^3\over {3^{5/2}\lambda_{j}^{3/2}}}-3\Big({\eta\over3}-{\beta\over 2}\Big)^2{1\over\lambda_{j}} \nonumber\\ && +{2\over\sqrt{3}}\Big({\eta\over 3}-{\beta\over 2}\Big)\Big(\eta^2+{27\over8}\beta^2-{9\over 2}\eta\beta\Big){1\over\lambda_{j}^{3/2}} +O\Big({1\over{\lambda_{j}^{2}}}\Big)\Big] \nonumber\\ &=& \sqrt{{\lambda_{j}\over3}}-{\eta\over3}+{\beta\over 2}+{\beta\over2\sqrt 3}\Big({3\over4}\beta-\eta \Big){1\over\sqrt\lambda_{j}}
+\Big({5\over {27}}\eta^3-\frac43\eta^2\beta+\frac94\eta\beta^2-{9\over8}\beta^3\Big){1\over\lambda_{j}} +O\Big({1\over{\lambda_{j}^{3/2}}}\Big) \,,\quad j\to\infty\,.\nonumber \end{eqnarray}
In a similar way we get \begin{eqnarray*} &&-{q_{j}\over 2}-\sqrt{{q_{j}^{2}\over 4}+{{p_{j}^{3}}\over {27}}}\\ &=&-{\eta^3\over {27}}-\Big({\eta\over 3}-{\beta\over 2}\Big)\lambda_{j}-\sqrt{{{\lambda_{j}^{3}}\over{27}}+ \Big({2\over{27}}\eta^2+{\beta^2\over4}-{\eta\beta\over 3}\Big)\lambda_{j}^{2} +{\eta^3\over 27}(\eta-\beta)\lambda_{j} }\\ &=&-{\eta^3\over {27}}-\Big({\eta\over 3}-{\beta\over 2}\Big)\lambda_{j}-\Big({{\lambda_{j}}\over{3}}\Big)^{3/2}\sqrt{1+\Big({2\over{27}}\eta^2+{\beta^2\over4}-{\eta\beta\over 3}\Big){27\over\lambda_{j}}+\eta^3(\eta-\beta){1\over\lambda_{j}^2}}\\ &=&-\Big({{\lambda_{j}}\over{3}}\Big)^{3/2}\left[{\eta^3\over {\sqrt{27}\lambda_{j}^{3/2}}}+\Big({\eta\over 3}-{\beta\over 2}\Big){\sqrt{27}\over\sqrt{\lambda_{j}}}+\sqrt{1+\Big({2\over{27}}\eta^2+{\beta^2\over4}-{\eta\beta\over 3}\Big){27\over\lambda_{j}}+\eta^3(\eta-\beta){1\over\lambda_{j}^2}}\ \right] \\ &=&-\Big({{\lambda_{j}}\over{3}}\Big)^{3/2}\Big[{\eta^3\over {\sqrt{27}\lambda_{j}^{3/2}}}+\Big({\eta\over 3}-{\beta\over 2}\Big){\sqrt{27}\over\sqrt{\lambda_{j}}}+1+{27\over2}\Big({2\over{27}}\eta^2+{\beta^2\over4}-{\eta\beta\over 3}\Big){1\over\lambda_{j}}+O\Big({1\over{\lambda_{j}^{2}}}\Big)\Big] \\ &=&-\Big({{\lambda_{j}}\over{3}}\Big)^{3/2}\Big[1+\Big({\eta\over 3}-{\beta\over 2}\Big){\sqrt{27}\over\sqrt{\lambda_{j}}}+\Big(\eta^2+{27\over8}\beta^2-{9\over 2}\eta\beta\Big){1\over\lambda_{j}}+{\eta^3\over {\sqrt{27}\lambda_{j}^{3/2}}}+O\Big({1\over{\lambda_{j}^{2}}}\Big)\Big]\,,\qquad j\to \infty\,. \end{eqnarray*} Therefore, using again (\ref{eq:1/3}), we have \begin{eqnarray}\label{eq:zj} z_{j}&=& \left(-{q_{j}\over 2}-\sqrt{{q_{j}^{2}\over 4}+{{p_{j}^{3}}\over{27}}}\ \right)^{1/ 3}\\ &=& -\sqrt{{\lambda_{j}\over3}} \Big[1+\Big({\eta\over 3}-{\beta\over 2}\Big){3^{3/2}\over\sqrt{\lambda_{j}}}+\Big(\eta^2+{27\over8}\beta^2-{9\over 2}\eta\beta\Big){1\over\lambda_{j}}+{\eta^3\over {(3\lambda_{j})^{3/2}}}+O\Big({1\over{\lambda_{j}^{2}}}\Big)\Big]^{1/ 3} \nonumber\\ &=& -\sqrt{{\lambda_{j}\over3}}\Big[1+\Big({\eta\over3}-{\beta\over 2}\Big)\sqrt{{3\over\lambda_{j}}}+{1\over3}\Big(\eta^2+{27\over8}\beta^2-{9\over 2}\eta\beta\Big){1\over\lambda_{j}}+{\eta^3\over {3^{5/2}\lambda_{j}^{3/2}}}-3\Big({\eta\over3}- {\beta\over2}\Big)^2{1\over\lambda_{j}} \nonumber\\ && -{2\over\sqrt{3}}\Big({\eta\over 3}-{\beta\over 2}\Big)\Big(\eta^2+{27\over8}\beta^2-{9\over 2}\eta\beta\Big){1\over\lambda_{j}^{3/2}} +O\Big({1\over{\lambda_{j}^{2}}}\Big)\Big] \nonumber\\ &=& -\sqrt{{\lambda_{j}\over3}}-{\eta\over3}+{\beta\over 2}-{\beta\over2\sqrt 3}\Big({3\over4}\beta-\eta \Big){1\over\sqrt\lambda_{j}}
+\Big({5\over {27}}\eta^3-\frac43\eta^2\beta+\frac94\eta\beta^2-{9\over8}\beta^3\Big){1\over\lambda_{j}} +O\Big({1\over{\lambda_{j}^{3/2}}}\Big) \,,\quad j\to\infty\,.\nonumber \end{eqnarray} By (\ref{eq:yj}) and (\ref{eq:zj}) it follows
$$y_{j}+z_{j}=-{2\over 3}\eta+ \beta+2\Big({5\over {27}}\eta^3-\frac43\eta^2\beta+\frac94\eta\beta^2-{9\over8}\beta^3\Big){1\over\lambda_{j}} +O\Big({1\over{\lambda_{j}^{3/2}}}\Big)\,,\quad j\to\infty\,,$$
$$ y_{j}-z_{j}={2\over \sqrt{3}}\sqrt{\lambda_{j}}+{\beta\over\sqrt 3}\Big({3\over4}\beta-\eta \Big){1\over\sqrt\lambda_{j}}+O\Big({1\over{\lambda_{j}^{3/2}}}\Big)\,,\quad j\to\infty\,. $$ In virtue of (\ref{eq:si1})--(\ref{eq:si3}), the above relationships yield $$\sigma_{1,j}=-{2\over 3}\eta+ \beta+2\Big({5\over {27}}\eta^3-\frac43\eta^2\beta+\frac94\eta\beta^2-{9\over8}\beta^3\Big){1\over\lambda_{j}} +O\Big({1\over{\lambda_{j}^{3/2}}}\Big)\,,\quad j\to\infty\,,$$
\begin{eqnarray*} \sigma_{2,j}&=&{\eta\over 3}- {\beta\over 2}-\Big({5\over {27}}\eta^3-\frac43\eta^2\beta+\frac94\eta\beta^2-{9\over8}\beta^3\Big){1\over\lambda_{j}} +O\Big({1\over{\lambda_{j}^{3/2}}}\Big) \\ && + i\Big[\sqrt{\lambda_{j}}+{\beta\over2}\Big({3\over4}\beta-\eta \Big){1\over\sqrt\lambda_{j}}+O\Big({1\over{\lambda_{j}^{3/2}}}\Big)\Big]\,,\quad j\to\infty\,, \end{eqnarray*} \begin{eqnarray*} \sigma_{3,j}&=&{\eta\over 3}- {\beta\over 2}-\Big({5\over {27}}\eta^3-\frac43\eta^2\beta+\frac94\eta\beta^2-{9\over8}\beta^3\Big){1\over\lambda_{j}} +O\Big({1\over{\lambda_{j}^{3/2}}}\Big)\\ && - i\Big[\sqrt{\lambda_{j}}+{\beta\over2}\Big({3\over4}\beta-\eta \Big){1\over\sqrt\lambda_{j}}+O\Big({1\over{\lambda_{j}^{3/2}}}\Big)\Big]\,,\quad j\to\infty\,. \end{eqnarray*} Finally, using also the condition $\Lambda=\sigma-\eta/ 3$, we are able to write the solutions of equation (\ref{eq:char}), that is \begin{eqnarray}\label{eq:lambda1} \Lambda_{1,j}&=&\beta-\eta+2\Big({5\over {27}}\eta^3-\frac43\eta^2\beta+\frac94\eta\beta^2-{9\over8}\beta^3\Big){1\over\lambda_{j}} +O\Big({1\over{\lambda_{j}^{3/2}}}\Big)\nonumber\\ &=&\beta-\eta +O\Big({1\over{\lambda_{j}}}\Big)\,,\quad j\to\infty\,, \end{eqnarray} \begin{eqnarray}\label{eq:lambda2} \Lambda_{2,j}&=&- {\beta\over 2}-\Big({5\over {27}}\eta^3-\frac43\eta^2\beta+\frac94\eta\beta^2-{9\over8}\beta^3\Big){1\over\lambda_{j}} +O\Big({1\over{\lambda_{j}^{3/2}}}\Big)\nonumber\\ && + i\Big[\sqrt{\lambda_{j}}+{\beta\over2}\Big({3\over4}\beta-\eta \Big){1\over\sqrt\lambda_{j}}+O\Big({1\over{\lambda_{j}^{3/2}}}\Big)\Big] \nonumber\\ &=&- {\beta\over 2} +O\Big({1\over{\lambda_{j}}}\Big) + i\Big[\sqrt{\lambda_{j}}+{\beta\over2}\Big({3\over4}\beta-\eta \Big){1\over\sqrt\lambda_{j}}+O\Big({1\over{\lambda_{j}^{3/2}}}\Big)\Big]\,,\quad j\to\infty\,, \end{eqnarray} \begin{eqnarray}\label{eq:lambda3} \Lambda_{3,j}&=&- {\beta\over 2}-\Big({5\over {27}}\eta^3-\frac43\eta^2\beta+\frac94\eta\beta^2-{9\over8}\beta^3\Big){1\over\lambda_{j}} +O\Big({1\over{\lambda_{j}^{3/2}}}\Big)\nonumber\\ && - i\Big[\sqrt{\lambda_{j}}+{\beta\over2}\Big({3\over4}\beta-\eta \Big){1\over\sqrt\lambda_{j}}+O\Big({1\over{\lambda_{j}^{3/2}}}\Big)\Big]\nonumber\\ &=&- {\beta\over 2} +O\Big({1\over{\lambda_{j}}}\Big) - i\Big[\sqrt{\lambda_{j}}+{\beta\over2}\Big({3\over4}\beta-\eta \Big){1\over\sqrt\lambda_{j}}+O\Big({1\over{\lambda_{j}^{3/2}}}\Big)\Big]\,,\quad j\to\infty\,. \end{eqnarray}
Therefore, we can write the solution of (\ref{eq:third}) in the following way \begin{equation}\label{eq:fj} f_{j}(t)=C_{1,j}e^{t\Lambda_{1,j}}+C_{2,j}e^{t\Lambda_{2,j}}+C_{3,j}e^{t\Lambda_{3,j}}\,, \end{equation} where $C_{k,j}$, $k=1,2,3$, are complex numbers to determine. To find the coefficients $C_{k,j}$ we impose that $f_{j}$ verifies the initial conditions \begin{equation}\label{ini_con1} f_{j}(0)=\alpha_{j}\,, \qquad f_{j}^{'}(0)=\gamma_{j}\,,\qquad f_{j}^{''}(0)=-\alpha_{j}\lambda_{j}\,, \end{equation} so we obtain the system \begin{equation}\label{vandermonde} \left \{\begin{array}{l} \phantom{\Lambda_{1,j}}C_{1,j}+\phantom{\Lambda_{1,j}}C_{2,j}+\phantom{\Lambda_{1,j}}C_{3,j}=\alpha_{j}\,,\\ \Lambda_{1,j}C_{1,j}+\Lambda_{2,j}C_{2,j}+\Lambda_{3,j}C_{3,j}=\gamma_{j}\,,\\ \Lambda_{1,j}^2C_{1,j}+\Lambda_{2,j}^2C_{2,j}+\Lambda_{3,j}^2C_{3,j}=-\alpha_{j}\lambda_{j}\,. \end{array}\right . \end{equation} The matrix $C$ of the coefficients of system (\ref{vandermonde}) has determinant given by $$ \mbox{det}(C)=(\Lambda_{2,j}-\Lambda_{1,j})(\Lambda_{3,j}-\Lambda_{1,j})(\Lambda_{3,j}-\Lambda_{2,j})\,, $$ so we obtain \begin{eqnarray} \nonumber C_{1,j} &=&{\alpha_{j}\Lambda_{2,j}\Lambda_{3,j}(\Lambda_{3,j}-\Lambda_{2,j})-\gamma_{j}(\Lambda_{3,j}^2-\Lambda_{2,j}^2)- \alpha_{j}\lambda_{j}(\Lambda_{3,j}-\Lambda_{2,j})\over(\Lambda_{2,j}-\Lambda_{1,j})(\Lambda_{3,j}-\Lambda_{1,j})(\Lambda_{3,j}-\Lambda_{2,j})}\\\nonumber &=&{\alpha_{j}\Lambda_{2,j}\Lambda_{3,j}-\gamma_{j}(\Lambda_{3,j}+\Lambda_{2,j})- \alpha_{j}\lambda_{j}\over(\Lambda_{2,j}-\Lambda_{1,j})(\Lambda_{3,j}-\Lambda_{1,j})}\,, \end{eqnarray} \begin{eqnarray} \nonumber C_{2,j} &=&{\gamma_{j}(\Lambda_{3,j}^2-\Lambda_{1,j}^2)+\alpha_{j}\lambda_{j}(\Lambda_{3,j}-\Lambda_{1,j})-\alpha_{j}\Lambda_{1,j}\Lambda_{3,j}(\Lambda_{3,j}-\Lambda_{1,j}) \over(\Lambda_{2,j}-\Lambda_{1,j})(\Lambda_{3,j}-\Lambda_{1,j})(\Lambda_{3,j}-\Lambda_{2,j})}\\\nonumber &=&{\gamma_{j}(\Lambda_{3,j}+\Lambda_{1,j})+\alpha_{j}\lambda_{j}-\alpha_{j}\Lambda_{1,j}\Lambda_{3,j}\over(\Lambda_{2,j}-\Lambda_{1,j})(\Lambda_{3,j}-\Lambda_{2,j})}\,, \end{eqnarray} \begin{eqnarray} \nonumber C_{3,j} &=&{-\alpha_{j}\lambda_{j}(\Lambda_{2,j}-\Lambda_{1,j})-\gamma_{j}(\Lambda_{2,j}^2-\Lambda_{1,j}^2)+\alpha_{j}\Lambda_{1,j}\Lambda_{2,j}(\Lambda_{2,j}-\Lambda_{1,j}) \over(\Lambda_{2,j}-\Lambda_{1,j})(\Lambda_{3,j}-\Lambda_{1,j})(\Lambda_{3,j}-\Lambda_{2,j})}\\\nonumber &=&{-\alpha_{j}\lambda_{j}-\gamma_{j}(\Lambda_{2,j}+\Lambda_{1,j})+\alpha_{j}\Lambda_{1,j}\Lambda_{2,j}\over(\Lambda_{3,j}-\Lambda_{1,j})(\Lambda_{3,j}-\Lambda_{2,j})} \,. \end{eqnarray}
Plugging \eqref{eq:lambda1}--\eqref{eq:lambda3} into the above identities, we obtain the expressions of coefficients $ C_{k,j}$. Indeed, \begin{eqnarray}\label{eq:Cj1} \nonumber C_{1,j} &=&{\alpha_{j}\left\{\left[-{\beta\over 2}+O\Big({1\over{\lambda_{j}}}\Big)\right]^2+ \Big[\sqrt{\lambda_{j}}+{\beta\over2}\Big({3\over4}\beta-\eta \Big){1\over\sqrt\lambda_{j}}+O\Big({1\over{\lambda_{j}^{2}}}\Big)\Big]^2\right\} +\gamma_{j}\beta+\gamma_jO\Big({1\over{\lambda_{j}}}\Big)-\alpha_{j}\lambda_{j}\over \left[\eta-{3\over2}\beta+O\Big({1\over{\lambda_{j}}}\Big)\right]^2+ \Big[\sqrt{\lambda_{j}}+{\beta\over2}\Big({3\over4}\beta-\eta \Big){1\over\sqrt\lambda_{j}}+O\Big({1\over{\lambda_{j}^{2}}}\Big)\Big]^2}\\\nonumber &=&{\alpha_{j}\beta^2- \alpha_{j}\eta\beta+\gamma_{j}\beta+O\Big({1\over{\lambda_{j}}}\Big) \over\Big(\eta-{3\over 2}\beta\Big)^2+ {\lambda_{j}}+{3\over4}\beta^2-\eta\beta+O\Big({1\over{\lambda_{j}}}\Big)} ={\alpha_{j}\beta^2- \alpha_{j}\eta\beta+\gamma_{j}\beta+O\Big({1\over{\lambda_{j}}}\Big) \over {\lambda_{j}}+\eta^2+3\beta^2-4\eta\beta+O\Big({1\over{\lambda_{j}}}\Big)} \\ &=&{\alpha_{j}\beta^2- \alpha_{j}\eta\beta+\gamma_{j}\beta+O\Big({1\over{\lambda_{j}}}\Big) \over {1+(\eta^2+3\beta^2-4\eta\beta)\frac{1}{\lambda_{j}}+O\Big({1\over{\lambda^2_{j}}}\Big)}}\ \frac{1}{\lambda_{j}}\,. \end{eqnarray} We note that $C_{1,j}\in{\mathbb R}$. To write explicitly $C_{2,j}$ we observe that \begin{eqnarray*} &&(\Lambda_{2,j}-\Lambda_{1,j})(\Lambda_{3,j}-\Lambda_{2,j}) \\\nonumber &=& -2i\Big[\sqrt{\lambda_{j}}+{\beta\over2}\Big({3\over4}\beta-\eta \Big){1\over\sqrt\lambda_{j}}+O\Big({1\over{\lambda_{j}^{2}}}\Big)\Big]\Big\{\eta-{3\over2}\beta +O\Big({1\over{\lambda_{j}}}\Big) + i\Big[\sqrt{\lambda_{j}}+{\beta\over2}\Big({3\over4}\beta-\eta \Big){1\over\sqrt\lambda_{j}}+O\Big({1\over{\lambda_{j}^{2}}}\Big)\Big]\Big\}\\\nonumber &=& 2\lambda_{j}+{3\over2}\beta^2-2\beta\eta+O\Big({1\over{\lambda_{j}}}\Big)-i\Big[(2\eta-3\beta)\sqrt{\lambda_{j}}+O\Big({1\over{\sqrt{\lambda_{j}}}}\Big)\Big] \,, \end{eqnarray*} whence \begin{eqnarray*} C_{2,j} &=&{\gamma_{j}\left\{{\beta\over 2}-\eta+O\Big({1\over{\lambda_{j}}}\Big)- i\Big[\sqrt{\lambda_{j}}+{\beta\over2}\Big({3\over4}\beta-\eta \Big){1\over\sqrt\lambda_{j}}+O\Big({1\over{\lambda_{j}^{2}}}\Big)\Big]\right\}+\alpha_{j}\lambda_{j}\over 2\lambda_{j}+{3\over2}\beta^2-2\beta\eta+O\Big({1\over{\lambda_{j}}}\Big)-i\Big[(2\eta-3\beta)\sqrt{\lambda_{j}}+O\Big({1\over{\sqrt{\lambda_{j}}}}\Big)\Big]} \\ &&-{\alpha_{j}\Big[\beta-\eta+O\Big({1\over{\lambda_{j}}}\Big)\Big]\left\{- {\beta\over 2} +O\Big({1\over{\lambda_{j}}}\Big) - i\Big[\sqrt{\lambda_{j}}+{\beta\over2}\Big({3\over4}\beta-\eta \Big){1\over\sqrt\lambda_{j}}+O\Big({1\over{\lambda_{j}^{2}}}\Big)\Big]\right\}\over 2\lambda_{j}+{3\over2}\beta^2-2\beta\eta+O\Big({1\over{\lambda_{j}}}\Big)-i\Big[(2\eta-3\beta)\sqrt{\lambda_{j}}+O\Big({1\over{\sqrt{\lambda_{j}}}}\Big)\Big]}\,. \end{eqnarray*} Therefore, \begin{eqnarray}\label{eq:Cj2} \nonumber C_{2,j} &=&{ \alpha_{j}\lambda_{j}+\gamma_{j}({\beta\over 2}-\eta)+\alpha_{j}(\beta-\eta){\beta\over 2} +\alpha_{j}O\Big({1\over{\lambda_{j}}}\Big) +\gamma_{j}O\Big({1\over{\lambda_{j}}}\Big) \over 2\lambda_{j}+{3\over2}\beta^2-2\beta\eta+O\Big({1\over{\lambda_{j}}}\Big)-i\Big[(2\eta-3\beta)\sqrt{\lambda_{j}}+O\Big({1\over{\sqrt{\lambda_{j}}}}\Big)\Big]} \nonumber\\ && -{i\Big[(\gamma_{j}-\alpha_{j}\beta+\alpha_{j}\eta)\sqrt{\lambda_{j}} +\alpha_{j}O\Big({1\over{\sqrt{\lambda_{j}}}}\Big) +\gamma_{j}O\Big({1\over{\sqrt{\lambda_{j}}}}\Big)\Big] \over 2\lambda_{j}+{3\over2}\beta^2-2\beta\eta+O\Big({1\over{\lambda_{j}}}\Big)-i\Big[(2\eta-3\beta)\sqrt{\lambda_{j}}+O\Big({1\over{\sqrt{\lambda_{j}}}}\Big)\Big]}\,, \end{eqnarray} from which it follows that there exist some $c_1\,,c_2>0$ such that
\begin{eqnarray}\label{eq:|Cj2|} c_1\Big(\alpha_{j}^2+\gamma_{j}^2O\Big({1\over{\lambda_{j}}}\Big)\Big)\le
|C_{2,j}|^2 \le c_2\Big(\alpha_{j}^2+\gamma_{j}^2O\Big({1\over{\lambda_{j}}}\Big)\Big)\,. \end{eqnarray} Similarly, \begin{eqnarray*} &&(\Lambda_{3,j}-\Lambda_{1,j})(\Lambda_{3,j}-\Lambda_{2,j}) = 2\lambda_{j}+{3\over2}\beta^2-2\beta\eta+O\Big({1\over{\lambda_{j}}}\Big)+i\Big[(2\eta-3\beta)\sqrt{\lambda_{j}}+O\Big({1\over{\sqrt{\lambda_{j}}}}\Big)\Big] \,, \end{eqnarray*} and hence \begin{eqnarray*} C_{3,j} &=&{-\alpha_{j}\lambda_{j}-\gamma_{j}\left\{{\beta\over 2}-\eta+O\Big({1\over{\lambda_{j}}}\Big)+ i\Big[\sqrt{\lambda_{j}}+{\beta\over2}\Big({3\over4}\beta-\eta \Big){1\over\sqrt\lambda_{j}}+O\Big({1\over{\lambda_{j}^{2}}}\Big)\Big]\right\}\over 2\lambda_{j}+{3\over2}\beta^2-2\beta\eta+O\Big({1\over{\lambda_{j}}}\Big)+i\Big[(2\eta-3\beta)\sqrt{\lambda_{j}}+O\Big({1\over{\sqrt{\lambda_{j}}}}\Big)\Big]}\\\nonumber &&+{\alpha_{j}\Big[\beta-\eta+O\Big({1\over{\lambda_{j}}}\Big)\Big]\left\{-{\beta\over 2}+O\Big({1\over{\lambda_{j}}}\Big)+ i\Big[\sqrt{\lambda_{j}}+{\beta\over2}\Big({3\over4}\beta-\eta \Big){1\over\sqrt\lambda_{j}}+O\Big({1\over{\lambda_{j}^{2}}}\Big)\Big]\right\}\over 2\lambda_{j}+{3\over2}\beta^2-2\beta\eta+O\Big({1\over{\lambda_{j}}}\Big)+i\Big[(2\eta-3\beta)\sqrt{\lambda_{j}}+O\Big({1\over{\sqrt{\lambda_{j}}}}\Big)\Big]} \,. \end{eqnarray*} Moreover, \begin{eqnarray}\label{eq:Cj3} \nonumber C_{3,j} &=&-{ \alpha_{j}\lambda_{j}+\gamma_{j}({\beta\over 2}-\eta)+\alpha_{j}(\beta-\eta){\beta\over 2} +\alpha_{j}O\Big({1\over{\lambda_{j}}}\Big) +\gamma_{j}O\Big({1\over{\lambda_{j}}}\Big) \over 2\lambda_{j}+{3\over2}\beta^2-2\beta\eta+O\Big({1\over{\lambda_{j}}}\Big)+i\Big[(2\eta-3\beta)\sqrt{\lambda_{j}}+O\Big({1\over{\sqrt{\lambda_{j}}}}\Big)\Big]} \nonumber\\ && -{i\Big[(\gamma_{j}-\alpha_{j}\beta+\alpha_{j}\eta)\sqrt{\lambda_{j}} +\alpha_{j}O\Big({1\over{\sqrt{\lambda_{j}}}}\Big) +\gamma_{j}O\Big({1\over{\sqrt{\lambda_{j}}}}\Big)\Big] \over 2\lambda_{j}+{3\over2}\beta^2-2\beta\eta+O\Big({1\over{\lambda_{j}}}\Big)+i\Big[(2\eta-3\beta)\sqrt{\lambda_{j}}+O\Big({1\over{\sqrt{\lambda_{j}}}}\Big)\Big]}\,, \end{eqnarray} and
\begin{eqnarray}\label{eq:|Cj3|} c_1\Big(\alpha_{j}^2+\gamma_{j}^2O\Big({1\over{\lambda_{j}}}\Big)\Big)\le
|C_{3,j}|^2 \le c_2\Big(\alpha_{j}^2+\gamma_{j}^2O\Big({1\over{\lambda_{j}}}\Big)\Big)\,. \end{eqnarray} By
\eqref{eq:Cj1}, \eqref{eq:|Cj2|} and \eqref{eq:|Cj3|}, one deduces that there exists a positive constant $c$ such that for any $j\in{\mathbb N}$ we have \begin{equation}\label{eq:C1overC2} {{\vert C_{1,j}\vert}\over{\vert C_{2,j}\vert}}\le {c \over {{\lambda_j}}}\,, \qquad {{\vert C_{1,j}\vert}\over{\vert C_{3,j}\vert}}\le {c \over {{\lambda_j}}} \,. \end{equation} In conclusion, thanks into account \eqref{eq:fj} we have proved that the solution $v(t)$ of the Cauchy problem \eqref{eq:v}--\eqref{eq:incond} can be written as \begin{equation*} v(t)=\sum_{j=1}^{\infty}\big(C_{1,j}e^{t\Lambda_{1,j}}+C_{2,j}e^{t\Lambda_{2,j}}+C_{3,j}e^{t\Lambda_{3,j}}(t)\big)w_{j}\qquad t\ge 0\,, \end{equation*} where $\Lambda_{k,j}$ and $C_{k,j}$ are given by \eqref{eq:lambda1}--\eqref{eq:lambda3} and \eqref{eq:Cj1}--\eqref{eq:Cj3} respectively, and condition \eqref{eq:C1overC2} holds.
We will show as the function $v$ can be written in the form \begin{equation}\label{eq:vsum}
v(t)=\sum_{n=-\infty}^{\infty}\big(C_ne^{i\omega_nt}+R_ne^{r_nt}\big)w_{|n|}\qquad t\ge 0\,, \end{equation} where $C_n\,,\omega_n\in{\mathbb C}$ and $R_n\,,r_n\in{\mathbb R}$. Indeed, we define $\omega_n$ as the complex numbers having real and imaginary parts given by \begin{equation*} \Re\omega_n:=
\mbox{sign}(n)\sqrt{\lambda_{|n|}}+\mbox{sign}(n){\beta\over2}\Big({3\over4}\beta-\eta
\Big){1\over\sqrt{\lambda_{|n|}}}+O\Big({1\over{\lambda_{|n|}^{3/2}}}\Big)
\,\qquad\qquad\ |n|\ge1\,, \end{equation*}
\begin{equation*} \Im\omega_n:=
{\beta\over 2}+\Big({5\over {27}}\eta^3-\frac43\eta^2\beta+\frac94\eta\beta^2-{9\over8}\beta^3\Big){1\over\lambda_{|n|}}
+O\Big({1\over{\lambda_{|n|}^{3/2}}}\Big)
\,\qquad\qquad\ |n|\ge1\,. \end{equation*} Moreover, we set \begin{equation*}
r_n:=\beta-\eta+2\Big({5\over {27}}\eta^3-\frac43\eta^2\beta+\frac94\eta\beta^2-{9\over8}\beta^3\Big){1\over\lambda_{|n|}}
+O\Big({1\over{\lambda_{|n|}^{3/2}}}\Big)\,,\qquad \ |n|\ge1\,, \end{equation*}
\begin{equation*} C_n:=\left \{\begin{array}{l} C_{2,n}\,\qquad\qquad \mbox{if}\ n\ge1\,,\\ \\ C_{3,-n}\,\qquad\quad \mbox{if}\ n\le-1\,, \end{array}\right . \end{equation*} $$ R_n:=C_{1,n}\quad\,n\ge1\,,\qquad w_0=C_0=R_n=0\qquad n\le 0. $$ \vskip0.5cm Finally, applying the abstract results of sections \ref{se:dir} and \ref{se:inv} we can show our reachability result. \begin{theorem}\label{th:reachres} Let $\eta>3\beta/2$. For any $T>2\pi$, $u_0\in L^{2}(0,\pi)$ and $u_1\in H^{-1}(0,\pi)$ there exists $g\in L^2(0,T)$ such that the weak solution $u$ of problem \begin{equation}\label{eq:problem-usix} \begin{cases} \displaystyle u_{tt}(t,x) - u_{xx}(t,x)+\beta\int_0^t\ e^{-\eta(t-s)} u_{xx}(s,x)ds= 0\,, \qquad t\in (0,T),\,\, x\in (0,\pi), \\ u(0,x)=u_{t}(0,x)=0,\qquad x\in (0,\pi), \\ u(t,0)=0\quad u(t,\pi)=g(t)\,,\quad t\in (0,T), \end{cases} \end{equation} verifies the final conditions \begin{equation}\label{eq:findataT} u(T,x)=u_0(x)\,,\qquad u_{t}(T,x)=u_1(x)\,,\quad x\in (0,\pi)\,. \end{equation} \end{theorem} \begin{Proof}
To prove our claim, we apply the HUM method described in section \ref{se:HUM}. Let $ X= L^2(0,\pi ) $ be endowed with the usual scalar product and norm $$
\|u\|:=\left(\int_0^\pi |u(x)|^{2}\ dx\right)^{1/2}\qquad u\in L^2(0,\pi)\,. $$ We consider the operator $A:D(A)\subset X\to X$ defined by $$ \begin{array}{l} D(A)=H^2(0,\pi )\cap H_0^1(0,\pi ) \\ \\ Au=\displaystyle -u_{xx}\qquad u\in D(A)\,. \end{array} $$ It is well-known that $A$ is a self-adjoint positive
operator on $X$ with dense domain $D(A)$, $\{j^2\}_{j\ge1}$ is the sequence of eigenvalues for $A$ and the sequence of the corresponding eigenvectors is $\{\sin(jx)\}_{j\ge1}$. The fractional power $\sqrt A$ of $A$ is well defined and $D(\sqrt A) = H^1_0(0,\pi)$. Therefore, we can apply our spectral analysis to the adjoint problem of (\ref{eq:problem-usix}). Indeed, the solution $z$ of the adjoint problem can be written in the form \eqref{eq:vsum}, that is \begin{equation*}
z(t,x)=\sum_{n=-\infty}^{\infty}\big(C_ne^{i\omega_n(T-t)}+R_ne^{r_n(T-t)}\big)\sin(|n|x)\qquad (t,x)\in [0,T]\times [0,\pi]\,, \end{equation*} whence \begin{equation*}
z_x(t,\pi)=\sum_{n=-\infty}^{\infty}(-1)^n|n|\big(C_ne^{i\omega_n(T-t)}+R_ne^{r_n(T-t)}\big)\qquad (t,x)\in [0,T]\times [0,\pi]\,. \end{equation*}
Since $\eta>3\beta/2$ we can apply theorems \ref{th:diringham} and \ref{th:inv.ingham1} to function $z_x(t,\pi)$. Therefore, thanks to inequalities \eqref{eq:diringham} and \eqref{eq:inv.ingham} the uniqueness theorem \ref{th:uniqueness} holds true. In addition, by estimates \eqref{eq:|Cj2|} and \eqref{eq:|Cj3|} we have that $$
c_1(\|v_0\|^2_{H^1_0}+\|v_1\|^2) \le
\int_0^T |z_x(t,\pi)|^2\ dt
\le c_2(\|v_0\|^2_{H^1_0}+\|v_1\|^2)\,, $$ so the space $F$ introduced at the end of section \ref{se:HUM} is $H^1_0(0,\pi)\times L^2(0,\pi)$. So, our proof is complete. \end{Proof}
\appendix \section{Appendix}\label {se:app}
To prove theorem \ref{th:dir.ingham} we need to introduce an auxiliary function. Let $T>0$. We define \begin{equation}\label{eq:kcos} k^*(t):=\left \{\begin{array}{l}
\cos \frac{\pi t}{2T}\,\qquad\qquad \mbox{if}\ |t|\le T\,,\\ \\
0\,\qquad\qquad\quad\ \ \ \ \mbox{if}\ |t|>T\,. \end{array}\right . \end{equation} For the reader's convenience, we list some easy to check properties of $k^*$ in the following lemma. \begin{lemma} \label{th:k} Set \begin{equation*}\label{} K^*(u):=\frac{4T\pi}{\pi^2-4T^2u^2}\,,\qquad u\in {\mathbb C}\,, \end{equation*} the following properties hold for any $u\in {\mathbb C}$ \begin{equation}\label{eqn:k1} \int_{-\infty}^{\infty} k^*(t)e^{iu t}dt=\cos(uT)K^*(u)\,, \end{equation} \begin{equation}\label{eqn:k2bis} \overline{K^*(u)}=K^*(\overline{u})\,, \end{equation} \begin{equation}\label{eqn:k2}
\big|K^*(u)\big|=\big|K^*(\overline{u})\big|\,, \end{equation} \begin{equation}\label{eqn:k3}
\big|K^*(u)\big|\le \frac{4T\pi}{|4T^2(\Re u)^2-4T^2(\Im u)^2-\pi^2|}\,. \end{equation} \end{lemma}
\begin{Proofv} First, without loss of generality, it may be assumed that the sequence $\{{\Im}\omega_n\}$ converges to $0$, that is \begin{equation}\label{eq:dnull} \alpha=0\,. \end{equation} Indeed, suppose for a moment that we have proved inequality (\ref{eq:dir-cosine}) under this extra condition. For the general case $\alpha\not=0$, we consider the function \begin{equation*} g(t):=e^{\alpha t}f(t)=\sum_{n=-\infty}^{\infty}\Big(C_ne^{i\omega_n't}+R_ne^{(r_n+\alpha)t}\Big)\,, \end{equation*} where $\omega_n'=\omega_n-i\alpha$ and
$\displaystyle\lim_{|n|\to\infty}{\Im}\omega_n'=0$. So, inequality (\ref{eq:dir-cosine}) holds for $g$, that is \begin{equation*}
\int_{-T}^{T} |g(t)|^2 d t\le c_2(T)\sum_{|n|\ge n_0}|C_n|^2\,. \end{equation*} Since $f(t)=e^{-\alpha t}g(t)$, we have $$
\vert f(t)\vert\le \max \{e^{\alpha T},e^{-\alpha T}\}|g(t)|\,,\qquad\forall t\in[-T,T]\,, $$ whence it follows $$ \int_{-T}^{T} \vert f(t)\vert ^2 d t
\le \max \{e^{2\alpha T},e^{-2\alpha T}\}\int_{-T}^{T} |g(t)|^2 d t
\le \max \{e^{2\alpha T},e^{-2\alpha T}\}c_2(T)\sum_{|n|\ge n_0}|C_n|^2\,, $$ that is (\ref{eq:dir-cosine}) also holds for $f$.
Let $k^*(t)$ be the function defined by (\ref{eq:kcos}). If we use (\ref{eqn:k1}), then we have \begin{eqnarray}\label{eqn:sum-c}
&&\int_{-\infty}^{\infty} k^*(t)|f(t)|^2\ dt \\ &=& \int_{-\infty}^{\infty} k^*(t)\sum_{n}\Big(C_ne^{ i\omega_nt}+R_ne^{r_nt}\Big) \sum_{m}\Big(\overline{C}_me^{-i\overline{\omega}_mt}+R_me^{r_mt}\Big)\ dt\nonumber\\ &=&\sum_{n, m}C_n\overline{C}_m \cos(( \omega_n-\overline{\omega}_m)T)K^*( \omega_n-\overline{\omega}_m)+ \sum_{n, m}C_nR_m\cosh(( i\omega_n+r_m)T)K^*( \omega_n-ir_m) \nonumber\\ && +\sum_{n, m}R_n\overline{C}_m\cosh(( r_n-i\overline{\omega}_m)T)K^*(ir_n+\overline{\omega}_m) \nonumber\\ && +\sum_{n, m}R_nR_m\cosh((r_n+r_m)T)K^*(i(r_n+r_m))\nonumber\,. \end{eqnarray} We may write the first sum on the right-hand side as follows \begin{eqnarray*} &&\sum_{n, m}C_n\overline{C}_m \cos(( \omega_n-\overline{\omega}_m)T)K^*( \omega_n-\overline{\omega}_m)\\ &=&
\sum_{n}|C_n|^2 \cosh(2\Im \omega_nT)K^*( \omega_n-\overline{\omega}_n) +\sum_{n, m,\,n\not=m}C_n\overline{C}_m\cos(( \omega_n-\overline{\omega}_m)T)K^*( \omega_n-\overline{\omega}_m) \,. \nonumber\\ \end{eqnarray*} Plugging the above identity into (\ref{eqn:sum-c}) and using (\ref{eqn:k2bis}), we obtain \begin{eqnarray*}
&&\int_{-\infty}^{\infty} k^*(t)|f(t)|^2\ dt \\
&=&\sum_{n}|C_n|^2 \cosh(2\Im \omega_nT)K^*( \omega_n-\overline{\omega}_n) +\sum_{n, m,\,n\not=m}C_n\overline{C}_m\cos(( \omega_n-\overline{\omega}_m)T)K^*( \omega_n-\overline{\omega}_m) \nonumber\\ && + 2\sum_{n, m}R_m\Re \big[C_n\cosh(( i\omega_n+r_m)T)K^*( \omega_n-ir_m)\big] \nonumber\\ && +\sum_{n, m}R_nR_m\cosh((r_n+r_m)T)K^*(i(r_n+r_m))\nonumber\,. \end{eqnarray*}
Notice that the terms on the right-hand side of the previous identity are real. Therefore, applying the elementary estimates $\theta\le |\theta|$, $\theta\in{\mathbb R}$, and
$|\cosh z|\le \cosh (\Re z)$, $z\in{\mathbb C}$, we obtain \begin{eqnarray}\label{eqn:sum-c2}
&&\int_{-\infty}^{\infty} k^*(t)|f(t)|^2\ dt\\
&\le&\sum_{n}|C_n|^2\cosh(2\Im \omega_nT)K^*( \omega_n-\overline{\omega}_n)
+\sum_{n, m,\,n\not=m}|C_n| |C_m|\cosh((\Im \omega_n+\Im \omega_m)T)\ |K^*( \omega_n-\overline{\omega}_m)| \nonumber\\ &&
+2\sum_{n, m}|C_n|\ |R_m|\cosh((\Im \omega_n-r_m)T) |K^*( \omega_n-ir_m)|
+\sum_{n, m}|R_n|\ |R_m|\cosh((r_n+r_m)T) K^*(i(r_n+r_m))\,. \nonumber \end{eqnarray} Since the sequences $\{{\Im}\omega_n\}$ and $\{r_n\}$ are bounded, there exists a positive constant $c(T)$ such that for any $n,m\in{\mathbb Z}$ we have $$ \cosh((\Im \omega_n+\Im \omega_m)T)+\cosh((\Im \omega_n-r_m)T)+\cosh((r_n+r_m)T)\le c(T)\,, $$ and hence from (\ref{eqn:sum-c2}) it follows \begin{eqnarray*}
&&\int_{-\infty}^{\infty} k^*(t)|f(t)|^2\ dt\\
&\le&c(T)\sum_{n}|C_n|^2K^*( \omega_n-\overline{\omega}_n)
+c(T)\sum_{n, m,\,n\not=m}|C_n| |C_m|\ |K^*( \omega_n-\overline{\omega}_m)| \nonumber\\ &&
+2c(T)\sum_{n, m}|C_n|\ |R_m| |K^*( \omega_n-ir_m)|
+c(T)\sum_{n, m}|R_n|\ |R_m| K^*(i(r_n+r_m))\,. \nonumber \end{eqnarray*} In virtue of the definition of $K^*$ we have $$ K^*( \omega_n-\overline{\omega}_n)=\frac{4T\pi}{\pi^2+16T^2(\Im\omega_n)^2}\le\frac{4T}{\pi}\,, $$ whence \begin{eqnarray}\label{eq:sum-dc3}
&&\int_{-\infty}^{\infty} k^*(t)|f(t)|^2\ dt\\
&\le&\frac{4T}{\pi}c(T)\sum_{n}|C_n|^2
+c(T)\sum_{n, m,\,n\not=m}|C_n| |C_m|\ |K^*( \omega_n-\overline{\omega}_m)| \nonumber\\ &&
+2c(T)\sum_{n, m}|C_n|\ |R_m| |K^*( \omega_n-ir_m)|
+c(T)\sum_{n, m}|R_n|\ |R_m| K^*(i(r_n+r_m))\,. \nonumber \end{eqnarray}
To evaluate the second sum on the right-hand side of the above inequality, we note that, in virtue of (\ref{eqn:k2}), we have $$
|K^*( \omega_n-\overline{\omega}_m)|=|K^*( \overline{\omega}_n-\omega_m)|\,, $$ whence \begin{eqnarray}\label{eq:sum-c4d} &&
\sum_{n, m,\,n\not=m}|C_n|\ |C_m|\ |K^*( \omega_n-\overline{\omega}_m)|\\ &\le&
\frac{1}{2}\sum_{n, m,\,n\not=m}\ \big(|C_n|^2+ |C_m|^2\big)|K^*( \omega_n-\overline{\omega}_m)|\nonumber\\ &=&
\frac{1}{2}\sum_{n}\ |C_n|^2\sum_{m,m\not=n}\ |K^*( \omega_n-\overline{\omega}_m)|
+\frac{1}{2}\sum_{m}\ |C_m|^2\sum_{n,n\not=m}\ |K^*( \omega_n-\overline{\omega}_m)|\nonumber\\ &=&
\frac{1}{2}\sum_{n}\ |C_n|^2\sum_{m,m\not=n}\ |K^*( \omega_n-\overline{\omega}_m)|
+\frac{1}{2}\sum_{m}\ |C_m|^2\sum_{n,n\not=m}\ |K^*( \omega_m-\overline{\omega}_n)| \nonumber\\
&=& \sum_{n}\ |C_n|^2\sum_{m,m\not=n}\ |K^*( \omega_n-\overline{\omega}_m)|\nonumber \,. \end{eqnarray} Now, using (\ref{eqn:k3}) we get
\begin{eqnarray}\label{eq:modulo1}
&&\sum_{m,m\not=n} |K^*( \omega_n-\overline{\omega}_m)|\\\nonumber &\le& 4T\pi\sum_{m,m\not=n}\frac{1}
{\Big|4T^2(\Re \omega_n-\Re \omega_m)^2-4T^2(\Im \omega_n+\Im \omega_m)^2-\pi^2\Big|}. \end{eqnarray} From assumption (\ref{eq:h1d}) it follows \begin{equation}\label{eq:gapd}
|\Re \omega_n-\Re \omega_m|\ge \gamma|n-m|\,,\qquad\forall |n|\,,|m|\ge n'\,. \end{equation}
Fix $0<\varepsilon<1$, thanks to (\ref{eq:dnull}), there exists $n_1\in{\mathbb N}$, $n_1\ge n'$, such that for any $n\in{\mathbb Z}$, $|n|\ge n_1$\,, \begin{equation*}
|\Im \omega_n|<\frac{\gamma\sqrt{\varepsilon}}{4}\,. \end{equation*}
Therefore, for any $n\,,m\in{\mathbb Z}$, $|n|,|m|\ge n_1$\,, we have \begin{equation*}\label{eq:} 4T^2(\Re \omega_n-\Re \omega_m)^2-4T^2(\Im \omega_n+\Im \omega_m)^2-\pi^2 \ge 4T^2\gamma^2(n-m)^2-T^2\gamma^2\varepsilon-\pi^2\,. \end{equation*} Now, for any $T>\frac{\pi}{\gamma\sqrt{1-\varepsilon}}$ we have $T^2\gamma^2\varepsilon+\pi^2<T^2\gamma^2$, so from the above inequality it follows \begin{eqnarray*}\label{eq:} 4T^2(\Re \omega_n-\Re \omega_m)^2-4T^2(\Im \omega_n+\Im \omega_m)^2-\pi^2 \ge 4T^2\gamma^2(n-m)^2-T^2\gamma^2>0 \,,\quad\hbox{for}\,\,m\not=n\,. \end{eqnarray*} Putting the previous formula into (\ref{eq:modulo1}), we obtain \begin{eqnarray*}
&&\sum_{|m|\ge n_1,m\not=n}\ |K^*( \omega_n-\overline{\omega}_m)|\nonumber\\ &\le& 4T\pi\sum_{m,m\not=n}\ \frac{1}{4T^2\gamma^2(m-n)^2-T^2\gamma^2}
=\frac{4\pi}{T\gamma^2}\sum_{m,m\not=n}\ \frac{1}{4(m-n)^2-1}\nonumber\\ &\le& \frac{8\pi}{T\gamma^2}\sum_{j=1}^{\infty}\ \frac{1}{4j^2-1}=\frac{4\pi}{T\gamma^2}\sum_{j=1}^{\infty}\ \Big(\frac{1}{2j-1}-\frac{1}{2j+1}\Big)= \frac{4\pi}{T\gamma^2}\,. \end{eqnarray*}
Assuming $C_n=0$ for $|n|\le n_1$ and putting the above formula into (\ref{eq:sum-c4d}), we get \begin{eqnarray}\label{eq:sum-dc6}
\sum_{|n|, |m|\ge n_1,\,n\not=m}|C_n| |C_m|\ |K^*( \omega_n-\overline{\omega}_m)| \le
\frac{4\pi }{T\gamma^2}\sum_{|n|\ge n_1}\ |C_n|^2 \,. \end{eqnarray}
Notice that, thanks to (\ref{eq:h3dbis}), we have $R_n=0$ for $|n|\le n_1$. Therefore, from \eqref{eq:sum-dc3} and \eqref{eq:sum-dc6} it follows \begin{eqnarray}\label{eq:sum-c3}
&&\int_{-\infty}^{\infty} k^*(t)|f(t)|^2\ dt\\
&\le&c(T)\left(\frac{4T}{\pi}+\frac{4\pi }{T\gamma^2}\right)\sum_{|n|\ge n_1}|C_n|^2
+2c(T)\sum_{|n|,|m|\ge n_1}|C_n|\ |R_m| |K^*( \omega_n-ir_m)| \nonumber\\ &&
+c(T)\sum_{|n|,|m|\ge n_1}|R_n|\ |R_m| K^*(i(r_n+r_m))\,. \nonumber \end{eqnarray} To estimate the second term on the right-hand side, we use (\ref{eq:h3dbis}) to obtain \begin{eqnarray}\label{eqn:misticos}
&&2\sum_{|n|,|m|\ge n_1}|C_n||R_m|\ |K^*( \omega_n-ir_m)| \\\nonumber &\le&2\mu
\sum_{\ |n|, |m|\ge n_1 }|C_n|\frac{|C_m|}{|m|^\nu} \ |K^*( \omega_n-ir_m)| \\\nonumber &\le&
\mu\sum_{|n|\ge n_1 }|C_n|^2\sum_{|m|\ge n_1 }\frac{|K^*( \omega_n-ir_m)|}{m^{2\nu}}
+\mu\sum_{ |m|\ge n_1 }|C_m|^2\sum_{|n|\ge n_1 }|K^*( \omega_n-ir_m)| \,.\\ \nonumber \end{eqnarray} Applying (\ref{eqn:k3}), one gets \begin{eqnarray}\label{eqn:denom1d}
|K^*( \omega_n-ir_m)| \le \frac{{4T\pi}}
{\big|4T^2(\Re \omega_n)^2-4T^2(\Im \omega_n-r_m)^2-\pi^2\big|}\,. \end{eqnarray} Now, we observe that, by (\ref{eq:gapd}) it follows $$
|\Re \omega_n|\ge\gamma |n-n'|-|\Re \omega_{n'}|\,,\qquad\forall n\in{\mathbb Z}\,,|n|\ge n'\,, $$ whence \begin{eqnarray*}\label{}
|\Re \omega_n|\ge\frac{\gamma}{ 2} |n|\,,\qquad\forall |n|\ge\ 2n'+ 2\left[\frac{ |\Re \omega_{n'}|}{\gamma}\right]+1\,. \end{eqnarray*} Therefore, since the sequences $\{{\Im}\omega_n\}$, $\{r_n\}$ are bounded, there exists $n_0\in{\mathbb N}$, $$
n_0\ge \max\Big\{n_1,2n'+ 2\left[\frac{ |\Re \omega_{n'}|}{\gamma}\right]+1\Big\} $$ such that for any
$n\,,m\in{\mathbb Z}$, $|n|,|m|\ge n_0$\,, we have \begin{equation*} 4T^2(\Re \omega_n)^2-4T^2(\Im \omega_n-r_m)^2-\pi^2 \ge \frac12T^2\gamma^2 n^2\,; \end{equation*} so, plugging the above inequality into (\ref{eqn:denom1d}) we have \begin{eqnarray*}
|K^*( \omega_n-ir_m)| \le \frac{{8\pi}}{T\gamma^2 n^2}\,. \end{eqnarray*}
Assuming $C_n=0$ for $|n|\le n_0$, and hence also $R_n=0$ for $|n|\le n_0$, by (\ref{eqn:misticos}) it follows \begin{eqnarray}\label{eq:sum-c7}
&&2\sum_{|n|,|m|\ge n_0}|C_n||R_m|\ |K^*( \omega_n-ir_m)| \\\nonumber &\le&
\frac{{8\pi\mu}}{T\gamma^2 }\sum_{|n|\ge n_0}|C_n|^2\sum_{m\not=0}\frac{1}{m^{2\nu}}
+\frac{{8\pi\mu}}{T\gamma^2 }\sum_{ |m|\ge n_0}|C_m|^2\sum_{n\not=0}\frac{1}{n^2}\\\nonumber
&=&\frac{{16\pi\mu}}{T\gamma^2 }\left(\sum_{j=1}^\infty\frac{1}{j^{2\nu}}+\sum_{j=1}^\infty\frac{1}{j^{2 }}\right)\sum_{ |n|\ge n_0}|C_n|^2 \,.\\ \nonumber \end{eqnarray} At last, we must consider the term $$
\sum_{|n|,|m|\ge n_0}|R_n|\ |R_m| K^*(i(r_n+r_m))\,. $$ Recalling the definition of $K^*$ we have $$ K^*(i(r_n+r_m))= \frac{4T\pi}{\pi^2+4T^2(r_n+r_m)^2}\le \frac{4T}{\pi}\,, $$ so, in virtue of (\ref{eq:h3dbis}) we get \begin{eqnarray}\label{eq:sum-c8}
&&\sum_{|n|,|m|\ge n_0}|R_n|\ |R_m| K^*(i(r_n+r_m))\\
&\le& \frac{4T\mu^2}{\pi}\sum_{|n|,|m|\ge n_0}\frac{|C_n|}{|m|^{\nu}}\ \frac{|C_m|}{|n|^{\nu}} \nonumber\\ &\le&
\frac{2T\mu^2}{\pi}\sum_{ m\not=0}\frac{1}{m^{2\nu}}\sum_{|n|\ge n_0}|C_n|^2+\frac{2T\mu^2}{\pi}\sum_{n\not=0}\frac{1}{n^{2\nu}}\sum_{|m|\ge n_0}|C_m|^2
\nonumber\\ &=&\frac{4T\mu^2}{\pi}\sum_{n\not=0}\frac{1}{n^{2\nu}}\sum_{|n|\ge n_0}|C_n|^2
=\frac{8T\mu^2}{\pi}\sum_{j=1}^\infty\frac{1}{j^{2\nu}}\sum_{|n|\ge n_0}|C_n|^2\,. \nonumber \end{eqnarray} Putting (\ref{eq:sum-c7}) and (\ref{eq:sum-c8}) into (\ref{eq:sum-c3}), we obtain \begin{eqnarray*}
&&\int_{-\infty}^{\infty} k^*(t)|f(t)|^2\ dt\\ &\le&c(T)\left(\frac{4T}{\pi}+\frac{4\pi }{T\gamma^2}+\frac{16\pi\mu }{T\gamma^2}\sum_{j=1}^{\infty}\ \frac{1}{j^2}+ 8\mu\Big(\frac{{2\pi}}{T\gamma^2 }
+\frac{T\mu}{\pi}\Big)\sum_{j=1}^\infty\frac{1}{j^{2\nu}}\right)\sum_{|n|\ge n_0}|C_n|^2 \,. \nonumber \end{eqnarray*} Now, if we consider the auxiliary function $k^*$ defined by (\ref{eq:kcos}) with $T$ replaced by $2T$, then from the above inequality we get \begin{eqnarray*}
\int_{-2T}^{2T} \cos \frac{\pi t}{4T}|f(t)|^2\ dt \le c(2T)\left(\frac{8T}{\pi}+\frac{2\pi }{T\gamma^2}+\frac{8\pi\mu }{T\gamma^2}\sum_{j=1}^{\infty}\ \frac{1}{j^2}+ 8\mu\Big(\frac{{\pi}}{T\gamma^2 }
+\frac{2T\mu}{\pi}\Big)\sum_{j=1}^\infty\frac{1}{j^{2\nu}}\right)\sum_{|n|\ge n_0}|C_n|^2 \,, \nonumber \end{eqnarray*} whence \begin{equation*}
\int_{-T}^{T} |f(t)|^2\ dt \le\sqrt2c(2T)\left(\frac{8T}{\pi}+\frac{2\pi }{T\gamma^2}+\frac{8\pi\mu }{T\gamma^2}\sum_{j=1}^{\infty}\ \frac{1}{j^2}+ 8\mu\Big(\frac{{\pi}}{T\gamma^2 }
+\frac{2T\mu}{\pi}\Big)\sum_{j=1}^\infty\frac{1}{j^{2\nu}}\right)\sum_{|n|\ge n_0}|C_n|^2 \,. \end{equation*} So, the proof is complete. \end{Proofv}
As for the direct inequality, to prove theorem \ref{th:inv.ingham} we need to introduce an auxiliary function. We define \begin{equation}\label{eq:k} k(t):=\left \{\begin{array}{l} \displaystyle\sin \frac{\pi t}{T}\,\qquad\qquad \mbox{if}\,\, t\in\ [0,T]\,,\\ \\ 0\,\qquad\qquad\quad\ \ \ \ \mbox{otherwise}\,. \end{array}\right . \end{equation} For the reader's convenience, we list some easy to check properties of $k$ in the following lemma.
\begin{lemma} \label{th:k} Set \begin{equation}\label{eqn:K} K(u):=\frac{T\pi}{\pi^2-T^2u^2}\,,\qquad u\in {\mathbb C}\,, \end{equation} the following properties hold for any $u\in {\mathbb C}$ \begin{equation}\label{eqn:sinek1} \int_{-\infty}^{\infty} k(t)e^{iu t}dt = (1+e^{iu T})K(u)
\,, \end{equation} \begin{equation}\label{eqn:sinek2bis} \overline{K(u)}=K(\overline{u})\,, \end{equation} \begin{equation}\label{eqn:sinek2}
\big|K(u)\big|=\big|K(\overline{u})\big|\,, \end{equation} \begin{equation}\label{eq:sinek3}
\big|K(u)\big|\le \frac{T\pi}{|T^2(\Re u)^2-T^2(\Im u)^2-\pi^2|}\,. \end{equation} \end{lemma}
\begin{Proofy} As in the proof of theorem \ref{th:dir.ingham}, without loss of generality, it may be assumed that \begin{equation}\label{eq:null} \alpha=0\,. \end{equation} Indeed, suppose for a moment that we have proved inequality (\ref{eq:inv-sine}) under this extra condition. For the general case $\alpha\not=0$, we consider the function \begin{equation*} g(t):=e^{\alpha t}f(t)=\sum_{n=-\infty}^{\infty}\Big(C_ne^{i\omega_n't}+R_ne^{(r_n+\alpha)t}\Big)\,, \end{equation*} where $\omega_n'=\omega_n-i\alpha$ and
$\displaystyle\lim_{|n|\to\infty}{\Im}\omega_n'=0$. So, inequality (\ref{eq:inv-sine}) holds for $g$, that is \begin{equation*}
\int_{0}^{T} |g(t)|^2 d t\ge \Big( \frac{T\pi}{\pi^2+T^2\gamma^2\varepsilon/8} -\frac{4\pi}{T\gamma^2}(1+\varepsilon)\Big)
\sum_{|n|\ge n_0}(1+e^{-2{\Im}\omega_n' T})|C_n|^2\,. \end{equation*} Since $f(t)=e^{-\alpha t}g(t)$, we have $$
\vert f(t)\vert\ge \min \{1,e^{-\alpha T}\}|g(t)|\,,\qquad\forall t\in[0,T]\,, $$ whence it follows $$ \int_{0}^{T} \vert f(t)\vert ^2 d t
\ge \min \{1,e^{-2\alpha T}\}\int_{0}^{T} |g(t)|^2 d t
\ge c_1(T,\varepsilon)\sum_{|n|\ge n_0}(1+e^{-2({\Im}\omega_n -\alpha)T})|C_n|^2\,, $$ that is (\ref{eq:inv-sine}) also holds for $f$.
Let $k(t)$ be the function defined by (\ref{eq:k}). If we use (\ref{eqn:sinek1}), then we have \begin{eqnarray}\label{eqn:sum-s}
&&\int_{-\infty}^{\infty} k(t)|f(t)|^2\ dt \nonumber\\ &=& \int_{-\infty}^{\infty} k(t)\sum_{n}\Big(C_ne^{ i\omega_nt}+R_ne^{r_nt}\Big) \sum_{m}\Big(\overline{C}_me^{-i\overline{\omega}_mt}+R_me^{r_mt}\Big)\ dt\nonumber\\ &=&\sum_{n, m}C_n\overline{C}_m (1+e^{i( \omega_n-\overline{\omega}_m) T})K( \omega_n-\overline{\omega}_m)+ \sum_{n, m}C_nR_m(1+e^{( i\omega_n+r_m) T})K( \omega_n-ir_m) \nonumber\\ && +\sum_{n, m}R_n\overline{C}_m(1+e^{( r_n-i\overline{\omega}_m) T})K(ir_n+\overline{\omega}_m)+
\int_{-\infty}^{\infty} k(t)\ \Big|\sum_{n}R_ne^{r_nt}\Big|^2\ dt\,. \end{eqnarray} We may write the first sum on the right-hand side as follows \begin{eqnarray*} &&\sum_{n, m}C_n\overline{C}_m (1+e^{i( \omega_n-\overline{\omega}_m) T})K( \omega_n-\overline{\omega}_m)\\ &=&
\sum_{n}|C_n|^2 (1+e^{-2\Im \omega_n T})K( \omega_n-\overline{\omega}_n) +\sum_{n, m,\,n\not=m}C_n\overline{C}_m(1+e^{i( \omega_n-\overline{\omega}_m) T})K( \omega_n-\overline{\omega}_m) \,. \nonumber\\ \end{eqnarray*} Plugging the above identity into (\ref{eqn:sum-s}) and using (\ref{eqn:sinek2bis}), we obtain \begin{eqnarray*}
&&\int_{-\infty}^{\infty} k(t)|f(t)|^2\ dt \\
&=&\sum_{n}|C_n|^2(1+e^{-2\Im \omega_n T})K( \omega_n-\overline{\omega}_n) +\sum_{n, m,\,n\not=m}C_n\overline{C}_m(1+e^{i( \omega_n-\overline{\omega}_m) T})K( \omega_n-\overline{\omega}_m) \nonumber\\ && + 2\sum_{n, m}R_m\Re \big[C_n(1+e^{( i\omega_n+r_m) T})K( \omega_n-ir_m)\big]+
\int_{-\infty}^{\infty} k(t)\ \Big|\sum_{n}R_ne^{r_nt}\Big|^2\ dt\,. \nonumber \end{eqnarray*}
Notice that, by difference, the second term on the right-hand side of the previous identity is real. Therefore, using the elementary estimate $\theta\ge -|\theta|$, $\theta\in{\mathbb R}$, we obtain \begin{multline}\label{eqn:sum-s2}
\int_{-\infty}^{\infty} k(t)|f(t)|^2\ dt \\
\ge \sum_{n}|C_n|^2(1+e^{-2\Im \omega_n T})K( \omega_n-\overline{\omega}_n)
-\sum_{n, m,\,n\not=m}|C_n| |C_m|(1+e^{-(\Im \omega_n+\Im \omega_m) T})\ |K( \omega_n-\overline{\omega}_m)| \\
-2\sum_{n, m}|C_n|\ |R_m|(1+e^{(r_m-\Im \omega_n) T}) |K( \omega_n-ir_m)| +
\int_{-\infty}^{\infty} k(t)\ \Big|\sum_{n}R_ne^{r_nt}\Big|^2\ dt\,. \end{multline} Now, arguing as in the proof of (\ref{eq:sum-c4d}) and using $
|K( \omega_n-\overline{\omega}_m)|=|K( \overline{\omega}_n-\omega_m)|\,, $ we have \begin{eqnarray}\label{eq:sum-s3}
\sum_{n, m,\,n\not=m}|C_n| |C_m|\ |K( \omega_n-\overline{\omega}_m)| \le
\sum_{n}\ |C_n|^2\sum_{m,m\not=n}\ |K( \omega_n-\overline{\omega}_m)| \,. \end{eqnarray} Similarly, we get \begin{equation}\label{eq:sum-s4}
\sum_{n, m,\,n\not=m}|C_n| |C_m|e^{-(\Im \omega_n+\Im \omega_m) T}\ |K( \omega_n-\overline{\omega}_m)| \le
\sum_{n}\ |C_n|^2e^{-2\Im \omega_n T}\sum_{m,m\not=n}\ |K( \omega_n-\overline{\omega}_m)| \,. \end{equation} Therefore, plugging (\ref{eq:sum-s3}) and (\ref{eq:sum-s4}) into (\ref{eqn:sum-s2}) and being $k$ a non-negative function, we have \begin{multline}\label{eqn:sum-s5}
\int_{-\infty}^{\infty} k(t)|f(t)|^2\ dt \ge
\sum_{n}|C_n|^2 (1+e^{-2\Im \omega_n T})\Big(K( \omega_n-\overline{\omega}_n)
-\sum_{m,m\not=n} |K( \omega_n-\overline{\omega}_m)| \Big) \\
-2\sum_{n, m}|C_n|\ |R_m|(1+e^{(r_m-\Im \omega_n) T}) |K( \omega_n-ir_m)|\,. \end{multline} Now, fixed $n\in{\mathbb Z}$, we have to estimate the sum $$
\sum_{m,m\not=n}\ |K( \omega_n-\overline{\omega}_m)|\,. $$ Using (\ref{eq:sinek3}), we get
\begin{eqnarray}\label{eq:modulo}
\sum_{m,m\not=n} |K( \omega_n-\overline{\omega}_m)| \le T\pi\sum_{m,m\not=n}\frac{1}
{\Big|T^2(\Re \omega_n-\Re \omega_m)^2-T^2(\Im \omega_n+\Im \omega_m)^2-\pi^2\Big|}. \end{eqnarray} From assumption (\ref{eq:h1}) it follows \begin{equation}\label{eq:gap}
|\Re \omega_n-\Re \omega_m|\ge \gamma|n-m|\,,\qquad\forall |n|\,,|m|\ge n'\,. \end{equation}
Moreover, if we fix $0<\varepsilon<1$, then, thanks to (\ref{eq:null}), there exists $n_1\in{\mathbb N}$, $n_1\ge n'$, such that for any $n\in{\mathbb Z}$, $|n|\ge n_1$\,, \begin{equation}\label{eq:lim0}
|\Im \omega_n|<\frac{\gamma}{4}\sqrt\frac{\varepsilon}{2}\,. \end{equation}
Therefore, for any $n\,,m\in{\mathbb Z}$, $|n|,|m|\ge n_1$\,, we have \begin{equation*}\label{eq:} T^2(\Re \omega_n-\Re \omega_m)^2-T^2(\Im \omega_n+\Im \omega_m)^2-\pi^2 \ge T^2\gamma^2(n-m)^2-T^2\gamma^2\frac{\varepsilon}{4}-\pi^2\,. \end{equation*} Now, for any $T>\frac{2\pi}{\gamma\sqrt{1-\varepsilon}}$ we have $T^2\gamma^2\varepsilon+4\pi^2<T^2\gamma^2$, so from the above inequality it follows \begin{eqnarray*}\label{eq:} T^2(\Re \omega_n-\Re \omega_m)^2-T^2(\Im \omega_n+\Im \omega_m)^2-\pi^2 \ge T^2\gamma^2(n-m)^2-\frac14T^2\gamma^2>0 \,,\quad\hbox{for}\,\,m\not=n\,. \end{eqnarray*} Putting the previous formula into (\ref{eq:modulo}), we obtain \begin{multline*}
\sum_{|m|\ge n_1,m\not=n}\ |K( \omega_n-\overline{\omega}_m)|\\ \le 4T\pi\sum_{m,m\not=n}\ \frac{1}{4T^2\gamma^2(m-n)^2-T^2\gamma^2} =\frac{4\pi}{T\gamma^2}\sum_{m,m\not=n}\ \frac{1}{4(m-n)^2-1}\\ \le \frac{8\pi}{T\gamma^2}\sum_{j=1}^{\infty}\ \frac{1}{4j^2-1}=\frac{4\pi}{T\gamma^2}\sum_{j=1}^{\infty}\ \Big(\frac{1}{2j-1}-\frac{1}{2j+1}\Big)= \frac{4\pi}{T\gamma^2}\,. \end{multline*}
If we assume $C_n=0$ for $|n|\le n_1$, then due to (\ref{eq:h3}) we also have $R_n=0$ for $|n|\le n_1$. Therefore, putting the above estimate into (\ref{eqn:sum-s5}), for $T>\frac{2\pi}{\gamma\sqrt{1-\varepsilon}}$ we get \begin{multline}\label{eqn:sum-s6}
\int_{-\infty}^{\infty} k(t)|f(t)|^2\ dt \ge
\sum_{|n|\ge n_1}|C_n|^2 (1+e^{-2\Im \omega_n T})\Big(K( \omega_n-\overline{\omega}_n) -\frac{4\pi}{T\gamma^2}\Big) \\
-2\sum_{\ |n|,|m|\ge n_1}|C_n|\ |R_m|(1+e^{(r_m-\Im \omega_n) T}) |K( \omega_n-ir_m)|\,. \end{multline} It remains to estimate the second sum on the right-hand side. Thanks to (\ref{eq:h3}) we have \begin{multline}\label{eqn:mistisin}
2\sum_{|n|,|m|\ge n_1}|C_n||R_m|\ |K( \omega_n-ir_m)| \le 2\mu
\sum_{|n|,|m|\ge n_1}|C_n|\frac{|C_m|}{|m|^\nu} \ |K( \omega_n-ir_m)|\\ \le
\mu\sum_{|n|\ge n_1}|C_n|^2\sum_{|m|\ge n_1}\frac{|K( \omega_n-ir_m)|}{|m|^{2\nu}}
+\mu\sum_{|m|\ge n_1}|C_m|^2\sum_{|n|\ge n_1}|K( \omega_n-ir_m)| \,. \end{multline} Again by (\ref{eq:sinek3}) we have \begin{eqnarray}\label{eqn:denom}
|K( \omega_n-ir_m)| \le \frac{{T\pi}}
{\big|T^2(\Re \omega_n)^2-T^2(\Im \omega_n-r_m)^2-\pi^2\big|}\,. \end{eqnarray} Now, we observe that, by (\ref{eq:gap}) it follows $$
|\Re \omega_n|\ge\gamma |n-n'|-|\Re \omega_{n'}|\,,\qquad\forall n\in{\mathbb Z}\,,|n|\ge n'\,, $$ whence \begin{eqnarray*}\label{}
|\Re \omega_n|\ge\frac{\gamma}{\sqrt 2} |n|\,,\qquad\forall |n|\ge\ \left[\frac{\gamma n'+ |\Re \omega_{n'}|}{\gamma(1-1/\sqrt 2)}\right]+1=:n_2\,. \end{eqnarray*}
Therefore, for any $n\in{\mathbb Z}$, $|n|\ge\ n_2$, we get \begin{multline}\label{eqn:denom1} T^2(\Re \omega_n)^2-T^2(\Im \omega_n-r_m)^2-\pi^2 \\ \ge T^2\Big( \frac{1}{ 2}\gamma^2n^2-(\Im \omega_n-r_m)^2\Big)-\pi^2 \ge T^2\gamma^2n^2\Big( \frac{1}{ 2}-\frac{(\Im \omega_n-r_m)^2}{\gamma^2n^2}\Big)-\pi^2\,. \end{multline} Since the sequences $\{{\Im}\omega_n\}$ and $\{r_n\}$ are bounded, there exists $n_3\in{\mathbb N}$, such that \begin{eqnarray}\label{eqn:denom1bis}
\frac{1}{ 2}-\frac{(\Im \omega_n-r_m)^2}{\gamma^2n^2}\ge\frac{1}{ 4}\,,\qquad\forall |n|\,, |m|\ge n_3 \,. \end{eqnarray} Choosing $n_0\in{\mathbb N}$ such that \begin{equation}\label{eq:n_0} n_0\ge\max\Big\{n_1,n_2,n_3,2\Big\}\,, \end{equation}
and putting (\ref{eqn:denom1bis}) into (\ref{eqn:denom1}), for any $|n|,|m|\ge n_0$ we have \begin{eqnarray*}\label{} T^2(\Re \omega_n)^2-T^2(\Im \omega_n-r_m)^2-\pi^2 \ge \frac{1}{ 4}(T^2\gamma^2n^2 -4\pi^2)\,. \end{eqnarray*} Moreover, since $T>2\pi/\gamma$ we have $4\pi^2<T^2\gamma^2n_0^{1/2}$, so \begin{eqnarray*} T^2(\Re \omega_n)^2-T^2(\Im \omega_n-r_m)^2-\pi^2 \ge \frac{1}{ 4}T^2\gamma^2(n^2 -n_0^{1/2})
\ge \frac{1}{ 4}T^2\gamma^2n_0^{1/2}(|n|^{3/2}-1)\,. \end{eqnarray*} Therefore from (\ref{eqn:denom}), thanks to the above inequality, we get \begin{eqnarray}\label{eqn:denom2}
|K( \omega_n-ir_m)| \le \frac{{4\pi}}
{T\gamma^2n_0^{1/2}(|n|^{3/2 }-1)}\,,\qquad\forall |n|\,,|m|\ge n_0 \,, \end{eqnarray}
and hence, assuming $C_n=0$ for $|n|\le n_0$, (\ref{eqn:mistisin}) can be written as \begin{multline}\label{eqn:mistisin1}
2\sum_{|n|, |m|\ge n_0}|C_n||R_m|\ |K( \omega_n-ir_m)| \\ \le
\frac{4\pi\mu}{T\gamma^2n_0^{1/2}}\sum_{|n|\ge n_0}|C_n|^2\sum_{m\not= 0}\frac{1}{|m|^{2\nu}}
+\frac{4\pi\mu}{T\gamma^2n_0^{1/2}}\sum_{ |m|\ge n_0}|C_m|^2\sum_{|n|\ge2}\frac{1}{|n|^{3/2 }-1}\\ =
\frac{8\pi\mu }{T\gamma^2n_0^{1/2}}\left(\sum_{j=1}^\infty\frac{1}{j^{2\nu}}+\sum_{j=2}^\infty\frac{1}{j^{3/2 }-1}\right)\sum_{|n|\ge n_0}|C_n|^2 \,. \end{multline} Moreover, by (\ref{eq:h2}) and (\ref{eq:h3}) we have \begin{multline*}
2\sum_{|n|, |m|\ge n_0}|C_n||R_m|e^{(r_m-\Im \omega_n) T}\ |K( \omega_n-ir_m)| \\ \le 2\mu
\sum_{|n|, |m|\ge n_0}|C_n|e^{-\Im \omega_n T}\frac{|C_m|e^{-\Im \omega_m T}}{|m|^\nu}\ |K( \omega_n-ir_m)| \\ \le
\mu\sum_{|n|\ge n_0}|C_n|^2e^{-2\Im \omega_n T}\sum_{|m|\ge n_0}\frac{|K( \omega_n-ir_m)|}{|m|^{2\nu}}
+\mu\sum_{|m|\ge n_0}|C_m|^2e^{-2\Im \omega_m T}\sum_{|n|\ge n_0}|K( \omega_n-ir_m)| \,. \end{multline*} If we use again (\ref{eqn:denom2}), then, reasoning as in (\ref{eqn:mistisin1}), we obtain \begin{multline}\label{eqn:mistisin2}
2\sum_{|n|, |m|\ge n_0}|C_n||R_m|\ e^{(r_m-\Im \omega_n) T}\ |K( \omega_n-ir_m)| \\ \le
\frac{8\pi \mu}{T\gamma^2n_0^{1/2}}\left(\sum_{j=1}^\infty\frac{1}{j^{2\nu}}+\sum_{j=2}^\infty\frac{1}{j^{3/2 }-1}\right)\sum_{|n|\ge n_0}|C_n|^2e^{-2\Im \omega_n T} \,. \end{multline} Set $$ S:=2\mu\left(\sum_{j=1}^\infty\frac{1}{j^{2\nu}}+\sum_{j=2}^\infty\frac{1}{j^{3/2 }-1}\right)\,, $$ (\ref{eqn:mistisin1}) and (\ref{eqn:mistisin2}) yield \begin{eqnarray*}
2\sum_{|n|, |m|\ge n_0}|C_n|\ |R_m|(1+e^{(-\Im \omega_n+r_m) T})\ |K( \omega_n-ir_m)|
\le \frac{4\pi S}{T\gamma^2n_0^{1/2}}\sum_{|n|\ge n_0}|C_n|^2 (1+e^{-2\Im \omega_n T})\,. \end{eqnarray*} Plugging the above formula into (\ref{eqn:sum-s6}), we get \begin{eqnarray*}
\int_{-\infty}^{\infty} k(t)|f(t)|^2\ dt \ge
\sum_{|n|\ge n_0}|C_n|^2 (1+e^{-2\Im \omega_n T})\Big(K( \omega_n-\overline{\omega}_n) -\frac{4\pi}{T\gamma^2}\Big(1+\frac{ S}{n_0^{1/2}}\Big)\Big) \,. \end{eqnarray*} Now, in virtue of (\ref{eqn:K}) we note that $$ K( \omega_n-\overline{\omega}_n)=\frac{T\pi}{\pi^2+4T^2(\Im\omega_n)^2}\,, $$ so \begin{eqnarray}\label{eqn:sum-s7}
\int_{-\infty}^{\infty} k(t)|f(t)|^2\ dt \ge
\sum_{|n|\ge n_0}|C_n|^2 (1+e^{-2\Im \omega_n T})\Big(\frac{T\pi}{\pi^2+4T^2(\Im\omega_n)^2} -\frac{4\pi}{T\gamma^2}\Big(1+\frac{ S}{n_0^{1/2}}\Big)\Big) \,. \end{eqnarray} If we use (\ref{eq:lim0}) and take $$
n_0\ge S^2/\varepsilon^2\,, $$
then we get, for any $|n|\ge n_0$, \begin{eqnarray}\label{eqn:sum-s8} \frac{T\pi}{\pi^2+4T^2(\Im\omega_n)^2} -\frac{4\pi}{T\gamma^2}\Big(1+\frac{ S}{n_0^{1/2}}\Big) \ge
\frac{T\pi}{\pi^2+T^2\gamma^2\varepsilon/8} -\frac{4\pi}{T\gamma^2}(1+\varepsilon) \,. \end{eqnarray} Now, we prove that for $T>\frac{2\pi}{\gamma}\sqrt{\frac{1+\varepsilon}{1-\varepsilon}}$ $$ \frac{T\pi}{\pi^2+T^2\gamma^2\varepsilon/8} -\frac{4\pi}{T\gamma^2}(1+\varepsilon)>0\,. $$ Indeed, \begin{multline*}
\frac{T\pi}{\pi^2+T^2\gamma^2\varepsilon/8} -\frac{4\pi}{T\gamma^2}(1+\varepsilon) \\ = \pi\ \frac{T^2\gamma^2-4(1+\varepsilon)(\pi^2+T^2\gamma^2\varepsilon/8)}{(\pi^2+T^2\gamma^2\varepsilon/8)T\gamma^2} = \pi\ \frac{T^2\gamma^2(1-(1+\varepsilon)\varepsilon/2)-4\pi^2(1+\varepsilon)}{(\pi^2+T^2\gamma^2\varepsilon/8)T\gamma^2}\,. \end{multline*} Since $\varepsilon<1$, we have $(1+\varepsilon)\varepsilon/2<\varepsilon$, whence for $T>\frac{2\pi}{\gamma}\sqrt{\frac{1+\varepsilon}{1-\varepsilon}}$ \begin{eqnarray*}
\frac{T\pi}{\pi^2+T^2\gamma^2\varepsilon/8} -\frac{4\pi}{T\gamma^2}(1+\varepsilon) > \pi\ \frac{T^2\gamma^2(1-\varepsilon)-4\pi^2(1+\varepsilon)}{(\pi^2+T^2\gamma^2\varepsilon/8)T\gamma^2}>0\,.
\nonumber\\ \end{eqnarray*} Finally, by (\ref{eqn:sum-s7}), (\ref{eqn:sum-s8}) and the definition of $k(t)$ we obtain \begin{eqnarray*}\label{}
&&\int_{0}^{T}|f(t)|^2\ dt\ge \Big( \frac{T\pi}{\pi^2+T^2\gamma^2\varepsilon/8} -\frac{4\pi}{T\gamma^2}(1+\varepsilon) \Big)
\sum_{|n|\ge n_0}|C_n|^2 (1+e^{-2\Im \omega_n T})\,, \end{eqnarray*} so the proof is complete. \end{Proofy}
To prove proposition \ref{pr:haraux-inv}, we first introduce some auxiliary tools. Indeed, we introduce a family of operators, which will be needed to annihilate a finite number of terms in the Fourier series. Our operators are slightly different from those introduced in \cite{Ha} and \cite{KL1}. For that reason and for the reader's convenience, we then proceed to recall and prove some of their properties.
Given $\delta >0$ and $\omega\in{\mathbb C}$ arbitrarily, we define the linear operator $I_{\delta,\omega}$ as follows: for every continuous function $u:{\mathbb R}\to{\mathbb C}$ the function $I_{\delta,\omega}u:{\mathbb R}\to{\mathbb C}$ is given by the formula \begin{equation}\label{eq:defI} I_{\delta,\omega}u(t):=u(t)-\frac 1\delta\int_0^\delta e^{-i\omega s} u(t+s)\ ds\,,\qquad t\in{\mathbb R}\,. \end{equation} The following result states some properties connected with operators $I_{\delta,\omega}$. \begin{lemma} \label{le:opI1} {\rm (a)} If $u(t)= e^{i\omega t} $, then $I_{\delta,\omega}u= 0 $.
{\rm (b)} If $u(t)= e^{i\omega' t} $ with $\omega'\not=\omega$, then \begin{equation*} I_{\delta,\omega}u(t)= \Big(1-\frac{e^{i(\omega'-\omega) \delta}-1}{i(\omega'-\omega) \delta}\Big)u(t)
\,. \end{equation*}
{\rm (c)} The linear operators $I_{\delta,\omega}$ commute, that is \begin{equation*} I_{\delta,\omega}I_{\delta',\omega'}u=I_{\delta',\omega'}I_{\delta,\omega}u \end{equation*} for all $\delta,\omega,\delta',\omega'$ and $u$. \end{lemma} \begin{Proof} (a) By definition, we have \begin{equation*} I_{\delta,\omega}u(t)=u(t)-\frac 1\delta\int_0^\delta e^{-i\omega s} e^{i\omega (t+s)}\ ds=u(t)-e^{i\omega t}=0\,. \end{equation*} (b) Again by definition, we obtain \begin{eqnarray*} I_{\delta,\omega}u(t) &=&u(t)-\frac 1\delta\int_0^\delta e^{-i\omega s} e^{i\omega' (t+s)}\ ds =u(t)-\frac 1\delta\Big[\frac{e^{i(\omega'-\omega) s}}{i(\omega'-\omega)}\Big]_0^\delta e^{i\omega' t} =\Big(1-\frac{e^{i(\omega'-\omega) \delta}-1}{i(\omega'-\omega) \delta}\Big)u(t)
\,. \end{eqnarray*} (c) It follows at once by definition of operators $I_{\delta,\omega}$. \end{Proof} \begin{lemma} For any $T>0$ and every continuous function $u:{\mathbb R}\to{\mathbb C}$ we have \begin{equation}\label{eq:boundedI}
\int_0^T|I_{\delta,\omega}u(t)|^2\ dt\le 2(1+e^{2|{\Im}\omega| \delta})\int_0^{T+\delta}|u(t)|^2\ dt \,,\qquad \delta\in (0,T),\,\omega\in{\mathbb C}\,. \end{equation} \end{lemma} \begin{Proof} For every $t\in [0,T]$, by (\ref{eq:defI}) one has \begin{eqnarray*}
|I_{\delta,\omega}u(t)|^2 &\le&
2|u(t)|^2+2\Big|\frac 1\delta\int_0^\delta e^{-i\omega s} u(t+s)\ ds\Big|^2\nonumber\\ &\le&
2|u(t)|^2+\frac 2{\delta^2}\int_0^\delta |e^{-i\omega s}|^2\ ds \int_0^\delta |u(t+s)|^2\ ds\nonumber\\ &\le&
2|u(t)|^2+\frac 2{\delta^2}\int_0^\delta e^{2{\Im}\omega s}\ ds \int_0^\delta |u(t+s)|^2\ ds\nonumber\\ &\le&
2|u(t)|^2+\frac 2{\delta} e^{2|{\Im}\omega| \delta} \int_t^{t+\delta} |u(x)|^2\ dx\,. \end{eqnarray*} Integrating the above inequality from $0$ to $T$, we obtain \begin{eqnarray}\label{eq:int}
\int_0^{T}|I_{\delta,\omega}u(t)|^2\ dt
\le 2\int_0^{T}|u(t)|^2\ dt+\frac 2{\delta} e^{2|{\Im}\omega| \delta} \int_0^{T}\int_t^{t+\delta} |u(x)|^2\ dx\ dt\,. \end{eqnarray} Since $\delta\in (0,T)$ we have that \begin{eqnarray*}
\int_0^{T}\int_t^{t+\delta} |u(x)|^2\ dx\ dt &=&
\int_0^{\delta}|u(x)|^2\int_0^{x} \ dt\ dx
+\int_\delta^{T}|u(x)|^2\int_{x-\delta}^{x} \ dt\ dx
+\int_T^{T+\delta}|u(x)|^2\int_{x-\delta}^{T} \ dt\ dx \nonumber\\ &=&
\int_0^{T+\delta}|u(x)|^2\int_{\max\{0,x-\delta\}}^{\min\{x,T\}} \ dt\ dx \nonumber\\ &\le&
\int_0^{T+\delta}|u(x)|^2\int_{x-\delta}^{x} \ dt\ dx
=\delta \int_0^{T+\delta}|u(x)|^2\ dx\,. \end{eqnarray*} Plugging this inequality into (\ref{eq:int}), we get \begin{eqnarray*}
\int_0^{T}|I_{\delta,\omega}u(t)|^2\ dt &\le&
2\int_0^{T}|u(t)|^2\ dt+2 e^{2|{\Im}\omega| \delta} \int_0^{T+\delta}|u(x)|^2\ dx \nonumber\\ &\le&
2(1+e^{2|{\Im}\omega| \delta})\int_0^{T+\delta}|u(t)|^2\ dt \,, \end{eqnarray*} that is (\ref{eq:boundedI}). \end{Proof} We now proceed to define another operator, namely: \begin{equation}\label{eq:defI1} I_{\delta,\omega, r}:=I_{\delta,\omega}\circ I_{\delta, -ir}\,, \qquad\delta >0\,, \omega\in{\mathbb C}\,, r\in{\mathbb R}\,. \end{equation} Some properties of that operator are collected in the following results. \begin{lemma} \label{le:opI2} {\rm (a)} If $u(t)= e^{i\omega t}$ or $u(t)= e^{r t}$, then $I_{\delta,\omega,r}u= 0 $.
{\rm (b)} If $u(t)= e^{i\omega' t}$ with $\omega'\not=\omega$ and $\omega'\not=-ir$, then \begin{equation*} I_{\delta,\omega,r}u(t)=\Big(1-\frac{e^{i(\omega'-\omega) \delta}-1}{i(\omega'-\omega) \delta}\Big)\Big(1-\frac{e^{(i\omega'-r) \delta}-1}{(i\omega'-r) \delta}\Big)u(t)\,. \end{equation*}
{\rm (c)} If $u(t)= e^{r' t} $ with $r'\not=r$ and $r'\not=i\omega$, then \begin{equation*} I_{\delta,\omega,r}u(t)=\Big(1-\frac{e^{(r'-r) \delta}-1}{(r'-r) \delta}\Big)\Big(1-\frac{e^{(r'-i\omega) \delta}-1}{(r'-i\omega) \delta}\Big) u(t)\,. \end{equation*}
{\rm (d)} The linear operators $I_{\delta,\omega,r}$ commute, that is \begin{equation*} I_{\delta,\omega,r}I_{\delta',\omega',r'}u=I_{\delta',\omega',r'}I_{\delta,\omega,r}u \end{equation*}
for all $\delta,\omega,r,\delta',\omega',r'$ and $u$. \end{lemma} \begin{Proof} (a) Thanks to (c) and (a) of lemma \ref{le:opI1}, we have \begin{equation*} I_{\delta,\omega,r} (e^{i\omega t})=I_{\delta,\omega}( I_{\delta, -ir}(e^{i\omega t})) =I_{\delta, -ir}( I_{\delta,\omega}(e^{i\omega t}))=I_{\delta, -ir}(0)=0\,, \end{equation*} \begin{equation*} I_{\delta,\omega,r}(e^{r t})=I_{\delta,\omega}( I_{\delta, -ir}(e^{r t}))=I_{\delta,\omega}( 0)=0\,. \end{equation*} (b) By lemma \ref{le:opI1}-(a) we get \begin{eqnarray*} I_{\delta,\omega,r}u(t) &=&I_{\delta, -ir}( I_{\delta,\omega}(e^{i\omega 't})) =\Big(1-\frac{e^{i(\omega'-\omega) \delta}-1}{i(\omega'-\omega) \delta}\Big)I_{\delta, -ir}( e^{i\omega 't}) \\ &=&\Big(1-\frac{e^{i(\omega'-\omega) \delta}-1}{i(\omega'-\omega) \delta}\Big)\Big(1-\frac{e^{i(\omega'+ir) \delta}-1}{i(\omega'+ir) \delta}\Big)e^{i\omega 't} =\Big(1-\frac{e^{i(\omega'-\omega) \delta}-1}{i(\omega'-\omega) \delta}\Big)\Big(1-\frac{e^{(i\omega'-r) \delta}-1}{(i\omega'-r) \delta}\Big)e^{i\omega 't} \,. \end{eqnarray*} (c) It follows by (b) with $\omega'=-ir'$.
\noindent (d) It is a consequence of lemma \ref{le:opI1}-(c). \end{Proof}
\begin{corollary}\label{co:boundedI} For any $T>0$ and every continuous function $u:{\mathbb R}\to{\mathbb C}$ we have \begin{equation}\label{eq:boundedI1}
\int_0^T|I_{\delta,\omega,r}u(t)|^2\ dt\le 4(1+e^{2|{\Im}\omega| \delta})(1+e^{2|r| \delta})\int_0^{T+2\delta}|u(t)|^2\ dt \,,\qquad \delta\in (0,T),\,\omega\in{\mathbb C}\,, r\in{\mathbb R}\,. \end{equation} \end{corollary} \begin{Proof} Applying (\ref{eq:boundedI}) two times, first to function $I_{\delta, -ir}u(t)$ and next to $u(t)$, we obtain \begin{eqnarray*}
\int_0^T|I_{\delta,\omega,r}u(t)|^2\ dt
&=&\int_0^T|I_{\delta,\omega}I_{\delta, -ir}u(t)|^2\ dt
\le 2(1+e^{2|{\Im}\omega| \delta})\int_0^{T+\delta}|I_{\delta, -ir}u(t)|^2\ dt\\ &\le&
4(1+e^{2|{\Im}\omega| \delta})(1+e^{2|r| \delta})\int_0^{T+2\delta}|u(t)|^2\ dt\,, \end{eqnarray*} that is (\ref{eq:boundedI1}). \end{Proof}
\begin{Proof2} To begin with, we will transform the function $$ f(t)=\sum_{n=-\infty}^{\infty}\big(C_ne^{i\omega_nt}+R_ne^{r_nt}\big) $$ in a series such that the terms corresponding to indices in ${\cal F}$ are null,
so we can apply assumption \eqref{eq:haraux-inv}.
To this end, we fix $\varepsilon>0$ and choose $\delta\in (0,\frac{\varepsilon}{2|{\cal F}|}\wedge T)$, where $|{\cal F}|$ indicates the number of elements in the set ${\cal F}$. Let us denote by $I$ the composition of all linear operators $I_{\delta,\omega_j,r_j}$, where $j\in {\cal F}$; by lemma \ref{le:opI2}-(d) the definition of $I$ does not depend on the order of the operators $I_{\delta,\omega_j,r_j}$. Therefore,
we can use lemma \ref{le:opI2} to get \begin{eqnarray*} If(t) &=& \sum_{n\not\in {\cal F}}C_n \prod_{j\in {\cal F}}\Big(1-\frac{e^{i(\omega_n-\omega_j) \delta}-1}{i(\omega_n-\omega_j) \delta}\Big)\Big(1-\frac{e^{(i\omega_n-r_j)\delta}-1}{(i\omega_n-r_j)\delta}\Big)e^{i\omega_nt} \\ & & +\sum_{n\not\in {\cal F}}R_n \prod_{j\in {\cal F}}\Big(1-\frac{e^{(r_n-r_j) \delta}-1}{(r_n-r_j) \delta}\Big)\Big(1-\frac{e^{(r_n-i\omega_j) \delta}-1}{(r_n-i\omega_j) \delta}\Big)e^{r_nt}\,. \end{eqnarray*} If we define for any $n\not\in {\cal F}$ $$ C'_n:=C_n \prod_{j\in {\cal F}}\Big(1-\frac{e^{i(\omega_n-\omega_j) \delta}-1}{i(\omega_n-\omega_j) \delta}\Big)\Big(1-\frac{e^{(i\omega_n-r_j)\delta}-1}{(i\omega_n-r_j)\delta}\Big)
\,, $$ $$ R'_n:=R_n \prod_{j\in {\cal F}}\Big(1-\frac{e^{(r_n-r_j) \delta}-1}{(r_n-r_j) \delta}\Big)\Big(1-\frac{e^{(r_n-i\omega_j) \delta}-1}{(r_n-i\omega_j) \delta}\Big)
\,, $$ then we have $$ If(t)=\sum_{n\not\in {\cal F}}\big(C'_ne^{i\omega_nt}+R'_ne^{r_nt}\big)\,. $$ Therefore, applying estimate (\ref{eq:haraux-inv}) to $If(t)$ we obtain \begin{equation}\label{eq:haraux-inv2}
\int_{0}^{T}|If(t)|^2 d t \ge c'_1
\sum_{n\not\in {\cal F}}|C'_n|^2\,. \end{equation}
Next, we choose $\delta\in (0,\frac{\varepsilon}{2|{\cal F}|}\wedge T)$ such that none of the products $$ \prod_{j\in {\cal F}}\Big(1-\frac{e^{i(\omega_n-\omega_j) \delta}-1}{i(\omega_n-\omega_j) \delta}\Big)\Big(1-\frac{e^{(i\omega_n-r_j)\delta}-1}{(i\omega_n-r_j)\delta}\Big) \qquad n\not\in {\cal F} \,, $$ vanishes. This is possible because the analytic function $\displaystyle1-\frac{e^z-1}{z}$ does not vanish identically and, since the numbers $\omega_n-\omega_j$ and $i\omega_n-r_j$ are all different from zero, we have to exclude only a countable set of values of $\delta$.
Now, we note that there exists a constant $c'>0$ such that \begin{equation}\label{eq:C'}
\left|\prod_{j\in {\cal F}}\Big(1-\frac{e^{i(\omega_n-\omega_j) \delta}-1}{i(\omega_n-\omega_j)
\delta}\Big)\Big(1-\frac{e^{(i\omega_n-r_j)\delta}-1}{(i\omega_n-r_j)\delta}\Big)\right|^2\ge c' \qquad\forall n\not\in {\cal F} \,. \end{equation} Indeed, it is sufficient to observe that for any fixed $j\in {\cal F}$ we have $$
\Big|\frac{e^{i(\omega_n-\omega_j) \delta}-1}{i(\omega_n-\omega_j) \delta}\Big|\le\frac{e^{-{\Im}(\omega_n-\omega_j) \delta}+1}{|\omega_n-\omega_j| \delta} \to 0\qquad
\mbox{as}\quad |n|\to \infty\,, $$ $$
\Big|\frac{e^{(i\omega_n-r_j)\delta}-1}{(i\omega_n-r_j)\delta}\Big|\le\frac{e^{-({\Im}\omega_n+r_j)\delta}+1}{|\omega_n+ir_j|\delta} \to 0\qquad
\mbox{as}\quad |n|\to \infty\,, $$ in view of \eqref{eq:ha1}. As a result, the product $$ \prod_{j\in {\cal F}}\Big(1-\frac{e^{i(\omega_n-\omega_j) \delta}-1}{i(\omega_n-\omega_j) \delta}\Big)\Big(1-\frac{e^{(i\omega_n-r_j)\delta}-1}{(i\omega_n-r_j)\delta}\Big) $$
tends to $1$ as $|n|\to \infty$, so that it is minorized, e.g., by $1/2$ for all sufficiently large $|n|$.
Therefore, \eqref{eq:haraux-inv2} and \eqref{eq:C'} yield \begin{equation}\label{eq:haraux-inv3}
\int_{0}^{T}|If(t)|^2 d t \ge c'_1c'
\sum_{n\not\in {\cal F}}|C_n|^2\,. \end{equation} On the other hand, applying (\ref{eq:boundedI1}) repeatedly with $\omega=\omega_j$ and $r=r_j$, $j\in {\cal F}$, we have \begin{equation*}
\int_{0}^{T}|If(t)|^2 d t
\le 4^{|{\cal F}|}\prod_{j\in {\cal F}}(1+e^{2|{\Im}\omega_j| \delta})(1+e^{2|r_j| \delta})\int_0^{T+2|{\cal F}|\delta}|f(t)|^2\ dt \,, \end{equation*}
from which, using \eqref{eq:haraux-inv3} and $2|{\cal F}|\delta<\varepsilon$, it follows \begin{equation*}\label{eq:C''}
\sum_{n\not\in {\cal F}}|C_n|^2 \le
\frac{4^{|{\cal F}|}}{c'_1c'}\prod_{j\in {\cal F}}(1+e^{|{\Im}\omega_j| \varepsilon/{|{\cal F}|}})(1+e^{|r_j| \varepsilon/{|{\cal F}|}})\int_0^{T+\varepsilon}|f(t)|^2\ dt \,, \end{equation*} whence \begin{equation}\label{eq:C''}
\sum_{n\not\in {\cal F}}|C_n|^2 \le
\frac{4^{2|{\cal F}|}}{c'_1c'}\int_0^{T}|f(t)|^2\ dt \,. \end{equation} In addition, thanks to the triangle inequality, \eqref{eq:haraux-inv1} and \eqref{eq:C''} we get \begin{eqnarray}\label{eq:1+C2C''}
\int_0^T\Big|\sum_{n\in {\cal F}}\Big(C_ne^{i\omega_nt}+R_ne^{r_nt}\Big)\Big|^2 d t &=&
\int_0^T\Big|f(t)-\sum_{n\not\in {\cal F}}\Big(C_ne^{i\omega_nt}+R_ne^{r_nt}\Big)\Big|^2 d t\nonumber\\ &\le&
2\int_0^T |f(t)|^2 d t+2\int_0^T\Big|\sum_{n\not\in {\cal F}}\Big(C_ne^{i\omega_nt}+R_ne^{r_nt}\Big)\Big|^2 d t\nonumber\\ &\le&
2\int_0^T |f(t)|^2 d t+2 c'_2\sum_{n\not\in {\cal F}}|C_n|^2\nonumber\\ &\le&
2\Big(1+c'_2\frac{4^{2|{\cal F}|}}{c'_1c'}\Big)\int_0^{T} |f(t)|^2 d t\,. \end{eqnarray} Let us note that the expression \begin{eqnarray*}
\int_0^T\Big|\sum_{n\in {\cal F}}\big(C_ne^{i\omega_nt}+R_ne^{r_nt}\big)\Big|^2 d t \end{eqnarray*}
is a positive semidefinite quadratic form of the variable $\big(\{C_n\}_{n\in {\cal F}},\{R_n\}_{n\in {\cal F}}\big)\in{\mathbb C}^{|{\cal F}|}\times {\mathbb R}^{|{\cal F}|}$. Moreover, it is positive {\it definite}, because the functions $e^{i\omega_nt}$, $e^{r_nt}$, $n\in {\cal F}$, are linearly independent. Hence, there exists a constant $c''>0$ such that \begin{eqnarray*}
\int_0^T\Big|\sum_{n\in {\cal F}}\big(C_ne^{i\omega_nt}+R_ne^{r_nt}\big)\Big|^2 d t\ge c''\sum_{n\in {\cal F}}\big(|C_n|^2+|R_n|^2\big)\,, \end{eqnarray*} so, from \eqref{eq:1+C2C''} and the above inequality we deduce that \begin{eqnarray*}
\sum_{n\in {\cal F}}|C_n|^2\le
\frac{2}{c''}\Big(1+c'_2\frac{4^{2|{\cal F}|}}{c'_1c'}\Big)\int_0^{T} |f(t)|^2 d t \,. \end{eqnarray*} Finally, from the above estimate and \eqref{eq:C''} the desired inequality (\ref{eq:haraux-inv2}) follows with
$
c_1=\frac{2}{c''}\Big(1+c'_2\frac{4^{2|{\cal F}|}}{c'_1c'}\Big)+\frac{4^{2|{\cal F}|}}{c'_1c'}\,. $ \end{Proof2}
To begin, we need a simple lemma.
\begin{lemma} \label{th:exp} Assume that there exist two positive constants $c_1(T)$ and $c_2(T)$ such that
$$c_1(T)\sum_{n=-\infty}^{\infty} |C_n|^2 \leq \int_{0}^{T} |g(t)|^2 d t \leq c_2(T)\sum_{n=-\infty}^{\infty} |C_n|^2\,.$$ If $f(t)=e^{-\alpha t}g(t)$, $\alpha\in{\mathbb R}$, then we have
$$\min (1,e^{-2\alpha T})\ c_1(T)\sum_{n=-\infty}^{\infty} |C_n|^2 \leq \int_{0}^{T} \vert f(t)\vert ^2 d t
\leq \max (1,e^{-2\alpha T})\ c_2(T)\sum_{n=-\infty}^{\infty} |C_n|^2\,.$$
\end{lemma}
\begin{Proof} If we suppose $\alpha>0$, then we infer
$$ \int_{0}^{T} \vert f(t)\vert ^2 d t = \int_{0}^{T} e^{-2\alpha t} \vert g(t)\vert ^2 d t
\ge e^{-2\alpha T}\int_{0}^{T} |g(t)|^2 d t \geq
e^{-2\alpha T}c_1(T)\sum_{n=-\infty}^{\infty} |C_n|^2 $$
and
$$ \int_{0}^{T} \vert f(t)\vert ^2 d t = \int_{0}^{T} e^{-2\alpha t} \vert g(t)\vert ^2 d t
\le \int_{0}^{T} |g(t)|^2 d t
\leq c_2(T)\sum_{n=-\infty}^{\infty} |C_n|^2\,.$$ In the case $\alpha<0$, in a similar way we obtain
$$ \ c_1(T)\sum_{n=-\infty}^{\infty} |C_n|^2 \leq \int_{0}^{T} \vert f(t)\vert ^2 d t
\leq e^{-2\alpha T}c_2(T)\sum_{n=-\infty}^{\infty} |C_n|^2\,.$$ So the proof is complete. \end{Proof}
Trigonometric inequalities, as Ingham ty pe inequalities, play a central role in the achievement of the result. To achieve these inequalities we need an asymptotic estimation of the gap between the eigeinvalues of the spatial operator associated to the system. Starting from phisical problems, we arrive to consider different situations where the original assumptions of the Ingham theorem could be satisfied or not. In the last case, the original result of Ingham has to be generalized, in order to take care of the new behaviour of the eigeinvalues. Taking into account this point of view, we consider coupled systems with variable coefficients, representing the displacement of a thin, spherical elastic shell [1]. In this case the assumptions on the parameters permit to apply the original Ingham result. Hence we analize the behaviour when the thinness paremeter is zero (the membrane approximation) [2]. The limit behaviour of the shell model implies an essential spectrum. The accumutation point in the limit process, leads to a new version of Ingham type inequalities, in order to get a partial controllability result. Next, we consider other type of coupled system. In this case a vectorial generalization of Ingham type theoreom is needed [3]. We discuss about a recent improvement of the last result to repeated eigeivalues [4],
and on possible applications. The latter resultis obtained combining results in [5] and in [3].
\begin{enumerate} \item A. E. Ingham{ Some trigonometrical inequalities with applications to the theory of series, Math. Z. 41 (1936), 367-379
\item G. Geymonat, P. Loreti, e V. Valente, {\it Spectral problems for thin shells and exact controllability}, in Spectral Analysis of Complex Structures, Collection Travaux en cours,
n. 49 (1995), Hermann ed., 35-57.
\item P. Loreti e V. Valente, {\it Partial Exact Controllability for Spherical Membranes}, SIAM J. Control Optim. 35 (1997), 641-653.
\item V. Komornik e P.Loreti, {\it Ingham type theorems for vector-valued functions and observability of coupled linear system},
SIAM J. Control Optim. 37 (1998), 461-485
\item C. Baiocchi, V.Komornik, P. Loreti, {\it work in preparation}
\item C. Baiocchi, V.Komornik, P. Loreti,{\it Ingham type theorems and applications to control theory} B.U.M.I. B (8), II-B, n.1 Febbraio 1999, 33-63
$$ \theta=\frac{\sum_{n,m}{'}\ C_n\overline{C}_m K( c_n+\overline{c}_m)+ \sum_{n, m}C_nR_mK( c_n+r_m)+\sum_{n, m}R_n\overline{C}_mK(r_n+\overline{c}_m)}
{\sum{'}_{n,m}\ \frac{|C_n|^2+ |C^c_m|^2}{2}\big| K( c_n+\overline{c}_m)\big|
+\big|\sum_{n, m}C_n R_m K( c_n+r_m)\big|
+\big|\sum_{n, m}R_n\overline{C}_mK(r_n+\overline{c}_m)\big|} $$
\begin{eqnarray*}
\big|\sum_{n, m}C_n R_m K( c_n+r_m)\big|
&\le&\int_{-\infty}^{\infty}\sqrt{k(t)}\ |\sum_{n}C_ne^{ c_nt}|\sqrt{k(t)}\ \Big|\sum_{m}R_me^{r_mt}\Big|\ dt\\\nonumber &\le&
\left(\int_{-\infty}^{\infty}k(t)\ |\sum_{n}C_ne^{ c_nt}|^2\ dt\right)^{1/2}
\left(\int_{-\infty}^{\infty} k(t)\ \Big|\sum_{n}R_ne^{r_nt}\Big|^2\ dt\right)^{1/2} \end{eqnarray*}
\begin{theorem} \begin{equation}\label{eq:hyplambda} \sqrt{\lambda_n}-\sqrt{\lambda_{n-1}}\ge\gamma>0\,,\qquad \sqrt{\lambda_1}\ge\gamma\qquad\forall n\ge 2 \end{equation} \begin{equation}\label{eq:coeff}
|R_n|\le C|C_n|\,, \end{equation}
For any $\varepsilon>0$ there exists $n_0=n_0(\varepsilon)$ such that for all coefficients satisfying $C^+_n=0$, $|n|\le n_o$, and for any $T>\frac{\pi}{\sqrt{\gamma^2-\varepsilon}}?$ we have that
$$c(T)\sum_n |c_n^+|^2 \leq \int_{-T}^{T} |f|^2 d t\,. $$ \end{theorem} \begin{proof} Thanks to lemma we set $\beta =0$ \begin{eqnarray}\label{eqn:sum}
&&\int_{-\infty}^{\infty} k(t)|f(t)|^2\ dt \nonumber\\ &=& \int_{-\infty}^{\infty} k(t)\sum_{n}\Big(C_ne^{ c_nt}+R_ne^{r_nt}\Big) \sum_{m}\Big(\overline{C}_me^{\overline{c}_mt}+R_me^{r_mt}\Big)\ dt\nonumber\\ &=&\sum_{n, m}C_n\overline{C}_m K( c_n+\overline{c}_m)+ \sum_{n, m}C_nR_mK( c_n+r_m) +\sum_{n, m}R_n\overline{C}_mK(r_n+\overline{c}_m)\nonumber\\ && +
\int_{-\infty}^{\infty} k(t)\ \Big|\sum_{n}R_ne^{r_nt}\Big|^2\ dt\nonumber\\
&=&\sum_{n}|C_n|^2 K\Big(o\big({1\over{\sqrt\lambda_{n}}}\big)\Big)+ \sum_{n,m}{'}\ C_n\overline{C}_m K( c_n+\overline{c}_m)+ \sum_{n, m}C_nR_mK( c_n+r_m) \nonumber\\ && +\sum_{n, m}R_n\overline{C}_mK(r_n+\overline{c}_m)
+\int_{-\infty}^{\infty} k(t)\ \Big|\sum_{n}R_ne^{r_nt}\Big|^2\ dt\nonumber\\ &=&
\sum_{n}|C_n|^2 K\Big(o\big({1\over{\sqrt\lambda_{n}}}\big)\Big)+
\theta \sum_{n,m}{'}\ \frac{|C_n|^2+ |C^c_m|^2}{2}\big| K( c_n+\overline{c}_m)\big| \nonumber\\ &&
+2\theta \sum_{n, m}|C_n||R_m|\big|K( c_n+r_m)\big|
+\int_{-\infty}^{\infty} k(t)\ \Big|\sum_{n}R_ne^{r_nt}\Big|^2\ dt \,, \end{eqnarray} where $$ \theta:=\frac{\sum_{n,m}{'}\ C_n\overline{C}_m K( c_n+\overline{c}_m)+ \sum_{n, m}C_nR_mK( c_n+r_m)+\sum_{n, m}R_n\overline{C}_mK(r_n+\overline{c}_m)}
{\sum_{n,m}{'}\ \frac{|C_n|^2+ |C^c_m|^2}{2}\big| K( c_n+\overline{c}_m)\big|
+2\sum_{n, m}|C_n||R_m|\big|K( c_n+r_m)\big|}\,, $$ $$
\theta\in{\mathbb R}\,,\qquad |\theta|\le1\,. $$ Since $$
\big| K( c_n+\overline{c}_m)\big|=\big| K(\overline{ c_n}+\Lambda_m^c)\big|\,, $$ we have \begin{eqnarray}
&&\sum_{n,m}{'}\ |C_n|\ |C^c_m|\big| K( c_n+\overline{c}_m)\big|\le
\sum_{n,m}{'}\ \frac{|C_n|^2+ |C^c_m|^2}{2}\big| K( c_n+\overline{c}_m)\big|\nonumber\\ &=&
\frac{1}{2}\sum_{n}\ |C_n|^2\sum_{m}{'}\ \big| K( c_n+\overline{c}_m)\big|
+\frac{1}{2}\sum_{m}\ \frac{|C^c_m|^2}{2}\sum_{n}{'}\ \big| K( c_n+\overline{c}_m)\big|\nonumber\\ &=&
\frac{1}{2}\sum_{n}\ |C_n|^2\sum_{m}{'}\ \big| K( c_n+\overline{c}_m)\big|
+\frac{1}{2}\sum_{m}\ |C^c_m|^2\sum_{n}{'}\ \big| K(\Lambda_m^c+\overline{ c_n})\big|= \,, \end{eqnarray} and so by (\ref{eqn:sum}) we obtain \begin{eqnarray}\label{eqn:sum1}
&&\int_{-\infty}^{\infty} k(t)|f(t)|^2\ dt \\ \nonumber&\ge&
\sum_{n}|C_n|^2 \Big[K\Big(o\big({1\over{\sqrt\lambda_{n}}}\big)\Big)+
\theta \sum_{m}{'}\ \big| K( c_n+\overline{c}_m)\big|\Big]
+2\theta \sum_{n, m}|C_n||R_m|\big|K( c_n+r_m)\big|
\\ \nonumber&\ge&
\sum_{n}|C_n|^2 \Big[K\Big(o\big({1\over{\sqrt\lambda_{n}}}\big)\Big)
- \sum_{m}{'}\ \big| K( c_n+\overline{c}_m)\big|\Big]
-2 \sum_{n, m}|C_n||R_m|\big|K( c_n+r_m)\big|
\,. \end{eqnarray} \begin{eqnarray*}
&&\sum_{m} {'}\ \big| K( c_n+\overline{c}_m)\big|\\ &\le& 4T\pi\sum_{m}{'}\frac{\cosh(o\Big({1\over{\sqrt\lambda_{n}}}\Big)T+o\Big({1
\over{\sqrt\lambda_{m}}}\Big)T)}{|4T^2(o\Big({1\over{\sqrt\lambda_{n}}}\Big)+o\Big({1\over{\sqrt\lambda_{m}}}\Big))^2
-4T^2(\Im( c_n+\overline{c}_m))^2+\pi^2|}\\ &=& 4T\pi\sum_{m}{'}\ \frac{\cosh(o\Big({1\over{\sqrt\lambda_{n}}}\Big)T +o\Big({1\over{\sqrt\lambda_{m}}}\Big)T)}{4T^2(\Im( c_n+\overline{c}_m))^2-4T^2(o\Big({1\over{\sqrt\lambda_{n}}}\Big)+o\Big({1 \over{\sqrt\lambda_{m}}}\Big))^2-\pi^2}\\ &\le& 4T\pi\cosh(o\Big({1\over{\sqrt\lambda_{n}}}\Big)T\Big)(1+\varepsilon)\sum_{m}{'}\ \frac{1}{4T^2(\Im( c_n+\overline{c}_m))^2-4T^2(o\Big({1\over{\sqrt\lambda_{n}}}\Big)+o\Big({1 \over{\sqrt\lambda_{m}}}\Big))^2-\pi^2}\\ &\le& 4T\pi\cosh(o\Big({1\over{\sqrt\lambda_{n}}}\Big)T\Big)(1+\varepsilon)\sum_{m}{'}\ \frac{1}{4T^2\gamma^2(n-m)^2-4T^2\varepsilon'-\pi^2}\\ &\le& 4T\pi\cosh(o\Big({1\over{\sqrt\lambda_{n}}}\Big)T\Big)(1+\varepsilon)\sum_{m}{'}\ \frac{1}{4T^2\gamma^2(n-m)^2-T^2\gamma^2} \qquad\mbox{if}\quad T^2\gamma^2>4T^2\varepsilon'+\pi^2\\ &\le& \frac{4\pi}{T\gamma^2}\cosh(o\Big({1\over{\sqrt\lambda_{n}}}\Big)T\Big)(1+\varepsilon)\sum_{m}{'}\ \frac{1}{4(n-m)^2-1}\\ &\le& \frac{8\pi}{T\gamma^2}\cosh(o\Big({1\over{\sqrt\lambda_{n}}}\Big)T\Big)(1+\varepsilon)\sum_{r=1}^{\infty}\ \frac{1}{4r^2-1}=\frac{4\pi}{T\gamma^2}\cosh(o\Big({1\over{\sqrt\lambda_{n}}}\Big)T\Big)(1+\varepsilon)\,. \end{eqnarray*} In addition, thanks to (\ref{eq:coeff}) we have that \begin{eqnarray*}
&&\sum_{n, m}|C_n||R_m|\big|K( c_n+r_m)\big|\le C
\sum_{n, m}|C_n||C^c_m|\big|K( c_n+r_m)\big|\\\nonumber &\le& C
\sum_{n, m}\frac{|C^c_m|^2}{2}\big|K( c_n+r_m)\big|
+C\sum_{n, m}\frac{|C_n|^2}{2}\big|K( c_n+r_m)\big|\\\nonumber &\le& C
\sum_{ m}|C^c_m|^2\sum_{n}\big|K( c_n+r_m)\big|\,; \end{eqnarray*} \begin{eqnarray*}\label{}
\sum_{n}\big|K( c_n+r_m)\big| \le 4T\pi\sum_{n}\frac{\cosh\Big(-\eta T+o\big({1\over{\sqrt\lambda_{m}}}\big) T+o\big({1\over{\sqrt\lambda_{n}}}\big)T\Big)}
{\Big|4T^2\Big(r_m+o\big({1\over{\sqrt\lambda_{n}}}\big)\Big)^2- 4T^2(\Im c_n)^2+\pi^2\Big|} \end{eqnarray*} Since $$
\big|\Im c_n\big|\ge\gamma |n|\,,\qquad\forall n\,, $$ it follows $$
\big|\Im c_n-r_m\big|\ge\gamma_2 |n|\,,\qquad\forall |n|\,, |m| \ge n_0\,, $$ $$
\big|\Im c_n+r_m\big|\ge\gamma_2|n|\,,\qquad\forall |n|\,, |m| \ge n_0\,, $$ for a suitable $\gamma_2>0$, and hence \begin{eqnarray*}\label{}
&&\Big|4T^2\Big(r_m+o\big({1\over{\sqrt\lambda_{n}}}\big)\Big)^2-4T^2(\Im c_n)^2+\pi^2\Big|\\ &\le&
4T^2\Big|r_m+o\big({1\over{\sqrt\lambda_{n}}}\big)-\Im c_n\Big|\
\Big|r_m+o\big({1\over{\sqrt\lambda_{n}}}\big)+\Im c_n\Big|-\pi^2\\ &\ge& 4T^2\gamma_2^2 \ n^2-\pi^2\ge T^2\gamma_2^2(4n^2-1)\,,\qquad\mbox{for any }\,T>\frac{\pi}{\gamma_2}\,. \end{eqnarray*} Therefore, \begin{eqnarray*}\label{}
&&\sum_{n}\big|K( c_n+r_m)\big|\\\nonumber &\le& \frac{4\pi}{T\gamma_2^2}\sum_{n}\frac{\cosh\Big(-\eta T+o\big({1\over{\sqrt\lambda_{m}}}\big) T+o\big({1\over{\sqrt\lambda_{n}}}\big)T\Big)} {4n^2-1} \\\nonumber &\le& \frac{8\pi}{T\gamma_2^2}\cosh(\eta T+\varepsilon)\sum_{r=1}^\infty\frac{1}{4r^2-1}=\frac{4\pi}{T\gamma_2^2}\cosh(\eta T+\varepsilon) \end{eqnarray*}
\begin{eqnarray}\label{eqn:sum2}
&&\int_{-\infty}^{\infty} k(t)|f(t)|^2\ dt \\ \nonumber&\ge&
\sum_{n}|C_n|^2 \Big[K\Big(o\big({1\over{\sqrt\lambda_{n}}}\big)\Big) - \frac{4\pi}{T\gamma^2}\cosh(o\Big({1\over{\sqrt\lambda_{n}}}\Big)T\Big)(1+\varepsilon) -\frac{8\pi}{T\gamma_2^2}\cosh(\eta T+\varepsilon)\Big] \,. \end{eqnarray}
\end{proof}
\end{document} | arXiv |
\begin{document}
\title{The first law of general quantum resource theories}
\author{Carlo Sparaciari}\email{[email protected]} \affiliation{Department of Physics and Astronomy, University College London, London WC1E 6BT, United Kingdom}
\author{L\'idia del Rio}
\affiliation{Institute for Theoretical Physics, ETH Zurich, 8093 Z{\"u}rich, Switzerland}
\author{Carlo Maria Scandolo}
\affiliation{Department of Computer Science, University of Oxford, Oxford OX1 3QD, UK}
\author{Philippe Faist}
\affiliation{Dahlem Center for Complex Quantum Systems, Freie Universit\"at Berlin, 14195 Berlin, Germany} \affiliation{Institute for Quantum Information and Matter, Caltech, Pasadena CA, 91125 USA}
\author{Jonathan Oppenheim}
\affiliation{Department of Physics and Astronomy, University College London, London WC1E 6BT, United Kingdom}
\date{20-04-2020}
\begin{abstract} We extend the tools of quantum resource theories to scenarios in which multiple quantities (or resources) are present, and their interplay governs the evolution of physical systems. We derive conditions for the interconversion of these resources, which generalise the first law of thermodynamics. We study reversibility conditions for multi-resource theories, and find that the relative entropy distances from the invariant sets of the theory play a fundamental role in the quantification of the resources. The first law for general multi-resource theories is a single relation which links the change in the properties of the system during a state transformation and the weighted sum of the resources exchanged. In fact, this law can be seen as relating the change in the relative entropy from different sets of states. In contrast to typical single-resource theories, the notion of free states and invariant sets of states become distinct in light of multiple constraints. Additionally, generalisations of the Helmholtz free energy, and of adiabatic and isothermal transformations, emerge. We thus have a set of laws for general quantum resource theories, which generalise the laws of thermodynamics. We first test this approach on thermodynamics with multiple conservation laws, and then apply it to the theory of local operations under energetic restrictions. \end{abstract}
\maketitle
\tableofcontents
\section{Introduction} \label{intro} \textbf{Resource theories.} Resource theories are a versatile set of tools developed in quantum information theory. They are used to describe the physical world from the perspective of an agent, whose ability to modify a quantum system is restricted by either practical or fundamental constraints. These limitations mean that while some states can still be created under the restricted class of operations (the {\it free} or {\it invariant set} of states), other state transformations can only be done with the help of additional resources. The goal of resource theories is then to quantify this cost, and to consequently assign a price to every state of the system, from the most expensive to the free ones. Because of their very general structure, which only involves the set of states describing a quantum system and a given set of allowed operations for acting on such system, resource theories can be used to study many different branches of quantum physics, from entanglement theory~\cite{bennett_mixed-state_1996,rains_entanglement_1997, vedral_entanglement_1998,rains_bound_1999,horodecki_quantum_2009} to thermodynamics~\cite{janzing_thermodynamic_2000,horodecki_reversible_2003, rio_thermodynamic_2011,workvalue,brandao_resource_2013,horodecki_fundamental_2013, skrzypczyk_work_2014,gallego_thermodynamic_2016}, from asymmetry~\cite{gour_resource_2008, gour_measuring_2009,marvian_theory_2013} to the theory of magic states~\cite{mari_positive_2012, veitch_resource_2014,veitch_negative_2012}. Additionally, these theories can often be formulated within more abstract, axiomatic frameworks~\cite{lieb_physics_1999,lieb_entropy_2013, weilenmann_axiomatic_2016,fritz_resource_2015,del_rio_resource_2015,coecke_mathematical_2016, anshu_quantifying_2017}. \par Thanks to the underlying common structure present in all the theories described within this framework, one can find general results which apply to all. For example, a resource theory may be equipped with a zeroth, second, and even third law, i.e., relations that regulate the different aspects of the theory, which are reminiscent of the Laws of Thermodynamics. In fact, we have that the \emph{zeroth law} for resource theories states that there exists equivalence classes of free states, and that states from one of these classes are the only ones that can be freely added to the system without trivialising the theory~\cite{brandao_second_2015}. The \emph{second law} of resource theories states that some quantities, linked to the amount of resource contained in a system, never increase under the action of the allowed operations~\cite{popescu_thermodynamics_1997}, and for reversible resource theories satisfying modest assumptions, this quantity is unique~\cite{horodecki_are_2002,brandao_entanglement_2008, brandao_reversible_2010,horodecki_quantumness_2012,brandao_reversible_2015} --- an example of this is the free energy, which is a monotone in thermodynamics as it decreases in any cyclic process, and the local entropy for pure state entanglement theory. Finally, one might have a generalisation of the third law which places limitations on the time needed to reach a state when starting from another one, rather than simply telling us whether such transformation is possible or not~\cite{masanes_general_2017}. With the present work, we aim to derive the \emph{first law} for resource theories, and to do so we will have to extend the framework so as to include multiple resources. The law we derive connects the amount of different resources exchanged during a state transformation to the change, quantified by a specific monotone, between the initial and final state of the system. When considering thermodynamics, this law connects the amount of work and heat exchanged during a process to the internal energy of the systems. \par \textbf{Multiple resources.} It is often the case that many resources are needed to perform a given task. For instance, thermodynamics can be understood as a resource theory with multiple resources~\cite{sparaciari_resource_2016, bera_thermodynamics_2017}, where in order to transform the state of the system we need both \emph{energy} and \emph{information}, or equivalently, work and heat. As another example, some quantum computational schemes consider the idealized case in which the input qubits are pure, and the gates acting on them create coherence. In order to better understand the role played in quantum computation by these two resources, \emph{coherence} and \emph{purity}, a possible approach might consist in combining the resource theories of purity~\cite{horodecki_reversible_2003, gour_resource_2015} and coherence~\cite{aberg_quantifying_2006,baumgratz_quantifying_2014,winter_operational_2016} together. Other examples of theories in which multiple resources are considered can be found in the literature~\cite{slepian_noiseless_1973,horodecki_partial_2005, ahmadi_wignerarakiyanase_2013,singh_maximally_2015,streltsov_entanglement_2016,chitambar_relating_2016, sparaciari_resource_2016,erker_autonomous_2017,bera_thermodynamics_2017}. Given the success of resource theories to describe physical situations where only one resource is involved, it seems natural to ask the question whether the framework can be extended to the case in which more resources are involved. For example, it is known that the resource theoretic approach to thermodynamics allows us to derive a second law relation even in the case in which many (commuting, non-commuting) conserved quantities are present~\cite{guryanova_thermodynamics_2016, yunger_halpern_microcanonical_2016,yunger_halpern_beyond_2016,lostaglio_thermodynamic_2017, halpern_beyond_2018}, and one can consider trade-offs of these \cite{popescu_quantum_2018}. We are thus interested in understanding if one can extend these results to other resource theories, and whether a first law of general resource theories exists.
\begin{figure}\label{fig:slizard_boxes}
\end{figure}
\par \textbf{Contribution of this work.} In this paper we present a framework for resource theories with multiple resources, introduced in Sec.~\ref{multi_res_framework}. In our framework we first consider the different constraints and conservation laws that the model needs to satisfy, and for each of these constraints, we introduce the corresponding single-resource theory. Then, we define the class of allowed operations of the multi-resource theory as the set of maps lying in the intersection of all the classes of allowed operations of the single-resource theories. Due to this construction, we find that a multi-resource theory with $m$ resources has at least $m$ invariant sets (i.e., sets of states that are mapped into themselves by the action of the allowed operations of the theory), each of them corresponding to the set of free states of one of the $m$ single-resource theories. In order to make the paper self-contained, in Sec.~\ref{multi_res_framework} we also provide a brief review of the resource theoretic formalism (see Ref.~\cite{horodecki_quantumness_2012,chitambar_quantum_2018} for reviews on this topic). \par We then study, in Sec.~\ref{rev_theory_mult}, the properties of general multi-resource theories in the asymptotic limit, that is, when the agent is allowed to act globally over many identical copies of the system. This limit is of fundamental importance in resource theories since it allows us to investigate reversibility and the emergence of unique measures for quantifying different resources~\cite{horodecki_quantumness_2012}. In a reversible theory, we have that the resources consumed to perform a given state transformation can always be completely recovered with the reverse transformation, so that no resource is ever lost. In single-resource theories, we can rephrase this notion of reversibility in terms of rates of conversion, but for general multi-resource theories this is not always possible. As a result, we focus our study on multi-resource theories that satisfy an additional property, which we refer to as the \emph{asymptotic equivalence property}~\cite{fritz_resource_2015,sparaciari_resource_2016}, see Def.~\ref{def:asympt_equivalence_multi} below. We show that, when a multi-resource theory satisfies the asymptotic equivalence property, there is a unique measure associated with each resource present in the theory. Furthermore, when the invariant sets of the theory satisfy some natural properties, we find that the unique measures are given by the (regularised) relative entropy distances from these sets, each of those associated with a different resource. Finally, we show that when a resource theory satisfies asymptotic equivalence, it is also reversible in the sense that resources are never lost during a state transformation, and they can be recovered. This result can be seen as the extension of what has already been shown for reversible single-resource theories~\cite{popescu_thermodynamics_1997,horodecki_are_2002,horodecki_quantumness_2012, brandao_reversible_2015}. \par In Sec.~\ref{interconv} we address the question of whether it is possible to exchange resources. We consider the case in which different resources are individually stored in separate systems, which we call \emph{batteries}. Then, we investigate under which conditions it is possible to find an additional system, which we refer to as a \emph{bank}\footnote{We apologise in advance for introducing this terminology into the field of resource theories, but the banks considered here exchange resources without charging interest or fees, and are thus more akin to community cooperative banks than their more exploitative cousins.}, that allows us to reduce the amount of resource contained in one battery while simultaneously increasing the amount of resource in another battery. During such conversion, we ask the bank not to change its properties -- with respect to a specific measure defined in Eq.~\eqref{f3_monotone} -- so as to be able to use this system again. For example, in thermodynamics the thermal bath plays the role of the bank, as it allows us to exchange energy for information and vice versa, see Fig.~\ref{fig:slizard_boxes}. In order to study interconversion, we demand the invariant sets of the theory to satisfy an "additivity" condition, which is satisfied by some resource theories, for example by thermodynamics and purity theory. We find that a multi-resource theory needs to have an empty set of free states for a bank to exist, and when this condition is satisfied we derive an interconversion relation, see Thm.~\ref{thm:interconvert_relation}, which defines the rates at which resources are exchanged. \par We additionally show that, when the agent is allowed to use batteries and bank, they can perform any state transformation using variable amounts of resources. Indeed, since the agent can use the bank to inter-convert between resources, they can decide to invest a higher amount of one resource to save on the others. This freedom is reflected in our framework by a single relation, the first law of resource theories, which connects the different resources, each of them weighted by the corresponding exchange rate, to the change of a particular monotone between the initial and final state of the system, see Cor.~\ref{coro:first_law}. This equality is a generalisation of the first law of thermodynamics, where the sum of the work performed on the system and the heat absorbed from the environment is equal to the change in internal energy of the system. In fact, the first law of thermodynamics can be understood as equating various relative entropy distances which quantify different types of resources, as we discuss at the beginning of Sec.~\ref{interconv}. \par Finally, in Sec.~\ref{examples} we provide two examples of multi-resource theories which admit an interconversion relation between their resources. The first example concerns thermodynamics of multiple conserved quantities, for which the interconversion of resources was shown in Ref.~\cite{guryanova_thermodynamics_2016}. The second example concerns the theory of local control under energy restrictions. Here we consider a system with a non-local Hamiltonian, and we assume that the experimentalists acting on this system only have access to a portion of the system. In this scenario, the entanglement between the different portions of the system and the overall energy of the global system are the main resources of the theory, and we study under which conditions we can inter-convert energy and entanglement. For a summary of how to apply our work to an arbitrary resource theory, see the flowchart in Fig.~\ref{fig:flowchart}.
\section{Framework for multi-resource theories} \label{multi_res_framework} Let us now introduce the framework for multi-resource theories. A multi-resource theory is useful when we need to describe a physical task or process which is subjected to different constraints and conservation laws. The first step consists in associating each of these constraints with a single-resource theory, whose class of allowed operations satisfies the specific constraint or conservation law. The multi-resource theory is then obtained by defining its class of allowed operations as the intersection between the sets of allowed operations of the different single-resource theories previously defined. In this way, we are sure of acting on the quantum system with operations that do not violate the multiple constraints imposed on the task.
\subsection{Single-resource theory} \label{single_resource}
For simplicity, we restrict ourselves to the study of finite-dimensional quantum systems. Therefore, the system under investigation is described by a Hilbert space $\mathcal{H}$ with dimension $d$. The state-space of this quantum system is given by the set of density operators acting on the Hilbert space, $\mathcal{S} \left( \mathcal{H} \right) = \left\{ \rho \in \mathcal{B} \left( \mathcal{H} \right) \ | \ \rho \geq 0, \ \tr{\rho} = 1 \right\}$, where $\mathcal{B} \left( \mathcal{H} \right)$ is the set of bounded operators acting on $\mathcal{H}$. A single-resource theory for the quantum system under examination is defined through a class of allowed operations $\mathcal{C}$, that is, a constrained set of completely positive maps acting on the state-space $\mathcal{S} \left( \mathcal{H} \right)$\footnote{Although the operations we consider are endomorphisms of a given state space, our formalism is still able to describe the general case in which the agent modifies the quantum system. If the agent's action transforms the state of the original system, associated with $\mathcal{H}_1$, into the state of a final system $\mathcal{H}_2$, we can model this action with a map acting on the state space of $\mathcal{H} = \mathcal{H}_1 \otimes \mathcal{H}_2$. Suppose the operation maps $\rho_1 \in \mathcal{S} \left( \mathcal{H}_1 \right)$ into $\sigma_2 \in \mathcal{S} \left( \mathcal{H}_2 \right)$. Then, the map acting on $\mathcal{S} \left( \mathcal{H} \right)$ takes the state $\rho_1 \otimes \gamma_2$ and outputs the state $\gamma_1' \otimes \sigma_2$, where $\gamma_1$ and $\gamma_2'$ are free states for the systems described by $\mathcal{H}_2$ and $\mathcal{H}_1$, respectively.} \cite{horodecki_are_2002}. The constraints posed on the set of allowed operations are specific to the resource theory under consideration. For example, in the theories that study entanglement it is often the case that we constrain the set of allowed operations to be composed by the maps that are local, and only make use of classical communication~\cite{bennett_mixed-state_1996}. In asymmetry theory, instead, we only allow the maps whose action is covariant with respect to the elements of a given group~\cite{gour_resource_2008}. Furthermore, in the resource theoretic approach to thermodynamics we can, without loss of generality, constrain this set to those operations, known as Thermal Operations, which preserve the energy of a closed system, and can thermalise the system with respect to a background temperature~\cite{janzing_thermodynamic_2000, brandao_resource_2013,horodecki_fundamental_2013,renes_work_2014}. Once the set of allowed operations is defined, it is usually possible to identify which states in $\mathcal{S} \left( \mathcal{H} \right)$ are resourceful, and which ones are not. In particular, the set of \emph{free states} for a single-resource theory, $\mathcal{F} \subset \mathcal{S} \left( \mathcal{H} \right)$, is composed of those states that can always be prepared using the allowed operations, no matter the initial state of the system. Mathematically, this set of states is defined as \begin{equation} \label{free_state_set}
\mathcal{F} = \left\{ \sigma \in \mathcal{S} \left( \mathcal{H} \right) \ | \ \forall \, \rho \in \mathcal{S} \left( \mathcal{H} \right), \exists \, \mathcal{E} \in \mathcal{C} : \mathcal{E}(\rho) = \sigma \right\}. \end{equation} For example, in entanglement theory the free states are the separable states, in asymmetry theory they are the ones that commute with the elements of the considered group, and in thermodynamics they are the thermal states at the background temperature. \par An \emph{invariant set} is a set of states that is preserved under action of any allowed operation. From the definition of free states in Eq.~\eqref{free_state_set}, it is easy to show that $\mathcal{F}$ is an invariant set, and we write this as $\mathcal{E}(\mathcal{F}) \subseteq \mathcal{F}$ for all $\mathcal{E} \in \mathcal{C}$. It is worth noting that while the set of free states is invariant, the opposite clearly does not need to be true. In particular, when we study multi-resource theory, we will see that several invariant sets can be found, and still there might be no free set for the theory. Due to the invariant property of free states, we can also define the class of allowed operations in a different way. Instead of considering the specific constraints defining the set of allowed operations $\mathcal{C}$, we can simply assume that this set is a subset of the bigger class of completely positive and trace preserving (CPTP) maps \begin{equation} \label{maps_single_copy}
\tilde{\mathcal{C}} = \left\{ \mathcal{E} : \mathcal{B} \left( \mathcal{H} \right) \rightarrow \mathcal{B} \left( \mathcal{H} \right) \ | \ \mathcal{E} \left( \mathcal{F} \right) \subseteq \mathcal{F} \right\}, \end{equation} that is, the set of maps for which the free states $\mathcal{F}$ form an invariant set. It is worth noting that $\mathcal{C}$ is often a proper subset of $\tilde{\mathcal{C}}$. For example, in entanglement theory, we have that $\mathcal{C}$ might be composed by local operations and classical communication (LOCC), which is a proper subset of the set of all quantum channels which preserve the separable states. Indeed, the map that swaps between the local states describing the quantum system is clearly not LOCC, but it preserves separable states~\cite{bennett_quantum_1999}. \par We can also extend the single-resource theory to the case in which we consider $n \in \mathbb{N}$ copies of the quantum system. The class of allowed operations, which in this case we refer to as $\mathcal{C}^{(n)}$, is still defined by the same constraints, but now acts on $\SHn{n}$, the state-space of $n$ copies of the system. For example, in the resource theory of thermodynamics with Thermal Operations we have that the energy of a closed system needs to be exactly conserved. For a single system, this implies that the operations need to commute with the Hamiltonian $H^{(1)}$. For $n$ non-interacting copies of the system, instead, the operations commute with the global Hamiltonian $H_n = \sum_{i=1}^n H^{(1)}_i$. Within the state-space $\SHn{n}$, we can find the set of free states, $\mathcal{F}^{(n)} \subset \SHn{n}$. It is worth noting that the set of free states for $n$ copies of the system is such that $\mathcal{F}^{\otimes n} \subseteq \mathcal{F}^{(n)}$, that is, it contains more states than just the tensor product of $n$ states in $\mathcal{F}$. This is the case, for example, of entanglement theory, where among the free states for two copies of the system we can find states that are locally entangled, since each agent is allowed to entangle the partitions of the system they own. On the contrary, the two sets coincide for any $n \in \mathbb{N}$ for the resource theory of thermodynamics, where the free state is the Gibbs state of a given Hamiltonian. Anyway, it is still the case that $\mathcal{F}^{(n)}$ is invariant under the class $\mathcal{C}^{(n)}$, and therefore we can think of the set of allowed operations acting on $n$ copies of the system as a subset of the bigger set of CPTP maps \begin{equation} \label{maps_n_copies}
\tilde{\mathcal{C}}^{(n)} = \left\{ \mathcal{E}_n : \mathcal{B} \left( \mathcal{H}^{\otimes n} \right) \rightarrow \mathcal{B} \left( \mathcal{H}^{\otimes n} \right) \ | \ \mathcal{E}_n \left( \mathcal{F}^{(n)} \right) \subseteq \mathcal{F}^{(n)} \right\}. \end{equation} Thus, in order to extend a single-resource theory to the many-copy case, we need to take into account the sequence of all sets of allowed operations $\mathcal{C}^{(n)}$, where $n \in \mathbb{N}$ is the number of copies of the system the maps are acting on. \par It is worth noting that the allowed operations we have introduced keep the number of copies of the system fixed, see Eq.~\eqref{maps_n_copies}. Indeed, we only consider these maps because, when the number of input and output systems of a quantum channel changes, the internal structure of the channel involves the discarding (or the addition) of some of these systems. However, in a (reversible) resource theory, one can perform such operations only if the amount of resources is kept constant. This is certainly possible if we are to add or trace out some free states of the theory (which do not contain any resource), but as we will see in the next section, multi-resource theory not always have any free states. For this reason, we decide to only focus on maps that conserve the number of systems, even for single-resource theories. \par We can now address the problem of quantifying the amount of resource associated with different states of the quantum system. In resource theories, a resource quantifier is called \emph{monotone}. This object is a function $f$ from the state-space $\mathcal{S} \left( \mathcal{H} \right)$ to the set of real numbers $\mathbb{R}$, which satisfies the following property, \begin{equation} \label{second_law} f \left( \mathcal{E}(\rho) \right) \leq f \left( \rho \right) , \qquad \forall \, \rho \in \mathcal{S} \left( \mathcal{H} \right), \ \forall \, \mathcal{E} \in \mathcal{C}. \end{equation} The above inequality can be interpreted as a ``second law'' for the resource theory, since there is a quantity (the monotone) that never increases as we act on the system with allowed operations. In the thermodynamic case, in fact, we know that the Second Law of Thermodynamics imposes that the entropy of a closed system can never decrease as time goes by. We can extend the definition of monotones to the case in which we consider $n$ copies of the system. In this case, the function $f$ maps states in $\SHn{n}$ into $\mathbb{R}$, and an analogous relation to the one of Eq.~\eqref{second_law} holds, this time for states in $\SHn{n}$ and the set of allowed operations $\mathcal{C}^{(n)}$. Finally, we can also define the \emph{regularisation} of a monotone $f$ as \begin{equation} f^{\infty}(\rho) = \lim_{n \rightarrow \infty} \frac{f \left( \rho^{\otimes n} \right)}{n}, \end{equation} where $\rho \in \mathcal{S} \left( \mathcal{H} \right)$, and $\rho^{\otimes n} \in \SHn{n}$. Notice that, given a generic monotone $f$, we need the above limit to exist and be finite in order to define its regularisation. \par For each resource theory there exists several monotones, and we can always build one out of a \emph{contractive distance}~\cite{brandao_reversible_2015}. Consider the distance $C \left( \cdot, \cdot \right) : \mathcal{S} \left( \mathcal{H} \right) \times \mathcal{S} \left( \mathcal{H} \right) \rightarrow \mathbb{R}$ such that \begin{equation} C \left( \mathcal{E}(\rho) , \mathcal{E}(\sigma) \right) \leq C \left( \rho, \sigma \right) , \qquad \forall \, \rho, \sigma \in \mathcal{S} \left( \mathcal{H} \right), \ \forall \, \mathcal{E} \ \text{CPTP map}. \end{equation} Then, a monotone for the single-resource theory with allowed operations $\mathcal{C}$ and free states $\mathcal{F}$ is \begin{equation} \label{monotone_contract} M_{\mathcal{F}} (\rho) = \inf_{\sigma \in \mathcal{F}} C \left( \rho, \sigma \right), \end{equation} where it is easy to show that $M_{\mathcal{F}}$ satisfies the property of Eq.~\eqref{second_law}, which follows from the fact that $\mathcal{F}$ is invariant under the set of allowed operations $\mathcal{C}$, and from the contractivity of $C \left( \cdot, \cdot \right)$ under any CPTP map. A specific example of a monotone obtained from a contractive distance is the relative entropy distance from the set $\mathcal{F}$. Consider two states $\rho, \sigma \in \mathcal{S} \left( \mathcal{H} \right)$, such that $\supp{\rho} \subseteq \supp{\sigma}$. Then, we define the relative entropy between these two states as \begin{equation} \label{rel_entr} \re{\rho}{\sigma} = \tr{\rho \, \left( \log \rho - \log \sigma \right)}. \end{equation} The relative entropy is contractive under CPTP maps~\cite{lindblad_completely_1975}, and even if it does not satisfy all the axioms to be a metric\footnote{The relative entropy is non-negative for any two inputs, and zero only when the two inputs coincide, but it is not symmetric, nor does it satisfy the triangular inequality.} over $\mathcal{S} \left( \mathcal{H} \right)$, we can still obtain a monotone out of this quantity, building it as in Eq.~\eqref{monotone_contract}. This monotone is \begin{equation} \label{rel_entr_dist} E_{\mathcal{F}}( \rho ) = \inf_{\sigma \in \mathcal{F}} \re{\rho}{\sigma}, \end{equation} and is known as the relative entropy distance from $\mathcal{F}$. When the separable states form the set $\mathcal{F}$, for example, the monotone is the relative entropy of entanglement~\cite{vedral_entanglement_1998}. It is worth noting that, in order for $E_{\mathcal{F}}$ to be well-defined, the set $\mathcal{F}$ has to contain at least one full-rank state.
\subsection{Multi-resource theory} \label{multi_resource} Let us consider the case in which we can identify in the theory a number $m > 1$ of resources, which can arise from some conservation laws, or from some constraints. We now introduce a multi-resource theory with these $m$ resources. The quantum system under investigation is the same as in the previous section, described by the states in the state-space $\mathcal{S} \left( \mathcal{H} \right)$. For the $i$-th resource of interest, where $i = 1, \ldots, m$, we consider the corresponding single-resource theory $\text{R}_i$, defined by the set of allowed operations $\mathcal{C}_i$ acting on the state-space $\mathcal{S} \left( \mathcal{H} \right)$. We denote the set of free states of this single-resource theory as $\mathcal{F}_i \subset \mathcal{S} \left( \mathcal{H} \right)$, and we recall that any allowed operation in $\mathcal{C}_i$ leaves this set invariant. Therefore, we can consider the class of allowed operation as a subset of the set of CPTP maps \begin{equation} \label{maps_single_copy_i} \tilde{\mathcal{C}}_i = \left\{ \mathcal{E}_i : \mathcal{B} \left( \mathcal{H} \right) \rightarrow \mathcal{B} \left( \mathcal{H} \right)
\ | \ \mathcal{E}_i \left( \mathcal{F}_i \right) \subseteq \mathcal{F}_i \right\}. \end{equation} We can also extend the resource theory $\text{R}_i$ to the case in which we consider more than one copy of the system, following the same procedure used in the previous section. Then, the class of allowed operations $\mathcal{C}_i^{(n)}$ acting on $n$ copies of the system is a subset of the set of operations which leave $\mathcal{F}_i^{(n)} \subset \SHn{n}$ invariant, see Eq.~\eqref{maps_n_copies}. \par Once all the single-resource theories $\text{R}_i$'s are defined, together with their sets of allowed operations, we can build the multi-resource theory $\text{R}_{\text{multi}}$ for the quantum system described by the Hilbert space $\mathcal{H}$. The set of allowed operations for this theory is given by the maps contained in the intersection\footnote{While other multi-resource theory constructions can be imagined, the one we use in this paper provides the certainty that no resource can be created out of free states.} between the classes of allowed operations of the $m$ single-resource theories, that is \begin{equation} \label{all_ops_multi} \mathcal{C}_{\text{multi}} = \overset{m}{\underset{i=1}{\cap}} \mathcal{C}_i. \end{equation} Notice that, alternatively, one can define the set of allowed operations $\mathcal{C}_{\text{multi}}$ as a subset of the bigger set $\cap_{i=1}^m \tilde{\mathcal{C}}_i$, where $\tilde{\mathcal{C}}_i$ is the set of all the CPTP maps for which $\mathcal{F}_i$ is invariant, see Eq.~\eqref{maps_single_copy_i}. When $n$ copies of the system are considered, the class of allowed operations for the multi-resource theory, $\mathcal{C}_{\text{multi}}^{(n)}$, is obtained by the intersection between the sets of allowed operations $\mathcal{C}_i^{(n)}$ of the different single-resource theories, that is, $\mathcal{C}_{\text{multi}}^{(n)} = \cap_{i=1}^m \mathcal{C}_i^{(n)}$.
\begin{figure}
\caption{The structure of the sets of free states for two single-resource theories which compose a multi-resource theory. These sets are invariant under the allowed operations of the resulting multi-resource theory. For theories with $m > 2$ resources, the structure of the free sets can be obtained by composing the three fundamental scenarios presented here. {\bf Left.} The invariant set $\mathcal{F}_2$ is a subset of $\mathcal{F}_1$. This multi-resource theory has a set of free states, which coincides with $\mathcal{F}_2$. An example of such a theory is that of coherence~\cite{baumgratz_quantifying_2014} and purity~\cite{gour_resource_2015}, where the invariant sets are incoherent states with respect to a given basis and the maximally-mixed state, respectively. {\bf Centre.} The two invariant sets intersect each other. This theory has a set of free states which coincides with the intersection, $\mathcal{F}_1 \cap \mathcal{F}_2$. An example of multi-resource theory with this structure concerns tripartite entanglement for systems $A$, $B$, and $C$. The allowed operations of this theory are defined by the intersection of the operations associated with the theories of bipartite entanglement for systems $AB$ and $C$, systems $AC$ and $B$, and systems $A$ and $BC$. Notice that this theory does not coincide with the theory of tripartite LOCC, since some of the free states are entangled~\cite{bennett_unextendible_1999}. {\bf Right.} The two invariant sets are separated. Consequently, the theory does not have any free states. In this situation, one can find an interconversion relation between the resources, as shown in Sec.~\ref{sec:first_law}. An example of a multi-resource theory with this structure is thermodynamics of closed systems. If the agent does not have perfect control on the reversible operations they implement, and the closed system is coupled to a sink of energy (an ancillary system which can only absorb energy), then the allowed operations are given by the intersection between the set of mixtures of unitary operations, and the set of average-energy-non-increasing maps. In this case, the maximally-mixed state and the ground state of the Hamiltonian are the two invariant sets of the theory. Notice that the set of energy-preserving unitary operations, considered in Ref.~\cite{sparaciari_resource_2016}, is a subset of this bigger set.}
\label{fig:invariant_sets_structure}
\end{figure}
\par We can now consider the invariant sets of this multi-resource theory. Clearly, each set of free states $\mathcal{F}_i$ associated with the single-resource theory $\text{R}_i$ is an invariant set for the class of operations $\mathcal{C}_{\text{multi}}$. However, it is worth noting that the states contained in the $\mathcal{F}_i$'s might not be free when the multi-resource theory is considered, where a free state is (as we pointed out in the previous section) a state that does not contain any resource and can be realised using the allowed operations. Indeed, the states contained in the set $\mathcal{F}_i$ might be resourceful states for the single-resource theory $\text{R}_j$, and therefore we would not be able to realise such states with the class of operations $\mathcal{C}_{\text{multi}}$. In Fig.~\ref{fig:invariant_sets_structure} we show the different configurations for the invariant sets of a multi-resource theory with two resources. While in the left and central panels the theory has free states, in the right panel no free states can be found, a noticeable difference from the framework for single-resource theories. \par The multi-resource theory $\text{R}_{\text{multi}}$ also inherits the monotones of the single-resource theories that compose it. This follows trivially from the choice we made in defining the class of allowed operations $\mathcal{C}_{\text{multi}}$, see Eq.~\eqref{all_ops_multi}. Furthermore, other monotones, that are only valid for the multi-resource theory, can be obtained from the ones inherited from the single-resource theories $\text{R}_i$'s. For example, if $f_i$ is a monotone for the single-resource theory $\text{R}_i$, and $f_j$ is a monotone for the theory $\text{R}_j$, their linear combination, where the linear coefficients are positive, is a monotone for the multi-resource theory $\text{R}_{\text{multi}}$. Interestingly, in Sec.~\ref{interconv} we will see that a specific linear combination of the monotones of the different single-resource theories plays an important role in the interconversion of resources. \par Examples of multi-resource theories that can be described within our formalism are already present in the literature. In Ref.~\cite{streltsov_entanglement_2016}, for instance, the authors study the problem of state-merging when the parties can only use local operations and classical communication (LOCC), and they restrict the local operations to be incoherent operations, that is, operations that cannot create coherence (in a given basis). This theory coincides with the multi-resource theory obtained from combining two single-resource theories, the one of entanglement, whose set of allowed operations only contains quantum channels built out of LOCC, and the one of coherence, whose set of allowed operations only contains maps which do not create coherence. In this case, the structure of the invariant sets is given by the central panel of Fig.~\ref{fig:invariant_sets_structure}. Another example is the one of Ref.~\cite{sparaciari_resource_2016}, where thermodynamics is obtained as a multi-resource theory whose class of allowed operations is a subset of the one obtained by taking the intersection of energy-non-increasing maps (operations which do not increase the average energy of the quantum system, see Sec.~\ref{average_non_increasing}), and mixtures of unitary operations. In this case the resources are, respectively, average energy and entropy, and the structure of the invariant sets is given by the right panel of Fig.~\ref{fig:invariant_sets_structure}, where $\mathcal{F}_1$ coincides with the ground state of the Hamiltonian (if the Hamiltonian is non-degenerate), while $\mathcal{F}_2$ coincides with the maximally-mixed state. Other examples of multi-resource theories can be found, and in future work~\cite{multi_resource_paper} we will present the general properties of multi-resource theories with different invariant sets structures.
\section{Reversible multi-resource theories} \label{rev_theory_mult} In this section we study reversibility in the context of multi-resource theories. We first introduce a property, which we refer to as the \emph{asymptotic equivalence property}, for multi-resource theories. We then show that, when a resource theory satisfies this property, we can (uniquely) quantify the amount of resources needed to perform an asymptotic state transformation. This allows us to introduce the notion of \emph{batteries}, i.e., systems where each individual resource can be stored, and to keep track of the changes of the resources during a state transformation. Furthermore, we show that a theory which satisfies the asymptotic equivalence property is also reversible, that is, the amount of resources exchanged with the batteries during an asymptotic state transformation mapping $\rho$ into $\sigma$ is equal, with negative sign, to the amount of resources exchanged when mapping $\sigma$ into $\rho$. Finally, we show that, when the invariant sets of the theory satisfy some general properties, and the theory satisfies asymptotic equivalence, then the relative entropy distances from the different invariant sets are the unique measures of the resources. This result is a generalisation of the one obtained in single-resource theories, see Ref.~\cite{horodecki_are_2002,horodecki_quantumness_2012,brandao_reversible_2015}.
\subsection{Asymptotic equivalence property} \label{asympt_eqiv_prop} Let us consider the multi-resource theory $\text{R}_{\text{multi}}$ introduced in Sec.~\ref{multi_resource}. This theory has $m$ resources, its set of allowed operations $\mathcal{C}_{\text{multi}}$ is defined in Eq.~\eqref{all_ops_multi}, and its invariant sets are the $\mathcal{F}_i$'s, that is, the sets of free states of the different single-resource theories composing it. The multi-resource theory $\text{R}_{\text{multi}}$ is \emph{reversible} if the amount of resources spent to perform an asymptotic state transformation is equal to the amount of resources gained when the inverse state transformation is performed. In this way, performing a cyclic state transformation over the system (which recovers its initial state at the end of the transformation) never consumes any of the $m$ resources initially present in the system. \par For a single-resource theory, the notions of reversibility and state transformation are usually associated with the \emph{rates of conversion}. Suppose that we are given $n \gg 1$ copies of a state $\rho \in \mathcal{S} \left( \mathcal{H} \right)$, and we want to find out the maximum number of copies of the state $\sigma \in \mathcal{S} \left( \mathcal{H} \right)$ that can be obtained by acting on the system with the allowed operations. If $k$ is the maximum number of copies of $\sigma$ achievable, then the rate of conversion is defined as $R(\rho \rightarrow \sigma) = \frac{k}{n}$, see Def.~\ref{def:rate_conversion} in appendix~\ref{rev_theory_sing}. Reversibility is then defined by asking that, for all $\rho, \sigma \in \mathcal{S} \left( \mathcal{H} \right)$, the rates of conversion associated to the forward and backward state transformations are such that $R(\rho \rightarrow \sigma) R(\sigma \rightarrow \rho) = 1$, see Def.~\ref{def:reversibility} in the appendix. It is worth noting that, when considering rates of conversion, one is in general allowed to trace out part of the system, or to add ancillary systems in a free state. For example, being able to map $n$ copies of $\rho$ into $k$ copies of $\sigma$, with $n < k$, implies that we have the possibility to add $k-n$ copies in a free state to the initial $n$ copies of $\rho$, and to act globally to produce $k$ copies of $\sigma$. This is certainly possible for single-resource theories, where free states always exists, but not always possible for multi-resource theories, see the invariant set structure of the right panel of Fig.~\ref{fig:invariant_sets_structure}. \par Due to the possible absence of free states in a generic multi-resource theory, we first need to introduce the following definition\footnote{Notice that this definition is analogous to the notion of ``seed regularisation'' in Ref.~\cite[Sec.~6]{fritz_resource_2015}, although in our case we are solely focused on reversible transformations and on equalities of monotones.}, which will then allow us to study reversibility. \begin{definition} \label{def:asympt_equivalence_multi} Consider a multi-resource theory $\text{R}_{\mathrm{multi}}$. We say that $\text{R}_{\mathrm{multi}}$ satisfies the \emph{asymptotic equivalence property} with respect to the set of monotones $\left\{ f_i \right\}_{i=1}^m$, where each $f_i$ is a monotone for the corresponding single-resource theory $\text{R}_i$ whose regularisation is not identically zero, if for all $\rho, \sigma \in \mathcal{S} \left( \mathcal{H} \right)$ we have that the following two statements are equivalent, \begin{itemize} \item $f_i^{\infty}(\rho) = f_i^{\infty}(\sigma)$ for all $i = 1, \ldots , m$. \item There exist a sequence of maps $\left\{ \tilde{\mathcal{E}}_n : \SHn{n} \rightarrow \SHn{n} \right\}_n$ such that \begin{equation}
\lim_{n \rightarrow \infty} \left\| \tilde{\mathcal{E}}_n(\rho^{\otimes n}) - \sigma^{\otimes n} \right\|_1 = 0, \end{equation} as well as a sequence of maps performing the reverse process. The maps $\left\{\tilde{\mathcal{E}}_n\right\}$ are defined as \begin{equation} \label{allowed_ancilla} \tilde{\mathcal{E}}_n(\cdot) = \Tr{A}{\mathcal{E}_n(\cdot \otimes \eta^{(A)}_n)}, \end{equation} where $A$ is an ancilla composed by a sub-linear number $o(n)$ of copies of the system, and it is described by an arbitrary state $\eta^{(A)}_n \in \SHn{o(n)}$, such that $f_i(\eta^{(A)}_n) = o(n)$ for all $i = 1, \ldots , m$. The map $\mathcal{E}_n \in \mathcal{C}_{\mathrm{multi}}^{(n+o(n))}$ is an allowed operation of the multi-resource theory. \end{itemize}
Here, $f_i^{\infty}$ is the regularisation of the monotone $f_i$, $\| \cdot \|_1$ is the trace norm, define as $\| O \|_1 = \tr{\sqrt{O^{\dagger}O}}$ for $O \in \mathcal{B} \left( \mathcal{H} \right)$, and we are using the little-o notation, where $g(n) = o(n)$ means $\lim_{n \rightarrow \infty} \frac{g(n)}{n} = 0$. \end{definition}
An example of a multi-resource theory that satisfies the above property is thermodynamics (even in the case in which multiple conserved quantities are present), as shown in Refs.~\cite{sparaciari_resource_2016,bera_thermodynamics_2017}. In this example the monotones for which asymptotic equivalence is satisfied are the average energy and the Von Neumann entropy of the system. Notice that the above property implicitly assumes that the monotones $f_i$'s can be regularised, that is, that the limit involved in the regularisation is always finite. Furthermore, in this property we are allowing the agent to act over many copies of the system with more than just the set of allowed operations; we assume the agent to be able to use a small ancillary system, sub-linear in the number of copies of the main system. Roughly speaking, the role of this ancilla is to absorb the fluctuations in the monotones $f^{\infty}_i$'s during the asymptotic state transformation. It is important to notice that this ancillary system only contributes to the transformation by exchanging a sub-linear amount of resources. Thus, its contribution per single copy of the system is negligible when $n \gg 1$, which justifies the use of this additional tool. \par The asymptotic equivalence property essentially states that the multi-resource theory can reversibly map between any two states with the same values of the monotones $f_i$'s. In particular, transforming between such two states comes at no cost, since we can do so by using the allowed operations of the theory, $\mathcal{C}_{\text{multi}}$. It is worth noting that, when the number of considered resources is $m = 1$, that is, our theory is a single-resource theory, the notion of asymptotic equivalence given in Def.~\ref{def:asympt_equivalence_multi} corresponds to the one given in terms of rates of conversion, Def.~\ref{def:reversibility}. We prove this equivalence in appendix~\ref{rev_theory_sing}, see Thm.~\ref{thm:reversible_asympt_equiv}. The set of monotones in Def.~\ref{def:asympt_equivalence_multi} is not a priori unique; however, in the following section we identify the properties that the monotones need to satisfy for this set to be unique, see Thm.~\ref{thm:reversible_multi}. Finally, notice that the asymptotic equivalence property does not say anything about the state transformations which involve states with different values of the monotones $f_i$'s. To include these transformations in the theory, we will have to add a bit more structure to the current framework, by considering some additional systems that can store a single type of resource each, which we refer to as \emph{batteries}~\cite{kraemer_currencies_2016}.
\subsection{Quantifying resources with batteries} \label{quant_res} When a multi-resource theory satisfies the asymptotic equivalence property of Def.~\ref{def:asympt_equivalence_multi}, we have that states with the same values of a specific set of monotones can be inter-converted between each others. In this section, we show that these monotones actually quantify the amount of resources contained in the system. To do so, we need to introduce some additional systems, which can only store a single kind of resource each, and can be independently addressed by the agent. These additional systems are referred to as batteries. Let us suppose that the multi-resource theory $\text{R}_{\text{multi}}$ satisfies the asymptotic equivalence property with respect to the set of monotones $\left\{ f_i \right\}_{i=1}^m$, and that the quantum system over which the theory acts is actually divided into $m+1$ partitions. The first partition is the main system $S$, and the remaining ones are the batteries $B_i$'s. Then, the Hilbert space under consideration is $\mathcal{H} = \mathcal{H}_S \otimes \mathcal{H}_{B_1} \otimes \ldots \otimes \mathcal{H}_{B_m}$. \par Let us now introduce some properties the monotones need to satisfy in order for the resources to be quantified in a meaningful way. Since each resource is associated to a different monotone, we can forbid a battery to store more than one resource by constraining the set of states describing it to those ones with a fixed value of all but one monotones. \begin{description} \item[M1\label{item:M1}] Consider two states $\omega_i, \omega'_i \in \mathcal{S} \left( \mathcal{H}_{B_i} \right)$ describing the battery $B_i$. Then, the value of the regularisation of any monotone $f_{j}$ (where $j \neq i$) over these two states is fixed, \begin{equation} \label{eq:batt_mon} f^{\infty}_j(\omega'_i) = f^{\infty}_j(\omega_i ) , \quad \forall j \neq i. \end{equation} \end{description} In this way, the battery $B_i$ is only able to store and exchange the resource associated with the monotone $f_i$. It would be natural to extend the condition of Eq.~\eqref{eq:batt_mon} to the monotones themselves, rather than to use their regularisations. However, this condition is not required for deriving our results, and to use it in our proofs we would need an additional assumption, namely the additivity of the monotones. \par In order to address each battery as an individual system, we ask the value of the monotones over the global system to be given by the sum of their values over the individual components, \begin{description} \item[M2\label{item:M2}] The regularisations of the monotones $f_i$'s can be separated between main system and batteries, \begin{equation} f^{\infty}_i(\rho \otimes \omega_{1} \otimes \ldots \otimes \omega_{m}) = f^{\infty}_i(\rho) + f^{\infty}_i(\omega_{1}) + \ldots + f^{\infty}_i(\omega_{m}), \end{equation} where $\rho \in \mathcal{S} \left( \mathcal{H}_S \right)$ is the state of the main system, and $\omega_{i} \in \mathcal{S} \left( \mathcal{H}_{B_i} \right)$ is the state of the battery $B_i$. \end{description} The above property allows us to separate the contribution given by each subsystem to the amount of $i$-th resource present in the global system. It is important to stress that we are here requiring additivity for the regularisation of the monotones between system and batteries, and between batteries, but we are not requiring the regularised monotones to be additive in general. \par We then ask the monotones to satisfy an additional property, so as to simplify the notation. Namely, we ask the zero of each monotone $f_i$ to coincide with its value over the states in $\mathcal{F}_i$, \begin{description} \item[M3\label{item:M3}] For each $n \in \mathbb{N}$ and $i \in \left\{ 1, \ldots , m \right\}$, the monotone $f_i$ is equal to $0$ when computed over the states of $\mathcal{F}_i^{(n)}$, that is \begin{equation} f_i(\gamma_{i,\,n}) = 0 , \qquad \forall \, \gamma_{i,\,n} \in \mathcal{F}_i^{(n)}. \end{equation} \end{description} This property serves as a way to ``normalise'' the monotone, setting its value to $0$ over the states that were free for the specific single-resource theory the monotone is linked to. Notice that property~\ref{item:M3} is trivially satisfied by any monotone after a translation. The next property requires that tracing out part of the system does not increase the value of the monotones $f_i$'s, \begin{description} \item[M4\label{item:M4}] For all $n, k \in \mathbb{N}$ where $k < n$, the monotones $f_i$'s are such that \begin{equation} f_i(\Tr{k}{\rho_n}) \leq f_i(\rho_n) , \qquad \forall \, i \in \left\{ 1, \ldots , m \right\}. \end{equation} where $\rho_n \in \SHn{n}$ and $\Tr{k}{\rho_n} \in \SHn{n-k}$. \end{description} This property implies that the resources contained in a system cannot increase if we discard/forget part of it. \par We require our monotones to satisfy sub-additivity, namely \begin{description} \item[M5\label{item:M5}] For all $n, k \in \mathbb{N}$, the monotones $f_i$'s are such that \begin{equation} f_i(\rho_n \otimes \rho_k) \leq f_i(\rho_n) + f_i(\rho_k) , \qquad \forall \, i \in \left\{ 1, \ldots , m \right\}. \end{equation} where $\rho_n \in \SHn{n}$ and $\rho_k \in \SHn{k}$. \end{description} That is, the amount of resources contained in two uncorrelated systems, when measured on the two systems independently, is bigger or equal to the value measured on the two systems together. This is the case, for example, of the relative entropy of entanglement~\cite{vollbrecht_entanglement_2001}. Notice that sub-additivity is here explicity required since, as we stressed before, property~\ref{item:M2} only requires additivity between system and different batteries, but not between different partitions of the individual system or battery. Another property we require is for the monotones $f_i$'s to be sub-extensive, \begin{description} \item[M6\label{item:M6}] Given any sequence of states $\left\{ \rho_n \in \SHn{n} \right\}$, the monotones $f_i$'s are such that \begin{equation} f_i(\rho_n) = O(n) , \qquad \forall \, i \in \left\{ 1, \ldots , m \right\}. \end{equation} where we are using the big-O notation. \end{description} This property is satisfied, for example, if the monotones scale extensively, that is, if they scale linearly in the number of systems considered. In the next section we will encounter a family of monotones which indeed satisfy this property, namely the relative entropy distance from a given set of free states, when some fairly generic conditions are satisfied by such set (see Prop.~\ref{thm:properties_rel_ent}). However, it is worth noting that property~\ref{item:M6} is not equivalent to extensivity, since a monotone scaling sub-linearly in the number of systems would still satisfy it. We demand that our monotones satisfy this property so as to be able to regularise them (although their regularisation might be identically zero on the whole state space). The last property we ask concerns a particular kind of continuity the monotones need to satisfy, \begin{description}
\item[M7\label{item:M7}] The monotones $f_i$'s are \emph{asymptotic continuous}, that is, for all sequences of states $\rho_n, \sigma_n \in \SHn{n}$ such that $\left\| \rho_n - \sigma_n \right\|_1 \rightarrow 0$
for $n \rightarrow \infty$, where $\| \cdot \|_1$ is the trace norm, we have \begin{equation}
\frac{\left| f_i \left( \rho_n \right) - f_i \left( \sigma_n \right) \right|}{n} \rightarrow 0 \ \text{for} \ n \rightarrow \infty , \qquad \forall \, i \in \left\{ 1, \ldots , m \right\}. \end{equation} This notion of asymptotic continuity coincides with condition (C2) given in Ref.~\cite{horodecki_entanglement_2001}. \end{description} This property implies that the monotones are physically meaningful, since their values over sequences of states converge if the sequences of states converge asymptotically. In Thm.~\ref{thm:reversible_multi} we show that, when the monotones satisfy asymptotic continuity, they are the unique quantifiers of the amount of resources contained in the main system. \par We can now use this formalism to discuss how resources can be quantified in a multi-resource theory, and consequently how the asymptotic equivalence property implies that the theory is reversible. Let us consider any two states $\rho, \sigma \in \mathcal{S} \left( \mathcal{H}_S \right)$, that do not need to have the same values for the monotones $f_i$'s. Then, we choose the initial and final states of each battery $B_i$ such that \begin{equation} \label{eq:fi_condition_Ri} f^{\infty}_i \left( \rho \otimes \omega_1 \otimes \ldots \otimes \omega_m \right) = f^{\infty}_i \left( \sigma \otimes \omega'_1 \otimes \ldots \otimes \omega'_m \right) , \qquad \forall \, i = 1, \ldots, m, \end{equation} where $\omega_i$, $\omega'_i \in \mathcal{S} \left( \mathcal{H}_{B_i} \right)$, for $i = 1, \ldots, m$. Under these conditions, due to the asymptotic equivalence property of $\text{R}_{\text{multi}}$, we have that the two global states can be asymptotically mapped one into the other in a reversible way, using the allowed operations of the theory, that is \begin{equation} \label{eq:transform} \rho \otimes \omega_1 \otimes \ldots \otimes \omega_m \xleftrightarrow{\text{asympt}} \sigma \otimes \omega'_1 \otimes \ldots \otimes \omega'_m, \end{equation} where the symbol $\xleftrightarrow{\text{asympt}}$ means that there exists two allowed operations that maps $n \gg 1$ copies of the state on the lhs into the state of the rhs, and viceversa, while satisfying the condition in the second statement of Def.~\ref{def:asympt_equivalence_multi}. \par We can now properly define the notion of resources in this framework. The resource associated with the monotone $f_i$ is the one exchanged by the battery $B_i$ during a state transformation.
\begin{definition} Consider a multi-resource theory $\text{R}_{\text{multi}}$ with $m$ resources, satisfying the asymptotic equivalence property with respect to the set of monotones $\left\{ f_i \right\}^m_{i=1}$. For a state transformation of the form given in Eq.~\eqref{eq:transform}, we define the amount of $i$-th resource exchanged between the system $S$ and the battery $B_i$ as \begin{equation} \label{work_Ri} \Delta W_i := f^{\infty}_i(\omega'_i) - f^{\infty}_i(\omega_i), \end{equation} where $\omega_i, \omega'_i \in \mathcal{S} \left( \mathcal{H}_{B_i} \right)$ are, respectively, the initial and final state of the battery $B_i$. \end{definition}
It is now possible to compute the amount of the $i$-th resource $\Delta W_i$ needed to map the state of the main system $\rho$ into $\sigma$.
\begin{prop} Consider a theory $\text{R}_{\text{multi}}$ with $m$ resources and allowed operations $\mathcal{C}_{\text{multi}}$, equipped with batteries $B_1$, $\ldots$, $B_m$. If the theory satisfies the asymptotic equivalence property with respect to the set of monotones $\left\{ f_i \right\}^m_{i=1}$, and these monotones satisfy the properties~\ref{item:M1} and~\ref{item:M2}, then the amount of $i$-th resource needed to perform the asymptotic state transformation $\rho \rightarrow \sigma$ is equal to \begin{equation} \label{resource_work_i} \Delta W_i = f_i^{\infty}(\rho) - f_i^{\infty}(\sigma). \end{equation} \end{prop}
\begin{proof} Due to asymptotic equivalence, a transformation mapping the global state $\rho \otimes \omega_1 \otimes \ldots \otimes \omega_m$ into $\sigma \otimes \omega'_1 \otimes \ldots \otimes \omega'_m$ exists iff the conditions in Eq.~\eqref{eq:fi_condition_Ri} are satisfied. For a given $i$, using the property~\ref{item:M2} of the monotone $f_i$, we can re-write the condition as \begin{equation} f^{\infty}_i \left( \rho \right) + f^{\infty}_i \left( \omega_1 \right) + \ldots + f^{\infty}_i \left( \omega_m \right) = f^{\infty}_i \left( \sigma \right) + f^{\infty}_i \left( \omega'_1 \right) + \ldots + f^{\infty}_i \left( \omega'_m \right). \end{equation} Then, we can use the property~\ref{item:M1}, which guarantees that the only systems for which $f_i$ changes are the main system and the battery $B_i$. Thus, we find that \begin{equation} f^{\infty}_i \left( \rho \right) + f^{\infty}_i \left( \omega_i \right) = f^{\infty}_i \left( \sigma \right) + f^{\infty}_i \left( \omega'_i \right). \end{equation} By rearranging the factors in the above equation, and using the definition of $\Delta W_i$ given in Eq.~\eqref{work_Ri}, we prove the proposition. \end{proof}
It is now easy to show that, if $\text{R}_{\text{multi}}$ satisfies the asymptotic equivalence property, any state transformation on the main system $S$ is reversible. Indeed, from Eq.~\eqref{resource_work_i} it follows that the amount of resources used to map the state of this system from $\rho$ to $\sigma$ is equal, but with negative sign, to the amount of resources used to perform the reverse transformation, from $\sigma$ to $\rho$. Therefore, any cyclic state transformation over the main system leaves the amount of resources contained in the batteries unchanged. \par The above formalism also provides us with a way to quantify the amount of resources contained in the main system. Indeed, if the system is described by the state $\rho \in \mathcal{S} \left( \mathcal{H}_S \right)$, the amount of $i$-th resource contained in the system is given by the amount of $i$-th resource exchanged, $\Delta W_i$, while mapping $\rho$ into a state contained in $\mathcal{F}_i$. Using property~\ref{item:M3} and Prop.~\eqref{resource_work_i} it follows that
\begin{coro} \label{cor:quantifier} Consider a theory $\text{R}_{\text{multi}}$ with $m$ resources and allowed operations $\mathcal{C}_{\text{multi}}$, equipped with batteries $B_1$, $\ldots$, $B_m$. If the theory satisfies the asymptotic equivalence property with respect to the set of monotones $\left\{ f_i \right\}^m_{i=1}$, and these monotones satisfy the properties~\ref{item:M1}, \ref{item:M2}, and~\ref{item:M3}, then the amount of the $i$-th resource contained in the main system, when described by the state $\rho$, is given by $f_i^{\infty}(\rho)$. \end{coro}
It is worth noting that, in general, one cannot extract all the resources contained in the main system at once. Indeed, this is only possible when the multi-resource theory contains free states, like for example in the cases depicted in the left and centre panels of Fig.~\ref{fig:invariant_sets_structure}. Furthermore, the process of resources extraction is in general non-trivial, since property~\ref{item:M1} forbids each battery from storing more than one kind of resource. As a result, it is not possible to simply perform a swap operation which exchanges the state of the system with one of the batteries, see Sec.~\ref{control_theory_ex} for an example involving the theory of local control.
\begin{figure}\label{fig:resource_diagram}
\end{figure}
\par Being able to quantify the amount of resources contained in a given quantum state allows us to represent the whole state-space of the theory in a \emph{resource diagram}~\cite{fritz_resource_2015,sparaciari_resource_2016}. In fact, from the definition of asymptotic equivalence it follows that, if two states contain the same amount of resources, i.e., if they have the same values of the monotones $f^{\infty}_i$'s, then we can map between them using the allowed operations $\mathcal{C}_{\text{multi}}$. This property implies that we can divide the entire state-space into equivalence classes, that is, sets of states with same value of the $m$ monotones (where we recall that $m$ is the number of resources, or batteries, in the theory). Then, we can represent each equivalence class as a point in a $m$-dimensional diagram, with coordinates given by the values of the monotones. By considering all the different equivalence classes, we can finally represent the state-space of the main system in the diagram, see for example Fig.~\ref{fig:resource_diagram}, where the state-space of a two-resource theory is shown.
\subsection{Reversibility implies a unique measure for each resource} \label{multi_rev_unique} We now show that, when a multi-resource theory satisfies the asymptotic equivalence property with respect to a set of monotones $\left\{ f_i \right\}_{i=1}^m$, and these monotones satisfy the properties~\ref{item:M1} -- \ref{item:M7}, then there exists a unique quantifier for each resource contained in the main system. In particular, when the $i$-th resource is considered, this quantifier coincides with $f_i^{\infty}$, modulo a multiplicative factor which sets the scale. It is worth noting that such multiplicative factor can be different for each resource. Indeed, the resources are generally independent of each other, since they are quantified by different measures. Each measure can have a different unit, which corresponds to an individual rescaling factor applied to each resource measure independently. \par In the previous section, Cor.~\ref{cor:quantifier}, we showed that a quantifier exists if the monotones satisfy the first three properties~\ref{item:M1}, \ref{item:M2}, and \ref{item:M3}. However, when these monotones are also asymptotic continuous, property~\ref{item:M7}, we can prove that they \emph{uniquely} quantify the amount of resources contained in the main system. This means that one cannot find other monotones $g_i$'s that give the same equivalence classes of the $f_i$'s, but order them in a different way. Asymptotic continuity was used in Ref.~\cite{horodecki_quantumness_2012} to show that the relative entropy distance from the set of free states of a reversible single-resource theory is the unique measure of resource. Thus, the following theorem (whose proof can be found in appendix~\ref{main_results}) can be understood as a generalisation of the above result to multi-resource theories,
\begin{restatable}{thm}{uniquemeas} \label{thm:reversible_multi} Consider the resource theory $\text{R}_{\text{multi}}$ with $m$ resources, equipped with the batteries $B_i$'s, where $i = 1, \ldots, m$. Suppose the theory satisfies the asymptotic equivalence property with respect to the set of monotones $\left\{ f_i \right\}_{i=1}^m$. If these monotones satisfy the properties~\ref{item:M1} -- \ref{item:M7}, then the amount of $i$-th resource contained in the main system $S$ is uniquely quantified by the regularisation of the monotone $f_i$ (modulo a multiplicative constant). \end{restatable}
In particular, we now consider the case of a multi-resource theory $\text{R}_{\text{multi}}$ that satisfies the asymptotic equivalence property of Def.~\ref{def:asympt_equivalence_multi} with respect to the relative entropy distances from the invariant sets $\mathcal{F}_i$'s. We refer to the relative entropy distance from the set $\mathcal{F}_i$ as $E_{\mathcal{F}_i}$, whose definition can be found in Eq.~\eqref{rel_entr_dist}. Since the multi-resource theory we consider is equipped with batteries, and we want to be able to measure the amount of resources they contain independently of the other subsystems, we ask the invariant sets to be of the form \begin{equation} \label{eq:independent_free} \mathcal{F}_i = \mathcal{F}_{i,S} \otimes \mathcal{F}_{i,B_1} \otimes \ldots \otimes \mathcal{F}_{i,B_m}, \end{equation} so that the main system $S$ and the batteries $B_i$'s all have they own independent invariant sets. We now show that, under very general assumptions over the properties of the invariant sets, the regularised relative entropy distances from these sets are the unique quantifiers of the resources, provided that these quantities are not identically zero over the whole state space\footnote{An example where the regularised relative entropy from an invariant set is identically zero for all states in $\mathcal{S} \left( \mathcal{H} \right)$ is the resource theory of asymmetry, see Ref.~\cite{gour_measuring_2009}.}. This result follows from Thm.~\ref{thm:reversible_multi}, and from the fact that these monotones satisfy the properties~\ref{item:M1}, \ref{item:M2}, \ref{item:M3}, and \ref{item:M7} listed in the previous sections. The properties we are interested in for the invariant sets $\left\{ \mathcal{F}_i \right\}_{i=1}^m$ of the theory are very general, and they are satisfied in most of the known resource theories, see Refs.~\cite{brandao_reversible_2010, brandao_generalization_2010}. \begin{description} \item[F1\label{item:F1}] The sets $\mathcal{F}_i$'s are closed sets. \item[F2\label{item:F2}] The sets $\mathcal{F}_i$'s are convex sets. \item[F3\label{item:F3}] Each set $\mathcal{F}_i$ contains at least one full-rank state. \item[F4\label{item:F4}] The sets $\mathcal{F}_i$'s are closed under tensor product, that is, $\mathcal{F}_i^{(k)} \otimes \mathcal{F}_i^{(n)} \subseteq \mathcal{F}_i^{(n+k)}$ for all $i = 1, \ldots , m$. \item[F5\label{item:F5}] The sets $\mathcal{F}_i$'s are closed under partial tracing, that is, $\Tr{k}{\mathcal{F}_i^{(n)}} \subseteq \mathcal{F}_i^{(n-k)}$ for all $i = 1, \ldots , m$. \end{description} Let us briefly comment on the above properties. Property~\ref{item:F1} requires that any converging sequence in the set converges to an element in the set. This property is necessary for the continuity of the resource theory. Property~\ref{item:F2}, instead, tells us that we are allowed to forget the exact state describing the system, and therefore we can have mixture of states. Property~\ref{item:F3} is necessary for the relative entropy distance to be physically natural, since the quantity $\re{\rho}{\sigma}$, see Eq.~\eqref{rel_entr}, diverges when $\text{supp}(\rho) \not\subseteq \text{supp}(\sigma)$. Finally, property~\ref{item:F4} implies that composing two systems that do not contain any amount of $i$-th resource is not going to increase that resource, and similarly, property~\ref{item:F5} implies that forgetting about part of a system which does not contain resources will not create resources. \par When the invariant sets satisfy the above properties, the relative entropy distances $E_{\mathcal{F}_i}$'s satisfy the same properties discussed in the previous section,
\begin{restatable}{prop}{relentproperties} \label{thm:properties_rel_ent} Consider a resource theory $\text{R}_{\text{multi}}$ with $m$ resources, equipped with the batteries $B_i$'s, where $i = 1, \ldots, m$. Suppose the class of allowed operations is $\mathcal{C}_{\text{multi}}$ and the invariant sets are $\left\{ \mathcal{F}_i \right\}_{i=1}^m$. If the invariant set $\mathcal{F}_i$ is of the form of Eq.~\eqref{eq:independent_free}, and it satisfies the properties~\ref{item:F1} -- \ref{item:F5}, then the relative entropy distances from this set, $E_{\mathcal{F}_i}$, is a regularisable monotone under the class of allowed operations, and it obeys the properties~\ref{item:M1} -- \ref{item:M7}. \end{restatable}
This result is known in the literature, see Refs.~\cite{synak-radtke_asymptotic_2006, brandao_generalization_2010}, but we nevertheless provide a proof in appendix~\ref{additional} to make the paper self-contained. By virtue of Thm.~\ref{thm:reversible_multi} it then follows that, if $E^{\infty}_{\mathcal{F}_i}$ has a positive value over the states that are not in $\mathcal{F}_i$, then it is the unique quantifier of the amount of $i$-th resource contained in the system for a multi-resource theory that satisfies the asymptotic equivalence property with respect to these monotones. Furthermore, the amount of $i$-th resource used to map the main system from the state $\rho$ into the state $\sigma$ is then equal to \begin{equation} \label{resource_work_ent} \Delta W_i = E_{\mathcal{F}_i}^{\infty}(\rho) - E_{\mathcal{F}_i}^{\infty}(\sigma), \end{equation} for all $i = 1, \ldots , m$.
\subsection{Relaxing the conditions on the monotones} \label{average_non_increasing} There are situations, when we consider specific resource theories, in which some of the properties of the set of free states are not satisfied. In particular, we can have that the set of free states does not contain a full-rank state, that is, property~\ref{item:F3} is not satisfied. An example would be the resource theory of energy-non-increasing maps for a system with Hamiltonian $H$, \begin{equation} \label{non_increasing_energy}
\mathcal{C}_{H} = \left\{ \mathcal{E}_H \ : \ \mathcal{B} \left( \mathcal{H} \right) \rightarrow \mathcal{B} \left( \mathcal{H} \right) | \ \tr{\mathcal{E}_{H} (\rho) H} \leq \tr{\rho H} \ \forall \rho \in \mathcal{S} \left( \mathcal{H} \right) \right\}. \end{equation} An example of a subset of $\mathcal{C}_{H}$ are unitary operations which commute with the Hamiltonian $H$ (as in the resource theory of Thermal Operations). If the Hamiltonian $H$ has a non-degenerate ground state $\ket{\text{g}}$, then it is easy to show that this state is fixed, that is, \begin{equation} \mathcal{E}_H \left( \ket{\text{g}}\bra{\text{g}} \right) = \ket{\text{g}}\bra{\text{g}}. \end{equation} In fact, the operation $\mathcal{E}_{\text{g}} (\cdot) = \Tr{A}{S ( \cdot \otimes \ket{\text{g}}\bra{\text{g}}_A ) S^{\dagger}}$, where $S$ is the unitary operation implementing the swap between the two states, belongs to $\mathcal{C}_{H}$ and maps all states into the ground state. Thus, the set of free states does not contain a full-rank state, which implies that the relative entropy distance from this set is ill-defined, and it is not asymptotic continuous. Notice that the above argument holds even in the case of a degenerate ground state, with the difference that the invariant set would be composed by any state with support on this degenerate subspace. \par We can introduce a different monotone for this kind of resource theory, that is, the average of the observable which is not increased by the allowed operations (modulo a constant factor). For the example we are considering, this monotone would be \begin{equation} \label{montone_average} M_H(\rho) = \tr{H \rho} - E_{\text{g}}, \end{equation} where $H$ is the Hamiltonian of the system, and $E_{\text{g}} = \tr{H \ket{\text{g}}\bra{\text{g}}}$ is the energy of the ground state. When $n$ copies of the system are considered, we define the total Hamiltonian as $H_n = \sum_{i=1}^{n} H^{(i)}$, where $H^{(i)}$ is the Hamiltonian acting on the $i$-th copy. In this case, it is easy to show that this quantity is equal to $0$ when evaluated on the fixed state $\ket{\text{g}}\bra{\text{g}}$, property~\ref{item:M3}, is monotonic under partial tracing, property~\ref{item:M4}, is additive (and therefore satisfies sub-additivity, property~\ref{item:M5}), and it scales extensively in the number of copies of the system, thus satisfying property~\ref{item:M6}. Furthermore, $M_H(\cdot)$ is monotonic under the class of operations (by definition of the class itself), and it is asymptotic continuous, property~\ref{item:M7}, as shown in Prop.~\ref{average_asymp_cont} in appendix~\ref{additional}. If batteries are introduced, we can define the operator $H$ is such a way that properties~\ref{item:M1} and \ref{item:M2} are satisfied, see for example Sec.~\ref{thermo_example}. \par Thus, if one (or more) of the monotones of the multi-resource theory is of the form given in Eq.~\eqref{montone_average}, we have that the results of the previous section still apply, particularly Thm.~\ref{thm:reversible_multi}. Furthermore, we can quantify the change in the resource associated with $M_H$ during a state transformation $\rho \rightarrow \sigma$ with
Eq.~\eqref{resource_work_ent}, where the regularised relative entropy distance $E_{\mathcal{F}_i}^{\infty}$ is replaced with the regularised monotone $M_H^{\infty}$. As a side remark, we notice that the monotone $M_H$ can be obtained as \begin{equation} M_H(\rho) = \lim_{\beta \rightarrow \infty} \frac{1}{\beta} \, \re{\rho}{\tau_{\beta}}, \end{equation} where $\tau_{\beta} = e^{- \beta H} / Z$ is the Gibbs state of the Hamiltonian $H$, and $Z = \tr{e^{- \beta H}}$ is the partition function of the system.
\section{Bank states, interconversion relations, and the first law} \label{interconv} Within certain types of multi-resource theories, it is possible to inter-convert the resources stored in the batteries, i.e., to exchange one resource for another at a given exchange rate. Examples of resource interconversion can be found in thermodynamics, where Landauer's principle~\cite{landauer_irreversibility_1961} tells us that energy can be exchanged for information, while a Maxwell's demon can trade information for energy~\cite{bennett_thermodynamics_1982}. In these examples, a thermal bath is necessary to perform the interconversion of resources. Indeed, in the following sections we show that in order to exchange between resources one always needs an additional system, which we refer to as a \emph{bank}, that captures the necessary properties of thermal baths in thermodynamics, and abstracts them so that they can be applied to other resource theories. When such a system exists, we can pay a given amount of one resource and gain a different amount of another resource, with an exchange rate that only depends on the state describing the bank, see Thm.~\ref{thm:interconvert_relation}. Within the thermodynamic examples we are considering, this corresponds to exchanging one bit of information for one unit of energy, and vice versa. The exchange rate of these processes is proportional to the temperature of the thermal bath. \par During a resource interconversion the state of the bank should not change its main properties, so that we can keep using it indefinitely. Furthermore, we should always have to invest one resource in order to gain the other. For these reasons the bank is taken to be of infinite size, and its state to be \emph{passive}, i.e., to always contain the minimum possible values of the resources. In fact, in the thermodynamic examples we are considering, the thermal bath has infinite size, and its state has maximum entropy for fixed energy, or equivalently minimum energy for fixed entropy~\cite{jaynes_information_1957}. We additionally show that the relative entropy distance from the set of bank states plays a fundamental role in quantifying the exchange rate at which resources are inter-converted, see Cor.~\ref{bank_equal_rel_ent}. For instance, in thermodynamics this quantity is proportional to the Helmholtz free energy $F = E - T S$, which links together the two resources, internal energy $E$ and information, which is proportional to $-S$. Through this quantity, one can define the exchange rate between energy and entropy, i.e., the temperature of the thermal bath $T$. Finally, we introduce a first-law-like relation for multi-resource theories. The first law consists of a single relation that regulates the state transformation of a system when the agent has access to a bank for exchanging the resources. In particular, this relation links the change in the relative entropy distance from the set of bank states over the main system to the amount of resources exchanged by the batteries during the transformations, see Cor.~\ref{coro:first_law}. In the example we are considering, this relation coincides with the First Law of thermodynamics, as it connects a change in the Helmholtz free energy $\Delta F$ of the system with the energy and information exchanged by the batteries, \begin{equation} \label{first_law} \Delta F = \Delta W_E + T \, \Delta W_I, \end{equation} where $\Delta W_E$ is the energy exchanged by the first battery, $\Delta W_I$ is the information exchanged by the second battery, and $T$ is the background temperature, describing the state of the bank. \par We now briefly discuss about the value that resources have in the different theories of thermodynamics, and the role of the first law in connecting these resources together. Let us first consider the single-resource theory of thermodynamics, where the system is in contact with an infinite thermal reservoir~\cite{brandao_resource_2013}. To perform a state transformation we need to provide only one kind of resource, known as athermality ($\Delta F$), or work. Since the thermal reservoir is present, it is easy to get close to the free state, i.e. to the thermal state at temperature $T$, because we can simply thermalise the system with the allowed operations. However, it is difficult to go in the opposite direction, unless we use part of the athermality stored in a battery. For this reason, a positive increment in the athermality of the battery is considered valuable, while a negative change is considered a cost. \par Let us now move to the multi-resource theory of thermodynamics, whose allowed operations are energy-preserving unitary operations~\cite{sparaciari_resource_2016}. In this case, it is easy to see that negative and positive contributions of energy and information are equally valuable, since these two quantities are conserved by the set of allowed operations. As a result, the agent cannot perform state transformations in any direction without having access to the batteries. If we now allow the agent to use a thermal bath as a bank, and we keep the system decoupled from it (so that the agent cannot perform operations that thermalise the system for free), we find that changing a single resource, either energy or information, is enough to perform a generic state transformation on the system. In fact, we can always inter-convert one resource for the other with the bank, and then change the state of the system accordingly. Notice that, however, we still have that negative and positive change in one resource are equally valuable. \par Thus, it seems that the advantage that multi-resource theories provide over single-resource theories is that they make explicit which resources are used during a state transformation. And the link between the single resource and the multiple ones is given by the first law. In thermodynamics, for example, we have that the first law, Eq.~\eqref{first_law}, indicates that the amount of athermality $\Delta F$ needed to transform a state can be actually divided in two contributions, energy $\Delta W_E$ and information $\Delta W_I$. Notice that all of these quantities can be understood in terms of the relative entropy distance to an invariant set of states. Athermality being measured by its relative entropy distance to the thermal state, information and energy being the relative entropy to the maximally mixed or ground state. As we will see, the generalised first law given in Eq.~\eqref{eq:first_law} also relates the relative entropy to the bank state, to the relative entropies to the invariant sets of the single resource theories.
\subsection{Banks and interconversion of resources} \label{bank_interconvert} We now introduce the bank system, and show how this additional tool allows us to perform interconversion between resources. To simplify the notation, we only focus on a theory with two resources. However, the results we obtain also apply to theories with more resources, since in that case we can just select two resources and perform interconversion while keeping the others fixed. Thus, in the following we consider a resource theory $\text{R}_{\text{multi}}$ with two invariant sets $\mathcal{F}_1$ and $\mathcal{F}_2$ (each of them associated with one of the resources), and allowed operations $\mathcal{C}_{\text{multi}}$. We assume the theory to satisfy the asymptotic equivalence property of Def.~\ref{def:asympt_equivalence_multi} with respect to the relative entropy distances from $\mathcal{F}_1$ and $\mathcal{F}_2$, and we ask the two invariant sets to satisfy the properties~\ref{item:F1}, \ref{item:F2}, and \ref{item:F3}, while we replace properties~\ref{item:F4} and ~\ref{item:F5} with the following, more demanding, property \begin{description} \item[F5b\label{item:F5b}] The invariant sets $\mathcal{F}_i$'s are such that $\mathcal{F}_i^{(n)} = \mathcal{F}_i^{\otimes n}$, for all $n \in \mathbb{N}$. \end{description} The above properties implies that the relative entropy distances $E_{\mathcal{F}_1}$ and $E_{\mathcal{F}_2}$ are the unique quantifiers for the two resources of our theory, as we have seen in Sec.~\ref{multi_rev_unique}. From property~\ref{item:F5b} it follows that these two monotones are additive, i.e., $E_{\mathcal{F}_i}(\rho \otimes \sigma) = E_{\mathcal{F}_i}(\rho) + E_{\mathcal{F}_i}(\sigma)$ for $i = 1,2$, and consequently that their regularisation $E^{\infty}_{\mathcal{F}_i}$ coincides with $E_{\mathcal{F}_i}$. Furthermore, the properties~\ref{item:F2} and \ref{item:F5b} together imply that the invariant sets are composed by a single state, i.e., $\mathcal{F}_i = \left\{ \rho_i \right\}$, where $\rho_i \in \mathcal{S} \left( \mathcal{H} \right)$, for $i = 1,2$. We make use of property~\ref{item:F5b} in Lem.~\ref{f_i_inequality}, shown in appendix~\ref{additional}, which itself is used to prove some essential properties of the set of bank states, see Def.~\ref{def:bank_state}. This property is ultimately used to show that the exchange rate between resources is given by the relative entropy distance from the set of states describing the bank, see Cor.~\ref{bank_equal_rel_ent}. \par It is important to stress that property~\ref{item:F5b} is not satisfied by every multi-resource theory. For example, this property is satisfied by the multi-resource theory of thermodynamics, but it is violated by other theories, like entanglement theory, where the set of free states is composed of separable states. We are currently working to weaken this property, following the ideas presented in Ref.~\cite{brandao_generalization_2010}, by requiring the invariant sets to be closed under permutations of copies. This less demanding property should allow us to use the approximate de Finetti's theorems~\cite{renner_symmetry_2007}, and to obtain similar conditions to those obtained with \ref{item:F5b}. To study the interconversion of entanglement with some other resource, however, one can think of restricting the state space of the theory in a way in which the resulting subset of separable states satisfies property~\ref{item:F5b}, see the example in Sec.~\ref{control_theory_ex}. Finally, it is worth noting that all the results we obtain in this section also apply if one of the monotones, or both, is of the form shown in Eq.~\eqref{montone_average}. Indeed, these monotones satisfy the same properties of the relative entropy distances, with the difference that the corresponding invariant set can be composed by multiple states, and these states do not need to have full rank. \par Let us now consider an example of resource interconversion which will highlight the properties that we are searching for in a bank system. Suppose we have a certain amount of euros and pounds in our wallet, and we want to convert one into the other, for example, from pounds to euros. In order to convert these two currencies we need to go to the bank, that we would expect to satisfy the following properties. The first property could be referred to as \emph{passivity} of the bank, and it is represented by the fact that if we do not provide some pounds, we cannot receive any euros (and vice versa). Second is the existence of an exchange rate, that is, the bank will convert the two currency at a certain exchange rate, and this rate can be different depending on the bank we use. The last property concerns the catalytic nature of the bank, since we would like a bank not to change the exchange rate between pounds and euros as a consequence of our transaction (this last property is approximately satisfied by real banks, at least for the amount exchanged by average costumers). \par The previous example shows that, in order to achieve resource interconversion, we need to introduce in our framework an additional system, the bank, with some specific properties. Within our formalism, we consider the same multi-partite system introduced in Sec.~\ref{quant_res}, with the main system $S$, and two batteries $B_1$ and $B_2$. The system $S$ is now used as a bank, which has to satisfy the three essential properties (passivity, existence of a rate, catalytic behaviour) that we have informally described in the previous paragraph, and that we are going to formalise in the following. First of all, we need the states describing the bank to be \emph{passive}, meaning that we should not be able to extract from this system both resources at the same time, since we always need to pay one resource to gain another one. Thus, the set of \emph{bank states} is defined as
\begin{definition} \label{def:bank_state} Consider a multi-resource theory $\text{R}_{\text{multi}}$ satisfying the asymptotic equivalence property with respect to the monotones $E_{\mathcal{F}_1}$ and $E_{\mathcal{F}_2}$. The \emph{set of bank states} of the theory is a subset of the state space $\mathcal{S} \left( \mathcal{H} \right)$ defined as, \begin{align} \label{set_f3}
\mathcal{F}_{\text{bank}} = \big\{ \rho \in \mathcal{S} \left( \mathcal{H} \right) \ | \ \forall \, \sigma \in \mathcal{S} \left( \mathcal{H} \right) , \ &E_{\mathcal{F}_1}(\sigma) > E_{\mathcal{F}_1}(\rho) \ \text{or} \nonumber \\ &E_{\mathcal{F}_2}(\sigma) > E_{\mathcal{F}_2}(\rho) \ \text{or} \nonumber \\ &E_{\mathcal{F}_1}(\sigma) = E_{\mathcal{F}_1}(\rho) \, \text{and} \, E_{\mathcal{F}_2}(\sigma) = E_{\mathcal{F}_2}(\rho) \, \big\}. \end{align} Within the set $\mathcal{F}_{\text{bank}}$ we can find different subsets of bank states with a fixed value of $E_{\mathcal{F}_1}$ and $E_{\mathcal{F}_2}$. We define each of these subsets as \begin{equation} \label{subset_bank} \mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right) =
\left\{ \rho \in \mathcal{F}_{\text{bank}} \ | \ E_{\mathcal{F}_1}(\rho) = \bar{E}_{\mathcal{F}_1} \, \text{and} \, \ E_{\mathcal{F}_2}(\rho) = \bar{E}_{\mathcal{F}_2} \right\}. \end{equation} \end{definition}
Notice that Eq.~\eqref{set_f3} implies that no state can be found with smaller values of both monotones $E_{\mathcal{F}_i}$'s. In this way, the agent is not able to transform the state of the bank in a way in which both resources are extracted from it and stored in the batteries. Instead, they always need to trade resources. The set of bank states $\mathcal{F}_{\mathrm{bank}}$ can be visualised in the resource diagram of the theory, see Fig.~\ref{fig:bank_subset}. This set is represented by a curve on the boundary of the state space, connecting the points associated with $\mathcal{F}_1$ to those associated with $\mathcal{F}_2$. In appendix~\ref{convex_bound} we show that, under the current assumptions, this curve is always convex, and in the following we focus our attention to those segments where the curve is strictly convex. \par The subsets $\mathcal{F}_{\mathrm{bank}} \left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$'s represent individual points in the resource diagram describing the multi-resource theory, and they obey many of the properties satisfied by the invariant sets $\mathcal{F}_i$'s. Indeed, one can show that \begin{itemize} \item For all $n \in \mathbb{N}$, we have that each subset of bank states is such that \begin{equation} \label{eq:add_bank} \mathcal{F}^{(n)}_{\mathrm{bank}} \left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right) = \mathcal{F}^{\otimes n}_{\mathrm{bank}} \left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2} \right), \end{equation} that is, these subsets satisfy property~\ref{item:F5b}. This equality is proved in Prop.~\ref{additive_f3} of appendix~\ref{additional}. \item Every subset $\mathcal{F}_{\mathrm{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$ is convex, property~\ref{item:F2}, as shown in Prop.~\ref{convex_f3} in appendix~\ref{additional}. \item Every subset $\mathcal{F}_{\mathrm{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$, and its extensions to the many-copy case, is invariant under the class of allowed operations $\mathcal{C}_{\mathrm{multi}}$ of the multi-resource theory, as shown in Lem.~\ref{lem:inv_f3} in appendix~\ref{additional}. \end{itemize}
\begin{figure}\label{fig:bank_subset}
\end{figure}
\par The second essential property for a bank is that the exchange rate needs only to depend on which state of the bank we choose to use. In our framework, it is the choice of the values $\bar{E}_{\mathcal{F}_1}$ and $\bar{E}_{\mathcal{F}_2}$, defining the subset $\mathcal{F}_{\text{bank}} \left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$, that determines the exchange rate at which the resources are converted. In order to obtain this exchange rate we introduce the following function, which quantifies how much the properties of the bank change during a transformation, and generalises the Helmholtz free energy used in thermodynamics. Given the subset of bank states $\mathcal{F}_{\text{bank}} \left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$, this function is defined as \begin{equation} \label{f3_monotone} f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}(\rho) := \alpha \, E_{\mathcal{F}_1}(\rho) + \beta \, E_{\mathcal{F}_2}(\rho) - \gamma, \end{equation} where $\alpha$, $\beta$, and $\gamma$ are non-negative constant factors, which depend on the subset of bank states we have chosen. In order to define the linear coefficients, we impose the following two properties for this function, \begin{description} \item[B1\label{item:B1}] The function $f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}$ is equal to zero over the subset $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$. \item[B2\label{item:B2}] The value of this function on the states contained in the subset $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}\right)$ is minimum. \end{description} Notice that property~\ref{item:B1} is there to set the zero of the function, and implies that \begin{equation} \gamma = \alpha \, \bar{E}_{\mathcal{F}_1} + \beta \, \bar{E}_{\mathcal{F}_2}. \end{equation} Property~\ref{item:B2}, instead, fixes the ratio between the constants $\alpha$ and $\beta$. This condition can be visualised in the resource diagram, and is equivalent to the request that, in such a diagram, the bank monotone is tangent to the state space, so that \begin{equation} \label{eq:tangent_f3} f_{\mathrm{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}(\rho) \geq f_{\mathrm{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}(\sigma) , \quad \forall \, \rho \in \mathcal{S} \left( \mathcal{H} \right) , \, \forall \, \sigma \in \mathcal{F}_{\mathrm{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right). \end{equation} The above property is always satisfied under our working assumptions, since the curve of bank states is convex, see Fig.~\ref{fig:bank_subset}. We refer to this function as the \emph{bank monotone}. \par The bank monotone can be easily extended to the state space of $n$ copies of the system. The main difference is that, when we consider states in $\SHn{n}$, the coefficient $\gamma$ is proportional to the number of copies $n$, and we write $\gamma = n \left( \alpha \, \bar{E}_{\mathcal{F}_1} + \beta \, \bar{E}_{\mathcal{F}_2} \right)$. This follows from property~\ref{item:B1}, together with the fact that the subset $\mathcal{F}_{\mathrm{bank}} \left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$ satisfies property~\ref{item:F5b}, see Eq.~\eqref{eq:add_bank}. Since the function in Eq.~\eqref{f3_monotone} is a linear combination of the monotones $E_{\mathcal{F}_1}$ and $E_{\mathcal{F}_2}$, it is easy to show (see also appendix~\ref{additional}) that it satisfies the properties listed in the following proposition
\begin{restatable}{prop}{bankproperties} \label{prop:properties_bank_mon} Consider a resource theory $\text{R}_{\text{multi}}$ with allowed operations $\mathcal{C}_{\text{multi}}$, satisfying asymptotic equivalence with respect to the monotones $E_{\mathcal{F}_1}$ and $E_{\mathcal{F}_2}$, i.e.~the relative entropy distances from the invariant sets of the theory. Suppose that these sets satisfy the properties~\ref{item:F1}, \ref{item:F2}, \ref{item:F3}, and \ref{item:F5b}. Then, the function $f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}$ introduced in Eq.~\eqref{f3_monotone} satisfies the following properties. \begin{description} \item[B3\label{item:B3}] The function $f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}$ is additive. \item[B4\label{item:B4}] The function $f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}$ is monotonic under partial tracing. \item[B5\label{item:B5}] The function $f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}$ is sub-extensive, i.e., this function scales at most linearly in the number of systems considered. More precisely, for any sequence of states $\left\{ \rho_n \in \SHn{n} \right\}$, we have that $f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}(\rho_n) = O(n)$. \item[B6\label{item:B6}] The function $f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}$ is asymptotic continuous. \item[B7\label{item:B7}] The function $f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}$ is monotonic under the set of allowed operations $\mathcal{C}_{\text{multi}}$, since $\alpha$ and $\beta$ are non-negative. \end{description} \end{restatable} \par The third and last property we demand from a bank concerns the back-reaction it experiences during interconversion of resources. We want that, after the transformation, the state of the bank only changes infinitesimally with respect to the bank monotone associated with it. If this is the case, we can show that the exchange rate only changes infinitesimally, and therefore we can keep using the bank to inter-convert between resources at the same exchange rate. More concretely, we now consider a tripartite system composed by a bank $S$ and and two batteries, $B_1$ and $B_2$. Each of these subsystems is composed by many copies of the same fundamental system described by $\mathcal{H}$, for which we defined the notion of bank states. Thus, the bank $S$ is described by $\mathcal{H}_S = \mathcal{H}^{\otimes n}$, with $n \in \mathbb{N}$, and its initial state is given by $n$ copies of the bank state $\rho \in \mathcal{F}_{\mathrm{bank}}\left(\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}\right)$. The batteries are described by $\mathcal{H}_{B_i} = \mathcal{H}^{\otimes m_i}$, $m_i \in \mathbb{N}$, where $i = 1,2$. The states describing the batteries are $\omega_1 \in \mathcal{S} \left( \mathcal{H}_{B_1} \right)$, and $\omega_2 \in \mathcal{S} \left( \mathcal{H}_{B_2} \right)$, respectively. \par A \emph{resource interconversion} is an asymptotically reversible transformation \begin{equation} \label{interconv_trasf} \rho^{\otimes n} \otimes \omega_1 \otimes \omega_2 \xleftrightarrow{\text{asympt}} \tilde{\rho}^{\otimes n} \otimes \omega'_1 \otimes \omega'_2, \end{equation} where $\tilde{\rho} \in \mathcal{S} \left( \mathcal{H} \right)$, $\omega'_1 \in \mathcal{S} \left( \mathcal{H}_{B_1} \right)$, and $\omega'_2 \in \mathcal{S} \left( \mathcal{H}_{B_2} \right)$, satisfying the following property, see also Fig.~\ref{fig:tangent_monotone}, \begin{description} \item[X1\label{item:X1}] The state of the bank changes infinitesimally during the resource interconversion.\\ If $\rho \in \mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right) \subset \mathcal{S} \left( \mathcal{H} \right)$, then the state $\tilde{\rho} \in \mathcal{S} \left( \mathcal{H} \right)$ is such that \begin{equation} \label{condition_x1} f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}(\tilde{\rho}^{\otimes n}) = f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}(\rho^{\otimes n}) + \delta_n, \end{equation} where $\delta_n > 0$ is such that $\delta_n \rightarrow 0$ as $n \rightarrow \infty$. \end{description} It is worth noting that, according to the above definition, the bank is here acting as a \emph{catalyst}, allowing for resource interconversion. Catalysts are used in resource theories to allow for state transformations which are otherwise impossible~\cite{gour_resource_2015,chitambar_quantum_2018}. These systems are generally described by resourceful states, and therefore are subject to strict constraints, for example the requirement that their initial state needs to be perfectly (or approximately) re-obtained at the end of the transformation~\cite{ng_limits_2015}. These constraints are required since, otherwise, one might act on the catalyst and extract resources from it, thus trivializing the theory~\cite{van_dam_universal_2003}. It is interesting to notice that our bank is similarly constrained, specifically by Eq.~\ref{condition_x1}. As we see in the following theorem, this constrain is enough to allow for resource interconversion, but also to ensure a non-trivial behaviour of the theory (no resource is extracted for free). \par We are now ready to introduce the interconversion relation which links the different amounts of resources exchanged, weighted by the exchange rate given by the bank. The theorem is proved in appendix~\ref{main_results}.
\begin{restatable}{thm}{interconvert} \label{thm:interconvert_relation} Consider a resource theory $\text{R}_{\text{multi}}$ with two resources, equipped with the batteries $B_1$ and $B_2$. Suppose the theory satisfies asymptotic equivalence with respect to the monotones $E_{\mathcal{F}_1}$ and $E_{\mathcal{F}_2}$, i.e.~the relative entropy distances from the invariant sets of the theory, and that these sets satisfy the properties~\ref{item:F1}, \ref{item:F2}, \ref{item:F3}, and \ref{item:F5b}. Then, the resource interconversion of Eq.~\eqref{interconv_trasf}, where the bank has to transform in accord to condition~\ref{item:X1}, is solely regulated by the following relation, \begin{equation} \label{finite_interconversion} \alpha \, \Delta W_1 = - \beta \, \Delta W_2 + \delta_n. \end{equation} Furthermore, when the number of copies of the bank system $n$ is sent to infinity, we have that the above equation reduces to the following one, which we refer to as the \emph{interconversion relation}, \begin{equation} \label{interconversion} \Delta W_1 = - \frac{\beta}{\alpha} \, \Delta W_2, \end{equation} where the amount of resources exchanged $\Delta W_i$ is non-zero. \end{restatable}
Let us highlight some properties that a bank state needs to satisfy in order to allow for interconversion of resources from one battery to another, and vice versa. We show that to interconvert between the resources in both directions we need a bank state containing a non-zero amount of both resources. First notice that, since both parameters $\alpha$ and $\beta$ are non-negative, whenever we exchange between resources, we increase the amount contained in one of the batteries (for example, $\Delta W_1 > 0$) while decreasing the amount contained in the other ($\Delta W_2 < 0$). However, the change in these two resources also depends on the transformation of the bank state, see Eq.~\eqref{resource_work_i}. Therefore, one has to consider the bank state used for interconversion, and the amount of resources contained in it. When the bank state $\rho$ is such that $E_{\mathcal{F}_1}(\rho) > 0$ and $E_{\mathcal{F}_2}(\rho) > 0$, then interconversion can be achieved (in both directions) between $\Delta W_1$ and $\Delta W_2$, at the rate specified by Eq.~\eqref{interconversion}. Moreover, as far as the amount of resources in the bank is non-zero, we can exchange any amount of one resource for the other (since we can take the number of copies of the bank to be infinite). This is the case of thermodynamics, where thermal states indeed contain a positive amount of both energy and entropy, the two resources of the theory, and Eq.~\eqref{interconversion} gives the conversion rate for Landauer's erasure.
\begin{figure}\label{fig:tangent_monotone}
\end{figure}
\par Finally, let us consider what would happen if we were to allow the states in $\mathcal{F}_1$ or $\mathcal{F}_2$ (or in their intersection) to describe the bank. If the bank state were such that $E_{\mathcal{F}_1}(\rho) > 0$ and $E_{\mathcal{F}_2}(\rho) = 0$ (or vice versa), then we could only exchange in one direction, since we could gain the first resource while paying the second resource (or vice versa). If the bank state did not contain any amount of resources, $E_{\mathcal{F}_1}(\rho) = 0$ and $E_{\mathcal{F}_2}(\rho) = 0$, then we could not perform interconversion at all, because we would have to reduce the amount of one of them within the bank. However, this would not be possible since the amount of resource stored in a (bank) state cannot be negative. As a result, the multi-resource theories in which an interesting interconversion relation can be found are the ones in which the invariant sets of the theory do not intercept, see the right panel of Fig.~\ref{fig:invariant_sets_structure}.
\subsection{Bank monotones and the relative entropy distance} \label{bank_monotone} We start this section with an example concerning different models to describe thermodynamics, and the connection between these models. In the last part of Sec.~\ref{multi_resource}, we have introduced a multi-resource theory whose resources are energy and entropy (or, information). For this theory, the bank states are thermal states at a given temperature $T$. We can move from this description of thermodynamics to a different one, based on a single-resource theory, by enlarging the class of operations in such a way that the agent can freely add ancillary systems in a thermal state with temperature $T$. This corresponds to the physical situation in which the system is put in contact with an infinite thermal bath. The single-resource theory we obtain is analogous to the one of Thermal Operations~\cite{brandao_resource_2013, horodecki_fundamental_2013}, and its resource quantifier is unique. In fact, we can show that the bank monotone of the multi-resource theory and the resource quantifier of the single resource theory both coincides (modulo a multiplicative factor) with $F - F_{\beta}$, where $F$ is the Helmholtz free energy of the state whose resource we are quantifying, and $F_{\beta}$ is the Helmholtz free energy of the thermal state with temperature $T = \beta^{-1}$. \par In the following we study the connection between a general multi-resource theory and the single-resource theory obtained by enlarging the allowed operations with the possibility of adding ancillary systems described by bank states in $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$. We find that the bank monotone of Eq.~\eqref{f3_monotone}, $f_{\text{bank}} ^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}$, coincides with the unique measure of resource for the obtained single-resource theory. As a result, we find that property~\ref{item:X1}, which regulates the exchange of resources in the multi-resource theory, can be understood as the condition that the resource characterising the bank does not increase during the transformation. Furthermore, we show that, when the subset of bank states $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$ contains a full-rank state, the monotone $f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}}$ is proportional to the relative entropy distance from this subset. Let us now introduce the single-resource theory which can be derived from $\text{R}_{\text{multi}}$ by allowing the possibility of adding ancillary systems described by specific bank states.
\begin{definition} \label{def:sing_res_constr} Consider the two-resource theory $\text{R}_{\text{multi}}$ with allowed operations $\mathcal{C}_{\text{multi}}$ and invariant sets $\mathcal{F}_1$ and $\mathcal{F}_2$ which satisfy the properties~\ref{item:F1}, \ref{item:F2}, \ref{item:F3}, and \ref{item:F5b}. Consider the bank set $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$ introduced in Eq.~\eqref{subset_bank}. We define the single-resource theory $\text{R}_{\text{single}}$ as that theory whose class of allowed operations $\mathcal{C}_{\text{single}}$ is composed by the following three fundamental operations, \begin{enumerate} \item Add an ancillary system described by $n \in \mathbb{N}$ copies of a bank state $\rho_P \in \mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$. \item Apply any operation $\mathcal{E} \in \mathcal{C}_{\text{multi}}$ to system and ancilla. \item Trace out the ancillary systems. \end{enumerate} The most general operation in $\mathcal{C}_{\text{single}}$ which does not change the number of systems between its input and output is \begin{equation} \label{sin_res_map} \mathcal{E}^{\text{(s)}}(\rho) = \Tr{P^{(n)}}{\mathcal{E} \left( \rho \otimes \rho_P^{\otimes n} \right)}, \end{equation} where we are partial tracing over the degrees of freedom $P^{(n)}$, that is, over the ancillary system initially in $\rho_P^{\otimes n}$. \end{definition}
The bank monotone associated with the bank set $\mathcal{F}_{\mathrm{bank}}\left( \bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}\right)$, see Eq.~\eqref{f3_monotone}, is the unique quantifier for the single-resource theory $\text{R}_{\mathrm{single}}$. In order to show the uniqueness of this monotone, we first have to show that the single-resource theory satisfies asymptotic equivalence.
\begin{restatable}{thm}{singres} \label{unique_f3} Consider the two-resource theory $\text{R}_{\text{multi}}$ with allowed operations $\mathcal{C}_{\text{multi}}$, and invariant sets $\mathcal{F}_1$ and $\mathcal{F}_2$ which satisfy the properties~\ref{item:F1}, \ref{item:F2}, \ref{item:F3}, and \ref{item:F5b}. Suppose the theory satisfies the asymptotic equivalence property with respect to the monotones $E_{\mathcal{F}_1}$ and $E_{\mathcal{F}_2}$. Then, given the subset of bank states $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}\right)$, the single-resource theory $\text{R}_{\text{single}}$ with allowed operations $\mathcal{C}_{\text{single}}$ satisfies the asymptotic equivalence property with respect to $f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}$. \end{restatable}
The proof of this theorem can be found in appendix~\ref{main_results}, and we provide a geometric sketch of it in Fig.~\ref{fig:geometric_proof}. As a side remark, notice that the functions $E_{\mathcal{F}_1}$ and $E_{\mathcal{F}_2}$ are not monotonic under the set of allowed operations $\mathcal{C}_{\text{single}}$. This follows from the fact that we can now replace any state of the system with a state in $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}\right)$, since we are free to add an ancillary system in such state, and to perform a swap between main system and ancilla (since this operation belongs to $\mathcal{C}_{\text{multi}}$). Then, if the bank state contains a non-zero amount of resources, meaning that $\bar{E}_{\mathcal{F}_i} > 0$ for $i=1,2$, we can always find a state in $\mathcal{S} \left( \mathcal{H} \right)$ with lower value of either $E_{\mathcal{F}_1}$ or $E_{\mathcal{F}_2}$ (but not both at the same time), and therefore the above transformation would increase the value of this monotone.
\begin{figure}\label{fig:geometric_proof}
\end{figure}
\par From the above theorem it follows an interesting link between the bank monotone $f_{\text{bank}} ^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}}$, defined in Eq.~\eqref{f3_monotone}, and the relative entropy distance from the set of bank states $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$. Indeed, when this set of states contains at least one full-rank state, we can prove that these two functions have to coincide, modulo a multiplicative factor. This is a consequence of the fact that $\text{R}_{\text{single}}$ satisfies asymptotic equivalence, which implies the uniqueness of the resource measure, and of the fact that both the bank monotone and the relative entropy distance from the bank set satisfy the same properties, in particular monotonicity under the operations in $\mathcal{C}_{\text{single}}$ and asymptotic continuity. We can express this fact in the following corollary, whose proof can be found in appendix~\ref{main_results}.
\begin{restatable}{coro}{bankmonrelent} \label{bank_equal_rel_ent} Consider the two-resource theory $\text{R}_{\text{multi}}$ with allowed operations $\mathcal{C}_{\text{multi}}$, and invariant sets $\mathcal{F}_1$ and $\mathcal{F}_2$ which satisfy the properties~\ref{item:F1}, \ref{item:F2}, \ref{item:F3}, and \ref{item:F5b}. Suppose the theory satisfies the asymptotic equivalence property with respect to the monotones $E_{\mathcal{F}_1}$ and $E_{\mathcal{F}_2}$. If the subset of bank states $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}\right)$ contains a full-rank state, then the bank monotone $f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}$ coincides with the relative entropy distance from this subset of states, modulo a multiplicative constant. \end{restatable}
\par We close the section with the remark that, in the currently known scenarios, the bank subsets always contain at least a full-rank state, and in fact we find that, for these theories, the above correspondence between the bank monotone of Eq.~\eqref{f3_monotone} and the relative entropy distance is satisfied. An example is the multi-resource theory of thermodynamics, in which the relative entropy distance from a thermal state at a given temperature is indeed equal to the linear combination of the average energy and the entropy of a system. Other examples can be found in Sec.~\ref{examples}.
\subsection{First law for multi-resource theories} \label{sec:first_law} We can now introduce a general first law for multi-resource theories with disjoint invariant sets, see the right panel of Fig.~\ref{fig:invariant_sets_structure}. In order for this law to be valid, we need access to the batteries, the bank, and the main system. Within this setting, the first law consists of a single relation which links the different amount of resources exchanged with the batteries, the $\Delta W_i$'s, with the change in bank monotone over the state of the main system. The idea is that, contrary to what seen in Sec.~\ref{quant_res}, a state transformation over the main system is possible, when a bank is present, if this single relation is satisfied. Indeed, we do not need to use a fixed amount of each resource, since they are inter-convertible using the bank system. \par In more detail, we consider a theory $\text{R}_{\text{multi}}$ that, for simplicity, has just two resources. The invariant sets are $\mathcal{F}_1$ and $\mathcal{F}_2$, they satisfy the properties~\ref{item:F1}, \ref{item:F2}, \ref{item:F3} and~\ref{item:F5b}, and the theory satisfies the asymptotic equivalence property with respect to the monotones $E_{\mathcal{F}_1}$ and $E_{\mathcal{F}_2}$. The global system is divided into four partitions, the main system $S$, the bank $P$, and the batteries $B_1$ and $B_2$. We assume the bank to be initially described by a state $\rho_P \in \mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}\right)$, where this subset contains at least one full-rank state. The relevant monotone for the interconversion of resources is then the relative entropy distance from the subset $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}\right)$, as shown in Cor.~\ref{bank_equal_rel_ent}. \par Suppose that the main system is initially described by the state $\rho \in \mathcal{S} \left( \mathcal{H}_S \right)$, and we want to map it into the state $\sigma \in \mathcal{S} \left( \mathcal{H}_S \right)$, with possibly a different value of $E_{\mathcal{F}_1}$ and $E_{\mathcal{F}_2}$. If we do not have access to the bank, then the amount of resources we need to exchange is given by the difference of the monotones $E_{\mathcal{F}_i}$'s between the initial and final state of the main system, see Eq.~\eqref{resource_work_ent} in Sec.~\ref{multi_rev_unique}. But since we have access to the battery, we can exchange between the resources, and we are not obliged any more to provide a fixed amount for each resource. In order to show this, consider the global initial state $\rho \otimes \rho_P \otimes \omega_1 \otimes \omega_2$, describing the main system, the bank, and the two batteries $B_1$ and $B_2$. Then, we (asymptotically) map this global state, using the allowed operations $\mathcal{C}_{\text{multi}}$, into the final state $\sigma \otimes \tilde{\rho}_P \otimes \omega'_1 \otimes \omega'_2$, where the final state of the bank is $\tilde{\rho}_P$, and the batteries $B_1$ and $B_2$ have final state $\omega'_1$ and $\omega'_2$, respectively. Due to asymptotic equivalence, this state transformation is possible only if the monotones $E_{\mathcal{F}_i}$'s are preserved. However, the final state of the bank only has to satisfy property~\ref{item:X1}, and we have shown in Sec.~\ref{bank_interconvert} that such constraint still allows us to exchange an arbitrary amount of resources, see Thm.~\ref{thm:interconvert_relation}. As a result, there is a single relation that regulates the state transformation over the main system,
\begin{coro} \label{coro:first_law} Consider the two-resource theory $\text{R}_{\text{multi}}$ with allowed operations $\mathcal{C}_{\text{multi}}$, and invariant sets $\mathcal{F}_1$ and $\mathcal{F}_2$ which satisfy the properties~\ref{item:F1}, \ref{item:F2}, \ref{item:F3}, and \ref{item:F5b}. Suppose the theory satisfies the asymptotic equivalence property with respect to the monotones $E_{\mathcal{F}_1}$ and $E_{\mathcal{F}_2}$, and that the global system is divided into a main system $S$, a bank described by the set of states $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}\right)$ (which contains at least one full-rank state), and two batteries $B_1$ and $B_2$. Then, a transformation which maps the state of the main system from $\rho$ into $\sigma$, where these states are completely general, only has to satisfy the following relation \begin{equation} \label{eq:first_law} \alpha \, \Delta W_1 + \beta \, \Delta W_2 = E_{\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}\right)}(\rho) - E_{\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}\right)}(\sigma), \end{equation} where each $\Delta W_i$ is defined as the difference in the monotone $E_{\mathcal{F}_i}$ over the final and initial state of the battery $B_i$, see Eq.~\eqref{work_Ri}, and $E_{\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}\right)}$ is the relative entropy distance from the set of states describing the bank. \end{coro}
\par We refer to Eq.~\eqref{eq:first_law} as the first law of multi-resource theories. Indeed, for the resource theory of thermodynamics, where energy and entropy are the two resources, and the bank is given by an infinite thermal reservoir with a given temperature $T$, this equation corresponds to the First Law of Thermodynamics. In fact, in the thermodynamic scenario we have that $\Delta W_1 = - \Delta U$, where $U$ is the internal energy of the system, while $\Delta W_2 = \Delta S$ is the change in entropy in the system. The change in relative entropy distance on the main system is proportional to the change in Helmholtz free-energy, which in turn is equal to the work extracted from the system, $W$. The linear coefficients in the equation can be computed from Eq.~\eqref{f3_monotone}, knowing that the bank monotone is equal to the relative entropy distance from the thermal state with temperature $T$. It is easy to show that $\alpha = T^{-1}$ and $\beta =1$ . If we re-arrange the equation, and we define $Q = T \, \Delta S$ as the amount of heat absorbed by the system, we obtain $\Delta U = Q - W$, that is, the First Law of Thermodynamics.
\section{Examples} \label{examples} In this section we present two examples of multi-resource theories where an interconversion relation can be derived. The first one is thermodynamics for multiple conserved quantities (even non-commuting ones), while the second one concerns local control under energetic restrictions. In both examples we describe the state-space (and we represent it with a resource diagram), we find the bank states of the theory, and we derive an interconversion relation for the different resources. Furthermore, in both cases we find that the bank monotone is proportional to the relative entropy distance from the given set of bank states, as expected from Cor.~\ref{bank_equal_rel_ent}.
\begin{figure}
\caption{How to apply the results of this paper to an arbitrary resource theory.}
\label{fig:flowchart}
\end{figure}
\par Before we introduce the examples, we provide a flowchart~\ref{fig:flowchart} that should help the reader in building a multi-resource theory. In particular, the flowchart clarifies in which situations each of the results we obtain hold for a specific theory. This tool should be used as follows,
\begin{itemize} \item The fundamental constraints and conservation laws of the task under consideration should be identified, and together with them the resources composing the theory. \item Given the set of resources for the theory, we define the class of allowed operations $\mathcal{C}_{\text{multi}}$ as in Eq.~\eqref{all_ops_multi}, and we identify the invariant sets of the theory $\left\{ \mathcal{F}_i \right\}_{i=1}^m$. \item Checking whether asymptotic equivalence holds for the multi-resource theory is the first step of the flowchart (box {\bf I} in Fig.~\ref{fig:flowchart}). To show that the theory satisfies this property, we need to find a protocol which maps between states with same values of a given set of monotones. \item If the theory satisfies asymptotic equivalence, we can focus on the properties of the monotones and of the invariant sets. Following the flowchart, we can then easily identify which properties and features hold for the theory under consideration. \end{itemize}
The flowchart here introduced is used in the first example to clarify how to characterise a multi-resource theory. \subsection{Thermodynamics of multiple-conserved quantities} \label{thermo_example} In this example we consider the resource theory of thermodynamics in the presence of multiple conserved quantities (even in the case in which these quantities do not commute)~\cite{guryanova_thermodynamics_2016,yunger_halpern_microcanonical_2016, lostaglio_thermodynamic_2017}. Our system is a $d$-level quantum system, and for simplicity, we only consider two conserved quantities $A$ and $B$. The allowed operations are Thermal Operations~\cite{brandao_resource_2013, horodecki_fundamental_2013}, composed by unitary operators which commute with both $A$ and $B$. This set of maps can be obtained as a proper subset of the intersection between the allowed operations of the following single-resource theories, \begin{itemize} \item The resource theory of the quantity $A$. The allowed operations are all the average-$A$-non-increasing maps, whose invariant set is composed by a single state, $\mathcal{F}_A = \left\{ \ket{a_0}\bra{a_0} \right\}$, the eigenstate of $A$ associated with its minimum eigenvalue $a_0$ (for simplicity, we here assume it to be non-degenerate). From Sec.~\ref{average_non_increasing} it follows that this theory has a monotone of the form $M_A(\rho) = \tr{A \rho} - a_0$. \item The resource theory of the quantity $B$. The allowed operations are all the average-$B$-non-increasing maps, whose invariant set is composed by a single state, $\mathcal{F}_B = \left\{ \ket{b_0}\bra{b_0} \right\}$, the eigenstate of $B$ associated with its minimum eigenvalue $b_0$ (for simplicity, we here assume it to be non-degenerate). From Sec.~\ref{average_non_increasing} it follows that this theory has a monotone of the form $M_B(\rho) = \tr{B \rho} - b_0$. \item The resource theory of purity, where the allowed operations are all the maps whose fix point is the maximally-mixed state $\mathcal{F}_S = \left\{ \frac{\mathbb{I}}{d} \right\}$ (unital maps). One monotone of the theory is the relative entropy distance from $\frac{\mathbb{I}}{d}$, that is, $E_{\mathcal{F}_S}(\rho) = \log d - S(\rho)$ where $S(\cdot)$ is the von Neumann entropy. \end{itemize} \par We can now make use of the flowchart to characterise the multi-resource theory. Box {\bf I} in the flowchart asks whether or not the considered multi-resource theory satisfies asymptotic equivalence. In Refs.~\cite{bera_thermodynamics_2017,sparaciari_resource_2016} it has been shown that, indeed, a resource theory of this kind does satisfy the asymptotic equivalence property of Def.~\ref{def:asympt_equivalence_multi} with respect to the monotones $M_A$, $M_B$ and $E_{\mathcal{F}_S}$. Furthermore, it is easy to see that these monotones are either relative entropy distances from the set of invariant states, or that they are of the form given in Eq.~\eqref{montone_average}. This implies that we can answer positively to box {\bf II} in the flowchart. \par We now need to consider the properties of the invariant sets of the theory, which in turn determine the properties of the monotones. It is easy to show that these sets are closed (property~\ref{item:F1}) and convex (property~\ref{item:F2}). Furthermore, $\mathcal{F}_S$ contains a full-rank state (property~\ref{item:F3}), that implies asymptotic continuity of the associated monotone, see Refs.~\cite{synak-radtke_asymptotic_2006, brandao_generalization_2010}. The fact that the other sets do not contain a full-rank state is not problematic since we are considering monotones of the form of Eq.~\eqref{montone_average}, that are nevertheless asymptotic continuous, see Prop.~\ref{average_asymp_cont}. Thus, the invariant sets satisfy all the properties required in box {\bf IV}, and we now need to construct batteries able to store the different resources separately (property~\ref{item:M1}). \par For the first kind of resource, this can be achieved by selecting two pure states with different average values of $A$, and same average values of $B$. The battery $B_A$, storing the first kind of resource, is then composed by a certain number of copies of these two states, where the number varies when we extract/store the resource. A similar construction can be done for the other battery $B_B$. For the purity battery, we can take a system with degenerate $A$ and $B$, and take states with a certain number of copies of a pure state and mixed state. If this construction is possible, then we can answer positively to box {\bf IV} in the flowchart. \par We can now study the properties of the invariant sets, specifically their closure with respect to tensor product (property~\ref{item:F4}) and partial trace (property~\ref{item:F5}). Since each invariant set is composed by a single state, we find that both these properties and property~\ref{item:F5b} are satisfied. Thus, from box {\bf VI} we can move to box {\bf VIII}, and therefore the theory can be studied with a resource diagram, see Fig.~\ref{fig:example_monotones_thermo} and the representation is unique. \par Let us now consider a reversible transformation, described by the following equation \begin{equation} \rho^{\otimes n} \otimes \omega_A \otimes \omega_B \otimes \omega_S \xleftrightarrow{\text{asympt}} \sigma^{\otimes n} \otimes \omega'_A \otimes \omega'_B \otimes \omega'_S, \end{equation} where the $n$ copies of $\rho$ and $\rho'$ describe the main system at the beginning and the end of the transformation, and the states $\omega_i$ and $\omega_i'$ are the initial and final states of the battery $B_i$, for $i = A,B,S$. According to asymptotic equivalence, the transformation is possible if \begin{align} \Delta W_A &= M_A^{\infty}(\rho) - M_A^{\infty}(\sigma) = \tr{A \left( \rho - \sigma \right)}, \\ \Delta W_B &= M_B^{\infty}(\rho) - M_B^{\infty}(\sigma) = \tr{B \left( \rho - \sigma \right)}, \\ \Delta W_S &= E_{\mathcal{F}_S}^{\infty}(\rho) - E_{\mathcal{F}_S}^{\infty}(\sigma) = S(\sigma) - S(\rho). \end{align} \par To answer the last box of the flowchart, box {\bf VIII}, we need to focus on the resources contained in the bank states. Indeed, in order to get an interconversion relation and a first law we need the bank states to contain a non-zero amount of each resource. This has to be the case for the current resource theory, since the invariant sets do not intercept each other. Therefore, this theory admits a first law, as we are going to show. It can be easily shown, using Jaynes principle~\cite{jaynes_information_1957}, that the bank states are of the following form \begin{equation} \label{GGS} \tau_{\beta_1,\beta_2} = \frac{e^{- \beta_1 A - \beta_2 B}}{Z}, \end{equation} where the parameters $\beta_1, \beta_2 \in [0, \infty)$, and $Z = \tr{e^{- \beta_1 A - \beta_2 B}}$ is the partition function of the system. These states are known in thermodynamics as the grand-canonical ensemble. Each $\tau_{\beta_1,\beta_2}$ is a bank state with a different value of resource $A$, resource $B$, and purity. The value of these three resources only depends on the parameters $\beta_1$ and $\beta_2$. In order to find the interconversion relation we need to construct the bank monotone \begin{equation} f_{\text{bank}}^{\bar{\beta}_1, \bar{\beta}_2}(\rho) = \alpha_{\bar{\beta}_1, \bar{\beta}_2} M_A(\rho) + \gamma_{\bar{\beta}_1, \bar{\beta}_2} M_B(\rho) + \delta_{\bar{\beta}_1, \bar{\beta}_2} E_{\mathcal{F}_S}(\rho) - \xi_{\bar{\beta}_1, \bar{\beta}_2} \end{equation} which is equal to zero over the bank state $\tau_{\bar{\beta}_1, \bar{\beta}_2}$. Properties~\ref{item:B1} and~\ref{item:B2} provide a geometrical way of building the monotone. If we represent the state space in a three-dimensional diagram (where the axes are given by $M_A$, $M_B$, and $E_{\mathcal{F}_S}$), then the hyperplane defined by the equation $f_{\text{bank}}^{\bar{\beta}_1, \bar{\beta}_2} = 0$ is tangent to the state space and only intercepts it in $\tau_{\bar{\beta}_1, \bar{\beta}_2}$, see Fig.~\ref{fig:example_monotones_thermo} for an example.
\begin{figure}\label{fig:example_monotones_thermo}
\end{figure}
\par The hyperplane defined by $f_{\text{bank}}^{\bar{\beta}_1, \bar{\beta}_2} = 0$ is identified by the normal vector \begin{equation} \hat{n} = \hat{r}_1 \times \hat{r}_2 , \quad \text{where} \ \hat{r}_i = \left( \frac{\partial M_A(\tau_{\bar{\beta}_1, \bar{\beta}_2})}{\partial \beta_i} ; \frac{\partial M_B(\tau_{\bar{\beta}_1, \bar{\beta}_2})}{\partial \beta_i} ; \frac{\partial E_{\mathcal{F}_S}(\tau_{\bar{\beta}_1, \bar{\beta}_2})}{\partial \beta_i} \right)^T \ \text{for} \ i = 1, 2. \end{equation} The parametric equation of the hyperplane then gives us the expression of the monotone, \begin{equation} f_{\text{bank}}^{\bar{\beta}_1, \bar{\beta}_2}(\rho) = n_1 \left( M_A(\rho) - M_A(\tau_{\bar{\beta}_1, \bar{\beta}_2}) \right) + n_2 \left( M_B(\rho) - M_B(\tau_{\bar{\beta}_1, \bar{\beta}_2}) \right) + n_3 \left( E_{\mathcal{F}_S}(\rho) - E_{\mathcal{F}_S}(\tau_{\bar{\beta}_1, \bar{\beta}_2}) \right), \end{equation} where $n_i$ is the $i$-th component of the normal vector $\hat{n}$. By evaluating the monotones $M_A$, $M_B$, $E_{\mathcal{F}_S}$, and their derivatives we find that $f_{\text{bank}}^{\bar{\beta}_1, \bar{\beta}_2}$ is equal (modulo a positive multiplicative factor depending on the parameters $\bar{\beta}_1$ and $\bar{\beta}_2$) to the relative entropy distance from $\tau_{\bar{\beta}_1, \bar{\beta}_2}$, \begin{equation} f_{\text{bank}}^{\bar{\beta}_1, \bar{\beta}_2}(\rho) \propto E_{\tau_{\bar{\beta}_1, \bar{\beta}_2}}(\rho) = \bar{\beta}_1 \, \tr{\rho A} + \bar{\beta}_2 \, \tr{\rho B} - S(\rho) + \log Z. \end{equation} Thus, the bank state $\tau_{\bar{\beta}_1, \bar{\beta}_2}$ allows us to obtain the following interconversion relation between the three resources, \begin{equation} \bar{\beta}_1 \, \Delta W_A + \bar{\beta}_2 \, \Delta W_B = \Delta W_S, \end{equation} while the state of the bank only changes by an infinitesimal amount in terms of $E_{\tau_{\bar{\beta}_1, \bar{\beta}_2}}$.
\subsection{Local control theory under energetic restrictions} \label{control_theory_ex} We now introduce a multi-resource theory describing local control under energetic restrictions. Specifically, we consider the situation in which a quantum system is divided into two well-defined partitions $A$ and $B$, and we can only act on the individual partitions with non-entangling operations, which furthermore need to not increase the energy of the overall system. This kind of simultaneous restrictions on locality and thermodynamics has also been considered in other previous works, see for example Refs.~\cite{hovhannisyan_entanglement_2013,huber_thermodynamic_2015,wilming_second_2016, beny_energy_2017,lekscha_quantum_2018}. The multi-resource theory is obtained by considering two single-resource theories, the one of entanglement and the one of energy. While this is a well-defined multi-resource theory, it is not straightforward to prove that it is also a reversible theory. Therefore, to provide a first law in this setting, we have to restrict the state-space to a subset of all bipartite density operators.
\subsubsection{Set-up} \label{setup_en_ent} Let us consider a bipartite system, whose partitions are labelled as $A$ and $B$, with a non-local Hamiltonian $H_{AB}$ (that is, the two partitions interact with each other, and the ground state of the system is an entangled state). The set of allowed operations of this multi-resource theory is obtained from the intersection of the allowed operations of the following single-resource theories, \begin{itemize} \item The resource theory of energy. The allowed operations are all the average-energy-non-increasing maps, defined in Sec.~\ref{average_non_increasing}. When the Hamiltonian has non-degenerate ground state $\ket{\text{g}}$, the fixed state of the maps is $\mathcal{F}_H = \ket{\text{g}}\bra{\text{g}}$. The monotone of this resource theory is $M_H(\rho) = \tr{H \rho} - E_{\text{g}}$, where $E_{\text{g}}$ is the eigenvalue associated with the ground state $\ket{\text{g}}$. \item The resource theory of entanglement. The allowed operations are the asymptotically non-entangling maps~\cite{brandao_reversible_2010}. These maps are relevant to us for two reasons. Firstly, all our results hold in the asymptotic limit, and therefore it is reasonable to consider the set of maps which do not create entanglement in this limit. Secondly, this is the only set of operations which provides a reversible theory for entanglement. The monotone is $E_{\mathcal{F}_{\text{sep}}}(\cdot)$, where $\mathcal{F}_{\text{sep}}$ is the set of separable states, invariant under the class of operations. \end{itemize} While the current multi-resource theory is well-defined and meaningful, it is not straightforward to prove whether it is reversible in the sense given in Def.~\ref{def:asympt_equivalence_multi}. Furthermore, it is known that the relative entropy of entanglement, $E_{\mathcal{F}_{\text{sep}}}$, is not additive (or even extensive) for all bipartite density operator. Therefore, if we want to study interconversion of resources in this setting, we need to consider a subset of the state-space (as well as of the invariant set $\mathcal{F}_{\text{sep}}$). \par In the following we will focus on the simplest example of a multi-resource theory of this kind. The bipartite system is composed by two qubits, so that its Hilbert space is $\mathcal{H}_{AB} = \mathbb{C}^2 \otimes \mathbb{C}^2$. The Hamiltonian of the system is \begin{equation} \label{non_loc_ham} H_{AB} = E_0 \ket{\Psi_\text{singlet}}\bra{\Psi_\text{singlet}} + E_1 \, \Pi_{\text{triplet}}, \end{equation} where $E_0 < E_1$, the ground state is the singlet state, \begin{equation} \ket{\Psi_\text{singlet}} = \frac{1}{\sqrt{2}} \left( \ket{01} - \ket{10} \right), \end{equation} and $\Pi_{\text{triplet}} = \sum_{i=1}^3 \ket{\Psi_\text{triplet}^{(i)}}\bra{\Psi_\text{triplet}^{(i)}}$ is the projector on the triplet subspace, where \begin{align} \ket{\Psi_\text{triplet}^{(1)}} &= \frac{1}{\sqrt{2}} \left( \ket{01} + \ket{10} \right), \\ \ket{\Psi_\text{triplet}^{(2)}} &= \frac{1}{\sqrt{2}} \left( \ket{00} - \ket{11} \right), \\ \ket{\Psi_\text{triplet}^{(3)}} &= \frac{1}{\sqrt{2}} \left( \ket{00} + \ket{11} \right). \end{align} In order to get a reversible multi-resource theory, and therefore to be able to define the interconversion relations, we consider a restricted state-space, given by the following subset of bipartite density operators, \begin{equation}
\mathcal{S}_1 = \left\{ \rho \in \mathcal{S}(\mathcal{H}_{AB}) \, | \, \rho = \mathrm{p}_0 \ket{\Psi_\text{singlet}}\bra{\Psi_\text{singlet}} + \sum_{i=1}^{3} \mathrm{p}_i \ket{\Psi_\text{triplet}^{(i)}}\bra{\Psi_\text{triplet}^{(i)}} \ , \ \text{with} \ \mathrm{p}_0 \geq \frac{1}{2} \right\}. \end{equation} There are two additional reasons why we are interested in this set of states. First of all, because the relative entropy of entanglement $E_{\mathcal{F}_{\text{sep}}}$ has an analytical expression for states which are diagonal in the Bell basis~\cite{vedral_quantifying_1997, audenaert_asymptotic_2001, miranowicz_closed_2008} (that here coincides with the energy eigenbasis). Secondly, because it is easy to show, see Eq.~\eqref{set_f3}, that $\mathcal{S}_1$ contains the bank states of the theory, that are the interesting ones when it comes to study interconversion. Finally, it is worth noting that the state-space $\mathcal{S}_1$ contains all the Gibbs states of the non-local Hamiltonian $H_{AB}$ with positive temperatures. Within this restricted state-space we find the following subset of separable states, \begin{equation} \label{css_invariant} \mathcal{F}_{\text{css}} = \left\{ \rho = \frac{1}{2} \ket{\Psi_\text{singlet}}\bra{\Psi_\text{singlet}} + \sum_{i=1}^{3} \mathrm{p}_i \ket{\Psi_\text{triplet}^{(i)}}\bra{\Psi_\text{triplet}^{(i)}} \right\}. \end{equation} It is worth noticing that the above subset $\mathcal{F}_{\text{css}}$ contains all the closest-separable states to the entangled states in our restricted state-space $\mathcal{S}_1$ (see Ref.~\cite{miranowicz_closed_2008}). As a result, for any state $\rho \in \mathcal{S}_1$ we have that \begin{equation} E_{\mathcal{F}_{\text{sep}}}(\rho) = E_{\mathcal{F}_{\text{css}}}(\rho) = 1 - \mathrm{h}\left( \bra{\Psi_\text{singlet}}\rho\ket{\Psi_\text{singlet}} \right), \end{equation} where $\mathrm{h}(\cdot)$ is the binary entropy function. Since our focus is restricted to the sole states in the subset $\mathcal{S}_1$, we will now re-define\footnote{The modified set of allowed operations makes it easier for us to find a protocol for inter-converting resources. However, we do not exclude the possibility of being able to perform interconversion with the original set of allowed operations, that preserve all separable states. However, finding this protocol might be non-trivial, and could be material of future work.} the set of allowed operations of the multi-resource theory as those energy-non-increasing maps which only preserve the subset of separable states $\mathcal{F}_{\text{css}} = \mathcal{F}_{\text{sep}} \cap \mathcal{S}_1$. We can define this class of operation as \begin{equation} \label{ent_en_all_ops} \mathcal{C}_{\text{multi}} = \left\{ \mathcal{E} : \mathcal{S}(\mathcal{H}_{AB}) \rightarrow \mathcal{S}(\mathcal{H}_{AB})
\ | \ \mathcal{E}(\mathcal{F}_{\text{css}}) \subseteq \mathcal{F}_{\text{css}} \ \text{and} \ \tr{ \mathcal{E}(\rho) H_{AB}} \leq \tr{ \rho \, H_{AB}} \ \forall \, \rho \in \mathcal{S}(\mathcal{H}_{AB}) \right\}, \end{equation} where each $\mathcal{E} \in \mathcal{C}_{\text{multi}}$ is a completely positive and trace preserving map.
\begin{figure}\label{state_space_energy_entanglement}
\end{figure}
\par The two batteries we use in the theory store, respectively, energy and entanglement. One can imagine different kinds of energy batteries. For example, we could have that only Alice (or Bob) has access to the battery, which would imply that only one of them can change the energy of the non-local system. However, we prefer to consider a symmetric situation in which both Alice and Bob can interact with the battery. Moreover, we chose the battery to be non-local, so that they are effectively using the same battery, and not two local batteries. Thus, the battery $B_W$ is composed by $m$ copies of a two-qubit system with the same Hamiltonian of the main system, that is, \begin{equation} H_W = E_0 \ket{\Psi_\text{singlet}}\bra{\Psi_\text{singlet}} + E_1 \, \Pi_{\text{triplet}}. \end{equation} The state of the battery is \begin{equation} \label{energy_batt} \omega_W(k) = \ket{\Psi_\text{singlet}}\bra{\Psi_\text{singlet}}^{\otimes k} \otimes \ket{\Psi_\text{triplet}^{(1)}}\bra{\Psi_\text{triplet}^{(1)}}^{\otimes m - k}, \end{equation} where the excited state $\ket{\Psi_\text{triplet}^{(1)}}$ could be replaced by any other triplet state. Notice that, in order to store/provide energy, we have to change the number of triplet and singlet states contained in the battery, and this can be done locally by both Alice and Bob. Moreover, even if we are changing the energy of the battery, we are not modifying its entanglement, in accord with property~\ref{item:M1}. \par The second battery $B_E$ is composed by $\ell$ copies of a two-qubit system with trivial Hamiltonian $H_E \propto \mathbb{I}$ (so as to be able to exchange entanglement while preserving the energy of the battery). We choose the state of the battery to be \begin{equation} \label{entang_batt} \omega_E(h) = \ket{\Psi_\text{singlet}}\bra{\Psi_\text{singlet}}^{\otimes h} \otimes \sigma_{\text{mm}}^{\otimes \ell - h}, \end{equation} where the state $\sigma_{\text{mm}} \in \mathcal{F}_{\text{css}}$, and we take it to be the maximally-mixed state on the subspace spanned by $\ket{\Psi_\text{singlet}}$ and $\ket{\Psi_\text{triplet}^{(1)}}$, that is \begin{equation} \label{max_mix_sin_trip} \sigma_{\text{mm}} = \frac{1}{2} \ket{\Psi_\text{singlet}}\bra{\Psi_\text{singlet}} + \frac{1}{2} \ket{\Psi_\text{triplet}^{(1)}}\bra{\Psi_\text{triplet}^{(1)}}. \end{equation} The change in entanglement is measured by the change in the number of singlet states $h$.
\subsubsection{Reversibility and the interconversion relation} In order for the present multi-resource theory to admit an interconversion relation, we first need to show that the asymptotic equivalence property of Def.~\ref{def:asympt_equivalence_multi} is satisfied. Let us consider the subset of states $\mathcal{S}_{\mathrm{p}_0} \subset \mathcal{S}_1$, where $\mathrm{p}_0 > \frac{1}{2}$, defined as \begin{equation} \label{bank_subset_en_ent}
\mathcal{S}_{\mathrm{p}_0} = \left\{ \rho \in \mathcal{S}_1 \ | \ \bra{\Psi_\text{singlet}}\rho\ket{\Psi_\text{singlet}} = \mathrm{p}_0 \right\}. \end{equation} It is easy to show that all the states in this subset have the same value of the energy and entanglement monotones, which we label $\bar{M}_H$ and $\bar{E}_{\mathcal{F}_{\text{css}}}$ respectively. Furthermore, for any two states in this set, we can find an allowed operation in $\mathcal{C}_{\text{multi}}$, see Eq.~\eqref{ent_en_all_ops}, which maps one into the other. Indeed, consider an ancillary qutrit system described by the state $\eta = \sum_{i=1}^3 \mathrm{q}_i \ket{\theta_i}\bra{\theta_i}$, and the global unitary operation $U$ acting on main system and ancilla. The unitary operation maps $\ket{\Psi_\text{triplet}^{(i)}} \ket{\theta_j}$ into $\ket{\Psi_\text{triplet}^{(j)}} \ket{\theta_i}$, for $i, j \in \{1,2,3\}$, and acts trivially on the remaining basis states. Then, the operation $\mathcal{E}_{\eta}(\cdot) = \Tr{A}{U \left( \cdot \otimes \eta_A \right) U^{\dagger}} \in \mathcal{C}_{\text{multi}}$ maps any state $\rho \in \mathcal{S}_{\mathrm{p}_0}$ into the state \begin{equation} \label{rev_ent_en} \mathcal{E}_{\eta}(\rho) = \mathrm{p}_0 \ket{\Psi_\text{singlet}}\bra{\Psi_\text{singlet}} + \left( 1 - \mathrm{p}_0 \right) \sum_{i=1}^{3} \mathrm{q}_i \ket{\Psi_\text{triplet}^{(i)}}\bra{\Psi_\text{triplet}^{(i)}}, \end{equation} where the probability distribution $\left\{ \mathrm{q}_i \right\}_{i=1}^3$ is defined by $\eta$. By choosing different ancillary states $\eta$, we can reach different states in $\mathcal{S}_{\mathrm{p}_0}$, proving in this way that the resource theory satisfies asymptotic equivalence\footnote{The operation $\mathcal{E}_{\eta}(\cdot)$ we introduce is allowed since we restricted the invariant set $\mathcal{F}_{\text{sep}}$ to $\mathcal{F}_{\text{css}}$. Indeed, the above map would not leave invariant the set of separable states $\mathcal{F}_{\text{sep}}$.}. \par We can now consider the interconversion of energy and entanglement. Together with the two batteries $B_W$ and $B_E$, one for energy and the other for entropy, we need to use a bank system. One can show that, when diagonal states in the energy eigenbasis are considered, bank states belongs to the set $\mathcal{S}_1$ introduced in the previous section. Thus, we describe the bank system using $n \gg 1$ copies of a state $\rho_{\text{in}} \in \mathcal{S}_{\mathrm{p}_0}$, where $\mathrm{p}_0 > \frac{1}{2}$ (the actual form of the state is not relevant, since we can use the allowed operation $\mathcal{E}_{\eta}$ to freely select any state in this set). In order to obtain an interconversion relation, we need to find an allowed operation in $\mathcal{C}_{\text{multi}}$, acting on the global state of bank and batteries, which modifies the state of the batteries (by exchanging resources) while leaving the state of the bank almost unchanged with respect to the relative entropy distance from $\mathcal{S}_{\mathrm{p}_0}$. \par In appendix~\ref{protocol_example} we provide a protocol which performs the following resource interconversion using an allowed operation $\mathcal{C}_{\text{multi}}$, \begin{equation} \rho_{\text{in}}^{\otimes n} \otimes \omega_W(k) \otimes \omega_E(h) \xleftrightarrow{\text{asympt}} \rho_{\text{fin}}^{\otimes n} \otimes \omega_W(k') \otimes \omega_E(h'). \end{equation} In the above transformation, the initial state of the bank $\rho_{\text{in}}$ is mapped into a state $\rho_{\text{fin}} \in \mathcal{S}_{\mathrm{p}'_0}$, where $\mathrm{p}'_0 = \mathrm{p}_0 + O(n^{-1})$. The energy battery $B_W$ is mapped from the initial state $\omega_W(k)$, containing $k$ copies of the ground state of $H_{AB}$, into the final state $\omega_W(k')$ with $k' = k + \Delta k$ copies of this ground state, where $\Delta k > 0$ is arbitrary big. Likewise, the entanglement battery $B_E$ changes from the initial state $\omega_E(h)$, containing $h$ singlets, to the final state $\omega_E(h')$ containing $h' = h - \log \frac{\mathrm{p}_0}{1-\mathrm{p}_0} \, \Delta k$ singlets. From the above transformation one is able to derive an interconversion relation between energy and entanglement, \begin{equation} \label{int_rel_ent_ene} \Delta W_W = - \frac{\Delta E}{\log \frac{\mathrm{p}_0}{1-\mathrm{p}_0}} \, \Delta W_E, \end{equation} where $\Delta W_W = M_H \left( \omega_W(k') \right) - M_H \left( \omega_W(k) \right)$ is the amount of energy exchanged, $\Delta W_E = E_{\mathcal{F}_{\text{css}}} \left( \omega_E(h') \right) - E_{\mathcal{F}_{\text{css}}} \left( \omega_E(h) \right)$ is the amount of entanglement exchanged, and $\Delta E = E_1 - E_0$ is the energy gap of the Hamiltonian $H_{AB}$. Additionally, we find that the change in monotone $E_{\mathcal{S}_{\mathrm{p}_0}}$ between the initial and final global state of the bank is negligible (for $n \rightarrow \infty$), in accord with property~\ref{item:X1}.
\section{Conclusions} \label{end} {\bf From multiple constraints to a resource theory.} With the present work we set the mathematical ground for the development of resource theories with multiple resources able to describe new physical scenarios. Our construction of multi-resource theories is based on the definition of their class of allowed operations. First, we pinpoint the resources that compose the theory, and we introduce the corresponding single-resource theories. Then, we define the set of allowed operations for the multi-resource theory as the one composed by the maps in the intersection of the different classes of allowed operations of each single-resource theory, Eq.~\eqref{all_ops_multi}. This construction leaves the theory with multiple invariant sets, some of which are the sets of free states of the relevant single-resource theories. It is worth remarking again that, in multi-constraint theories, there is a difference between the set of free states and the invariant sets (in contrast with the case of single-resource theories), and a multi-resource theory can have multiple invariant sets and no free states, Fig.~\ref{fig:invariant_sets_structure}. \par {\bf Reversibility.} Together with the introduction of a general framework for multi-resource theories, we have studied the properties of these reversible theories. In particular, to analyse reversibility when multiple resources are present, we have first introduced the asymptotic equivalence property, see Def.~\ref{def:asympt_equivalence_multi}. This property implies that a unique monotone can be used to quantify each resource. Furthermore, in the case of single-resource theories, it coincides with the usual notion of reversible rates of conversion. We know of multi-resource theories that satisfy this property, see the two examples provided in Sec.~\ref{examples}. However, it would be interesting to study which of the other, already existing, multi-resource theories satisfy the property of Def.~\ref{def:asympt_equivalence_multi}. Ultimately, one would hope to find some general condition according to which a multi-resource theory is reversible, similarly to what has been found in Ref.~\cite{brandao_reversible_2015}. \par {\bf The role of batteries.} A crucial feature of our framework is the presence of batteries, used to store and quantify the resources exchanged during a state transformation over the main system. While batteries can be defined for single-resource theories as well, they do not seem to play the same fundamental role in that case, since one can quantify the amount of resource contained in a system using the conversion rate, see Def.~\ref{def:rate_conversion} in appendix~\ref{rev_theory_sing}. However, the conversion rate is linked to a change in the number of copies, for example $\rho^{\otimes n} \to \sigma^{\otimes k}$, where it is implicitly assumed that the remaining
$|n-k|$ copies of the system are in a free state. Since the framework allows us to model theories with no free states, we cannot change the number of systems with the allowed operations, and therefore we need to use batteries to quantify the amount of resources. We have seen in this paper what are the main properties for these batteries, primarily property~\ref{item:M1}, which requires each battery to store one and only one of the resource. It would be interesting to study these systems more carefully, possibly linking them to the kind of batteries used for fluctuation theorems~\cite{alhambra_entanglement_2017,alhambra_fluctuating_2016-1,renes_relative_2016, morris_quantum_2018}, which are described by states in a big superposition, so as to always remain uncorrelated from the main system during a state transformation~\cite{van_dam_universal_2003, harrow_entanglement_2010}. A different line of research in this direction could involve the study of correlated and entangled batteries, already explored in the setting of the single-resource theory of thermodynamics~\cite{alicki_entanglement_2013,hovhannisyan_entanglement_2013}. \par {\bf Interconversion and further examples.} We have studied the interconversion of resources and we have introduced a first law for multi-resource theories, Eq.~\eqref{eq:first_law}, valid when the theories are reversible and the invariant sets are disjoint. We have provided two examples of theories with a first law, one related to thermodynamics, and the other concerning a theory of local control under energy restriction. In this latter example, we have studied an extremely simplified case, due to the fact that reversibility has not been proved in general for this theory. Due to the high importance of both non-locality and thermodynamics in the field of quantum technology and many-body physics, we believe that a complete analysis of this multi-resource theory would be useful. Furthermore, it would be interesting to know which other multi-resource theories allow for an interconversion relation, and whether it is possible to define interconversion for theories with a different structure of invariant sets, by for instance relaxing the assumptions made on the bank. For example, one could consider bank states from which both resources could in principle be extracted, and forbid such extraction by further constraining the class of allowed operations. \par {\bf Multiple ways to build a multiple-resource theory.} In general, there could be different ways to intersect constraints in order to obtain the same final resource theory, and some of these constructions are a better fit for the analysis presented here than others. For example, the resource theory of thermodynamics equipped with Thermal Operations can be built as the intersection of either (1) the resource theories of information and energy, as we have done in Sec.~\ref{bank_monotone}, or (2) the resource theories of athermality and coherence~\cite{aberg_catalytic_2014, lostaglio_description_2015, kwon_clock--work_2018}. However, the most convenient setting for the study of this latter construction is the single-copy regime, since in the many-copy scenario coherence is lost, as this quantity scales sub-linearly in the number of copies of the system considered. \par {\bf Beyond the asymptotic limit.} The concrete results presented here for reversibility and interconversion of resources are only valid in the asymptotic limit where many independent and identically distributed copies of a system are considered. However, the general framework we introduced to describe resource theories with multiple resources can also be applied to scenarios with a single system. Understanding how resources can be exchanged in the single-copy regime, and studying the corrections to the first law in such a regime are worthwhile questions to pursue. We believe that extending the notion of batteries to the single-shot regime should be the first step toward the definition of a complete framework for multi-resource theories. However, we anticipate that this will be a highly non-trivial task, since in the single-copy case a resource is not generally quantified by a single measure, which complicates the definition of batteries, currently given through property~\ref{item:M1}. These difficulties are exemplified by the single-shot version of the resource theory of thermodynamics, where the $\alpha$-R{\'e}nyi divergences from the thermal state are all valid resource measures. A possible way forward in this setting might be the definition of an arbitrary notion of resource, for instance in terms of the number of resourceful states contained in the battery. Otherwise, the fluctuation relations for arbitrary resources~\cite{alhambra_entanglement_2017} and their connection to majorization could be useful conecpts for quantifying resources in the single-shot regime.
\tocless{\section*}{Acknowledgement} We thank the anonymous TQC referees for feedbacks, and Tobias Fritz for detailed comments on a previous version of this manuscript, CS is supported by the EPSRC (grant number {EP/L015242/1}). LdR acknowledges support from the Swiss National Science Foundation through SNSF project No.~{200020$\_$165843} and through the National Centre of Competence in Research \emph{Quantum Science and Technology} (QSIT), and from the FQXi grant \emph{Physics of the observer}. CMS is supported by the Engineering and Physical Sciences Research Council (EPSRC) through the doctoral training grant {1652538}, and by Oxford-Google DeepMind graduate scholarship. CMS would like to thank the Department of Physics and Astronomy at UCL for their hospitality. PhF acknowledges support from the Swiss National Science Foundation (SNSF) through the Early PostDoc.Mobility Fellowship No. {P2EZP2$\_$165239} hosted by the Institute for Quantum Information and Matter (IQIM) at Caltech, from the IQIM which is a National Science Foundation (NSF) Physics Frontiers Center (NSF Grant {PHY-1733907}), from the Department of Energy Award {DE-SC0018407}, as well as from the Deutsche Forschungsgemeinschaft (DFG) Research Unit {FOR 2724}. JO is supported by the Royal Society, and by an EPSRC Established Career Fellowship. We thank the COST Network {MP1209} in Quantum Thermodynamics.
\tocless{\subsection*}{Author contributions} All authors contributed significantly to the ideas behind this work and to the development of the general framework (Sec.~\ref{multi_res_framework}). CS, LdR and JO developed the results on batteries, bank states and the first law (Secs.~\ref{rev_theory_mult}, \ref{interconv}, \ref{examples}). CS wrote the proofs and initial draft.
\appendix {\Large{\sc{Appendix}}}
\section{Reversibility and asymptotic equivalence for single-resource theories} \label{rev_theory_sing} In this section we show that, for a single-resource theory, the asymptotic equivalence property of Def.~\ref{def:asympt_equivalence_multi} is equivalent to the notion of reversibility given in terms of rates of conversion. Let us first introduce the concept of rate of conversion for a single-resource theory, see Ref.~\cite{horodecki_quantumness_2012}. The definition of rate we use coincides with the one used in the literature, with the difference that we are making explicit use of the partial trace and of the addition of free states. In fact, we prefer not to include these operations within the set $\mathcal{C}$, as we want the allowed operations to preserve the number of copies of the system they act over (with the exception of sub-linear ancillae).
\begin{definition} \label{def:rate_conversion} Consider a single-resource theory with allowed operations $\mathcal{C}$ and free states $\mathcal{F}$, and two states $\rho, \sigma \in \mathcal{S} \left( \mathcal{H} \right)$. We define the \emph{rate of conversion} from $\rho$ to $\sigma$ as \begin{align}
R(\rho \rightarrow \sigma) = \sup \bigg\{ \frac{k_n}{n} \ | \ &\text{{\rm either}} \ \lim_{n \rightarrow \infty} \left( \min_{\tilde{\mathcal{E}}_n}
\left\| \Tr{n - k_n}{ \tilde{\mathcal{E}}_n (\rho^{\otimes n}) } - \sigma^{\otimes k_n} \right\|_1 \right) = 0 \nonumber \\ &\text{{\rm or}} \ \lim_{n \rightarrow \infty} \left( \min_{\tilde{\mathcal{E}}_{k_n}}
\left\| \tilde{\mathcal{E}}_{k_n} (\rho^{\otimes n} \otimes \gamma_{k_n - n}) - \sigma^{\otimes k_n} \right\|_1 \right) = 0 \ , \nonumber \\ &\text{{\rm where}} \ \gamma_{k_n - n} \in \mathcal{F}^{(k_n - n)} \bigg\}. \end{align} where the maps $\tilde{\mathcal{E}}_n$ have been defined in Eq.~\eqref{allowed_ancilla}, and they are of the form $\tilde{\mathcal{E}}_n (\cdot) = \Tr{A}{\mathcal{E}_n ( \cdot \otimes \eta^{(A)}_n ) }$, with $\mathcal{E}_n \in \mathcal{C}^{(n+o(n))}$ and $\eta^{(A)}_n \in \SHn{o(n)}$. \end{definition}
\par Now that the notion of rate is defined, we introduce the concept of \emph{reversible} single-resource theory,
\begin{definition} \label{def:reversibility} A single-resource theory with allowed operations $\mathcal{C}$ and free states $\mathcal{F}$ is \emph{reversible} if, given any non-free states $\rho, \sigma \in \mathcal{S} \left( \mathcal{H} \right)$, the rate of conversion from $\rho$ to $\sigma$ is such that $R(\rho \rightarrow \sigma) \in (0 , \infty)$, and $R(\rho \rightarrow \sigma) R(\sigma \rightarrow \rho) = 1$. \end{definition}
The above notion of reversibility is based on the rates of conversion between two resourceful states. However, it is not clear how to extend Def.~\ref{def:rate_conversion} to the case of multiple resources, since the set of free states might be empty for multi-resource theories. For this reason, we have introduced the property of asymptotic equivalence in Sec.~\ref{asympt_eqiv_prop}. This property also apply to the single-resource theory case, when $m = 1$. \par Now we want to show that Defs.~\ref{def:asympt_equivalence_multi} and \ref{def:reversibility}, for a single-resource theory, coincide. First, let us introduce a function $f : \SHn{n} \rightarrow \mathbb{R}$ (more formally, a family of functions) with the following properties, \begin{description} \item[SM1\label{item:SM1}] For each $n \in \mathbb{N}$, the function $f$ is monotonic under the set of allowed operations $\mathcal{C}^{(n)}$, that is \begin{equation} f \left( \mathcal{E}_n \left( \rho_n \right) \right) \leq f \left( \rho_n \right) , \qquad \forall \, \rho_n \in \SHn{n} \ , \ \forall \, \mathcal{E}_n \in \mathcal{C}^{(n)}. \end{equation} \item[SM2\label{item:SM2}] For each $n \in \mathbb{N}$, the function $f$ is equal to $0$ for all states $\gamma_n \in \mathcal{F}^{(n)}$, that is \begin{equation} f \left( \gamma_n \right) = 0 , \qquad \forall \, \gamma_n \in \SHn{n}. \end{equation} \item[SM3\label{item:SM3}] The function $f$ is asymptotic continuous. \item[SM4\label{item:SM4}] The function $f$ is monotonic under partial tracing, that is \begin{equation} f \left( \Tr{k}{\rho_n} \right) \leq f \left( \rho_n \right) , \qquad \forall \, n, k \in \mathbb{N} \ , \ k < n \ , \ \forall \, \rho_n \in \SHn{n}. \end{equation} \item[SM5\label{item:SM5}] For each $n,k \in \mathbb{N}$, the function $f$ is sub-additive, that is \begin{equation} f \left( \rho_n \otimes \rho_k \right) \leq f \left( \rho_n \right) + f \left( \rho_k \right) , \qquad \forall \, \rho_n \in \SHn{n} \ , \ \forall \, \rho_k \in \SHn{k}. \end{equation} \item[SM6\label{item:SM6}] For any given sequence of states $\left\{ \rho_n \in \SHn{n} \right\}$, the function $f$ scales sub-extensively, that is, $f \left( \rho_n \right) = O(n)$. \end{description} Notice that property~\ref{item:SM6} implies that the function $f$ is regularisable. Furthermore, the value of $f$ is preserved if we add free states, that is, \begin{equation} \label{free_zero} f(\rho_n \otimes \gamma_k) = f(\rho_n) , \qquad \forall \, \rho_n \in \SHn{n} \ , \ \forall \, \gamma_k \in \mathcal{F}^{(k)}, \end{equation} which follows from properties~\ref{item:SM2}, \ref{item:SM4}, and \ref{item:SM5}. \par The first lemma we introduce show that the rate of conversion of a reversible single-resource theory is linked to the function $f$ satisfying the above properties. Notice that this proof is analogous to the one of Ref.~\cite{horodecki_entanglement_2001}, with the difference that we are allowing for the presence of a sub-linear ancilla in the definition of rate, following the notion of ``seed regularisation'' introduced in Ref.~\cite[Sec.~9]{fritz_resource_2015}.
\begin{lem} \label{lem:reversible_rate} Consider a reversible resource theory with allowed operations $\mathcal{C}$ and free states $\mathcal{F}$, and the function $f$ satisfying~\ref{item:SM1} -- \ref{item:SM6}. Then, for all non-free states $\rho, \sigma \in \mathcal{S} \left( \mathcal{H} \right)$, we have that \begin{equation} R(\rho \rightarrow \sigma) = \frac{f^{\infty}(\rho)}{f^{\infty}(\sigma)} \end{equation} \end{lem}
\begin{proof} Let consider $\rho$ and $\sigma$ such that $R(\rho \rightarrow \sigma) \leq 1$ (the proof of the other case is equivalent). Then, there exists a sequence of operations $\left\{ \tilde{\mathcal{E}}_n \right\}$ of the form given in Eq.~\eqref{allowed_ancilla} such that \begin{equation} \lim_{n \rightarrow \infty}
\left\| \Tr{n - k_n}{ \tilde{\mathcal{E}}_n (\rho^{\otimes n}) } - \sigma^{\otimes k_n}
\right\|_1 = 0 \end{equation} where $\lim_{n \rightarrow \infty} \frac{k_n}{n} = R( \rho \rightarrow \sigma )$. If we use the asymptotic continuity of the function $f$, property~\ref{item:SM3}, we obtain \begin{equation} f \left( \Tr{n - k_n}{ \tilde{\mathcal{E}}_n (\rho^{\otimes n}) } \right) = f \left( \sigma^{\otimes k_n} \right) + o ( k_n ). \end{equation} Let us now consider the lhs of the above equation. Using the properties of the monotone $f$, together with the definition of $\tilde{\mathcal{E}}_n$ in terms of sub-linear ancillae and allowed operations, we can prove the following chain of inequalities, \begin{align} f \left( \Tr{n - k_n}{ \tilde{\mathcal{E}}_n (\rho^{\otimes n}) } \right) &\leq f \left( \tilde{\mathcal{E}}_n (\rho^{\otimes n}) \right) = f \left( \Tr{A}{\mathcal{E}_n ( \rho^{\otimes n} \otimes \eta^{(A)}_n ) } \right) \leq f \left( \mathcal{E}_n ( \rho^{\otimes n} \otimes \eta^{(A)}_n ) \right) \nonumber \\ &\leq f \left( \rho^{\otimes n} \otimes \eta^{(A)}_n \right) \leq f \left( \rho^{\otimes n} \right) + f \left( \eta^{(A)}_n \right) \leq f \left( \rho^{\otimes n} \right) + o(n) \end{align} where the first and second inequalities follow from property~\ref{item:SM4}, the equality follows from the definition of $\tilde{\mathcal{E}}_n$, see Eq.~\eqref{allowed_ancilla}, the third inequality follows from monotonicity under allowed operations, property~\ref{item:SM1}, the forth inequality from sub-additivity, property~\ref{item:SM5}, and the last one from the fact that the ancillary system is sub-linear in $n$ together with property~\ref{item:SM6}. Thus, combining the last two equations, we get \begin{equation} f \left( \rho^{\otimes n} \right) \geq f \left( \sigma^{\otimes k_n} \right) + o ( n ). \end{equation} We can now divide the left and right hand side of the above equation by $n$, obtaining \begin{equation} \frac{1}{n} \, f \left( \rho^{\otimes n} \right) \geq \frac{k_n}{n} \, \frac{1}{k_n} \, f \left( \sigma^{\otimes k_n} \right) + o ( 1 ). \end{equation} By taking the limit of $n \rightarrow \infty$, and using the fact that $f$ is regularisable (which follows from property~\ref{item:SM6}) together with the definition of rate, we get \begin{equation} f^{\infty} \left( \rho \right) \geq R(\rho \rightarrow \sigma) \, f^{\infty} \left( \sigma \right). \end{equation} We can also consider the reverse transformation, mapping $n$ copies of the state $\sigma$ into $k'_n$ copies of $\rho$. Using the same steps used above, together with the fact that the monotone $f$ is equal to zero over free states, property~\ref{item:SM2}, we can show that \begin{equation} f^{\infty} \left( \sigma \right) \geq R(\sigma \rightarrow \rho) \, f^{\infty} \left( \rho \right). \end{equation} If we now use the reversibility property, which implies $R(\sigma \rightarrow \rho) = \frac{1}{R(\rho \rightarrow \sigma)}$, we find that \begin{equation} \frac{f^{\infty} \left( \rho \right)}{f^{\infty} \left( \sigma \right)} \geq R(\rho \rightarrow \sigma) \geq \frac{f^{\infty} \left( \rho \right)}{f^{\infty} \left( \sigma \right)} \end{equation} which proves the lemma. \end{proof}
Furthermore, we introduce a second small lemma, that can be found in Ref.~\cite[Prop.~13]{donald_uniqueness_2002},
\begin{lem} \label{lem:add_regularised_mon} Given a regularisable function $f : \SHn{n} \rightarrow \mathbb{R}$, the regularised version is extensive, \begin{equation} f^{\infty}(\rho^{\otimes k}) = k \, f^{\infty}(\rho) \ , \ \forall \, \rho \in \mathcal{S} \left( \mathcal{H} \right) \ , \ \forall \, k \in \mathbb{N}. \end{equation} \end{lem}
\begin{proof} Consider a function $h : \mathbb{R} \rightarrow \mathbb{R}$, such that $\lim_{n \rightarrow \infty} h(n) = L < \infty$. This is equivalent to say that \begin{equation} \label{limit_def}
\forall \, \epsilon > 0, \exists \, c \in \mathbb{R} \ : \ | h(n) - L | < \epsilon, \ \forall \, n > c. \end{equation} Let us now consider an invertible function $g : \mathbb{R} \rightarrow \mathbb{R}$, and consider $m \in \mathbb{R}$ such that $n = g(m)$. Then, we can rewrite Eq.~\eqref{limit_def} as \begin{equation}
\forall \, \epsilon > 0, \exists \, c \in \mathbb{R} \ : \ | h(g(m)) - L | < \epsilon, \ \forall \, g(m) > c, \end{equation} and by defining $\tilde{c} = g^{-1}(c)$, we get \begin{equation}
\forall \, \epsilon > 0, \exists \, \tilde{c} \in \mathbb{R} \ : \ | h(g(m)) - L | < \epsilon, \ \forall \, m > \tilde{c}. \end{equation} Therefore, we have $\lim_{m \rightarrow \infty} h(g(m)) = L $. \par If we choose $h(n) = \frac{1}{n} f(\rho^{\otimes n})$, whose limit is $L = f^{\infty} (\rho)$, and we use the reversible function $g(m) = k \cdot m$ where $k \in \mathbb{N}$ is fixed, we get \begin{equation} f^{\infty} (\rho) = \lim_{m \rightarrow \infty} \frac{1}{k \cdot m} f(\rho^{\otimes k \cdot m}) = \frac{1}{k} \lim_{m \rightarrow \infty} \frac{1}{m} f( ( \rho^{\otimes k})^{\otimes m} ) = \frac{1}{k} f^{\infty} (\rho^{\otimes k}), \end{equation} which proves the lemma. \end{proof}
We can now show that a single-resource theory which is reversible also satisfies the asymptotic equivalence property, and vice versa.
\begin{thm} \label{thm:reversible_asympt_equiv} Consider the resource theory with allowed operations $\mathcal{C}$ and free states $\mathcal{F}$. If the theory is reversible, then it satisfies the asymptotic equivalence property with respect to a function $f$ satisfying the properties~\ref{item:SM1} -- \ref{item:SM6}, and viceversa. \end{thm}
\begin{proof} {\bf (a)} Let us first assume that the theory is reversible. Then, if we consider two non-free states $\rho, \sigma \in \mathcal{S} \left( \mathcal{H} \right)$ such that $f^{\infty}(\rho) = f^{\infty}(\sigma)$, and we use Lem.~\ref{lem:reversible_rate}, we find that the rate of conversion is $R(\rho \rightarrow \sigma) = 1$. Then, there exists a sequence of operations $\left\{ \tilde{\mathcal{E}}_n \right\}$ that approach this limit in one of two ways. In one case, we have \begin{equation}
\left\| \Tr{n - k_n}{\tilde{\mathcal{E}}_n (\rho^{\otimes n})} - \sigma^{\otimes k_n} \right\|_1 \rightarrow 0. \end{equation} Notice that, since we have $\frac{k_n}{n} \rightarrow 1$, it follows that $n - k_n = o(n)$. Then, the above equation coincides with the second part of Def.~\ref{def:asympt_equivalence_multi}, where we are mapping $\rho^{\otimes k_n}$ into $\sigma^{\otimes k_n}$, and the sub-linear ancilla is $\eta'^{(A)}_n = \eta^{(A)}_n \otimes \rho^{n - k_n}$, where $\eta^{(A)}_n$ is completely arbitrary, and come from the definition of $\tilde{\mathcal{E}}_n$. Alternatively, we can have that the sequence of maps is such that \begin{equation}
\left\| \tilde{\mathcal{E}}_{k_n} (\rho^{\otimes n} \otimes \gamma_{k_n - n} ) - \sigma^{\otimes k_n}
\right\|_1 \rightarrow 0. \end{equation} We now use the monotonicity of the trace distance under discarding subsystems to obtain \begin{equation}
\left\| \Tr{k_n - n}{\tilde{\mathcal{E}}_{k_n} (\rho^{\otimes n} \otimes \gamma_{k_n - n} )} - \sigma^{\otimes n} \right\|_1 \rightarrow 0. \end{equation} Again, the above equation coincides with the second part of Def.~\ref{def:asympt_equivalence_multi}, where we are mapping $\rho^{\otimes n}$ into $\sigma^{\otimes n}$, and the sub-linear ancilla is $\eta'^{(A)}_n = \eta^{(A)}_n \otimes \gamma_{k_n - n}$. This proves the validity of one direction of the asymptotic equivalence property. To prove the other direction (existence of a sequence of maps implies same value of the monotone on the two states), we can use the fact that, if there exists a sequence of maps $\left\{ \tilde{\mathcal{E}}_n \right\}$ sending $\rho^{\otimes n}$ into $\sigma^{\otimes n}$, then the rate of conversion is $R(\rho \rightarrow \sigma) = 1$. Then, with the help of Lem.~\ref{lem:reversible_rate}, which is valid for reversible theories, we obtain that $f^{\infty}(\rho) = f^{\infty}(\sigma)$. This proves the other direction of the asymptotic equivalence property. \par {\bf (b)} Let now assume that the theory satisfies the asymptotic equivalence property. Consider any two non-free states $\rho, \sigma \in \mathcal{S} \left( \mathcal{H} \right)$, and suppose that $f^{\infty}(\rho) \leq f^{\infty}(\sigma)$ (in the other case, the proof would follow analogously to the one we are presenting). Take $n,k \in \mathbb{N}$ such that $n \, f^{\infty}(\rho) = k \, f^{\infty}(\sigma)$, and let us use the extensivity of $f^{\infty}$, Lem.~\ref{lem:add_regularised_mon}. Then, we have $f^{\infty}(\rho^{\otimes n}) = f^{\infty} (\sigma^{\otimes k})$. Using the property of the function $f$ shown in Eq.~\eqref{free_zero}, we find that \begin{equation} f^{\infty}(\rho^{\otimes n}) = f^{\infty}(\sigma^{\otimes k} \otimes \gamma_{n - k}), \end{equation} where we add the free state $\gamma_{n - k} \in \mathcal{F}^{(n - k)}$ to the right hand side since $n \geq k$. Then, we can use the asymptotic equivalence property, which implies the existence of a sequence of maps $\left\{ \tilde{\mathcal{E}}_{m \cdot n} \right\}_m$, see Eq.~\ref{allowed_ancilla}, such that \begin{equation} \lim_{m \rightarrow \infty}
\left\| \mathcal{E}_{m \cdot n} (\rho^{\otimes m \cdot n}) -
\sigma^{\otimes m \cdot k} \otimes \gamma_{n - k}^{\otimes m} \right\|_1 = 0. \end{equation} If we use the monotonicity of the trace distance under partial tracing, we find that \begin{equation} \lim_{m \rightarrow \infty}
\left\| \Tr{m \cdot (n-k)}{\mathcal{E}_{m \cdot n} (\rho^{\otimes m \cdot n})} -
\sigma^{\otimes m \cdot k} \right\|_1 = 0. \end{equation} The existence of this sequence of maps implies that the rate of conversion $R(\rho \rightarrow \sigma) \geq \frac{k}{n}$. At the same time, we can use asymptotic equivalence to find a sequence of maps $\left\{ \tilde{\mathcal{E}}'_{m \cdot n} \right\}_m$ performing the reverse process. Using a similar argument to the one presented above, we find that $R(\sigma \rightarrow \rho) \geq \frac{n}{k}$. As a result, we find that the product of the forward and reverse rates of conversion is $R(\rho \rightarrow \sigma) R(\sigma \rightarrow \rho) \geq 1$. However, this product cannot be higher than one, as otherwise we would be able to perform a cyclic transformation turning free states intro resourceful one, which is forbidden under allowed operations, see also Ref.~\cite{popescu_thermodynamics_1997}. Therefore, we find that $R(\rho \rightarrow \sigma) R(\sigma \rightarrow \rho) = 1$, which closes the proof. \end{proof}
\section{Convex boundary and bank states} \label{convex_bound} In the following, we consider the case of a two-resource theory $\text{R}_{\text{multi}}$ defined on the Hilbert space $\mathcal{H}$. The set of allowed operations is $\mathcal{C}_{\text{multi}} = \mathcal{C}_1 \cap \mathcal{C}_2$, where each $\mathcal{C}_i$ is a subset of the set of all CPTP maps that leave the set of states $\mathcal{F}_i$ invariant, $i = 1, 2$. We ask the resource theory $\text{R}_{\text{multi}}$ to satisfy the asymptotic equivalence property with respect to the monotones $E_{\mathcal{F}_1}$ and $E_{\mathcal{F}_2}$. Furthermore, we assume that the two invariant sets satisfy the properties~\ref{item:F1}--\ref{item:F5}. Thus, it follows from Thm.~\ref{thm:reversible_multi} and Prop.~\ref{thm:properties_rel_ent} that the two monotones $E^{\infty}_{\mathcal{F}_1}$ and $E^{\infty}_{\mathcal{F}_2}$ uniquely quantify the resources in this theory. As a result, we can represent the state-space of $\text{R}_{\text{multi}}$ in a two-dimensional diagram, as shown in Fig.~\ref{fig:resource_diagram}. \par We choose the two invariant sets of the theory to be disjoints, i.e., $\mathcal{F}_1 \cap \mathcal{F}_2 = \emptyset$, and we focus on the set of bank states $\mathcal{F}_{\text{bank}} \subset \mathcal{S} \left( \mathcal{H} \right)$. Since in this section we are not making any assumption on the additivity (or extensivity) of the monotones $E_{\mathcal{F}_i}$'s, we have that the set of bank states is here defined as \begin{align} \label{eq:bank_reg}
\mathcal{F}_{\text{bank}} = \big\{ \rho \in \mathcal{S} \left( \mathcal{H} \right) \ | \ \forall \, \sigma \in \mathcal{S} \left( \mathcal{H} \right) , \ &E^{\infty}_{\mathcal{F}_1}(\sigma) > E^{\infty}_{\mathcal{F}_1}(\rho) \ \text{or} \nonumber \\ &E^{\infty}_{\mathcal{F}_2}(\sigma) > E^{\infty}_{\mathcal{F}_2}(\rho) \ \text{or} \nonumber \\ &E^{\infty}_{\mathcal{F}_1}(\sigma) = E^{\infty}_{\mathcal{F}_1}(\rho) \, \text{and} \, E^{\infty}_{\mathcal{F}_2}(\sigma) = E^{\infty}_{\mathcal{F}_2}(\rho) \, \big\}. \end{align} Notice that this set coincides with the one of Eq.~\eqref{set_f3} when property~\ref{item:F5b} is satisfied, and therefore the results we obtain in this appendix apply to Sec.~\ref{interconv} as well. It is easy to show that $E^{\infty}_{\mathcal{F}_2}(\rho) > E^{\infty}_{\mathcal{F}_2}(\mathcal{F}_2) = 0 \ \forall \, \rho \in \mathcal{F}_1$, and similarly $E^{\infty}_{\mathcal{F}_1}(\rho) > E^{\infty}_{\mathcal{F}_1}(\mathcal{F}_1) = 0 \ \forall \, \rho \in \mathcal{F}_2$. Moreover, inside both invariant sets $\mathcal{F}_1$ and $\mathcal{F}_2$ we can find a subset of states with minimum value of the monotones $E^{\infty}_{\mathcal{F}_2}$ and $E^{\infty}_{\mathcal{F}_1}$, respectively. We define these sets as \begin{align} \label{f_1_min}
\mathcal{F}_{1, \min} &= \left\{ \sigma \in \mathcal{F}_1 \, | \, E^{\infty}_{\mathcal{F}_2}(\sigma) = \min_{\rho \in \mathcal{F}_1} E^{\infty}_{\mathcal{F}_2}(\rho) \right\} \subseteq \mathcal{F}_1, \\ \label{f_2_min}
\mathcal{F}_{2, \min} &= \left\{ \sigma \in \mathcal{F}_2 \, | \, E^{\infty}_{\mathcal{F}_1}(\sigma) = \min_{\rho \in \mathcal{F}_2} E^{\infty}_{\mathcal{F}_1}(\rho) \right\} \subseteq \mathcal{F}_2. \end{align} Given these two subsets, we can then define the following real intervals, \begin{align} \label{interval_1} I_1 &= \left[ E^{\infty}_{\mathcal{F}_1}(\mathcal{F}_1) = 0 \, ; \, E^{\infty}_{\mathcal{F}_1}(\mathcal{F}_{2, \min}) \right], \\ \label{interval_2} I_2 &= \left[ E^{\infty}_{\mathcal{F}_2}(\mathcal{F}_2) = 0 \, ; \, E^{\infty}_{\mathcal{F}_2}(\mathcal{F}_{1, \min}) \right]. \end{align} In what follows, we make use of the following two properties of the monotones $E^{\infty}_{\mathcal{F}_i}$'s, \begin{itemize} \item \emph{Asymptotic continuity}, which follows from the assumptions~\ref{item:F1}--\ref{item:F5} over the sets $\mathcal{F}_i$'s, as shown in Refs.~\cite{christandl_structure_2006,brandao_reversible_2015}. \item \emph{Convexity}, which follows from the assumptions~\ref{item:F2} and \ref{item:F4} over the sets $\mathcal{F}_i$'s, as shown in Ref.~\cite{donald_uniqueness_2002}, Prop.~13. \end{itemize} We can now state the following lemma, concerning the value of the monotones $E^{\infty}_{\mathcal{F}_i}$'s for bank states. \par
\begin{lem} \label{lem:closed_int} Consider the multi-resource theory $\text{R}_{\text{multi}}$ with allowed operations $\mathcal{C}_{\text{multi}}$, and invariant sets $\mathcal{F}_1$ and $\mathcal{F}_2$ which satisfy properties~\ref{item:F1}--\ref{item:F5}, and $\mathcal{F}_1 \cap \mathcal{F}_2 = \emptyset$. If the theory satisfies the asymptotic equivalence property with respect to the monotones $E_{\mathcal{F}_1}$ and $E_{\mathcal{F}_2}$, then for all bank states $\rho \in \mathcal{F}_{\text{bank}}$ we have that $E^{\infty}_{\mathcal{F}_1}(\rho) \in I_1$ and $E^{\infty}_{\mathcal{F}_2}(\rho) \in I_2$. \end{lem}
\begin{proof} Suppose, for example, that there exists a bank state $\rho \in \mathcal{F}_{\text{bank}}$ such that $E^{\infty}_{\mathcal{F}_1}(\rho) \notin I_1$, that is, $\exists \, \sigma \in \mathcal{F}_{2, \min}$ such that $E^{\infty}_{\mathcal{F}_1}(\sigma) < E^{\infty}_{\mathcal{F}_1}(\rho)$. By definition of $\mathcal{F}_2$ we also have that $E^{\infty}_{\mathcal{F}_2}(\sigma) \leq E^{\infty}_{\mathcal{F}_2}(\rho)$. These two inequalities, however, contradict the fact that $\rho$ is a bank state, see Eq.~\eqref{eq:bank_reg}, and conclude the proof. \end{proof} It is easy to show that for all $\bar{E}_{\mathcal{F}_1} \in I_1$ there exists (at least) one state $\rho \in \mathcal{S} \left( \mathcal{H} \right)$ such that $E^{\infty}_{\mathcal{F}_1}(\rho) = \bar{E}_{\mathcal{F}_1}$, and the same holds for $I_2$. The proof that $\forall \, \bar{E}_{\mathcal{F}_1} \in I_1, \ \exists \, \rho \in \mathcal{S} \left( \mathcal{H} \right) \, : \, E^{\infty}_{\mathcal{F}_1}(\rho) = \bar{E}_{\mathcal{F}_1}$ follows from two facts: ({\it i}) $\mathcal{S} \left( \mathcal{H} \right)$ is a compact and path-connected set, and therefore its image under the (asymptotic) continuous function $E^{\infty}_{\mathcal{F}_1}$ is a compact and path-connected set in $\mathbb{R}$, that is, a closed and bounded interval $I_{1,\mathcal{S} \left( \mathcal{H} \right)}$, and ({\it ii}) $I_1 \subseteq I_{1,\mathcal{S} \left( \mathcal{H} \right)}$. \par Let us now define, in the $E^{\infty}_{\mathcal{F}_1}$--$E^{\infty}_{\mathcal{F}_2}$ diagram, the curve of bank states, which lies on part of the boundary of the state-space, as per definition in Eq.~\eqref{eq:bank_reg}. The curve is defined as \begin{equation} \label{curve_bank}
\gamma_{\text{bank}} = \left\{ \left( E^{\infty}_{\mathcal{F}_1}(\rho) , E^{\infty}_{\mathcal{F}_2}(\rho) \right) \ | \ \rho \in \mathcal{F}_{\text{bank}} \right\}, \end{equation} where $\mathcal{F}_{\text{bank}}$ is the set of bank states of the theory. It is easy to see that this curve is completely contained within the subset of $\mathbb{R}^2$ given by $I_1 \times I_2$. Together with this curve, we can introduce the real-valued function $c_{\text{bank}} \, : \, I_1 \rightarrow I_2$, defined as \begin{equation} \label{bank_function} c_{\text{bank}}(E_{\mathcal{F}_1}) = \text{if} \, \left( \exists \, P \in \gamma_{\text{bank}} \ \text{such that} \ P[0] = E_{\mathcal{F}_1} \right) \ \text{return} \ P[1]. \end{equation} Essentially, this function checks the first element of the tuples in $\gamma_{\text{bank}}$, and returns the second element of the tuple whose first element is equal to $E_{\mathcal{F}_1}$. Since $I_1$ is a closed interval in $\mathbb{R}$, we have that for all $E_{\mathcal{F}_1} \in I_1$, the function $c_{\text{bank}}$ is well-defined. See Fig.~\ref{fig:bank_curve} for the representation of the above curve of bank states in the resource diagram of the theory.
\begin{figure}\label{fig:bank_curve}
\end{figure}
\par We will now prove the following two propositions, which assure that the monotone $f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}$ of Eq.~\eqref{f3_monotone} satisfies the property~\ref{item:B2}. This first proposition essentially tells us that the function $c_{\text{bank}}$ is monotonic decreasing.
\begin{prop} \label{monotone_curve} For all $P_A, P_B \in \gamma_{\text{bank}}$, where $P_A = \left( E_{\mathcal{F}_1}^{(A)} , E_{\mathcal{F}_2}^{(A)} \right)$ and $P_B = \left( E_{\mathcal{F}_1}^{(B)} , E_{\mathcal{F}_2}^{(B)} \right)$, we have that \begin{equation} E_{\mathcal{F}_1}^{(A)} < E_{\mathcal{F}_1}^{(B)} \Leftrightarrow E_{\mathcal{F}_2}^{(A)} > E_{\mathcal{F}_2}^{(B)}. \end{equation} \end{prop}
\begin{proof} We prove the propositions in a single direction, as the other follows in analogue manner. Suppose that $E_{\mathcal{F}_1}^{(A)} < E_{\mathcal{F}_1}^{(B)}$, and consider the states $\rho_A$, $\rho_B \in \mathcal{F}_{\text{bank}}$ such that $E^{\infty}_{\mathcal{F}_1}(\rho_A) = E_{\mathcal{F}_1}^{(A)}$, and $E^{\infty}_{\mathcal{F}_1}(\rho_B) = E_{\mathcal{F}_1}^{(B)}$. Since $\rho_B$ belongs to the set of bank states, we have that one of the following conditions, see Eq.~\eqref{set_f3}, has to be satisfied for all states $\sigma \in \mathcal{S} \left( \mathcal{H} \right)$, \begin{enumerate} \item $E^{\infty}_{\mathcal{F}_1}(\sigma) > E^{\infty}_{\mathcal{F}_1}(\rho_B)$. \item $E^{\infty}_{\mathcal{F}_2}(\sigma) > E^{\infty}_{\mathcal{F}_2}(\rho_B)$. \item $E^{\infty}_{\mathcal{F}_1}(\sigma) = E^{\infty}_{\mathcal{F}_1}(\rho_B)$ and $E^{\infty}_{\mathcal{F}_2}(\sigma) = E^{\infty}_{\mathcal{F}_2}(\rho_B)$. \end{enumerate} Let us then take $\sigma = \rho_A$. In this case, options 1 and 3 are not possible, since they contradict the hypothesis. Therefore, option 2 has to be valid, which implies that $E^{\infty}_{\mathcal{F}_2}(\rho_A) > E^{\infty}_{\mathcal{F}_2}(\rho_B)$. In a similar manner, if $E_{\mathcal{F}_1}^{(A)} = E_{\mathcal{F}_1}^{(B)}$, the only possible option for $\rho_B$ would have been $E^{\infty}_{\mathcal{F}_2}(\rho_A) = E^{\infty}_{\mathcal{F}_2}(\rho_B)$, which concludes the proof. \end{proof}
The second propositions tells us, instead, that the function $c_{\text{bank}}$ is convex.
\begin{prop} \label{convex_curve} For all $P_A, P_B \in \gamma_{\text{bank}}$, where $P_A = \left( E_{\mathcal{F}_1}^{(A)} , E_{\mathcal{F}_2}^{(A)} \right)$ and $P_B = \left( E_{\mathcal{F}_1}^{(B)} , E_{\mathcal{F}_2}^{(B)} \right)$, and for all $\lambda \in [0,1]$, there exists a $P_C \in \gamma_{\text{bank}}$, where $P_C = \left( E_{\mathcal{F}_1}^{(C)} , E_{\mathcal{F}_2}^{(C)} \right)$, such that \begin{align} E_{\mathcal{F}_1}^{(C)} &= \lambda \, E_{\mathcal{F}_1}^{(A)} + ( 1 - \lambda ) \, E_{\mathcal{F}_1}^{(B)}, \\ E_{\mathcal{F}_2}^{(C)} &\leq \lambda \, E_{\mathcal{F}_2}^{(A)} + ( 1 - \lambda ) \, E_{\mathcal{F}_2}^{(B)} \end{align} \end{prop}
\begin{proof} Let us consider, without losing in generality, that $E_{\mathcal{F}_1}^{(A)} < E_{\mathcal{F}_1}^{(B)}$, and take $\rho_C \in \mathcal{F}_{\text{bank}}$ such that $E^{\infty}_{\mathcal{F}_1}(\rho_C) = \lambda \, E_{\mathcal{F}_1}^{(A)} + ( 1 - \lambda ) \, E_{\mathcal{F}_1}^{(B)}$. This state always exists since $I_1$ is a closed interval (and therefore is path-connected). Let us now define $\rho_A$, $\rho_B \in \mathcal{F}_{\text{bank}}$ such that $E^{\infty}_{\mathcal{F}_1}(\rho_A) = E_{\mathcal{F}_1}^{(A)}$, and $E^{\infty}_{\mathcal{F}_1}(\rho_B) = E_{\mathcal{F}_1}^{(B)}$. By convexity of the regularised relative entropy distance $E^{\infty}_{\mathcal{F}_1}$, it follows that \begin{equation} E^{\infty}_{\mathcal{F}_1}(\rho_C) = \lambda \, E_{\mathcal{F}_1}^{(A)} + ( 1 - \lambda ) \, E_{\mathcal{F}_1}^{(B)} \geq E^{\infty}_{\mathcal{F}_1} \left( \lambda \, \rho_A + ( 1 - \lambda ) \, \rho_B \right). \end{equation} Then, it is easy to show that \begin{equation} E^{\infty}_{\mathcal{F}_2}(\rho_C) \leq E^{\infty}_{\mathcal{F}_2} \left( \lambda \, \rho_A + ( 1 - \lambda ) \, \rho_B \right) \leq \lambda \, E_{\mathcal{F}_2}^{(A)} + ( 1 - \lambda ) \, E_{\mathcal{F}_2}^{(B)}, \end{equation} where the first inequality follows from Prop.~\ref{monotone_curve}, and the second one from the convexity of $E^{\infty}_{\mathcal{F}_2}$. Since $\rho_C \in \mathcal{F}_{\text{bank}}$, the point $P_C = \left(E^{\infty}_{\mathcal{F}_1}(\rho_C), E^{\infty}_{\mathcal{F}_2}(\rho_C) \right)$ is a point on the curve $\gamma_{\text{bank}}$. \end{proof}
It is easy to see that the above propositions imply that $c_{\text{bank}}$ is (strictly) monotonic decreasing, and convex. Since this function is defined on the closed interval $I_1 \in \mathbb{R}$, we have that $c_{\text{bank}}$ is continuous (except, maybe, at its endpoints). Therefore, we can always define the monotone $f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}}$ of Eq.~\eqref{f3_monotone}, and it always satisfies condition~\ref{item:B2}. Finally, it is worth noticing that all the results apply if one (or both) the monotones are of the form of Eq.~\eqref{montone_average}, since they satisfy all the necessary properties, in particular they are linear in both the tensor product and the admixture of states.
\section{Energy-entanglement interconversion protocol} \label{protocol_example} In this section we provide a protocol, based on the compression theorems~\cite{schumacher_quantum_1995} known in quantum information theory, to perform interconversion of energy and entanglement using two batteries and a bank, see Sec.~\ref{setup_en_ent} for revising the set-up we use. In our protocol, we assume that the bank is initially described by $n \gg 1$ copies of a generic state $\rho \in \mathcal{S}_{\mathrm{p}_0}$, where $\mathrm{p}_0 > \frac{1}{2}$, see Eq.~\eqref{bank_subset_en_ent}, while the batteries $B_W$ and $B_E$ are initially in the states $\omega_W(k)$ and $\omega_E(h)$, respectively. \par Our first step consists in using the allowed operation $\mathcal{E}_{\eta} \in \mathcal{C}_{\text{multi}}$, see Eq.~\eqref{rev_ent_en}, with $\eta = \ket{\theta_1}\bra{\theta_1}$, to map the generic bank state $\rho$ into \begin{equation} \rho_{\text{in}} = \mathrm{p}_0 \ket{\Psi_\text{singlet}}\bra{\Psi_\text{singlet}} + \left( 1 - \mathrm{p}_0 \right) \ket{\Psi_\text{triplet}^{(1)}}\bra{\Psi_\text{triplet}^{(1)}}. \end{equation} Thus, the bank system is now described by $n$ copies of the state $\rho_{\text{in}}$. Due to the central limit theorem, we can well approximate the state of the bank with an ensemble of its typical states, and in the following we will focus on the strongly typical ensemble, \begin{equation} \Pi_{\text{st}} = \frac{1}{d_{\text{st}}} \sum_{i=1}^{d_{\text{st}}} \pi_i \left( \ket{\Psi_\text{singlet}}\bra{\Psi_\text{singlet}}^{\otimes n \, \mathrm{p}_0} \otimes \ket{\Psi_\text{triplet}^{(1)}}\bra{\Psi_\text{triplet}^{(1)}}^{\otimes n \, ( 1 - \mathrm{p}_0)} \right), \end{equation} where $d_{\text{st}} \approx 2^{n \mathrm{h}(\mathrm{p}_0)}$ is the number of states contained in the strongly typical set, the $\pi_i$'s are the elements of the symmetric group acting on $n$ copies of the two-qubit system, and $\mathrm{h}(\cdot)$ is the binary entropy. Then, we can use a unitary operation to re-order the states in $\Pi_{\text{st}}$ so as to obtain \begin{equation} \Pi_{\text{st}}' = \sigma_{\text{mm}}^{\otimes n \mathrm{h}(\mathrm{p}_0)} \otimes \ket{\Psi_\text{singlet}}\bra{\Psi_\text{singlet}}^{\otimes n \left( 1 - \mathrm{h}(\mathrm{p}_0) \right)}, \end{equation} where $\sigma_{\text{mm}}$ is the separable state introduced in Eq.~\eqref{max_mix_sin_trip}. It is easy to see that this transformation, while leaving the amount of entanglement in the bank constant, $E_{\mathcal{F}_{\text{css}}}(\rho_{\text{in}}^{\otimes n}) = E_{\mathcal{F}_{\text{css}}}(\Pi_{\text{st}}')$, might not preserve the average energy. For this reason, while transforming the bank we also transform the energy battery, mapping $\omega_W(k)$ into $\omega_W(k+\Delta k)$ to keep the energy fixed. \par We can now exchange some singlets with the entanglement battery. For example, we can perform a swap between the bank and the battery, moving in this way an integer number $r$ of singlets from the bank into the battery. This transformation maps the state of the bank into \begin{equation} \Pi_{\text{st}}'' = \sigma_{\text{mm}}^{\otimes n \mathrm{h}(\mathrm{p}_0) + r} \otimes \ket{\Psi_\text{singlet}}\bra{\Psi_\text{singlet}}^{\otimes n \left( 1 - \mathrm{h}(\mathrm{p}_0) \right) - r}, \end{equation} and transforms the state of the entanglement battery from $\omega_E(h)$ into $\omega_E(h + r)$. Furthermore, the transformation also modify the energy of the bank, so that we need to map the state of the energy battery from $\omega_W(k+\Delta k)$ to $\omega_W(k+\Delta k')$. It is then possible to map the state $\Pi_{\text{st}}''$ into \begin{equation} \Pi_{\text{st}}''' = \frac{1}{d'_{\text{st}}} \sum_{i=1}^{d'_{\text{st}}} \pi_i \left( \ket{\Psi_\text{singlet}}\bra{\Psi_\text{singlet}}^{\otimes n \, \mathrm{p}'_0} \otimes \ket{\Psi_\text{triplet}^{(1)}}\bra{\Psi_\text{triplet}^{(1)}}^{\otimes n \, ( 1 - \mathrm{p}'_0)} \right), \end{equation} where $\mathrm{p}'_0$ is chosen in order to satisfy the equality \begin{equation} n \mathrm{h}(\mathrm{p}_0) + r = n \mathrm{h}(\mathrm{p}'_0), \end{equation} and $d'_{\text{st}} = 2^{n \mathrm{h}(\mathrm{p}'_0)}$. The state $\Pi_{\text{st}}'''$ is the strongly typical ensemble associated with $n$ copies of the state \begin{equation} \rho_{\text{fin}} = \mathrm{p}'_0 \ket{\Psi_\text{singlet}}\bra{\Psi_\text{singlet}} + \left( 1 - \mathrm{p}'_0 \right) \ket{\Psi_\text{triplet}^{(1)}}\bra{\Psi_\text{triplet}^{(1)}}, \end{equation} where it is easy to show that the probability of occupation of the singlet is $\mathrm{p}'_0 \approx \mathrm{p}_0 - \frac{r}{n} \frac{1}{\log \frac{\mathrm{p}_0}{1-\mathrm{p}_0}}$ for $n \gg 1$. The transformation mapping $\Pi_{\text{st}}''$ into $\Pi_{\text{st}}'''$ preserves the entanglement of the bank, while changing its energy. Therefore, while acting on the bank we have to modify the state of the energy battery as well, from $\omega_W(k+\Delta k')$ to $\omega_W(k+\Delta k'')$. In this way, we have modified the bank system by mapping $n$ copies of $\rho_{\text{in}}$ into $n$ copies of $\rho_{\text{fin}}$, and we kept entanglement and energy fixed on the global system by modifying the states of the batteries. Notice that the protocol can be extended to the typical ensembles by using a sub-linear ancillary system, and by considering corrections to the exchanged energy and entanglement of order $O(\sqrt{n})$. \par During the protocol, the bank has exchanged $r$ singlets with the battery $B_E$, so that the gain in entanglement for this battery is \begin{equation} \Delta W_E = E_{\mathcal{F}_{\text{css}}} \left( \omega_E(h + r) \right) - E_{\mathcal{F}_{\text{css}}} \left( \omega_E(h) \right) = r. \end{equation} In order to compute the amount of energy exchanged between the bank and the battery $B_W$, we consider the difference in average energy between $\rho_{\text{in}}^{\otimes n}$ and $\rho_{\text{fin}}^{\otimes n}$. In this way, we find that the amount of energy exchanged is \begin{equation} \Delta W_W = M_H \left( \omega_W(k+\Delta k'') \right) - M_H \left( \omega_W(k) \right) = - \frac{\Delta E}{\log \frac{\mathrm{p}_0}{1-\mathrm{p}_0}} r, \end{equation} that is, energy has been paid in order to gain entanglement during the process. The interconversion relation between the two resources is given by \begin{equation} \label{int_rel_ent_ene_app} \Delta W_W = - \frac{\Delta E}{\log \frac{\mathrm{p}_0}{1-\mathrm{p}_0}} \, \Delta W_E, \end{equation} and we only need to show that the bank state has changed in a negligible way with respect to the related bank monotone. It is worth noting that, since the current theory satisfies all the properties we have considered in the main text, the bank monotone coincides, modulo a multiplicative constant, with the relative entropy distance from the set of states $\mathcal{S}_{\mathrm{p}_0}$ initially describing the bank. \par Indeed, it is easy to show that the relative entropy distance from this set is given by a linear combination of the monotones $E_{\mathcal{F}_{\text{css}}}$ and $M_H$. For $\rho \in \mathcal{S}_1$ we find that \begin{equation} \label{rel_ent_lin_ent_energy} E_{\mathcal{S}_{\mathrm{p}_0}}(\rho) = \inf_{\sigma \in \mathcal{S}_{\mathrm{p}_0}} \re{\rho}{\sigma} = \left( E_{\mathcal{F}_{\text{css}}}(\rho) - \bar{E}_{\mathcal{F}_{\text{css}}} \right) + \frac{\log \frac{\mathrm{p}_0}{1-\mathrm{p}_0}}{\Delta E} \left( M_H(\rho) - \bar{M}_H \right), \end{equation} where we recall that $\bar{E}_{\mathcal{F}_{\text{css}}} = E_{\mathcal{F}_{\text{css}}}(\sigma)$ and $\bar{M}_H = M_H(\sigma)$, for any state $\sigma \in \mathcal{S}_{\mathrm{p}_0}$. The linear coefficient in the rhs of Eq.~\eqref{rel_ent_lin_ent_energy} is the (inverse) exchange rate that we find in the interconversion relation, Eq.~\eqref{int_rel_ent_ene_app}. If we now consider the initial and final state of the bank, and we study how much the state is changed by the above protocol with respect to $E_{\mathcal{S}_{\mathrm{p}_0}}$, we find that \begin{equation} E_{\mathcal{S}_{\mathrm{p}_0}}(\rho_{\text{fin}}^{\otimes n}) - E_{\mathcal{S}_{\mathrm{p}_0}}(\rho_{\text{in}}^{\otimes n}) = O(n^{-1}), \end{equation} so that, when $n \rightarrow \infty$, we obtain that the state of the bank is only infinitesimally changed, and can be used again to perform another resource interconversion with the same initial exchange rate.
\section{Proofs}
\subsection{Main results} \label{main_results} In the first part of this appendix we provide the proofs of the results presented in the main text. We start with the proof of the following theorem, where it is shown that a multi-resource theory which satisfies the asymptotic equivalence property of Def.~\ref{def:asympt_equivalence_multi} has a unique quantifier for each of the resources present in the theory. This theorem is introduced in Sec.~\ref{multi_rev_unique}.
\uniquemeas*
\begin{proof} Let us prove that $f_1^{\infty}$ uniquely quantifies the amount of $1$-st resource contained in the main system (the proof for the other $f_{i \neq 1}$'s is analogous). We prove the theorem by contradiction. Suppose that there exists two monotones $f_1$ and $g_1$ satisfying the properties~\ref{item:M1} -- \ref{item:M7}, such that \begin{enumerate} \item $\exists \, \rho \in \mathcal{S} \left( \mathcal{H}_S \right)$, where $\rho \not\in \mathcal{F}_1$, for which $f_1^{\infty}(\rho) = g_1^{\infty}(\rho)$ (this is always possible by rescaling the monotone $g$). \item $\exists \, \sigma \in \mathcal{S} \left( \mathcal{H}_S \right)$, where $\sigma \not\in \mathcal{F}_1$, for which $f_1^{\infty}(\sigma) \neq g_1^{\infty}(\sigma)$ (that is, $f_1$ is not unique). \end{enumerate} Consider now the values of $f_1^{\infty}(\rho)$ and $f_1^{\infty}(\sigma)$. If these are equal, it is easy to see, using the asymptotic equivalence property, that $f_1$ is unique. Suppose instead that they are not equal. Then, there exists $n, k \in \mathbb{N}$\footnote{Where we assume that all physically meaningful values of the $f_i^{\infty}$'s are in $\mathbb{Q}$, which we recall is dense in $\mathbb{R}$.} such that \begin{equation} \label{relation_rho_sigma_f1} n \, f_1^{\infty}(\rho) = k \, f_1^{\infty}(\sigma). \end{equation} \par Let us consider the system together with the batteries $B_i$'s, initially in the state $\rho^{\otimes n} \otimes \omega_1 \otimes \ldots \otimes \omega_m$. Then, we take the states $\omega'_i \in \mathcal{S} \left( \mathcal{H}_{B_i} \right)$, where $i = 1 , \ldots , m$, such that \begin{align} \label{unique_condition_i} f_i^{\infty}( \rho^{\otimes n} \otimes \omega_1 \otimes \ldots \otimes \omega_m ) &= f_i^{\infty}( \gamma_n \otimes \omega'_1 \otimes \ldots \otimes \omega'_m ) \ , \ \forall \, i \in \left\{ 1 , \ldots , m \right\}, \\ f_j^{\infty}( \omega_i ) &= f_j^{\infty}( \omega'_i ) \ , \ \forall \, i, j \in \left\{ 1 , \ldots , m \right\}, \ i \neq j, \end{align} where $\gamma_n \in \mathcal{F}_1^{(n)}$. Due to the asymptotic equivalence property, the conditions in Eq.~\eqref{unique_condition_i} imply that there exists a sequence of maps $\left\{ \tilde{\mathcal{E}}_N \right\}_N$ of the form of Eq.~\eqref{allowed_ancilla} such that \begin{equation} \label{rev_map} \lim_{N \rightarrow \infty}
\left\| \tilde{\mathcal{E}}_N \left( \left( \rho^{\otimes n} \otimes \omega_1 \otimes \ldots \otimes \omega_m \right)^{\otimes N} \right) - \left(\gamma_n \otimes \omega'_1 \otimes \ldots \otimes \omega'_m \right)^{\otimes N}
\right\|_1 = 0, \end{equation} as well as another sequence of maps performing the reverse transformation. From the asymptotic continuity of $g_1$, property~\ref{item:M7}, it then follows that \begin{equation} g_1 \left( \tilde{\mathcal{E}}_N \left( \left( \rho^{\otimes n} \otimes \omega_1 \otimes \ldots \otimes \omega_m \right)^{\otimes N} \right) \right) = g_1 \left( \left(\gamma_n \otimes \omega'_1 \otimes \ldots \otimes \omega'_m \right)^{\otimes N} \right) + o(N). \end{equation} Let us consider the lhs of the above equation, and recall that the map $\tilde{\mathcal{E}}_N$ is obtained by applying an allowed operation to $N$ copies of the system together with a sub-linear ancilla $\eta^{(A)}_N$, see Eq.~\eqref{allowed_ancilla}. For simplicity, let us refer to $\rho^{\otimes n} \otimes \omega_1 \otimes \ldots \otimes \omega_m$ as $\Omega$ in the following chain of inequalities, \begin{align} g_1 \left( \tilde{\mathcal{E}}_N \left( \Omega^{\otimes N} \right) \right) &= g_1 \left( \Tr{A}{\mathcal{E}_N \left( \Omega^{\otimes N} \otimes \eta^{(A)}_N \right)} \right) \leq g_1 \left( \mathcal{E}_N \left( \Omega^{\otimes N} \otimes \eta^{(A)}_N \right) \right) \leq g_1 \left( \Omega^{\otimes N} \otimes \eta^{(A)}_N \right) \nonumber \\ &\leq g_1 \left( \Omega^{\otimes N} \right) + g_1 \left( \eta^{(A)}_N \right) \leq g_1 \left( \Omega^{\otimes N} \right) + o(N) \end{align} where the first inequality follows from property~\ref{item:M4}, the second one from the monotonicity of $g_1$ under allowed operations, the third one from the sub-additivity of $g_1$, property~\ref{item:M5}, and the last inequality from property~\ref{item:M6} and the fact that the ancilla is sub-linear in $N$. If we now combine this equation with the previous one, we divide both sides by $N$, and we send it to infinity, we obtain that the regularised version of $g_1$ is such that, \begin{equation} g_1^{\infty} \left( \rho^{\otimes n} \otimes \omega_1 \otimes \ldots \otimes \omega_m \right) \geq g_1^{\infty} \left( \gamma_n \otimes \omega'_1 \otimes \ldots \otimes \omega'_m \right). \end{equation} By using the same argument for the sequence of maps performing the reverse transformation, we find that the above equation needs to hold as an equality, that is, \begin{equation} g_1^{\infty} \left( \rho^{\otimes n} \otimes \omega_1 \otimes \ldots \otimes \omega_m \right) = g_1^{\infty} \left( \gamma_n \otimes \omega'_1 \otimes \ldots \otimes \omega'_m \right). \end{equation} We can now separate each contribution to $g_1$ thanks to the property~\ref{item:M2}, use the fact that the batteries $B_{i \neq 1}$'s are not changing their value of $g_1$, property~\ref{item:M1}, and the fact that the final state of the system does not contain any resource associated with $g_1$, property~\ref{item:M3}. Then, we find that \begin{equation} \label{g1_rho_battery} n \, g_1^{\infty} \left( \rho \right) = g_1^{\infty} \left( \omega'_1 \right) - g_1^{\infty} \left( \omega_1 \right), \end{equation} where we have also used Lem.~\ref{lem:add_regularised_mon}. The same result follows for $f_1$, so that we find that \begin{equation} \label{f1_rho_battery} n \, f_1^{\infty} \left( \rho \right) = f_1^{\infty} \left( \omega'_1 \right) - f_1^{\infty} \left( \omega_1 \right). \end{equation} \par If we now consider Eqs.~\eqref{relation_rho_sigma_f1} and \eqref{f1_rho_battery}, we find that \begin{equation} k \, f_1^{\infty} \left( \sigma \right) = f_1^{\infty} \left( \omega'_1 \right) - f_1^{\infty} \left( \omega_1 \right). \end{equation} We can add to the above equation the term $f_1^{\infty} \left( \gamma_k \right)$, where $\gamma_k \in \mathcal{F}_1^{(k)}$, since this term is equal to zero due to property~\ref{item:M3}. Then, we find \begin{equation} \label{f1_sigma_cont} k \, f_1^{\infty} \left( \sigma \right) + f_1^{\infty} \left( \omega_1 \right) = f_1^{\infty} \left( \gamma_{k} \right) + f_1^{\infty} \left( \omega'_1 \right). \end{equation} Now, we want to introduce the initial and final states of the batteries $B_{i\neq1}$'s, so as to be sure that the transformation from $\sigma^{\otimes k}$ into $\gamma_k$ does not violate the conservation of the other resources. Specifically, we introduce $\omega_{i}$, $\omega''_{i} \in \mathcal{S} \left( \mathcal{H}_{B_i} \right)$ for $i \neq 1$, such that \begin{align} f_i^{\infty} \left( \sigma^{\otimes k} \otimes \omega_1 \otimes \omega_2 \otimes \ldots \otimes \omega_m \right) &= f_i^{\infty} \left( \gamma_k \otimes \omega'_1 \otimes \omega''_2 \otimes \ldots \otimes \omega''_m \right) \ , \ \forall \, i \in \left\{ 2 , \ldots , m \right\}, \label{fi_sigma_gamma} \\ f_1^{\infty}( \omega_i ) &= f_1^{\infty}( \omega''_i ) \ , \ \forall \, i \in \left\{ 2 , \ldots , m \right\}, \label{f1_Bi} \\ f_j^{\infty}( \omega_i ) &= f_j^{\infty}( \omega''_i ) \ , \ \forall \, i, j \in \left\{ 2 , \ldots , m \right\}, \ i \neq j. \end{align} Then, using the constraints of Eq.~\eqref{f1_Bi} over the states of the $B_{i\neq1}$'s batteries, we can re-write Eq.~\eqref{f1_sigma_cont} as \begin{equation} k \, f_1^{\infty} \left( \sigma \right) + f_1^{\infty} \left( \omega_1 \right) + f_1^{\infty} \left( \omega_2 \right) + \ldots + f_1^{\infty} \left( \omega_m \right) = f_1^{\infty} \left( \gamma_{k} \right) + f_1^{\infty} \left( \omega'_1 \right) + f_1^{\infty} \left( \omega''_2 \right) + \ldots + f_1^{\infty} \left( \omega''_m \right). \end{equation} If we now use Lem.~\ref{lem:add_regularised_mon} and property~\ref{item:M1}, we find that \begin{equation} \label{f1_sigma_gamma} f_1^{\infty} \left( \sigma^{\otimes k} \otimes \omega_1 \otimes \omega_2 \otimes \ldots \otimes \omega_m \right) = f_1^{\infty} \left( \gamma_k \otimes \omega'_1 \otimes \omega''_2 \otimes \ldots \otimes \omega''_m \right) \end{equation} From Eqs.~\eqref{fi_sigma_gamma} and \eqref{f1_sigma_gamma} it follows, using the asymptotic equivalence property, that there exists a sequence of maps $\left\{ \tilde{\mathcal{E}}'_N \right\}_N$ such that \begin{equation} \lim_{N \rightarrow \infty}
\left\| \tilde{\mathcal{E}}'_N \left( \left( \sigma^{\otimes k} \otimes \omega_1 \otimes \omega_2 \otimes \ldots \otimes \omega_m \right)^{\otimes N} \right) - \left( \gamma_k \otimes \omega'_1 \otimes \omega''_2 \otimes \ldots \otimes \omega''_m \right)^{\otimes N}
\right\|_1 = 0, \end{equation} as well as a related sequence of maps performing the reverse transformation. Using the properties of $g_1$, as we did before, we find that \begin{equation} \label{g1_sigma_battery} k \, g_1^{\infty} \left( \sigma \right) = g_1^{\infty} \left( \omega'_1 \right) - g_1^{\infty} \left( \omega_1 \right). \end{equation} Then, combining Eqs.~\eqref{g1_rho_battery} and \eqref{g1_sigma_battery}, we obtain that \begin{equation} n \, g_1^{\infty} \left( \rho \right) = k \, g_1^{\infty} \left( \sigma \right). \end{equation} Finally, using Eq.~\eqref{relation_rho_sigma_f1} and the initial assumption on the state $\rho$, we find that \begin{equation} f_1^{\infty} \left( \sigma \right) = g_1^{\infty} \left( \sigma \right), \end{equation} which contradicts our initial assumption. Therefore, $f_1^{\infty}$ uniquely quantify the amount of $1$-st resource contained in the main system. \end{proof}
In the next theorem, first stated in Sec.~\ref{bank_interconvert}, we show that in the presence of a bank two resources can always be exchanged one for the other, while the state of the bank is only infinitesimally modified by the resource interconversion.
\interconvert*
\begin{proof} Let us consider the resource interconversion of Eq.~\eqref{interconv_trasf}, where a global operation is performed over bank and batteries, and the sole constraint over the bank system is given by condition~\ref{item:X1}. As we discussed in Sec.~\ref{quant_res}, in order for the transformation to happen, the conditions of Eq.~\eqref{eq:fi_condition_Ri} need to be satisfied for both monotones $E_{\mathcal{F}_1}$ and $E_{\mathcal{F}_2}$, which in particular implies that the amount of resources exchanged with the batteries is \begin{equation} \label{intconv_conds} \Delta W_i = n \left( E_{\mathcal{F}_i}(\rho) - E_{\mathcal{F}_i}(\tilde{\rho}) \right) , \qquad i = 1, 2, \end{equation} where we have used property~\ref{item:F5b}. Furthermore, since $f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}$ is monotonic under the set of allowed operations, property~\ref{item:B7}, we find that \begin{equation} f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}(\rho^{\otimes n} \otimes \omega_1 \otimes \omega_2) = f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}(\tilde{\rho}^{\otimes n} \otimes \omega'_1 \otimes \omega'_2). \end{equation} Then, since the global system is given by many copies of $\mathcal{H}$, and since the bank monotone is additive, property~\ref{item:B3}, we can separate the contribution given by bank and batteries. Furthermore, from the definition of bank monotone, Eq.~\eqref{f3_monotone}, and the main property of the batteries, condition~\ref{item:M1}, it follows that \begin{equation} \alpha \left( E_{\mathcal{F}_1}(\rho^{\otimes n}) + E_{\mathcal{F}_1}(\omega_1) \right) + \beta \left( E_{\mathcal{F}_2}(\rho^{\otimes n}) +E_{\mathcal{F}_2}(\omega_2) \right) = \alpha \left( E_{\mathcal{F}_1}(\tilde{\rho}^{\otimes n}) + E_{\mathcal{F}_1}(\omega'_1) \right) + \beta \left( E_{\mathcal{F}_2}(\tilde{\rho}^{\otimes n}) + E_{\mathcal{F}_2}(\omega'_2) \right). \end{equation} Now, if we re-order the terms in the above equation, and we use Eq.~\eqref{f3_monotone} again, we obtain \begin{equation} f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}(\rho^{\otimes n}) - f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}(\tilde{\rho}^{\otimes n}) = \alpha \left( E_{\mathcal{F}_1}(\omega'_1) - E_{\mathcal{F}_1}(\omega_1) \right) + \beta \left( E_{\mathcal{F}_2}(\omega'_2) - E_{\mathcal{F}_2}(\omega_2) \right). \end{equation} If we use property~\ref{item:X1} together with the definitions of $\Delta W_1$ and $\Delta W_2$ given in Eq.~\eqref{work_Ri}, we get that \begin{equation} \alpha \, \Delta W_1 = - \beta \, \Delta W_2 + \delta_n, \end{equation} where $\delta_n \rightarrow 0$ as $n$ tends to infinity. However, we are still left to show that, when $n \rightarrow \infty$, the amount of resources exchanged by the batteries remains finite. \par Let us first recall that the way in which the monotone $f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}}$ is built implies that this monotone is tangent to the state-space, see property~\ref{item:B2} and Fig.~\ref{fig:tangent_monotone}. As a result, we have that the curve of bank states, see Eq.~\eqref{curve_bank} in appendix~\ref{convex_bound}, can be approximate, in the neighbourhood of $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$, by a line. This implies that, if we take the state $\tilde{\rho}$ in the set of bank states $\mathcal{F}_{\text{bank}}$, such that \begin{equation} E_{\mathcal{F}_1}(\tilde{\rho}) = E_{\mathcal{F}_1}(\rho) - \epsilon, \end{equation} where we recall $\rho \in \mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$, and $\epsilon \ll 1$, we find that the value of the monotone $E_{\mathcal{F}_2}$ for this state is \begin{equation} E_{\mathcal{F}_1}(\tilde{\rho}) = E_{\mathcal{F}_2}(\rho) + \frac{\alpha}{\beta} \epsilon + O(\epsilon^2). \end{equation} Then, it is easy to see that, if we map $\rho$ into $\tilde{\rho}$ during the resource interconversion, we obtain the following \begin{equation} \label{rate_of_convergence} \Delta W_1 = n \, \epsilon \quad , \quad \Delta W_2 = - n \, \frac{\alpha}{\beta} \epsilon + O(n \, \epsilon^2) \quad , \quad \delta_n = O(n \, \epsilon^2), \end{equation} where the first two equations follow from Eq.~\eqref{intconv_conds}, while the last one is given by Eq.~\eqref{condition_x1}. Thus, if we take $\epsilon \propto \frac{1}{n}$, and we send $n$ to infinity, we get that the amount of resources $\Delta W_i$ exchanged during the transformations are finite and their value is arbitrary, while the change in the bank monotone over the bank system $\delta_n$ is infinitesimal. \end{proof}
The next theorem can be found in Sec.~\ref{bank_monotone}. The theorem states that, given a multi-resource theory with a non-empty set of bank states, we can always build a single-resource theory out of it, by extending the class of allowed operations with the possibility of adding ancillary systems described by the bank states, see Def.~\ref{def:sing_res_constr}. In particular, we show that if the multi-resource theory satisfies the asymptotic equivalence property, so does the single-resource theory with respect to the bank monotone of Eq.~\eqref{f3_monotone}.
\singres*
\begin{proof} {\bf (a)} We start the proof by showing that, for the single resource theory $\text{R}_{\mathrm{single}}$, the second statement in Def.~\ref{def:asympt_equivalence_multi} implies the first one. In other words, we want to show that for any two states $\rho, \sigma \in \mathcal{S} \left( \mathcal{H} \right)$ which can be asymptotically mapped into one another with the allowed operations $\mathcal{C}_{\mathrm{single}}$, the value of the bank monotone on the two states is the same. Suppose there exists a sequence of operations $\left\{ \tilde{\mathcal{E}}_N^{\mathrm{(s)}}
\right\}_N$ such that $\lim_{N \rightarrow \infty} \left\| \tilde{\mathcal{E}}_N^{\mathrm{(s)}} (\rho^{\otimes N})
- \sigma^{\otimes N} \right\|_1 = 0$, where these maps are of the form \begin{equation} \label{allowed_ancilla_single} \tilde{\mathcal{E}}_N^{\mathrm{(s)}} (\cdot) = \Tr{A}{ \mathcal{E}_N^{\mathrm{(s)}} ( \cdot \otimes \eta_N^{(A)} ) }, \end{equation} with $\eta_N^{(A)} \in \SHn{o(N)}$ an arbitrary state of a sub-linear ancilla, and $\mathcal{E}_N^{\mathrm{(s)}}$ an allowed operation for $\text{R}_{\mathrm{single}}$. Likewise, suppose there is a sequence of maps that perform the reverse transformation. If we use the asymptotic continuity of the bank monotone, property~\ref{item:B6}, it follows that \begin{equation} f_{\mathrm{bank}}^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}} \left( \tilde{\mathcal{E}}_N^{\mathrm{(s)}} (\rho^{\otimes N}) \right) = f_{\mathrm{bank}}^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}} \left( \sigma^{\otimes N} \right) + o(N). \end{equation} Then, by using the properties~\ref{item:B1} -- \ref{item:B7} of the bank monotone, we can prove the following chain of inequalities for the lhs of the above equation \begin{align} f_{\mathrm{bank}}^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}} \left( \tilde{\mathcal{E}}_N^{\mathrm{(s)}} (\rho^{\otimes N}) \right) &= f_{\mathrm{bank}}^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}} \left( \Tr{A}{ \mathcal{E}_N^{\mathrm{(s)}} ( \rho^{\otimes N} \otimes \eta_N^{(A)} ) } \right) \leq f_{\mathrm{bank}}^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}} \left( \mathcal{E}_N^{\mathrm{(s)}} ( \rho^{\otimes N} \otimes \eta_N^{(A)} ) \right) \nonumber \\ &\leq f_{\mathrm{bank}}^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}} \left( \rho^{\otimes N} \otimes \eta_N^{(A)} \right) = f_{\mathrm{bank}}^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}} \left( \rho^{\otimes N} \right) + f_{\mathrm{bank}}^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}} \left( \eta_N^{(A)} \right) \nonumber \\ &\leq f_{\mathrm{bank}}^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}} \left( \rho^{\otimes N} \right) + o(N) \end{align} where the first inequality follows from monotonicity under partial trace, property~\ref{item:B4}, the second one from monotonicity under the allowed operations $\mathcal{C}_{\mathrm{single}}$ (that we still need to show), the equality follows from additivity, property~\ref{item:B3}, and the last inequality from the sub-extensivity of the monotone, property~\ref{item:B5}. If we use the same argument for the sequence of maps performing the reverse transformation, and we regularise the monotones by dividing the equations by the number of copies $N$, and sending $N$ to infinity, we find that \begin{equation} f_{\mathrm{bank}}^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}} \left( \rho \right) = f_{\mathrm{bank}}^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}} \left( \sigma \right), \end{equation} which proves the asymptotic equivalence property in one direction. \par We still need to show that the bank monotone is monotonic under the allowed operations $\mathcal{C}_{\mathrm{single}}$ of the single-resource theory. Recall that the most general of these operations, Eq.~\eqref{sin_res_map}, is given by \begin{equation} \mathcal{E}^{\mathrm{(s)}} ( \rho ) = \Tr{P^{(n)}}{\mathcal{E} ( \rho \otimes \rho_P^{\otimes n} )}, \end{equation} where $\mathcal{E} \in \mathcal{C}_{\mathrm{multi}}$, and we add $n \in \mathbb{N}$ copies of the bank state $\rho_P \in \mathcal{F}_{\mathrm{bank}}\left(\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}\right)$. Then, using the properties of the bank monotone, we can show that \begin{align} f_{\mathrm{bank}}^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}} \left( \mathcal{E}^{\mathrm{(s)}} ( \rho ) \right) &= f_{\mathrm{bank}}^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}} \left( \Tr{P^{(n)}}{\mathcal{E} ( \rho \otimes \rho_P^{\otimes n} ) } \right) \leq f_{\mathrm{bank}}^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}} \left( \mathcal{E} ( \rho \otimes \rho_P^{\otimes n} ) \right) \nonumber \\ &\leq f_{\mathrm{bank}}^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}} \left( \rho \otimes \rho_P^{\otimes n} \right) = f_{\mathrm{bank}}^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}} \left( \rho \right) + f_{\mathrm{bank}}^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}} \left( \rho_P^{\otimes n} \right) \nonumber \\ &= f_{\mathrm{bank}}^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}} \left( \rho \right), \end{align} where the first inequality follows from property~\ref{item:B4}, the second one from the monotonicity under the allowed operations $\mathcal{C}_{\mathrm{multi}}$, property~\ref{item:B7}, and the last two equalities from additivity, property~\ref{item:B3}, and the fact that the bank monotone is equal to zero over the bank states, property~\ref{item:B1}, respectively. \par {\bf (b)} We now want to prove the other direction of the asymptotic equivalence property for the resource theory $\text{R}_{\mathrm{single}}$, i.e., that the first statement in Def.~\ref{def:asympt_equivalence_multi} implies the second one. In other words, we want to show that for all states $\rho$, $\sigma \in \mathcal{S} \left( \mathcal{H} \right)$ such that $f_{\mathrm{bank}}^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}} \left( \rho \right) = f_{\mathrm{bank}}^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}} \left( \sigma \right)$, there exists a sequence of operations $\left\{ \tilde{\mathcal{E}}^{(s)}_N \right\}_N$ of the form given in Eq.~\eqref{allowed_ancilla_single}, mapping $N$ copies of $\rho$ into $N$ copies of $\sigma$, where $N \rightarrow \infty$. Before proving this part of the theorem, we recall that, given the bank state $\rho_P \in \mathcal{F}_{\mathrm{bank}} \left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$, all other bank states $\tilde{\rho}_P \in \mathcal{F}_{\mathrm{bank}}$ are such that, if $E_{\mathcal{F}_1} (\tilde{\rho}_P) = E_{\mathcal{F}_1} (\rho_P) + \delta$ with $\delta \ll 1$, then \begin{equation} \label{first_order_approx} E_{\mathcal{F}_2} ( \tilde{\rho}_P ) = E_{\mathcal{F}_2} (\rho_P) - \frac{\alpha}{\beta} \, \delta + O(\delta^2), \end{equation} which follows from the fact that $f_{\mathrm{bank}}^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}} = 0$ parametrises the line which is tangent to the state space and passes through the point $\left( \bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2} \right)$, see appendix~\ref{convex_bound}. \par Given the two states $\rho, \sigma \in \mathcal{S} \left( \mathcal{H} \right)$ with same value of the monotone $f_{\mathrm{bank}} ^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}}$, let us introduce the sequences of states $\left\{ \sigma_n \in \mathcal{S} \left( \mathcal{H} \right) \right\}_n$ and $\left\{ \tilde{\rho}_{P,n} \in \mathcal{F}_{\mathrm{bank}} \right\}_n$ such that, for $n \in \mathbb{N}$ big enough, we have \begin{align} \label{constraint_sigma_1} E_{\mathcal{F}_1} (\sigma_n) &= E_{\mathcal{F}_1} (\sigma) \\ \label{constraint_rho_tilde} E_{\mathcal{F}_1} (\rho \otimes \rho_P^{\otimes n}) &= E_{\mathcal{F}_1} (\sigma_n \otimes ( \tilde{\rho}_{P,n} )^{\otimes n}), \\ \label{constraint_existence} E_{\mathcal{F}_2}(\rho \otimes \rho_P^{\otimes n}) &= E_{\mathcal{F}_2}(\sigma_n \otimes ( \tilde{\rho}_{P,n} )^{\otimes n}), \end{align} where $\rho_P \in \mathcal{F}_{\mathrm{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$. From the above equations, and from the additivity of $E_{\mathcal{F}_1}$, which follows from property~\ref{item:F5b}, we obtain that \begin{equation} E_{\mathcal{F}_1}(\tilde{\rho}_{P,n}) = E_{\mathcal{F}_1}(\rho_P) + \frac{1}{n} \left( E_{\mathcal{F}_1}(\rho) - E_{\mathcal{F}_1}(\sigma) \right). \end{equation} Notice that, for $n \rightarrow \infty$, we have that $\frac{1}{n} \left( E_{\mathcal{F}_1}(\rho) - E_{\mathcal{F}_1}(\sigma) \right) \rightarrow 0$, and therefore, for $n$ sufficiently big, it follows from Eq.~\eqref{first_order_approx} that \begin{equation} \label{first_order_passive} E_{\mathcal{F}_2}(\tilde{\rho}_{P,n}) = E_{\mathcal{F}_2}(\rho_P) - \frac{\alpha}{\beta} \, \frac{1}{n} \left( E_{\mathcal{F}_1}(\rho) - E_{\mathcal{F}_1}(\sigma) \right) + O(n^{-2}). \end{equation} If we now combine Eq.~\eqref{constraint_existence} and \eqref{first_order_passive} together, we use the additivity of $E_{\mathcal{F}_2}$, and we use the fact that $\rho$ and $\sigma$ have the same value of the bank monotone, we obtain the following \begin{equation} \label{constraint_sigma_2} E_{\mathcal{F}_2}(\sigma_n) = E_{\mathcal{F}_2}(\sigma) + O(n^{-1}). \end{equation} \par Let us now focus on the operations mapping $\rho$ into $\sigma$. We do this in two steps. First, we use the fact that the theory $\text{R}_{\mathrm{multi}}$ satisfies asymptotic equivalence, and we consider the Eqs.~\eqref{constraint_rho_tilde} and \eqref{constraint_existence}. These equations imply that, for all $n \in \mathbb{N}$, there exists of a sequence of maps $\left\{ \tilde{\mathcal{E}}_{N,n} \right\}_N$ such that \begin{equation} \label{asy_eq_mult}
\lim_{N \rightarrow \infty} \left\| \tilde{\mathcal{E}}_{N,n} \left( \left( \rho \otimes \rho_P^{\otimes n} \right)^{\otimes N} \right) - \left( \sigma_n \otimes ( \tilde{\rho}_{P,n} )^{\otimes n} \right)^{\otimes N}
\right\|_1 = 0. \end{equation} As per definition of asymptotic equivalence, the maps $\tilde{\mathcal{E}}_{N,n} \ : \ \SHn{N (n+1)} \rightarrow \SHn{N (n+1)}$ are of the form \begin{equation} \tilde{\mathcal{E}}_{N,n} ( \cdot ) = \Tr{A}{\mathcal{E}_{N,n} \left( \cdot \otimes \eta^{(A)}_N \right)} \end{equation} where the map $\mathcal{E}_{N,n}$ is an allowed operation of $\text{R}_{\mathrm{multi}}$ acting on system and ancilla, and the state of the ancilla is $\eta^{(A)}_N \in \mathcal{S} \left( \left( \mathcal{H}^{\otimes n+1} \right)^{\otimes f(N)} \right)$, where $f(N) = o(N)$. Notice that, in particular, we can take $n$ to be a monotonic function of $N$, $n = g(N)$, such that $\lim_{N \rightarrow \infty} g(N) = \infty$ and $f(N) g(N) = o(N)$. For example, if $f(N) \propto N^{1/2}$, we can chose $g(N) \propto N^{1/4}$, so that their product is $N^{3/4} = o(N)$. \par We can now define the sequence of maps $\left\{ \tilde{\mathcal{E}}^{\mathrm{(s)}}_{N} \right\}_N$ acting on $\SHn{N}$. These maps are defined as \begin{equation} \tilde{\mathcal{E}}^{\mathrm{(s)}}_{N} (\rho^{\otimes N}) = \Tr{P}{\tilde{\mathcal{E}}_{N,g(N)} \left(
\rho^{\otimes N} \otimes \rho_P^{\otimes N g(N)} \right)}, \end{equation} where we are tracing out the part of the system which was initially in the state $\rho_P^{\otimes N g(N)}$. It is interesting to notice that this system is super-linear in the number of copies $N$ of $\rho$, a condition that seems to be necessary to achieve the conversion, see Ref.~\cite{brandao_resource_2013} for an example in thermodynamics. We can re-write these maps as \begin{equation} \label{single_sequence_ancilla} \tilde{\mathcal{E}}^{\mathrm{(s)}}_{N} (\rho^{\otimes N}) = \Tr{A}{\mathcal{E}^{\mathrm{(s)}}_{N} \left(
\rho^{\otimes N} \otimes \eta^{(A)}_N \right)}, \end{equation} where we recall that the ancillary system still lives on a sub-linear number of copies of $\mathcal{H}$, due to our choice of the function $g(N)$, and the operation $\mathcal{E}^{\mathrm{(s)}}_{N}$ is an allowed operations for the theory $\text{R}_{\mathrm{single}}$ -- compare it with Eq.~\eqref{sin_res_map} -- defined as \begin{equation} \mathcal{E}^{\mathrm{(s)}}_{N} ( \cdot ) = \Tr{P}{ \mathcal{E}_{N,g(N)} \left( \cdot \otimes \rho_P^{\otimes N g(N)} \right) }. \end{equation} If we now use Eq.~\eqref{asy_eq_mult} together with the monotonicity of the trace distance under partial tracing, we find that \begin{equation}
\lim_{N \rightarrow \infty} \left\| \tilde{\mathcal{E}}^{\mathrm{(s)}}_{N} (\rho^{\otimes N}) - \left( \sigma_{g(N)} \right)^{\otimes N}
\right\|_1 = 0. \end{equation} \par To conclude the proof, we notice that the sequence of states $\left\{ \sigma_{g(N)} \right\}_N$ does not need to converge to $\sigma$ with respect to the trace distance. However, if we consider the regularisation of the $E_{\mathcal{F}_i}$'s on these states, we find that \begin{equation} \lim_{N \rightarrow \infty} \frac{1}{N} E_{\mathcal{F}_i}( \sigma_{g(N)}^{\otimes N} ) = E_{\mathcal{F}_i}(\sigma) , \quad i = 1,2, \end{equation} which follows from Eqs.~\eqref{constraint_sigma_1} and \eqref{constraint_sigma_2}. Then, we can use the asymptotic equivalence of $\text{R}_{\text{multi}}$, which tells us that there exists a second sequence of allowed operations, and a sub-linear ancilla, such that we can asymptotically transform the state of the system into $\sigma$. This concludes the proof. \end{proof}
The following corollary is stated in Sec.~\ref{bank_monotone}, and it shows that the bank monotone introduced in Eq.~\eqref{f3_monotone} coincides with the relative entropy distance from the set of bank states $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}\right)$.
\bankmonrelent*
\begin{proof} We first notice that Thm.~\ref{unique_f3} promises us that, under the current assumptions over the theory $\text{R}_{\text{multi}}$, we can construct a single-resource theory $\text{R}_{\text{single}}$ with allowed operations $\mathcal{C}_{\text{single}}$ as in Def.~\ref{def:sing_res_constr}, which satisfies asymptotic equivalence with respect to the bank monotone $f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}}$. Furthermore, since this monotone satisfies the properties~\ref{item:SM1} -- \ref{item:SM6} listed in appendix~\ref{rev_theory_sing}, we can use Thm.~\ref{thm:reversible_asympt_equiv} in the same appendix to prove that this single resource theory is reversible. If we then use the results of Ref.~\cite{horodecki_quantumness_2012}, we obtain that this monotone is the unique measure of resource for the theory $\text{R}_{\text{single}}$. \par What we need to show in this proof is that, actually, both the bank monotone defined in Eq.~\eqref{f3_monotone} and the relative entropy distance from the set of bank states $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}\right)$ satisfy the properties from~\ref{item:SM1} to \ref{item:SM6}, and therefore by uniqueness these two functions need to coincide (modulo a multiplicative constant). That the bank monotone satisfies these properties is easy to show. Indeed, its monotonicity under the class of operations $\mathcal{C}_{\text{single}}$, property~\ref{item:SM1}, is proved in part {\bf (a)} of Thm.~\ref{unique_f3}. Furthermore, all other properties directly follow from property~\ref{item:B1} and the ones listed in Prop.~\ref{prop:properties_bank_mon}. \par Showing that the relative entropy distance from the set of states $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}\right)$ satisfies the same properties is not difficult either. First, we recall that the invariant sets of the theory, $\mathcal{F}_1$ and $\mathcal{F}_2$, satisfy the properties~\ref{item:F1}, \ref{item:F2}, \ref{item:F3} and~\ref{item:F5b} by hypothesis. This in turn implies that the subset of bank states under consideration satisfies properties~\ref{item:F1}, \ref{item:F2} and~\ref{item:F5b}, as it follows from the Props.~\ref{convex_f3} and \ref{additive_f3} in appendix~\ref{additional}. That the subset $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}\right)$ contains a full-rank state, property~\ref{item:F3}, is an hypothesis of this corollary. \par With the help of the above properties we can show that the relative entropy distance from $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$ satisfies the same properties of the bank monotone. That this relative entropy is monotonic under the set of operations $\mathcal{C}_{\text{single}}$, property~\ref{item:SM1}, is shown in Prop.~\ref{monotonicity_passive}. Furthermore, property~\ref{item:SM2} follows from the definition of relative entropy distance, see Eq.~\eqref{rel_entr_dist}, while property~\ref{item:SM3} follows from the fact that $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}\right)$ satisfies the properties~\ref{item:F1}, \ref{item:F2}, and \ref{item:F3}. The properties~\ref{item:SM4} and \ref{item:SM5} follow from the additivity of the set of bank states, property~\ref{item:F5b}. Finally, the fact that the monotone scales sub-extensively, property~\ref{item:SM6}, is a consequences of the additivity of the set of bank states, as well as of the fact that a full-rank state is contained in this set, properties~\ref{item:F5b} and \ref{item:F3}, respectively. \end{proof}
\subsection{Technical results} \label{additional} In this section we provide some minor results that are used to prove some of the main theorems in the paper. In particular, the next proposition is used in Sec.~\ref{multi_rev_unique}, together with Thm.~\ref{thm:reversible_multi}, to show that a multi-resource theory satisfying asymptotic equivalence with respect to the relative entropy distances from its invariant sets has unique resource quantifiers. This proposition is already known in the literature, see the references inside the proof.
\relentproperties*
\begin{proof} Let us first show that the relative entropy distance $E_{\mathcal{F}_i}$ is a monotone for the multi-resource theory $\text{R}_{\text{multi}}$, and that its regularisation is well-defined. These are necessary assumptions we have made in Def.~\ref{def:asympt_equivalence_multi}. The fact that $E_{\mathcal{F}_i}$ is monotonic under the class of allowed operations $\mathcal{C}_{\text{multi}}$, and that in particular it is monotonic under the allowed operations in $\mathcal{C}_i$, follows from the argument provided in the last paragraph of Sec.~\ref{single_resource}, and from the fact that $\mathcal{C}_{\text{multi}}$ is obtained from the intersection of all the other classes of allowed operations, see Eq.~\eqref{all_ops_multi}. Furthermore, that the regularisation of $E_{\mathcal{F}_i}$ exists follows from the properties~\ref{item:F3} and \ref{item:F4}. In fact, for all $\rho \in \mathcal{S} \left( \mathcal{H} \right)$, we have that \begin{equation} \label{regularisation_proof} \frac{1}{n} E_{\mathcal{F}_i}(\rho^{\otimes n}) = \frac{1}{n} \inf_{\gamma_n \in \mathcal{F}^{(n)}} \re{\rho^{\otimes n}}{\gamma_n} \leq \frac{1}{n} \inf_{\gamma \in \mathcal{F}} \re{\rho^{\otimes n}}{\gamma^{\otimes n}} = \inf_{\gamma \in \mathcal{F}} \re{\rho}{\gamma} \leq \re{\rho}{\gamma_{\text{full-rank}}} \end{equation} where the first inequality follows from the fact that the invariant sets are closed under tensor product, property~\ref{item:F4}, and the second inequality from the fact that they contain at least one full-rank state $\gamma_{\text{full-rank}}$, property~\ref{item:F3}. Since the rhs of Eq.~\eqref{regularisation_proof} is finite, and independent of $n$, we have that the regularisation of the $E_{\mathcal{F}_i}$'s is well-defined. \par In order for the monotone to satisfy the property~\ref{item:M1}, we can simply choose the states of the battery $B_i$ to have a fixed value of the monotones $E_{\mathcal{F}_{j \neq i}}$, for all $j \in \left\{1, \ldots, m \right\}$. Property~\ref{item:M2}, instead, follows from the fact that we want the batteries to be independent from each other, so as to address them individually. As a result, we choose the global invariant sets to be of the form $\mathcal{F}_i = \mathcal{F}_{i,S} \otimes \mathcal{F}_{i,B_1} \otimes \ldots \otimes \mathcal{F}_{i,B_m}$, where $i = 1, \ldots, m$, the main system is $S$, and the $B_i$'s refer to the batteries. This implies that the relative entropy distances from these sets are additive over system and batteries. However, it is still possible for $\mathcal{F}_i^{\otimes n}$ to be a proper subset of $\mathcal{F}_i^{(n)}$, since on the main systems or batteries we do not ask any additivity property. The validity of property~\ref{item:M3} for $E_{\mathcal{F}_i}$ follows straightforwardly from the definition of relative entropy distance, see Eq.~\eqref{rel_entr_dist}. That $E_{\mathcal{F}_i}$ satisfies property~\ref{item:M4} follows from property~\ref{item:F5}, since for all $\rho_n \in \SHn{n}$ we have that \begin{align} E_{\mathcal{F}_i}(\Tr{k}{\rho_n}) &= \inf_{\gamma_{n-k} \in \mathcal{F}_i^{(n-k)}} \re{\Tr{k}{\rho_n}}{\gamma_{n-k}} \leq \inf_{\gamma_n \in \mathcal{F}_i^{(n)}} \re{\Tr{k}{\rho_n}}{\Tr{k}{\gamma_n}} \nonumber \\ &\leq \inf_{\gamma_n \in \mathcal{F}_i^{(n)}} \re{\rho_n}{\gamma_n} = E_{\mathcal{F}_i}(\rho_n), \end{align} where the first inequality follows from property~\ref{item:F5}, and the second one from the monotonicity of the relative entropy under CPTP maps. The monotones $E_{\mathcal{F}_i}$'s are also sub-additive, property~\ref{item:M5}, since for any two states $\rho_n \in \SHn{n}$ and $\rho_k \in \SHn{k}$ we have that \begin{align} E_{\mathcal{F}_i}(\rho_n \otimes \rho_k) &= \inf_{\gamma_{n+k} \in \mathcal{F}_i^{(n+k)}} \re{\rho_n \otimes \rho_k}{\gamma_{n+k}} \leq \inf_{\gamma_n \in \mathcal{F}_i^{(n)},\gamma_k \in \mathcal{F}_i^{(k)}} \re{\rho_n \otimes \rho_k}{\gamma_n \otimes \gamma_k} \nonumber \\ &= \inf_{\gamma_n \in \mathcal{F}_i^{(n)}} \re{\rho_n}{\gamma_n} + \inf_{\gamma_k \in \mathcal{F}_i^{(k)}} \re{\rho_k}{\gamma_k} = E_{\mathcal{F}_i}(\rho_n) + E_{\mathcal{F}_i}(\rho_k), \end{align} where the inequality follows from property~\ref{item:F4} of the set $\mathcal{F}_i$. Property~\ref{item:M6} for the relative entropy distance $E_{\mathcal{F}_i}$ follows from similar considerations to the one presented in Eq.~\eqref{regularisation_proof}. In fact, we have that for all $\rho_n \in \SHn{n}$, \begin{align} E_{\mathcal{F}_i}(\rho_n) &= \inf_{\gamma_n \in \mathcal{F}_i^{(n)}} \re{\rho_n}{\gamma_n} \leq \re{\rho_n}{\gamma_{\text{full-rank}}^{\otimes n}} = - S(\rho_n) - \tr{ \rho_n \log \gamma_{\text{full-rank}}^{\otimes n} } \nonumber \\ &\leq - \tr{ \rho_n \log \gamma_{\text{full-rank}}^{\otimes n} } \leq n \log \lambda_{\text{min}}^{-1}, \end{align} where the first inequality follows from the fact that $\mathcal{F}_i$ contains a full-rank state, property~\ref{item:F3}, the second one from the fact that the von Neumann entropy is non-negative, and the last one from the fact that the optimal case is obtained when $\rho_n$ is given by $n$ copies of the pure state associated with the minimum eigenvalue $\lambda_{\text{min}}$ of the full-rank state $\gamma_{\text{full-rank}}$. Finally, in Ref.~\cite{synak-radtke_asymptotic_2006}, Lem.~1, it was shown that the relative entropy distance from a set $\mathcal{F}$ satisfying properties~\ref{item:F1}, \ref{item:F2}, and \ref{item:F3} is asymptotic continuous. In the proof, it was required the set $\mathcal{F}$ to contain the maximally-mixed state. However, as it was noticed in Ref.~\cite{brandao_generalization_2010}, Lem.~C.3, one simply needs $\mathcal{F}$ to contain a full-rank state. Thus, under the above properties on the free set, we have that $E_{\mathcal{F}_i}$ satisfies the property~\ref{item:M7}. \end{proof}
The next proposition collects the properties of the bank monotone defined in Eq.~\eqref{f3_monotone}.
\bankproperties*
\begin{proof} Most of the properties listed above follows straightforwardly from the ones of the invariant sets $\mathcal{F}_i$'s. Here, we only focus on property~\ref{item:B4}, stating that \begin{equation} f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}(\Tr{k}{\rho_n}) \leq f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}(\rho_n) , \qquad \forall \, n, k \in \mathbb{N} \ , \ k < n \ , \ \forall \, \rho_n \in \SHn{n}. \end{equation} In order to prove the above property, we make use of Lem.~\ref{f_i_inequality} and of the definition of bank monotone, see Eq.~\eqref{f3_monotone}. First, let us divide the $n$ copies of the system into two sets, so that $\mathcal{H}^{\otimes n} = \mathcal{H}^{\otimes k} \otimes \mathcal{H}^{\otimes n-k}$, and in the following equation we refer to $S_1$ as the system described by the first $k$ copies, and to $S_2$ as the system described by the last $n-k$ copies. In particular, $\rho_{S_1} = \Tr{n-k}{\rho_n} \in \SHn{k}$, and $\rho_{S_2} = \Tr{k}{\rho_n} \in \SHn{n-k}$. Then, we have the following chain of inequalities \begin{align} f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}(\rho_n) &= \alpha \left( E_{\mathcal{F}_1}(\rho_n) - n \, \bar{E}_{\mathcal{F}_1} \right) + \beta \left( E_{\mathcal{F}_2}(\rho_n) - n \, \bar{E}_{\mathcal{F}_2} \right) \nonumber \\ &\geq \alpha \left( E_{\mathcal{F}_1}(\rho_{S_1}) + E_{\mathcal{F}_1}(\rho_{S_2}) - n \, \bar{E}_{\mathcal{F}_1} \right) + \beta \left( E_{\mathcal{F}_2}(\rho_{S_1}) + E_{\mathcal{F}_2}(\rho_{S_2}) - n \, \bar{E}_{\mathcal{F}_2} \right) \nonumber \\ &= \alpha \left( E_{\mathcal{F}_1}(\rho_{S_1}) - k \, \bar{E}_{\mathcal{F}_1} \right) + \beta \left( E_{\mathcal{F}_2}(\rho_{S_1}) - k \, \bar{E}_{\mathcal{F}_2} \right) \nonumber \\ &+ \alpha \left( E_{\mathcal{F}_1}(\rho_{S_2}) - (n-k) \bar{E}_{\mathcal{F}_1} \right) + \beta \left( E_{\mathcal{F}_2}(\rho_{S_2}) - (n-k) \bar{E}_{\mathcal{F}_2} \right) \nonumber \\ &= f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}(\rho_{S_1}) + f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}(\rho_{S_2}) \geq f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}(\rho_{S_2}) = f_{\text{bank}}^{\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}}(\Tr{k}{\rho_n}), \end{align} where the first inequality follows from Lem.~\ref{f_i_inequality}, and the second one from the fact that the bank monotone is non-negative, which itself follows from properties~\ref{item:B1} and \ref{item:B2}. \end{proof}
The following proposition is used in Sec.~\ref{average_non_increasing} to show that single-resource theories whose class of allowed operations does not increase the average value of a given observable admit a monotone that is asymptotic continuous, see property~\ref{item:M7}.
\begin{prop} \label{average_asymp_cont} Consider an Hilbert space $\mathcal{H}$ with dimension $d$, an Hermitian operator $A \in \mathcal{B} \left( \mathcal{H} \right)$, and the function $M_A \, : \, \mathcal{S} \left( \mathcal{H} \right) \rightarrow \mathbb{R}$ defined as \begin{equation} M_A(\rho) = \tr{A \rho} - a_0, \end{equation} where $\rho \in \mathcal{S} \left( \mathcal{H} \right)$ is an element of the state-space, and $a_0$ is the minimum eigenvalue of $A$. When $n$ copies of the Hilbert space are considered, $\mathcal{H}_n = \otimes_{i=1}^n \mathcal{H}^{(i)}$, the above operator is extended as $A_n = \sum_{i=1}^n A^{(i)}$, where $A^{(i)} \in \mathcal{B} \left( \mathcal{H} \right)$ acts on the $i$-th copy of the Hilbert space. Then, the function $M_A$ is asymptotic continuous. \end{prop}
\begin{proof}
Consider two states $\rho_n$, $\sigma_n \in \mathcal{S}(\mathcal{H}^{\otimes n})$, such that $\left\|
\rho_n - \sigma_n \right\|_1 \rightarrow 0$ for $n \rightarrow \infty$. We are interested in the difference between the value of the function $M_A$ evaluated on $\rho_n$ and $\sigma_n$. By definition, \begin{equation}
\left| M_A(\rho_n) - M_A(\sigma_n) \right| = \left| \tr{ \left( \rho_n - \sigma_n \right) A_n } \right|. \end{equation} Now, we can diagonalise the operator $\rho_n - \sigma_n = \sum_{\lambda} \lambda \ket{\psi_{\lambda}} \bra{\psi_{\lambda}}$. Then, we find \begin{equation}
\left| \tr{ \left( \rho_n - \sigma_n \right) A_n } \right| =
\left| \sum_{\lambda} \lambda \bra{\lambda} A_n \ket{\lambda} \right| \leq
\sum_{\lambda} \left| \lambda \right| \left| \bra{\lambda} A_n \ket{\lambda} \right| \leq
\sum_{\lambda} \left| \lambda \right| \left\| A_n \right\|_{\infty}, \end{equation}
where we are using the operator norm $\| O \|_{\infty} = \sup_{\ket{\psi} \in \mathcal{H}}
\frac{\| O \ket{\psi} \|}{\| \ket{\psi} \|}$, and the last inequality straightforwardly follows from the definition of operator norm. Then, due to the way in which we have defined
$A_n$, it is easy to show that $\| A_n \|_{\infty} = n \, \| A \|_{\infty}$, and therefore \begin{equation}
\sum_{\lambda} \left| \lambda \right| \left\| A_n \right\|_{\infty} =
n \left\| A \right\|_{\infty} \sum_{\lambda} \left| \lambda \right| =
n \left\| A \right\|_{\infty} \left\| \rho_n - \sigma_n \right\|_1. \end{equation} Finally, notice that $\dim \mathcal{H}_n = d^n$, where $d$ is fixed by the initial choice of $\mathcal{H}$. Then, we have, \begin{equation}
\left| M_A(\rho_n) - M_A(\sigma_n) \right| \leq n \, \log d \, \left\| A \right\|_{\infty} \left\| \rho_n - \sigma_n \right\|_1. \end{equation} If we now divide by $n$ both side of the inequality, we get that \begin{equation}
\frac{\left| M_A(\rho_n) - M_A(\sigma_n) \right|}{n} \leq
\log d \, \left\| A \right\|_{\infty} \left\| \rho_n - \sigma_n \right\|_1, \end{equation} and if we send $n \rightarrow \infty$, we obtain that $\frac{1}{n} \,
\left| M_A(\rho_n) - M_A(\sigma_n) \right| \rightarrow 0$, which proves the theorem. \end{proof}
\par The following lemma states that, when the sets $\mathcal{F}_i$'s are such that $\mathcal{F}_i^{(n)} = \mathcal{F}_i^{\otimes n}$ for all $n \in \mathbb{N}$, property~\ref{item:F5b}, the relative entropy distances from these sets are super-additive. This lemma is used in Prop.~\ref{additive_f3} and Thm.~\ref{unique_f3}.
\begin{lem} \label{f_i_inequality} Consider a state $\rho_{S_1,S_2} \in \SHn{2}$, and suppose that the sets $\mathcal{F}_1$ and $\mathcal{F}_2$ satisfy property~\ref{item:F5b}, that is, $\mathcal{F}_i^{(n)} = \mathcal{F}_i^{\otimes n}$ for all $n \in \mathbb{N}$, $i = 1,2$. Then, the relative entropy distances from these sets, $E_{\mathcal{F}_1}$ and $E_{\mathcal{F}_2}$, are such that \begin{equation} \label{eq:fi_ineq} E_{\mathcal{F}_i}(\rho_{S_1,S_2}) \geq E_{\mathcal{F}_i}(\rho_{S_1}) + E_{\mathcal{F}_i}(\rho_{S_2}) \ , \ i = 1,2, \end{equation} where $\rho_{S_1} = \Tr{S_2}{\rho_{S_1,S_2}}$, and similarly $\rho_{S_2} = \Tr{S_1}{\rho_{S_1,S_2}}$. Furthermore, the above inequality is saturated if and only if $\rho_{S_1,S_2} = \rho_{S_1} \otimes \rho_{S_2}$. The result extends trivially to the case in which $n > 2$ copies of the system are considered. \end{lem}
\begin{proof} Let us consider the monotone $E_{\mathcal{F}_1}$, as the following argument can be equally applied to $E_{\mathcal{F}_2}$. By definition of relative entropy distance, we have that \begin{equation} \label{part_1_fi_ineq}
E_{\mathcal{F}_1} (\rho_{S_1,S_2}) = \inf_{\sigma_{S_1,S_2} \in \mathcal{F}_1^{(2)}} D( \rho_{S_1,S_2} \| \sigma_{S_1,S_2} ) = - S(\rho_{S_1,S_2}) + \inf_{\sigma_{S_1,S_2} \in \mathcal{F}_1^{(2)}} \left( - \tr{\rho_{S_1,S_2} \log \sigma_{S_1,S_2}} \right), \end{equation} where $S(\rho_{S_1,S_2}) = - \tr{\rho_{S_1,S_2} \log \rho_{S_1,S_2}}$ is the Von Neumann entropy of the state $\rho_{S_1,S_2}$. From the sub-additivity of the Von Neumann entropy, we have that \begin{equation} \label{superadd_negent} - S(\rho_{S_1,S_2}) \geq - S(\rho_{S_1}) - S(\rho_{S_2}), \end{equation} while from the property~\ref{item:F5b} it follows that \begin{align} \inf_{\sigma_{S_1,S_2} \in \mathcal{F}_1^{(2)}} \left( - \tr{\rho_{S_1,S_2} \log \sigma_{S_1,S_2}} \right) &= \inf_{\sigma_{S_1}, \sigma_{S_2} \in \mathcal{F}_1} \left( - \tr{\rho_{S_1,S_2} \log \sigma_{S_1} \otimes \sigma_{S_2}} \right) \nonumber \\ &= \inf_{\sigma_{S_1}, \sigma_{S_2} \in \mathcal{F}_1} \left( - \tr{\rho_{S_1} \log \sigma_{S_1}} - \tr{\rho_{S_2} \log \sigma_{S_2}} \right) \nonumber \\ &= \inf_{\sigma_{S_1} \in \mathcal{F}_1} \left( - \tr{\rho_{S_1} \log \sigma_{S_1}} \right) + \inf_{\sigma_{S_2} \in \mathcal{F}_1} \left( - \tr{\rho_{S_2} \log \sigma_{S_2}} \right). \label{part_2_fi_ineq} \end{align} From Eqs.~\eqref{part_1_fi_ineq}, \eqref{superadd_negent}, and \eqref{part_2_fi_ineq} it follows that \begin{align} E_{\mathcal{F}_1} (\rho_{S_1,S_2}) &\geq \inf_{\sigma_{S_1} \in \mathcal{F}_1} \left( - S(\rho_{S_1}) - \tr{\rho_{S_1} \log \sigma_{S_1}} \right) + \inf_{\sigma_{S_2} \in \mathcal{F}_1} \left( - S(\rho_{S_2}) - \tr{\rho_{S_2} \log \sigma_{S_2}} \right) \nonumber \\ &= E_{\mathcal{F}_1} (\rho_{S_1}) + E_{\mathcal{F}_1} (\rho_{S_2}). \end{align} \end{proof}
The following proposition is used in Sec.~\ref{bank_interconvert}, in Prop.~\ref{monotonicity_passive}, and in Cor.~\ref{bank_equal_rel_ent}. The proposition states that, when the curve of bank states is strictly convex, and we consider $n$ copies of a bank system, the set of bank states $\mathcal{F}^{(n)}_{\text{bank}}$ is given by the tensor product of $n$ copies of states that are in the set $\mathcal{F}_{\text{bank}}$, each of them with the same value of monotones $E_{\mathcal{F}_1}$ and $E_{\mathcal{F}_2}$.
\begin{prop} \label{additive_f3} Suppose the sets $\mathcal{F}_1$ and $\mathcal{F}_2$ satisfy property~\ref{item:F5b}, that is, $\mathcal{F}_i^{(n)} = \mathcal{F}_i^{\otimes n}$ for all $n \in \mathbb{N}$, $i = 1,2$, and the set of bank states $\mathcal{F}_{\text{bank}}$ is represented by a strictly convex curve in the resource diagram. Consider the set of bank states $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2} \right)$ defined in Eq.~\eqref{subset_bank}, where $E_{\mathcal{F}_1}$ and $E_{\mathcal{F}_2}$ are the relative entropy distances from the sets $\mathcal{F}_1$ and $\mathcal{F}_2$, respectively. Then, when $n \in \mathbb{N}$ copies of the bank system are considered, we find that the set of bank states coincides with \begin{equation}
\mathcal{F}^{(n)}_{\text{bank}} = \left\{ \rho_1 \otimes \ldots \otimes \rho_n \in \SHn{n} \ | \ \exists \, \bar{E}_{\mathcal{F}_1} , \bar{E}_{\mathcal{F}_2} \ \text{such that} \ \rho_1, \ldots , \rho_n \in \mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right) \right\}. \end{equation} Furthermore, we have that for all subset of bank state $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right) \subset \mathcal{S} \left( \mathcal{H} \right)$, the corresponding bank subset in $\SHn{n}$ is such that \begin{equation} \label{additivity_bank} \mathcal{F}^{(n)}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right) = \mathcal{F}^{\otimes n}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right). \end{equation} \end{prop}
\begin{proof} We prove the theorem for $n = 2$, as the argument extends trivially for $n > 2$. Consider a state $\sigma_{S_1,S_2} \in \SHn{2}$. From Lem.~\ref{f_i_inequality}, it follows that \begin{equation} \label{correlat_ineq} E_{\mathcal{F}_i}(\sigma_{S_1,S_2}) \geq E_{\mathcal{F}_i}(\sigma_{S_1}) + E_{\mathcal{F}_i}(\sigma_{S_2}) \ , \ i = 1,2, \end{equation} where $\sigma_{S_1} = \Tr{S_2}{\sigma_{S_1,S_2}}$, $\sigma_{S_2} = \Tr{S_1}{\sigma_{S_1,S_2}}$, and the inequality is saturated iff $\sigma_{S_1,S_2} = \sigma_{S_1} \otimes \sigma_{S_2}$. Now, for both the states $\sigma_{S_1}, \sigma_{S_2} \in \mathcal{S} \left( \mathcal{H} \right)$, select the bank states $\rho_{P_1}, \rho_{P_2} \in \mathcal{F}_{\text{bank}}$ such that \begin{equation} \label{passiv_ineq} E_{\mathcal{F}_i}(\sigma_{S_j}) \geq E_{\mathcal{F}_i}(\rho_{P_j}) \ , \ i,j = 1,2. \end{equation} Recall now that, in the $E_{\mathcal{F}_1}$--$E_{\mathcal{F}_2}$ diagram, the curve of bank state is convex (see Prop.~\ref{convex_curve}), and therefore given $\rho_{P_1}, \rho_{P_2} \in \mathcal{F}_{\text{bank}}$, we can find another $\rho_{P_3} \in \mathcal{F}_{\text{bank}}$ such that \begin{equation} \label{convex_ineq} \frac{1}{2} E_{\mathcal{F}_i}(\rho_{P_1}) + \frac{1}{2} E_{\mathcal{F}_i}(\rho_{P_2}) \geq E_{\mathcal{F}_i}(\rho_{P_3}) \ , \ i = 1,2, \end{equation} where the inequality (when the curve is strictly convex) is saturated iff $\rho_{P_1}$, $\rho_{P_2}$, and $\rho_{P_3}$ all belong to the same subset $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$. By combining Eqs.~\eqref{correlat_ineq}, \eqref{passiv_ineq}, and \eqref{convex_ineq}, together with property~\ref{item:F5b} of the sets $\mathcal{F}_1$ and $\mathcal{F}_2$ (that implies the additivity of the corresponding relative entropy distances), we find that for all $\sigma_{S_1,S_2} \in \SHn{2}$, it exists a $\rho_{P_3} \in \mathcal{F}_{\text{bank}}$ such that \begin{equation} E_{\mathcal{F}_i}(\sigma_{S_1,S_2}) \geq E_{\mathcal{F}_i}(\rho_{P_3}^{\otimes 2}) \ , \ i = 1, 2 \end{equation} where the inequality is saturated iff $\sigma_{S_1,S_2} = \sigma_{S_1} \otimes \sigma_{S_2}$, and both $\sigma_{S_1}$ and $\sigma_{S_2}$ belong to the same subset $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}\right)$. Due to the definition of bank states given in Eq.~\eqref{set_f3}, the thesis of this proposition follows. \end{proof}
The next proposition shows that, when the invariant sets $\mathcal{F}_1$ and $\mathcal{F}_2$ are convex sets, the set of bank states $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2} \right)$, defined in Eq.~\eqref{subset_bank}, is convex as well. This proposition is used in Sec.~\ref{bank_interconvert}, as well as in Thm.~\ref{bank_equal_rel_ent}.
\begin{prop} \label{convex_f3} Suppose that $\mathcal{F}_1$ and $\mathcal{F}_2$ are convex sets, property~\ref{item:F2}, and consider the relative entropy distances from these two sets, $E_{\mathcal{F}_1}$ and $E_{\mathcal{F}_2}$. Then, the set of bank states $\mathcal{F}_{\mathrm{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$ is convex, as well as its extension to the $n$-copy case, $\mathcal{F}^{(n)}_{\mathrm{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$, defined in Eq.~\eqref{additivity_bank}. \end{prop}
\begin{proof} Let us consider two states $\rho_1$, $\rho_2 \in \mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$. For these two states, there exists $\sigma_1, \sigma_2 \in \mathcal{F}_1$ such that \begin{subequations} \label{add_init_cond} \begin{align} E_{\mathcal{F}_1}(\rho_1) = \re{\rho_1}{\sigma_1} = \bar{E}_{\mathcal{F}_1}, \\ E_{\mathcal{F}_1}(\rho_2) = \re{\rho_2}{\sigma_2} = \bar{E}_{\mathcal{F}_1}. \end{align} \end{subequations} Then, for all $\lambda \in [0,1]$, we have \begin{align} E_{\mathcal{F}_1} \big( \lambda \, \rho_1 + (1 - \lambda) \, \rho_2 \big) &= \inf_{\sigma \in \mathcal{F}_1} \re{\lambda \, \rho_1 + (1 - \lambda) \, \rho_2}{\sigma} \nonumber \\ &\leq \re{\lambda \, \rho_1 + (1 - \lambda) \, \rho_2}{\lambda \, \sigma_1 + (1 - \lambda) \, \sigma_2} \nonumber \\ &\leq \lambda \, \re{\rho_1}{\sigma_1} + (1 - \lambda) \, \re{\rho_2}{\sigma_2} = \bar{E}_{\mathcal{F}_1}, \end{align} where the first inequality follows from the fact that $\mathcal{F}_1$ is convex, property~\ref{item:F2}, and the second inequality from the joint convexity of the relative entropy. In the same way, it follows that \begin{equation} E_{\mathcal{F}_2} \big( \lambda \, \rho_1 + (1 - \lambda) \, \rho_2 \big) \leq \bar{E}_{\mathcal{F}_2}. \end{equation} Since $\rho_1$, $\rho_2 \in \mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$, they satisfy the properties of Eq.~\eqref{set_f3}, and therefore it has to be that, for all $\lambda \in [0,1]$, \begin{equation} E_{\mathcal{F}_1} \big( \lambda \, \rho_1 + (1 - \lambda) \, \rho_2 \big) = \bar{E}_{\mathcal{F}_1} \ \text{and} \ E_{\mathcal{F}_2} \big( \lambda \, \rho_1 + (1 - \lambda) \, \rho_2 \big) = \bar{E}_{\mathcal{F}_2}. \end{equation} Thus, we have that $\lambda \, \rho_1 + (1 - \lambda) \, \rho_2 \in \mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$. This result can be extended to the case of $n \in \mathbb{N}$ copies of the system, where the bank set $\mathcal{F}^{(n)}_{\mathrm{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$ is defined as in Eq.~\eqref{additivity_bank}. In this case, the proof is analogous to the one considered above, with the exception that in the rhs of Eqs.~\eqref{add_init_cond}, and of the following ones, we add the multiplicative factor $n$. \end{proof}
The next lemma is used in Prop.~\ref{monotonicity_passive}. The lemma states that, given the class of operations $\mathcal{C}_{\text{multi}}$ for which $\mathcal{F}_1$ and $\mathcal{F}_2$ are invariant sets, the set of bank states $\mathcal{F}_{\text{bank}}$, defined in Eq.~\eqref{set_f3}, is invariant as well.
\begin{lem} \label{lem:inv_f3} Consider a resource theory $\text{R}_{\text{multi}}$ with allowed operations $\mathcal{C}_{\text{multi}}$, and two invariant sets $\mathcal{F}_1$ and $\mathcal{F}_2$. Consider the subset of bank states $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}\right)$ as defined in Eq.~\eqref{subset_bank}. Then, for all $\mathcal{E} \in \mathcal{C}_{\text{multi}}$, we have that $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}\right)$ is an invariant set, that is \begin{equation} \mathcal{E} \left( \rho \right) \in \mathcal{F}_{\mathrm{bank}}\left(\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}\right) , \quad \forall \, \rho \in \mathcal{F}_{\mathrm{bank}}\left(\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}\right) \end{equation} Analogously, the set of bank states describing $n$ copies of the bank system is invariant under the class of allowed operations $\mathcal{C}^{(n)}_{\text{multi}}$. \end{lem}
\begin{proof} Let us consider $\rho \in \mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}\right)$, as well as the state $\mathcal{E}(\rho)$ obtained by applying the map $\mathcal{E} \in \mathcal{C}_{\text{multi}}$ to the bank state. Due to the monotonicity of $E_{\mathcal{F}_1}$ and $E_{\mathcal{F}_2}$, we have that $E_{\mathcal{F}_1}\left(\mathcal{E}(\rho)\right) \leq E_{\mathcal{F}_1}\left(\rho\right)$, and $E_{\mathcal{F}_2}\left(\mathcal{E}(\rho)\right) \leq E_{\mathcal{F}_2}\left(\rho\right)$. Recall now that $\rho$ is a bank state, which implies that $\forall \, \sigma \in \mathcal{S} \left( \mathcal{H} \right)$, one (or more) of the following options holds \begin{enumerate} \item \label{ineq_f1} $E_{\mathcal{F}_1}(\sigma) > E_{\mathcal{F}_1}(\rho)$. \item \label{ineq_f2} $E_{\mathcal{F}_2}(\sigma) > E_{\mathcal{F}_2}(\rho)$. \item \label{eq_f1_f2} $E_{\mathcal{F}_1}(\sigma) = E_{\mathcal{F}_1}(\rho)$ and $E_{\mathcal{F}_2}(\sigma) = E_{\mathcal{F}_2}(\rho)$. \end{enumerate} However, the monotonicity conditions given by $E_{\mathcal{F}_1}$ and $E_{\mathcal{F}_2}$ implies that $\mathcal{E}(\rho)$ violates options \ref{ineq_f1} and \ref{ineq_f2}, so that option \ref{eq_f1_f2} is the only possible one. But this implies that $E_{\mathcal{F}_1}(\mathcal{E}(\rho)) = E_{\mathcal{F}_1}(\rho)$ and $E_{\mathcal{F}_2}(\mathcal{E}(\rho)) = E_{\mathcal{F}_2}(\rho)$, meaning that $\mathcal{E}(\rho) \in \mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1}, \bar{E}_{\mathcal{F}_2}\right)$. The same argument applies to the set $\mathcal{F}_{\text{bank}}^{(n)}$, when $n$ copies of the system are considered. Indeed, this case is analogous to the one considered above, with the sole difference that now the state $\rho \in \mathcal{F}_{\text{bank}}^{(n)}$, the state $\sigma \in \SHn{n}$, and the operations we use are in the class $\mathcal{C}_{\text{multi}}^{(n)}$ defined in Sec.~\ref{multi_resource}. \end{proof}
The last proposition of this section shows that the relative entropy distance from the set $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$ is monotonic under the class of operations $\mathcal{C}_{\text{single}}$, introduced in Def.~\ref{def:sing_res_constr}. This proposition is used in Cor.~\ref{bank_equal_rel_ent}.
\begin{prop} \label{monotonicity_passive} Consider a multi-resource theory $\text{R}_{\text{multi}}$ with two resources, whose allowed operations $\mathcal{C}_{\text{multi}}$ leave the sets $\mathcal{F}_1$ and $\mathcal{F}_2$ invariant. Suppose these invariant sets satisfy the properties~\ref{item:F1}, \ref{item:F2}, \ref{item:F3}, and \ref{item:F5b}. Then, the relative entropy distance from the subset of bank states $\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$ is monotonic under both the class of operations $\mathcal{C}_{\text{multi}}$ and the class $\mathcal{C}_{\text{single}}$ introduced in Def.~\ref{def:sing_res_constr}. \end{prop}
\begin{proof} {\bf 1}. Here we show monotonicity of the relative entropy distance with respect to the addition of an ancillary system described by $n \in \mathbb{N}$ copies of a bank states. Consider the state $\rho \in \mathcal{S} \left( \mathcal{H} \right)$, and the bank state $\rho_P \in \mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$. Then, we have \begin{align} E_{\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)}(\rho \otimes \rho_P^{\otimes n}) &= \inf_{\sigma, \sigma_{P_1}, \ldots , \sigma_{P_n} \in \mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)} \re{\rho \otimes \rho_P^{\otimes n}}{\sigma \otimes \sigma_{P_1} \otimes \ldots \otimes \sigma_{P_n}} \nonumber \\ &= \inf_{\sigma \in \mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)} \re{\rho}{\sigma} + \sum_{i=1}^n \inf_{\sigma_{P_i} \in \mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)} \re{\rho_P}{\sigma_{P_i}} \nonumber \\ &= \inf_{\sigma \in \mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)} \re{\rho}{\sigma} = E_{\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)}(\rho), \end{align} where the first equality follows from Prop.~\ref{additive_f3}, and the last one from the fact that $\rho_P \in \mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)$. \par {\bf 2}. Now we show monotonicity of the relative entropy distance with respect to the allowed operations $\mathcal{C}_{\text{mulit}}$. Let us consider a state $\rho \in \mathcal{S} \left( \mathcal{H} \right)$, together with an operation $\mathcal{E} \in \mathcal{C}_{\text{multi}}$. Then, we have that \begin{align} \label{monotonicity_eq} E_{\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)}\big(\mathcal{E}(\rho)\big) &= \inf_{\sigma \in \mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)} \re{\mathcal{E}(\rho)}{\sigma} \leq \inf_{\sigma \in \mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)} \re{\mathcal{E}(\rho)}{\mathcal{E}(\sigma)} \nonumber \\ &\leq \inf_{\sigma \in \mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)} \re{\rho}{\sigma} = E_{\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)}(\rho), \end{align} where the first inequality follows from Lem.~\ref{lem:inv_f3}, and the second one from the monotonicity of the relative entropy under CPTP maps. This result trivially extends to the case in which we have multiple copies of the system, since in Lem.~\ref{lem:inv_f3} we have shown that $\mathcal{F}^{(n)}_{\text{bank}}$ is invariant under the allowed operations $\mathcal{C}_{\text{multi}}^{(n)}$ for all $n \in \mathbb{N}$. \par {\bf 3}. We show the monotonicity of the relative entropy with respect to partial tracing when the ancillary system is composed by just one copy. However, the result straightforwardly extends to the case in which the ancillary system is composed by $n \in \mathbb{N}$ copies. Let us consider the state $\rho_{S_1,S_2} \in \SHn{2}$. Then, we have that \begin{align} E_{\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)}(\Tr{S_2}{\rho_{S_1,S_2}}) &= \inf_{\sigma_{S_1} \in \mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)} \re{\Tr{S_2}{\rho_{S_1,S_2}}}{\sigma_{S_1}} \nonumber \\ &= \inf_{\sigma_{S_1}, \sigma_{S_2} \in \mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)} \re{\Tr{S_2}{\rho_{S_1,S_2}}}{\Tr{S_2}{\sigma_{S_1} \otimes \sigma_{S_2}}} \nonumber \\ &\leq \inf_{\sigma_{S_1}, \sigma_{S_2} \in \mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)} \re{\rho_{S_1,S_2}}{\sigma_{S_1} \otimes \sigma_{S_2}} \nonumber \\ &= E_{\mathcal{F}_{\text{bank}}\left(\bar{E}_{\mathcal{F}_1},\bar{E}_{\mathcal{F}_2}\right)}(\rho_{S_1,S_2}), \end{align} where the second equality follows from Prop.~\ref{additive_f3}, while the inequality follows from the monotonicity of the relative entropy distance under CPTP maps. \end{proof}
\end{document} | arXiv |
BMC Bioinformatics
Volume 20 Supplement 3
Selected articles from the 17th Asia Pacific Bioinformatics Conference (APBC 2019): bioinformatics
Prediction of drug-disease associations based on ensemble meta paths and singular value decomposition
Guangsheng Wu1,
Juan Liu1,2 &
Xiang Yue1,3
BMC Bioinformatics volume 20, Article number: 134 (2019) Cite this article
In the field of drug repositioning, it is assumed that similar drugs may treat similar diseases, therefore many existing computational methods need to compute the similarities of drugs and diseases. However, the calculation of similarity depends on the adopted measure and the available features, which may lead that the similarity scores vary dramatically from one to another, and it will not work when facing the incomplete data. Besides, supervised learning based methods usually need both positive and negative samples to train the prediction models, whereas in drug-disease pairs data there are only some verified interactions (positive samples) and a lot of unlabeled pairs. To train the models, many methods simply treat the unlabeled samples as negative ones, which may introduce artificial noises. Herein, we propose a method to predict drug-disease associations without the need of similarity information, and select more likely negative samples.
In the proposed EMP-SVD (Ensemble Meta Paths and Singular Value Decomposition), we introduce five meta paths corresponding to different kinds of interaction data, and for each meta path we generate a commuting matrix. Every matrix is factorized into two low rank matrices by SVD which are used for the latent features of drugs and diseases respectively. The features are combined to represent drug-disease pairs. We build a base classifier via Random Forest for each meta path and five base classifiers are combined as the final ensemble classifier. In order to train out a more reliable prediction model, we select more likely negative ones from unlabeled samples under the assumption that non-associated drug and disease pair have no common interacted proteins. The experiments have shown that the proposed EMP-SVD method outperforms several state-of-the-art approaches. Case studies by literature investigation have found that the proposed EMP-SVD can mine out many drug-disease associations, which implies the practicality of EMP-SVD.
The proposed EMP-SVD can integrate the interaction data among drugs, proteins and diseases, and predict the drug-disease associations without the need of similarity information. At the same time, the strategy of selecting more reliable negative samples will benefit the prediction.
De novo drug discovery is a complex systematic project which is expensive, time-consuming and with high failure risks. As reported, it will take 0.8–1.5 billion dollars and about 10–17 years to bring a small molecule drug into market, and during the development stage, almost 90% of the small molecules can not pass the Phase I clinical trial and finally be eliminated [1, 2]. For the approved drugs in market, their pharmacological and toxicological properties are clear and the drug safeties are often guaranteed, but only some of their indications are found. For example, there are 2589 approved small molecule drugs in DrugBank [3], and more than 25000 diseases in UMLS medical database [4], resulting in over 60 millions of drug-disease pairs. However, only less than 5% of the drug-disease pairs were identified to have therapeutic relationships, and most of the drug-disease relationships are unknown [5]. Therefore, to discover the new indications of approved drugs, known as drug repositioning, can greatly save money and time, especially can improve the success rate, has become a promising alternative for de novo drug development.
Historically, finding a new indiction of a drug is likely to be an accidental event with a bit of luck. For example, Minoxidil, originally for the treatment of hypertension, was found by chance to have the treatment efficacy for hair loss [6]; Sildenafil (trade name: Viagra), originally for the treatment of angina, was occasionally found to have the potential to treat erectile dysfunction [7]. Such occasional findings of the drugs' new indictions suggest a new methodology of drug development. However, the "pot-luck" approach can not promise drug repositioning effectively and efficiently. It is necessary to develop a computational method that helps to redirect approved drugs. Fortunately, with the accumulation of multiple omics data and the development of machine learning methods, it is possible to mine the drugs' potential indications in silico. Up to now, many computational methods have been proposed to find new indictions of drugs by predicting potential treatment relationships of drug-disease pairs.
Based on the hypothesis that the gene expression signature of a particular drug is opposite to the gene expression signature of a disease, some gene expression based methods [8, 9] have been proposed. Noticing that such kind of methods may fail to consider the different roles of genes and their dependencies at the system level, system-level based approach that integrates the gene expressions and related network has recently been proposed [10].
Recently, along with the increase of drugs and diseases related multi-omics data, many methods have been proposed to integrate multiple sources of data to predict the drug-disease interactions based on machine learning techniques. Gottlieb et al. proposed a method (PREDICT) to predict new associations between drugs and diseases by integrating five drug-drug similarities and two disease-disease similarities data [11]. Wang et al. proposed a computational framework based on a three-layer heterogeneous network model (TL-HGBI) by integrating similarities and interactions among diseases, drugs and drug targets [12]. Luo et al. utilized some comprehensive similarities about drugs and diseases, and proposed a Bi-Random walk algorithm (MBiRW) to predict potential drug-disease interactions [13]. Martinez et al. developed a method named DrugNet for drug-disease and disease-drug priorization by integrating heterogeneous data [14]. Wu et al. integrated comprehensive drug-drug and disease-disease similarities from chemical/phenotype layer, gene layer and treatment network layer, and proposed a semi-supervised graph cut method (SSGC) to predict the drug-disease associations [15]. Moghadam et al. adopted the kernel fusion technique to combine different drug features and disease features, and then built SVM models to predict novel drug indications [16]. Liang et al. integrated drug chemical information, target domain information and gene ontology annotation information, and proposed a Laplacian regularized sparse subspace learning method (LRSSL) to predict drug-disease associations [17]. Zhang et al. introduced a linear neighborhood similarity [18] and a network topological similarity [19], then proposed a similarity constrained matrix factorization method (SCMFDD) to predict drug-disease associations by making use of known drug-disease associations, drug features and disease semantic information [20].
However, most of the existed methods are facing two main problems: one is that most of them are based on the hypothesis that similar drugs treat similar diseases, thus they need the similarity information between drugs, proteins, diseases, and so on. However, the similarity data can be not easily obtained. People often need to customize a program to collect data and to calculate the similarities so as to satisfy their own needs. Moreover, the calculation of similarity scores depends on the adopted measures, which may lead that the similarity score of a pair varies dramatically from one method to another. For example, two proteins are similar according to their structures, while they may be dissimilar according to their sequences. Even worse, some features required for calculating the similarities may be unknown or unavailable, resulting that these methods fail to work [21]. The other problem is that supervised learning based methods usually need both positive and negative samples to train the prediction models, whereas the drug-disease pair data, like other biological data, is lack of experimental validated negative samples. To train the models, most of the existing methods randomly select some unlabeled samples as the negative ones. Obviously, such strategy is very rough, for we are not sure whether there are some positive samples uncovered in the unlabeled data.
In this paper, we propose a method, called EMP-SVD (Ensemble Meta Paths and Singular Value Decomposition), to detect drug-disease treatment relations by using drug-disease, drug-protein and disease-protein interaction data. Unlike other methods, EMP-SVD needs no similarity information at all. In order to integrate different kinds of interaction data and consider different dependencies, we introduce five meta paths. For each meta path, we first generate a commuting matrix based on the corresponding interaction data, and then get latent features of drugs and diseases by using SVD (Singular Value Decomposition). All drug-disease pairs can be represented by the features. Finally, we train a base classifier by using the Random Forest algorithm. Five base classifiers are combined as an ensemble model to predict the drug-disease interactions. The framework of our method is shown in Fig. 1. In order to train out a more reliable prediction model, we select more likely negative ones from unlabeled samples under the assumption that non-associated drug and disease pair have no common interacted proteins, which is different from other methods. To evaluate our proposed method, we will compare it with the state-of-the-art methods, and also do case studies by literature investigation.
The framework of our proposed EMP-SVD
In this paper, we mainly made use of the interaction data of drug-disease, drug-protein and disease-protein to build the prediction model. We collected such data from DrugBank [3, 22, 23], OMIM [24] and Gottlieb's data set [11]. Concretely, we collected 4642 drug-protein interaction data from DrugBank, involving 1186 drugs and 1147 proteins; 1365 disease-protein interactions from OMIM, involving 449 diseases and 1147 proteins; and 1827 drug-disease interactions from Gottlieb's data set, involving 302 disease, 551 drugs. Obviously, the heterogenous network composed of drugs, proteins, diseases and the known interactions is sparse. The statistic of the data is shown in Table 1.
Table 1 Statistic information of the drug-protein-disease heterogenous network
Although our method does not need the similarity information, most of other machine learning based methods do need. For the convenience of comparison, we still collected the chemical structure of drugs and the sequence data of proteins from DrugBank. We computed the drug-drug chemical similarities according to their SMILES strings [25] via Openbabel tool [26], and the protein-protein similarities according to the sequence data by Smith-Waterman algorithm [27]. Moreover, we directly downloaded the disease-disease similarities from MimMiner [28].
Definitions and notations
In this section, we will give the formal definitions and notations used in this paper.
Definition 1
(Heterogeneous drug-protein-disease network schema). For a given heterogenous drug-protein-disease network G=(V,E), where V=D∪P∪S, D, P and S are the sets of drug, protein, disease nodes in the network respectively, while E=Ed,p∪Ep,d∪Ep,s∪Es,p∪Ed,s∪Es,d are the sets of heterogeneous links in G, which include the "binds to" link between drugs and proteins, "causes/caused by" link between proteins and diseases, "treats/treated by" link between drugs and diseases. The schema of G can be defined as \(M_{G} = (\mathcal {T,R})\), where \(\mathcal {T}=\{Drug, Protein, Disease\}\), \(\mathcal {R}=\{binds~to, cuases, caused~by, treats, treated~by\}\), \(\mathcal {T}\) and \(\mathcal {R}\) are the sets of node types and link types in G, respectively.
The network schema MG severs as a template of a network G. For a drug-protein-disease heterogenous network, the network schema is shown in Fig. 2.
Schema of drug-protein-disease heterogeneous network
(Heterogenous network meta path) Based on a given heterogenous network schema \(M_{G} = (\mathcal {T,R})\), \(\mathcal {P} = T_{1} \xrightarrow {R_{1}} T_{2}\xrightarrow {R_{2}}...\xrightarrow {R_{k-1}} T_{k} \) is defined to be a heterogenous network meta path in network G, where \(T_{i} \in \mathcal {T}\), i∈{1,2,...,k} and \(R_{i} \in \mathcal {R}\), i∈{1,2,...,k−1} and if (T1,T2,...,Tk are not all the same) ∨ (R1,R2,...,Rk−1 are not all the same).
For simplicity, we also omit the link types in denoting the meta path if there is no multiple links between the two types, for examples, \(\mathcal {P} = T_{1} \xrightarrow {} T_{2}\xrightarrow {}...\xrightarrow {} T_{k} \) denotes the meta path \(\mathcal {P} = T_{1} \xrightarrow {R_{1}} T_{2}\xrightarrow {R_{2}}...\xrightarrow {R_{k-1}} T_{k} \). The length of \(\mathcal {P}\) is the number of links in \(\mathcal {P}\).
(Commuting matrix [29]) Given a network G=(V,E) and its network schema MG, a commuting matrix for a meta path \(\mathcal {P} = T_{1} \xrightarrow {} T_{2}\xrightarrow {}...\xrightarrow {} T_{k} \) is defined as \(X = A_{T_{1} T_{2}} A_{T_{2} T_{3}}...A_{T_{k-1} T_{k}}\), where \(A_{T_{i} T_{j}}\) is the adjacency (interaction) matrix between type Ti and type Tj. X(i,j) represents the number of path instances between object ui∈T1 and object vj∈Tk under meta path \(\mathcal {P}\).
Since we want to detect the interactions between the drugs and the diseases, we only consider the cases of T1=Drug and Tk=Disease.
Now that there are only three kinds of nodes (drug, protein and disease) in the heterogenous network, we think the meta path with length greater than three may be too long to contribute to the prediction. Sun's work also has shown that short meta paths are good enough, and long meta paths may even reduce the quality [29]. Therefore, in this work, we only selected meta paths with length no longer than three. As a result, we select five meta paths described below.
Let Ads be the drug-disease interaction matrix, Adp be the drug-protein interaction matrix, and Asp be the disease-protein interaction matrix, we can get the commuting matrices of the five meta paths as follows:
Meta-path-1: Drug \(\xrightarrow {{treats}}\) Disease. The commuting matrix of it, denoted as X1, can be obtained by:
$$ X1 = A_{ds} $$
Meta-path-2: Drug \(\xrightarrow {{binds\ to}}\) Protein \(\xrightarrow {{causes}}\) Disease. The commuting matrix of it, denoted as X2, can be obtained by :
$$ X2 = A_{dp} \times A_{sp}^{T} $$
By using meta-path-2, we can integrate the drug-protein interaction information and the disease-protein interaction information, that is to say, we easily take the protein related information into account.
Meta-path-3: Drug \(\xrightarrow {{binds\ to}}\) Protein \(\xrightarrow {{binds\ to}}\) Drug \(\xrightarrow {treats}\) Disease. The commuting matrix of it, denoted as X3, can be obtained by:
$$ X3 = A_{dp} \times A_{dp}^{T} \times A_{ds} $$
By using meta-path-3, we can integrate drug-protein interaction and drug-disease interaction information. What's more, meta-path-3 also indicates that if two drugs share some common proteins, they may have similar indications.
Meta-path-4: Drug \(\xrightarrow {{treats}}\) Disease \(\xrightarrow {{treated\ by}}\) Drug \(\xrightarrow {{treats}}\) Disease. The commuting matrix of it, denoted as X4, can be obtained by :
$$ X4 = A_{ds} \times A_{ds}^{T} \times A_{ds} $$
By using meta-path-4, we can integrate the drug-disease interaction information. Besides, meta-path-4 also indicates that if two drugs share some common indications, then the indication of one drug may also be the potential indication of another drug.
Meta-path-5: Drug \(\xrightarrow {treats}\) Disease \(\xrightarrow {caused\ by}\) Protein \(\xrightarrow {causes}\) Disease. The commuting matrix of it, denoted as X5, can be obtained by :
$$ X5 = A_{ds} \times A_{sp} \times A_{sp}^{T} $$
By using meta-path-5, we can integrate the drug-disease interaction and the disease-protein interaction information. What's more, meta-path-5 also indicates that if two disease share some common proteins, the drug for treating one disease may also be the potential therapeutical drug for another disease.
As the definition, the element X(i,j) of the commuting matrix X denotes the number of path instances from drug di to disease sj under the corresponding meta path. We show an example in Fig. 3. There are two path instances from drug d3 to disease s2 under Meta-path-2, \(d_{3} \xrightarrow {} p_{3} \xrightarrow {} s_{2}\) and \(d_{3} \xrightarrow {} p_{5} \xrightarrow {} s_{2}\), thus we have X2(3,2)=2 in commuting matrix X2.
An example of the meaning of commuting matrix
Feature extraction with singular value decomposition
Now that element X(i,j) in a commuting matrix X denotes the number of path instances from the drug di to disease sj, then row i in the commuting matrix can be used as features of drug di, and column j can be used as features of disease sj. And we can use the concatenation of them to represent the drug-disease pair. Suppose there are m drugs and n diseases, we will have m+n (In this work, m=1186,n=449) features to represent the drug-disease pair. By contrast, the number of drug-disease pairs is small (We only have 1827 known interactions in this work). Obviously, the feature dimension is relatively high, which is not proper to construct a robust prediction model. Now that the singular value decomposition (SVD) has been successfully used to reduce the dimension in many researches, we also employed SVD to extract small number of features in our work.
By using SVD, the commuting matrix \(X \in \mathbb {R}^{m \times n}\) can be factorized into U, Σ and V such that
$$ X = U \Sigma V^{T} $$
where \(U \in \mathbb {R}^{m \times m}\), \(\Sigma \in \mathbb {R}^{m \times n}\) and \(V \in \mathbb {R}^{n \times n}\). The diagonal entries of Σ are equal to the singular values of X (Other elements in Σ other than diagonal entries are 0). The columns of U and V are, respectively, left- and right- singular vectors for the corresponding singular values.
As is known to all, the magnitude of the singular values represents the importance of the corresponding vectors; and in Σ, the singular values are ordered in descending order. Moreover, in most cases, the sum of the first 10% or even 1% of the singular values is over 99% of the total sum of all singular values. Specifically in this drug-disease associations prediction problem, in the biomedical meaning, the most useful information about drug and disease features will be included in the first 10% even less singular values. In the process of dimensionality reduction, the useful data will not be lost, but the redundant information will be discarded. That is to say, we can use the top r singular values to approximate the matrix X:
$$ X \approx U_{m \times r} \Sigma_{r \times r} {V^{T}}_{r \times n} $$
where r≪min(m,n).
Row i in U can be used as latent features of drug di, and row j in V can be used as latent features of disease sj. As a result, the dimension of the latent feature vector of each drug-disease pair can be reduced to 2∗r. In this work, we will introduce a parameter latent_feature_percent far less than 1 (say 1%, 2%,...) to control the value of r such that r=latent_feature_percent×min(m,n).
Selection of likely negative samples from unlabeled drug-disease pairs
To build a prediction model by using supervised learning, we need both positive and negative samples. The known drug-disease treatment relations are positive samples. Being lack of validated negative samples, most methods simply select some of unlabeled samples as negative ones by random. However, the unlabeled samples are not necessarily negative, some of them may be positive samples that still remain uncovered by experiments [30]. Different with other methods, we try to find more reliable negative samples from the unlabeled ones in this work.
If a drug shares some proteins with a disease, then the drug may have potential to treat the disease. Intuitively, if a drug and a disease have no common related proteins, we can think the disease is not the indication of the drug, and thus the drug-disease pair is more likely a negative sample. By this means, we can select out more reliable negative samples from the unlabeled pairs based on the drug-protein and disease-protein interactions information. The procedure is listed in Algorithm 1.
Construction and ensemble of classifiers
The five meta paths we have selected to integrate heterogeneous data reflect different aspects of the drug-disease treatment relationship, such as two drugs with common proteins having similar indications, two drugs sharing one common indication also sharing another indication, and so on. Thus we can build five base classifiers for the prediction of drug-disease treatment relations from different sides. In our work, the base classifiers are built based on the Random Forest algorithm which was implemented by using the RandomForestClassifier function in the scikit-learn package [31], we set the number of trees as 256.
Since ensemble learning can often help to improve the performances [32, 33], after the five base classifiers are constructed, we can obtain an ensemble classifier. For an input of drug-disease pair, each base classifier outputs two probabilities indicating that the pair being negative and positive respectively. Since we want to know whether the pair has treatment relation, we only take the positive probability as considered in the ensemble model.
For a drug-disease pair x with unknown label, suppose the predicted score (probability) of each base classifier be hi(x),i=1,2,...5, we used average strategy to get the final score of the ensemble model:
$$ H(x) = \frac{1}{5} \sum\limits_{i=1}^{5} h_{i}(x) $$
If H(x) is greater than a predetermined threshold, then the sample x is predicted as the positive. Because F1-measure is a comprehensive metric, in this work, we let the program automatically determine the threshold value when F1-measure reaches the maximum value, which is the same strategy as the other researchers used.
Experiments and results
We perform 5-fold cross validation to evaluate our method. Since the filtered negative samples are more than the positive ones, we randomly select a subset from them that with size equal to the positives, and use the balanced data to train the models. We first select the appropriate number of features according to the relationship of the model performance and the feature number. Then we did three kinds of evaluation experiments: (1) We investigate whether our negative samples filtering strategy can help to improve the prediction performance; (2) We compare EMP-SVD with other state-of-the-art methods by using the same data; (3) We check the practicality of our method by doing case studies.
Just as most other work, we performed 5-fold cross validation in the experiments. To evaluate performance of a method, there are some common metrics: Precison (PRE), Recall (REC), Accuracy (ACC), Matthews Correlation Coefficient (MCC) and F1-measure (F1). They can be calculated according to the following equations:
$$\begin{array}{*{20}l} PRE &= \frac{TP}{TP+FP} \end{array} $$
$$\begin{array}{*{20}l} REC &= \frac{TP}{TP+FN} \end{array} $$
$$\begin{array}{*{20}l} ACC &= \frac{TP+TN}{TP+FP+TN+FN} \end{array} $$
$$\begin{array}{*{20}l} MCC &= \frac{TP \times TN - FP \times FN}{ \sqrt{ (TP+FP)(TP+FN)(TN+FP)(TN+FN) }} \end{array} $$
$$\begin{array}{*{20}l} F_{1} &= \frac{2 \times PRE \times REC}{PRE+REC} \end{array} $$
where TP, FP, TN and FN denote the number of true positive samples, false positive samples, true negative samples and false negative samples, respectively.
Since Precision(PRE) and Recall(REC) have some conflicts, in general, a classifier gets a higher PRE will have a lower REC, and vise versa. To get a comprehensive performance, Area Under Precison-Recall Curve(AUPR) and Area Under Receiver Operating Characteristic Curve(AUC) are often used. AUPR takes both PRE and REC into account, AUC takes both the true positive rate(TPR, the same as REC) and the false positive rate (FPR) into account, so they are comprehensive metrics. At the same time, with the help of the curves we can intuitively find which classifier is better. Therefore, in this work, we adopted AUPR and AUC as the main metrics.
Determination of appropriate number of features
Parameters are often used in existing computational methods, which limits the generalization of a model. So, it will be better to use fewer parameters or to get an analytical solution.
In this work, we just need to determine the number of singular values (corresponding to the feature number that is controlled by the parameter latent_feature_percent) during the model construction, which is very different with most state-of-the-art methods. Just mentioned above r≪min(m,n), so we set latent_feature_percent as 1%, 2%, 3%,......, 20% respectively, and the performance curves of five base classifiers and the ensemble one with different latent_feature_percent are shown in Fig. 4. The results have shown that the performances of the ensemble classifier are better than other five base classifiers, illustrating that our ensemble rule is effective. Moreover, the performances of the six classifiers are robust across different parameter settings. Anyway, we set latent_feature_percent as 3% according to the curves in this work.
Influence of different latent_feature_percent on the a AUPR b AUC
We also find that the performances of classifiers based on meta-path-1 and meta-path-4 are the worst. Noticing that both meta-path-1 and meta-path-4 just take drug-disease interactions into consideration, while the other three meta paths contain more information on drug-protein or protein-disease interactions, we think integrating more interaction information into the meta path can help to improve the performance of the classifier.
Investigation of the filtering strategy of negative samples
Being lack of validated negative samples, most of the other methods randomly select unlabeled samples to be negative ones. However, the unlabeled samples are not necessarily negative, some of them may be positive samples still uncovered by experiments. So in this work we selected out more likely negative samples from unlabeled ones according to the common protein information (as described in Algorithm 1). As shown in Table 2, all the classifiers achieve better performances in most metrics when using our negative samples filtering strategy. We also noted that the improvement is little, which may due to the fact that the known drug-protein interactions and disease-protein interactions are too few (with density of 0.0034 and 0.0027, as shown in Table 1), resulting that very few proteins could be used in the filtering process. Anyway, our strategy for selecting more reliable negative samples is useful, feasible and interpretable. We believe that along with the increase of interactions data, we will get more reliable negative samples and thus achieve more great performance improvements.
Table 2 Performances comparison with different negative samples selecting strategies (random strategy is denoted "random", our strategy is "reliable")
Comparison with other methods
In this section, we compare EMP-SVD with state-of-the-art methods to demonstrate the superior performance of our method. PREDICT [11] and TL-HGBI method [12] are classical methods used to predict the drug-target and drug-disease interactions. MBiRW [13], LRSSL [17] and SCMFDD [20] are the methods proposed in these two years, and achieved high performance in the prediction of drug-disease interaction. So we choose these state-of-the-art methods to compare.
PREDICT calculates the score of a given drug-disease pair (dr,di) according to all the known drug-disease pairs \(\left (d_{r}^{\prime },d_{i}^{\prime }\right)\) associated with that given pair by equation \(Score(d_{r},d_{i})= {\underset {d_{r}^{\prime },d_{i}^{\prime } \neq d_{r},d_{i}}{\max }} \sqrt { S \left (d_{r},d_{r}^{\prime }\right) \times S \left (d_{i},d_{i}^{\prime }\right) }\), where \(S \left (d_{r},d_{r}^{\prime }\right)\) is drug-drug similarity and \(S\left (d_{i},d_{i}^{\prime }\right)\) is disease-disease similarity. TL-HGBI is a three layer heterogenous network model, which makes use of the similarities and interactions of drugs, diseases and targets by iterative update. MBiRW adjusts the similarities of drugs and diseases by correlation analysis and known drug-disease associations, then uses Bi-random walk algorithm to predict the potential drug-disease associations. LRSSL is a Laplacian regularized sparse subspace learning method used to predict the drug-disease associations which integrates drug chemical information, drug target domain information and target annotation information. SCMFDD is a similarity constrained matrix factorization method for the prediction of drug-disease associations by using known drug-disease interactions, drug features and disease semantic information.
We obtained the source code of PREDICT, TL-HGBI and SCMFDD from the authors, the code of MBiRW, LRSSL are publicly available, and the parameters were set according to their papers. The parameter latent_feature_percent in EMP-SVD was set 3%. To be fair, the five parts data were kept the same division in all methods when conducting 5-fold cross validation.
As shown in Table 3, compared with other five state-of-the-art methods which make use of several kinds of similarities as well as the interaction data, the proposed classifier EMP-SVD only uses the known interaction data but achieves better performances in most metrics, especially the comprehensive metrics (AUPR and AUC). To make it more intuitively, we plotted the Precison-Recall Curve and ROC curve, which are shown in Fig. 5a and b, respectively. The AUPR and AUC of the proposed EMP-SVD are 0.956 and 0.951, respectively, better than the compared methods. Hence, it shows the simplicity and effectiveness of our method.
a Precision-Recall Curve b ROC Curve of EMP-SVD and compared methods
Table 3 Performances of proposed EMP-SVD and state-of-the-art methods
Here, we test the practicality of EMP-SVD for predicting unknown associations. Except for training set composing of the known 1827 drug-disease associations and randomly selected 1827 negative samples by using our strategy, we used the trained EMP-SVD model to predict the associations for other unknown drug-disease pairs, and validate the results by literature investigation.
The new predicted top 20 drug-disease associations are shown in Table 4. We checked them carefully by literature validation and found that 13 of the top 20 predicted associations have been reported in the literatures. And these predicted associations were not originally in our data set, but we could find it out by our method, thus showing the practicality of our proposed EMP-SVD.
Table 4 The predicted drug-disease associations (Top 20)
It should be noted that Triamcinolone (DrugBank ID: DB00620) and Betamethasone (DrugBank ID: DB00443), as glucocorticoid, are commonly used in the treatment of various skin diseases such as "Eczema" [34–36], and we find that their predicted associations include the disease "Growth Retardation, Small And Puffy Hands And Feet, And Eczema" (OMIM ID:233810). During the process of literature validation, we also find a case of growth retardation and Cushing's syndrome due to excessive application of betamethasone-17-valerate ointment [37]. In a responsible attitude, we think that whether they can be used to treat the disease "Growth Retardation, Small And Puffy Hands And Feet, And Eczema", or the usage and dosage should be further carefully studied by the chemists and doctors, especially should be with caution when used on children and pregnant women.
In more details, we checked the predicted potential indications of drug "Amitriptyline" (DrugBank ID: DB00321). Amitriptyline is a tricyclic antidepressant which is often used to treat symptoms of depression with the brand name: Vanatrip, Elavil, Endep. As shown in Table 5, we can find literature evidences to support 8 diseases in the top 10 predictions for Amitriptyline.
Table 5 Top 10 predictions for the drug "Amitriptyline"
Breast cancer is a relatively common malignant tumor for female, which seriously endangers women's health and life safety. To discover the potential drugs is of great value. So we also checked the drug list that have been predicted to treat the disease "Breast Cancer" (OMIM ID: 114480). In the top 10 drugs, as shown in Table 6, we found that 8 have been reported to be used in the clinical treatment.
Table 6 Top 10 predictions for the disease "Breast Cancer"
Therefore, the case studies have further shown the practicality of the proposed method EMP-SVD.
Conclusions and discussions
To uncover the potential drug-disease associations is an important step in drug development, but it is time-consuming and costly to uncover them by wet experiments. Along with the accumulation of drug and disease related multi-omics data, as well as the development of machine learning techniques, more and more computational methods have been proposed to predict the potential drug-disease associations. To help the prediction, many methods integrate multiple source of data, including drugs, diseases, targets, side effects, and so on. They achieved good performances and could provide a helpful reference to the drug development. Most of them need the similarities of drug and disease related data. However, the similarity data can not be easily obtained, and people often need to customize a program to crawl data and to compute the similarities to satisfy their own need. Even worse, some features needed to calculate the similarity are unknown or unavailable. These methods will not work facing the incomplete data. Besides, being lack of validated negative samples in the prediction of drug-disease associations, most of the machine learning based methods assume the unlabeled samples to be negative ones in the training of the model. Such strategy may input errors because there may be positive samples uncovered in the unlabeled samples. What's more, most of the existing methods use many parameters in the data integration and the model construction. The parameters are difficult to tune, which limits the generalization ability of the method.
In this work, we proposed a method named EMP-SVD to predict drug-disease interactions based on ensemble meta paths and singular value decomposition. Five meta paths from source node (drug) to end node (disease) were selected to integrate the interaction information of drugs, proteins and diseases. Then the commuting matrices of these meta paths were calculated out, each element indicates the number of path instances between the corresponding drug and disease pair. By using singular value decomposition on the commuting matrices, we can extract small number of latent features of drugs and diseases. In order to get reliable negative samples, we selected those unlabeled samples as negative under the assumption that if a drug and a disease have no common proteins, then there is smaller probability for them to be treatment relationship. Based on each meta path we first built a base classifier, and then combined them to get an ensemble classifier. The experiments results have shown that our proposed EMP-SVD method outperformed several state-of-the-art methods. Better than other methods, EMP-SVD has few parameters and very easy to set. Further more, case studies have shown the predicted new associations could be useful for further biomedical research, which demonstrate the practicality of our method.
Although there are meta path based methods in social network and some other networks, to the best of our knowledge, it is the first work in the prediction of drug-disease associations by using ensemble meta paths and singular value decomposition. Different with many existing methods, we do not need the similarity data which are not easily obtained or sometimes unavailable or unknown. Instead, we just use the interaction data which can be easily accessed in many databases to build the prediction model. The other advantage of method is that there is only one parameter that can easily set. Though we use ensemble strategy to improve the performance, each of the five base classifiers can independently act as the model as well to predict the drug-disease interactions. Since there are many computational methods to predict the target proteins for a new drug such as docking methods. For a new drug which has no known interactions with any diseases, we still can predict its interacted diseases by building classifier using meta-path-2 by making use of drug-protein and protein-disease interactions.
Though the results of our methods are promising, there are still some limitations. Firstly, we only use the information of drugs, proteins and diseases, there are many other information could also be integrated in the further work, such as the information of side effects, pathways, tissues, and so on. Secondly, we only make use of common proteins to select out the negative samples, some other information such as gene expression data can also be used for this purpose. Or we can directly build the model by positive and unlabeled samples based learning method. We will address these issues in the future study.
Area under ROC curve
AUPR:
Area under Precison-Recall curve
FN:
FPR:
False positive rate
MCC:
Matthews correlation coefficient
Online Mendelian inheritance in man
PRE:
REC:
ROC:
Receiver operating characteristic
Simplified molecular input line entry specification
SVD:
Singular value decomposition
SVM:
Support vector machine
TN:
True negative
TP:
True positive
TPR:
True positive rate
UMLS:
Unified medical language system
Adams CP, Brantner VV. Estimating the cost of new drug development: is it really 802 million?Health Aff. 2006; 25(2):420–8.
Wu G, Liu J, Wang C. Semi-supervised graph cut algorithm for drug repositioning by integrating drug, disease and genomic associations. In: 2016 IEEE International Conference on Bioinformatics and Biomedicine (BIBM). Institute of Electrical and Electronics Engineers Inc.: 2016. p. 223–228.
Wishart DS, Feunang YD, Guo AC, et al.Drugbank 5.0: a major update to the drugbank database for 2018. Nucleic Acids Res. 2017; 46(D1):1074–82.
Article CAS Google Scholar
Bodenreider O. The unified medical language system (umls): integrating biomedical terminology. Nucleic Acids Res. 2004; 32(suppl_1):267–70.
Hurle M, Yang L, Xie Q, et al.Computational drug repositioning: from data to therapeutics. Clin Pharmacol Ther. 2013; 93(4):335–41.
Varothai S, Bergfeld WF. Androgenetic alopecia: an evidence-based treatment update. Am J Clin Dermatol. 2014; 15(3):217–30.
Novac N. Challenges and opportunities of drug repositioning. Trends Pharmacol Sci. 2013; 34(5):267–72.
Sirota M, Dudley JT, Kim J, et al.Discovery and preclinical validation of drug indications using compendia of public gene expression data. Sci Transl Med. 2011; 3(96):96ra77.
Jahchan NS, Dudley JT, Mazur PK, et al.A drug repositioning approach identifies tricyclic antidepressants as inhibitors of small cell lung cancer and other neuroendocrine tumors. Cancer Discov. 2013; 3(12):1364–77.
Peyvandipour A, Saberian N, Shafi A, et al.A novel computational approach for drug repurposing using systems biology. Bioinformatics. 2018; 34(16):2817–2825.
PubMed Article CAS PubMed Central Google Scholar
Gottlieb A, Stein GY, Ruppin E, et al.Predict: a method for inferring novel drug indications with application to personalized medicine. Mol Syst Biol. 2011; 7(1):496.
Wang W, Yang S, Zhang X, et al.Drug repositioning by integrating target information through a heterogeneous network model. Bioinformatics. 2014; 30(20):2923–30.
Luo H, Wang J, Li M, et al.Drug repositioning based on comprehensive similarity measures and bi-random walk algorithm. Bioinformatics. 2016; 32(17):2664–2671.
Martínez V, Navarro C, Cano C, et al.Drugnet: Network-based drug–disease prioritization by integrating heterogeneous data. Artif Intell Med. 2015; 63(1):41–9.
Wu G, Liu J, Wang C. Predicting drug-disease interactions by semi-supervised graph cut algorithm and three-layer data integration. BMC Med Genet. 2017; 10(5):79.
Moghadam H, Rahgozar M, Gharaghani S. Scoring multiple features to predict drug disease associations using information fusion and aggregation. SAR QSAR Environ Res. 2016; 27(8):609–28.
Liang X, Zhang P, Yan L, et al.Lrssl: predict and interpret drug-disease associations based on data integration using sparse subspace learning. Bioinformatics (Oxford, England). 2017; 33(8):1187–1196.
Zhang W, Yue X, Liu F, et al.A unified frame of predicting side effects of drugs by using linear neighborhood similarity. BMC Syst Biol. 2017; 11(6):101.
Zhang W, Yue X, Huang F, et al.Predicting drug-disease associations and their therapeutic function based on the drug-disease association bipartite network. Methods. 2018; 145:51–59.
Zhang W, Yue X, Lin W, et al.Predicting drug-disease associations by using similarity constrained matrix factorization. BMC Bioinforma. 2018; 19(1):233.
Zhang W, Yue X, Chen Y, et al.Predicting drug-disease associations based on the known association bipartite network. In: 2017 IEEE International Conference on Bioinformatics and Biomedicine (BIBM). Institute of Electrical and Electronics Engineers Inc.: 2017. p. 503–9.
Knox C, Law V, Jewison T, et al.Drugbank 3.0: a comprehensive resource for 'omics' research on drugs. Nucleic Acids Res. 2011; 39(suppl 1):1035–41.
Law V, Knox C, Djoumbou Y, et al.Drugbank 4.0: shedding new light on drug metabolism. Nucleic Acids Res. 2013; 42(D1):1091–7.
Hamosh A, Scott AF, Amberger JS, et al.Online mendelian inheritance in man (omim), a knowledgebase of human genes and genetic disorders. Nucleic Acids Res. 2005; 33(suppl 1):514–7.
Weininger D. Smiles, a chemical language and information system. 1. introduction to methodology and encoding rules. J Chem Inf Comput Sci. 1988; 28(1):31–6.
O'Boyle NM, Banck M, James CA, et al.Open babel: An open chemical toolbox. J Cheminformatics. 2011; 3(1):33.
Smith TF, Waterman MS, Burks C. The statistical distribution of nucleic acid similarities. Nucleic Acids Res. 1985; 13(2):645–56.
Van Driel MA, Bruggeman J, Vriend G, et al.A text-mining analysis of the human phenome. Eur J Hum Genet. 2006; 14(5):535–42.
Sun Y, Han J, Yan X, et al.Pathsim: Meta path-based top-k similarity search in heterogeneous information networks. Proc VLDB Endowment. 2011; 4(11):992–1003.
Wu G, Liu J, Min W. Prediction of drug-disease treatment relations based on positive and unlabeled samples. J Intell Fuzzy Syst. 2018; 35(2):1363–73.
Pedregosa F, Varoquaux G, Gramfort A, et al.Scikit-learn: Machine learning in python. J Mach Learn Res. 2011; 12(Oct):2825–30.
Guan D, Yuan W, Lee YK, et al.A review of ensemble learning based feature selection. Iete Tech Rev. 2014; 31(3):190–8.
Zhang W, Shi J, Tang G, Wu W, Yue X, Li D. Predicting small rnas in bacteria via sequence learning ensemble method. In: 2017 IEEE International Conference on Bioinformatics and Biomedicine (BIBM). Institute of Electrical and Electronics Engineers Inc.: 2017. p. 643–7.
Keczkes K, Frain-Bell W, Honeyman A, Sprunt G. The effect on adrenal function of treatment of eczema and psoriasis with triamcinolone acetonide. Br J Dermatol. 1967; 79(8–9):475–86.
Schmied C, Piletta P-A, Saurat J-H. Treatment of eczema with a mixture of triamcinolone acetonide and retinoic acid a double-blind study. Dermatology. 1993; 187(4):263–7.
Granlund H, Erkko P, Eriksson E, Reitamo S. Comparison of cyclosporine and topical betamethasone-17, 21-dipropionate in the treatment of severe chronic hand eczema. Acta Derm Venereol. 1996; 76(5):371–6.
Vermeer B, Heremans G. A case of growth retardation and cushings syndrome due to excessive application of betamethasone-17-valerate ointment. Dermatology. 1974; 149(5):299–304.
Sillanpää M, Pihlaja T. Oxcarbazepine (gp 47 680) in the treatment of intractable seizures. Acta Paediatr Hung. 1988; 29(3-4):359–64.
Virdi SK, Kanwar AS. Generalized morphea, lichen sclerosis et atrophicus associated with oral submucosal fibrosis in an adult male. Indian J Dermatol Venereol Leprology. 2009; 75(1):56.
Willemze R, Peters W, Van Hennik M, et al.Intermediate and high-dose ara-c and m-amsa (or daunorubicin) as remission and consolidation treatment for patients with relapsed acute leukaemia and lymphoblastic non-hodgkin lymphoma. Scand J Haematol. 1985; 34(1):83–7.
Richardson DS, Kelsey SM, Johnson SA, et al.Early evaluation of liposomal daunorubicin (daunoxome Ⓡ, nexstar) in the treatment of relapsed and refractory lymphoma. Investig New Drugs. 1997; 15(3):247–53.
Gustafson MP, Lin Y, New KC, et al.Systemic immune suppression in glioblastoma: the interplay between cd14+ hla-drlo/neg monocytes, tumor factors, and dexamethasone. Neuro-Oncol. 2010; 12(7):631–44.
Hultén K, Jaup B, Stenquist B, Engstrand L. Combination treatment with ranitidine is highly eff icient against helicobacter pylori despite negative impact of macrolide resistance. Helicobacter. 1997; 2(4):188–93.
Greenbaum-Lefkoe B, Rosenstock JG, Belasco JB, et al.Syndrome of inappropriate antidiuretic hormone secretion. a complication of high-dose intravenous melphalan. Cancer. 1985; 55(1):44–6.
Winick NJ, McKenna RW, Shuster JJ, et al.Secondary acute myeloid leukemia in children with acute lymphoblastic leukemia treated with etoposide. J Clin Oncol. 1993; 11(2):209–17.
Marks DI, Forman SJ, Blume KG, et al.A comparison of cyclophosphamide and total body irradiation with etoposide and total body irradiation as conditioning regimens for patients undergoing sibling allografting for acute lymphoblastic leukemia in first or second complete remission. Biol Blood Marrow Transplant. 2006; 12(4):438–53.
Christiaens G, Nieuwenhuis H, von Dem Borne AK, et al.Idiopathic thrombocytopenic purpura in pregnancy: a randomized trial on the effect of antenatal low dose corticosteroids on neonatal platelet count. BJOG: Int J Obstet Gynaecol. 1990; 97(10):893–8.
Ratain M, Rowley J. Therapy-related acute myeloid leukemia secondary to inhibitors of topoisomerase ii: from the bedside to the target genes. Ann Oncol. 1992; 3(2):107–11.
Rivera GK, Pui C-H, Abromowitch M, et al.Improved outcome in childhood acute lymphoblastic leukaemia with reinforced early treatment and rotational combination chemotherapy. Lancet. 1991; 337(8733):61–6.
Kamp O, Sieswerda GT, Visser CA. Comparison of effects on systolic and diastolic left ventricular function of nebivolol versus atenolol in patients with uncomplicated essential hypertension. Am J Cardiol. 2003; 92(3):344–8.
Williams B, MacDonald TM, Morant S, et al.Spironolactone versus placebo, bisoprolol, and doxazosin to determine the optimal treatment for drug-resistant hypertension (pathway-2): a randomised, double-blind, crossover trial. Lancet. 2015; 386(10008):2059–68.
Allen RP, Bharmal M, Calloway M. Prevalence and disease burden of primary restless legs syndrome: results of a general population survey in the united states. Mov Disord. 2011; 26(1):114–20.
Barrickman LL, Perry PJ, Allen A, et al.Bupropion versus methylphenidate in the treatment of attention-deficit hyperactivity disorder. J Am Acad Child Adolesc Psychiatry. 1995; 34(5):649–57.
Sinha R, Raut S. Management of nocturnal enuresis-myths and facts. World J Nephrol. 2016; 5(4):328.
Jeyakumar A, Brickman TM, Haben M. Effectiveness of amitriptyline versus cough suppressants in the treatment of chronic cough resulting from postviral vagal neuropathy. Laryngoscope. 2006; 116(12):2108–12.
Di Lorenzo C, Ambrosini A, Coppola G, Pierelli F. Heat stress disorders and headache: a case of new daily persistent headache secondary to heat stroke. J Neurol Neurosurg Psychiatry. 2008; 79(5):610–1.
Gawin FH, Markoff RA. Panic anxiety after abrupt discontinuation of amitriptyline. Am J Psychiatry. 1981; 138(1):117–118.
Beckman TJ, Mynderse LA. Evaluation and medical management of benign prostatic hyperplasia. In: Mayo Clinic Proceedings, vol. 80. Elsevier Inc.: 2005. p. 1356–62.
Cassidy J, Merrick MV, Smyth JF, Leonard RC. Cardiotoxicity of mitozantrone assessed by stress and resting nuclear ventriculography. Eur J Cancer. 1988; 24(5):935–8.
Yap H-Y, Blumenschein GR, Tashima CK, et al.Combination chemotherapy with vincristine and methotrexate for advanced refractory breast cancer. Cancer. 1979; 44(1):32–4.
Holland JF, Scharlau C, Gailani S, et al.Vincristine treatment of advanced cancer: a cooperative study of 392 cases. Cancer Res. 1973; 33(6):1258–64.
Gnant M, Mlineritsch B, Schippinger W, et al.Endocrine therapy plus zoledronic acid in premenopausal breast cancer. N Engl J Med. 2009; 360(7):679–91.
Mego M, Sycova-Mila Z, Obertova J, et al.Intrathecal administration of trastuzumab with cytarabine and methotrexate in breast cancer patients with leptomeningeal carcinomatosis. Breast. 2011; 20(5):478–80.
Hines SL, Mincey BA, Sloan JA, et al.Risedronate, breast cancer. J Clin Oncol Off J Am Soc Clin Oncol. 2009; 27(7):1047.
Fenner MH, Elstner E. Peroxisome proliferator-activated receptor- γ ligands for the treatment of breast cancer. Expert Opin Investig Drugs. 2005; 14(6):557–68.
Ma CX, Ellis MJ, Petroni GR, et al.A phase ii study of ucn-01 in combination with irinotecan in patients with metastatic triple negative breast cancer. Breast Cancer Res Treat. 2013; 137(2):483–92.
Thamake SI, Raut SL, Gryczynski Z, et al.Alendronate coated poly-lactic-co-glycolic acid (plga) nanoparticles for active targeting of metastatic breast cancer. Biomaterials. 2012; 33(29):7164–73.
Vaughan W, Reed E, Edwards B, Kessinger A. High-dose cyclophosphamide, thiotepa and hydroxyurea with autologous hematopoietic stem cell rescue: an effective consolidation chemotherapy regimen for early metastatic breast cancer. Bone Marrow Transplant. 1994; 13(5):619–24.
The authors thank the editors and the anonymous reviewers for their helpful comments and suggestions on the quality improvement of our present paper.
Publication of this article was sponsored by grant from the National Science Foundation of China [61272274, 60970063]; the program for New Century Excellent Talents in Universities [NCET-10-0644]; and the National Science Foundation of Jiangsu Province [BK20161249].
The data supporting the results of this research paper are included within this article.
About this supplement
This article has been published as part of BMC Bioinformatics Volume 20 Supplement 3, 2019: Selected articles from the 17th Asia Pacific Bioinformatics Conference (APBC 2019): bioinformatics. The full contents of the supplement are available online at https://bmcbioinformatics.biomedcentral.com/articles/supplements/volume-20-supplement-3.
School of Computer Science, Wuhan University, Wuhan, 430072, People's Republic of China
Guangsheng Wu, Juan Liu & Xiang Yue
Suzhou Institute of Wuhan University, Suzhou, 215123, People's Republic of China
Juan Liu
Department of Computer Science and Engineering, The Ohio State University, Ohio, 43210, USA
Xiang Yue
Guangsheng Wu
GW, JL and XY developed the methodology. GW and XY executed the experiments, JL provided guidance and supervision. JL and GW wrote this paper. All authors read and approved the final manuscript.
Correspondence to Juan Liu.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Wu, G., Liu, J. & Yue, X. Prediction of drug-disease associations based on ensemble meta paths and singular value decomposition. BMC Bioinformatics 20, 134 (2019). https://doi.org/10.1186/s12859-019-2644-5
Drug repositioning
Meta path
Commuting matrix
Submission enquiries: [email protected] | CommonCrawl |
The two-component Novikov-type systems with peaked solutions and $ H^1 $-conservation law
The boundedness of multi-linear and multi-parameter pseudo-differential operators
Elliptic problems with rough boundary data in generalized Sobolev spaces
Anna Anop 1, , Robert Denk 2,, and Aleksandr Murach 1,
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01024, Ukraine
University of Konstanz, Department of Mathematics and Statistics, 78457 Konstanz, Germany
Fund Project: The publication contains the results of studies conducted by the joint grant F81 of the National Research Fund of Ukraine and the German Research Society (DFG); competitive project F81/41686. This work was supported by the Grant H2020-MSCA-RISE-2019, project number 873071 (SOMPATY: Spectral Optimization: From Mathematics to Physics and Advanced Technology). The first author was supported by President of Ukraine's grant for competitive project F82/45932
We investigate regular elliptic boundary-value problems in boun\-ded domains and show the Fredholm property for the related operators in an extended scale formed by inner product Sobolev spaces (of arbitrary real orders) and corresponding interpolation Hilbert spaces. In particular, we can deal with boundary data with arbitrary low regularity. In addition, we show interpolation properties for the extended scale, embedding results, and global and local a priori estimates for solutions to the problems under investigation. The results are applied to elliptic problems with homogeneous right-hand side and to elliptic problems with rough boundary data in Nikolskii spaces, which allows us to treat some cases of white noise on the boundary.
Keywords: Elliptic boundary-value problem, generalized Sobolev space, rough boundary data, Fredholm property, a priory estimate of solution, boundary white noise.
Mathematics Subject Classification: Primary:35J40;Secondary:35R60, 46E35, 60H40.
Citation: Anna Anop, Robert Denk, Aleksandr Murach. Elliptic problems with rough boundary data in generalized Sobolev spaces. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2020286
M. S. Agranovich, Elliptic boundary problems, Encyclopaedia Math. Sci., 79 (1997), 1-144. doi: 10.1007/978-3-662-06721-5_1. Google Scholar
A. V. Anop and T. M. Kasirenko, Elliptic boundary-value problems in Hörmander spaces, Methods Funct. Anal. Topology, 22 (2016), 295-310. Google Scholar
A. V. Anop, T. M. Kasirenko and O. O. Murach, Irregular elliptic boundary-value problems and Hörmander spaces, Ukranian Math. J., 70 (2018), 341-361. doi: 10.1007/s11253-018-1504-1. Google Scholar
A. V. Anop and A. A. Murach, Regular elliptic boundary-value problems in the extended Sobolev scale, Ukrainian Math. J., 66 (2014), 969-985. doi: 10.1007/s11253-014-0988-6. Google Scholar
A. V. Anop and A. A. Murach, To the theory of elliptic boundary-value problems in Hörmander spaces, Transactions of Institute of Mathematics of NAS of Ukraine, 12 (2015), 39-64. Google Scholar
A. V. Anop and A. A. Murach, Some semi-homogeneous elliptic boundary-value problems in complete extended Sobolev scale, Transactions of Institute of Mathematics of NAS of Ukraine, 13 (2016), 27-54. Google Scholar
A. V. Anop and A. A. Murach, Homogeneous elliptic equations in an extended Sobolev scale, Dopov. Nac. Akad. Nauk Ukr., (2018), 3–11. doi: 10.15407/dopovidi2018.03.003. Google Scholar
V. G. Avakumović, O jednom O-inverznom stavu, Rad Jugoslav. Akad. Znan. Umjet., 254 (1936), 167-186. Google Scholar
Yu. M. Berezansky, Expansions in Eigenfunctions of Selfadjoint Operators, American Mathematical Society, Providence, RI, 1968. Google Scholar
J. Behrndt, S. Hassi and H. de. Snoo, Boundary Value Problems, Weyl Functions, and Differential Operators, Springer, Cham, 2020. doi: 10.1007/978-3-030-36714-5. Google Scholar
[11] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1989. Google Scholar
V. V. Buldygin, K. H. Indlekofer, O. I. Klesov and J. G. Steinebach, Pseudo-Regularly Varying Functions and Generalized Renewal Processes, Springer, Cham, 2018. doi: 10.1007/978-3-319-99537-3. Google Scholar
J. Fageot, A. Fallah and M. Unser, Multidimensional Lévy white noise in weighted Besov spaces, Stochastic Process. Appl., 127 (2017), 1599-1621. doi: 10.1016/j.spa.2016.08.011. Google Scholar
M. Faierman, Fredholm theory for an elliptic differential operator defined on $\mathbb R^n$ and acting on generalized Sobolev spaces, Commun. Pure Appl. Anal., 19 (2020), 1463-1483. doi: 10.3934/cpaa.2020074. Google Scholar
C. Foiaş and J. L. Lions, Sur certains théorèmes d'interpolation, Acta Scient. Math. Szeged, 22 (1961), 269-282. Google Scholar
F. Gesztesy and M. Mitrea, Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains, Proc. Sympos. Pure Math., 79 (2008), 105-173. doi: 10.1090/pspum/079/2500491. Google Scholar
G. Geymonat, Sul problema di Dirichlet per le equazoni lineari ellittiche, Ann. Sci. Norm. Sup. Pisa, 16 (1962), 225-284. Google Scholar
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (revised 3rd edition), Springer, Berlin, 1998. Google Scholar
M. L. Gol'dman, Imbedding theorems for anisotropic Nikol'skiǐ–Besov spaces with moduli of continuity of a general type, Proc. Steklov Inst. Math., 170 (1987), 95-116. Google Scholar
L. Hörmander, On the theory of general partial differential equations, Acta Math., 94 (1955), 161-248. doi: 10.1007/BF02392492. Google Scholar
L. Hörmander, Linear Partial Differential Operators, Springer, Berlin, 1963. Google Scholar
L. Hörmander, The Analysis of Linear Partial Differential Operators. Vol. Ⅱ. Differential Operators with Constant Coefficients, Springer, Berlin, 1983. doi: 10.1007/978-3-642-96750-4. Google Scholar
L. Hörmander, The Analysis of Linear Partial Differential Operators. Vol. Ⅲ. Pseudo-Differential Operators, Springer, Berlin, 1985. Google Scholar
T. Kasirenko, V. Mikhailets and A. Murach, Sobolev-like Hilbert spaces induced by elliptic operators, Complex Anal. Oper. Theor., 13 (2019), 1431-1440. doi: 10.1007/s11785-018-00886-8. Google Scholar
V. A. Kozlov, V. G. Maz'ya and J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/052. Google Scholar
J. L. Lions and E. Magenes, Problémes aux limites non homogénes, Ⅱ, Ann. Inst. Fourier (Grenoble), 11 (1961), 137-178. Google Scholar
J. L. Lions and E. Magenes, Problémes aux limites non homogénes, V, Ann. Scuola Norm. Sup. Pisa (3), 16 (1962), 1-44. Google Scholar
J. L. Lions and E. Magenes, Problémes aux limites non homogénes, VI, J. Analyse Math., 11 (1963), 165-188. doi: 10.1007/BF02789983. Google Scholar
J. L. Lions and E. Magenes, Non-Homogeneous Boundary-Value Problems and Applications, vol. I, Springer, Berlin, 1972. Google Scholar
V. Los, V. A. Mikhailets and A. A. Murach, An isomorphism theorem for parabolic problems in Hörmander spaces and its applications, Commun. Pure Appl. Anal., 16 (2017), 69-97. doi: 10.3934/cpaa.2017003. Google Scholar
B. Malgrange, Sur une classe d'opératuers différentiels hypoelliptiques, Bull. Soc. Math., 85 (1957), 283-306. Google Scholar
E. Magenes, Spazi di interpolazione ed equazioni a derivate parziali, Atti Ⅶ Congr. Un. Mat., (1965), 134–197. Google Scholar
W. Matuszewska, On a generalization of regularly increasing functions, Studia Math., 24 (1964), 271-279. doi: 10.4064/sm-24-3-271-279. Google Scholar
V. A. Mikhailets and A. A. Murach, Elliptic operators in a refined scale of function spaces, Ukrainian. Math. J., 57 (2005), 817-825. doi: 10.1007/s11253-005-0231-6. Google Scholar
V. A. Mikhailets and A. A. Murach, Refined scales of spaces, and elliptic boundary-value problems. Ⅱ, Ukrainian Math. J., 58 (2006), 398-417. doi: 10.1007/s11253-006-0074-9. Google Scholar
V. A. Mikhailets and A. A Murach, Regular elliptic boundary-value problem for a homogeneous equation in a two-sided improved scale of spaces, Ukrainian Math. J., 58 (2006), 1748-1767. doi: 10.1007/s11253-006-0166-6. Google Scholar
V. A. Mikhailets and A. A. Murach, Refined scales of spaces, and elliptic boundary-value problems. Ⅲ, Ukrainian Math. J., 59 (2007), 744-765. doi: 10.1007/s11253-007-0048-6. Google Scholar
V. A. Mikhailets and A. A. Murach, Extended Sobolev scale and elliptic operators, Ukrainian Math. J., 65 (2013), 435-447. doi: 10.1007/s11253-013-0787-5. Google Scholar
V. A. Mikhailets and A. A. Murach, Hörmander Spaces, Interpolation and Elliptic Problems, De Gruyter, Berlin, 2014. doi: 10.1515/9783110296891. Google Scholar
V. A. Mikhailets and A. A. Murach, Interpolation Hilbert spaces between Sobolev spaces, Results Math., 67 (2015), 135-152. doi: 10.1007/s00025-014-0399-x. Google Scholar
A. A. Murach, Extension of some Lions–Magenes theorems, Methods Funct. Anal. Topology, 15 (2009), 152-167. Google Scholar
S. M. Nikol'skii, Approximation of Functions of Several Variables and Imbedding Theorems [2nd edition], Springer-Verlag, New York–Heidelberg, 1975. Google Scholar
V. I. Ovchinnikov, The methods of orbits in interpolation theory, in Mathematical Reports. Vol. 1, Part. 2 (ed. J. Peetre), Harwood Academic Publishers, London, 1984. Google Scholar
J. Peetre, Another approach to elliptic boundary problems, Commun. Pure Appl. Math., 14 (1961), 711-731. doi: 10.1002/cpa.3160140404. Google Scholar
J. Peetre, On interpolation functions. Ⅱ, Acta Sci. Math., 29 (1968), 91-92. Google Scholar
A. Pliś, A smooth linear elliptic differential equation without any solution in a sphere, Commun. Pure Appl. Math., 14 (1961), 599-617. doi: 10.1002/cpa.3160140331. Google Scholar
Ya. A. Roitberg, Elliptic problems with nonhomogeneous boundary conditions and local increase of smoothness up to the boundary for generalized solutions, Dokl. Math., 5 (1964), 1034-1038. Google Scholar
Ya. Roitberg, Elliptic Boundary Value Problems in the Spaces of Distributions, Kluwer Academic Publishers, Dordrecht, 1996. doi: 10.1007/978-94-011-5410-9. Google Scholar
Ya. Roitberg, Boundary Value Problems in the Spaces of Distributions, Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-94-015-9275-8. Google Scholar
R. Schnaubelt and M. Veraar, Stochastic equations with boundary noise, Progr. Nonlinear Differ. Equ. Appl., 80 (2011), 609-629. doi: 10.1007/978-3-0348-0075-4_30. Google Scholar
R. T. Seeley, Singular integrals and boundary-value problems, Amer. J. Math., 88 (1966), 781-809. doi: 10.2307/2373078. Google Scholar
E. Seneta, Regularly Varying Functions, Springer, Berlin, 1976. Google Scholar
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators [2nd edition], Johann Ambrosius Barth, Heidelberg, 1995. Google Scholar
M. Veraar, Regularity of Gaussian white noise on the $d$-dimensional torus, Banach Center Publ., 95 (2011), 385-398. doi: 10.4064/bc95-0-24. Google Scholar
L. R. Volevich and B. P. Paneah, Certain spaces of generalized functions and embedding theorems, Uspehi Mat. Nauk, 20 (1965), 3-74. Google Scholar
T. N. Zinchenko and A. A. Murach, Douglis–Nirenberg elliptic systems in Hörmander spaces, Ukrainian Math. J., 64 (2012), 1672-1687. doi: 10.1007/s11253-013-0743-4. Google Scholar
Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020340
Marek Macák, Róbert Čunderlík, Karol Mikula, Zuzana Minarechová. Computational optimization in solving the geodetic boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 987-999. doi: 10.3934/dcdss.2020381
Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381
Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020398
Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, 2021, 14 (1) : 149-174. doi: 10.3934/krm.2020052
Kazunori Matsui. Sharp consistency estimates for a pressure-Poisson problem with Stokes boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1001-1015. doi: 10.3934/dcdss.2020380
Xinfu Chen, Huiqiang Jiang, Guoqing Liu. Boundary spike of the singular limit of an energy minimizing problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3253-3290. doi: 10.3934/dcds.2020124
Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387
Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020453
Qianqian Hou, Tai-Chia Lin, Zhi-An Wang. On a singularly perturbed semi-linear problem with Robin boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 401-414. doi: 10.3934/dcdsb.2020083
Chueh-Hsin Chang, Chiun-Chuan Chen, Chih-Chiang Huang. Traveling wave solutions of a free boundary problem with latent heat effect. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021028
Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020442
Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1749-1762. doi: 10.3934/dcdsb.2020318
Xiaofeng Ren, David Shoup. The impact of the domain boundary on an inhibitory system: Interior discs and boundary half discs. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3957-3979. doi: 10.3934/dcds.2020048
Chang-Yeol Jung, Roger Temam. Interaction of boundary layers and corner singularities. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 315-339. doi: 10.3934/dcds.2009.23.315
Huijuan Song, Bei Hu, Zejia Wang. Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 667-691. doi: 10.3934/dcdsb.2020084
Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571
Anna Anop Robert Denk Aleksandr Murach | CommonCrawl |
Continuous Functions: Definition, Examples, and Properties
Rachel McLean
In this article, we'll discuss the definition of a continuous function, how to prove continuity, and learn the different properties of continuous functions. In addition, we'll review continuous graph examples to solidify your understanding of continuous and discontinuous functions.
What is a Continuous Function?
What is a Discontinuous Function?
Properties of Continuous Functions
Theorems for Continuous Functions
A function is continuous everywhere if you can trace its curve on a graph without lifting your pencil. A function is discontinuous at a point if you cannot trace its curve without lifting your pencil at that point; meaning it has a hole, break, jump, or vertical asymptote at that point.
For example, the function f(x)=2sin(x)f(x) = 2\sin{(x)}f(x)=2sin(x) is continuous everywhere. We can draw its curve without ever lifting our hand. By contrast, the function f(x)=1x−2f(x) = \frac{1}{x-2}f(x)=x−21 has a discontinuity at x=2x = 2x=2. We can't draw its curve without lifting our pencil at x=2x = 2x=2.
In differential calculus, it's important to understand the concept of continuity because functions that are not continuous are not differentiable.
Let's learn how to prove a function is continuous at a point. Here's the formal definition of continuity at a point.
A function fff is continuous at the point x=ax = ax=a if:
f(a)f(a)f(a) exists
limx→af(x)\lim_{x\to a}f(x)limx→af(x) exists
f(a)=limx→af(x)f(a) = \lim_{x\to a}f(x)f(a)=limx→af(x)
In order to show that a function is continuous at a point aaa, you must show that all three of the above conditions are true. To refresh your knowledge of evaluating limits, you can review How to Find Limits in Calculus and What Are Limits in Calculus.
For example, let's show that f(x)=x2−3f(x) = x^2 - 3f(x)=x2−3 is continuous at x=1x = 1x=1.
f(1)=12−3=−2f(1) = 1^2 - 3 = -2f(1)=12−3=−2
limx→1f(x)=−2\lim_{x\to 1}f(x) = -2limx→1f(x)=−2
f(1)=limx→1f(x)=−2f(1) = \lim_{x\to 1}f(x) = -2f(1)=limx→1f(x)=−2
Thus, f(x)=x2−3f(x) = x^2 - 3f(x)=x2−3 is continuous at x=1x = 1x=1. In fact, we could show that f(x)=x2−3f(x) = x^2 - 3f(x)=x2−3 is continuous everywhere.
Other functions might be continuous only over a specific interval of the real numbers. If a function is continuous on an open interval, that means that the function is continuous at every point inside the interval.
For example, f(x)=tan(x)f(x) = \tan{(x)}f(x)=tan(x) has a discontinuity over the real numbers at x=π2x = \frac{\pi}{2}x=2π, since we must lift our pencil in order to trace its curve. However, we can say that f(x)=tan(x)f(x) = \tan{(x)}f(x)=tan(x) is continuous on the open interval (−π2,π2)(-\frac{\pi}{2}, \frac{\pi}{2})(−2π,2π) since it is continuous at every point inside that specific interval.
We can also say that f(x)=tan(x)f(x) = \tan{(x)}f(x)=tan(x) is continuous over its own domain, which is any real number excluding odd multiples of π2\frac{\pi}{2}2π.
A discontinuity is a hole, jump, break, or vertical asymptote on a function's curve.
There are 3 types of discontinuities.
Jump Discontinuities
If a function has a jump discontinuity at some point aaa, then limx→af(x)\lim_{x\to a}f(x)limx→af(x) does not exist. Remember that in order for a limit to exist, its one-sided limits must exist and they must equal the same value. In other words, the limit as xxx approaches aaa from the left must equal the limit as xxx approaches aaa from the right.
In functions with jump discontinuities, limx→a+f(x)≠limx→a−f(x)\lim_{x\to a^+}f(x) \neq \lim_{x\to a^-}f(x)limx→a+f(x)=limx→a−f(x). For example, in the function above, limx→2+f(x)=5\lim_{x\to 2^+}f(x) = 5limx→2+f(x)=5 and limx→2−f(x)=7\lim_{x\to 2^-}f(x) = 7limx→2−f(x)=7. Thus, limx→2f(x)\lim_{x\to 2}f(x)limx→2f(x) does not exist, and so there is a discontinuity at x=2x = 2x=2.
Removable Discontinuities
If a function has a removable discontinuity at some point aaa, then limx→af(x)≠f(a)\lim_{x\to a}f(x) \neq f(a)limx→af(x)=f(a). On a graph, this looks like a hole. In these discontinuities, the one-sided limits as xxx approaches aaa always equal each other. However, the function's value at x=ax = ax=a equals something different or might not exist at all.
For example, in the function above, limx→2f(x)=4\lim_{x\to 2}f(x) = 4limx→2f(x)=4. However, f(2)=2f(2) = 2f(2)=2. Since limx→2f(x)≠f(2)\lim_{x\to 2}f(x) \neq f(2)limx→2f(x)=f(2), the function has a discontinuity at x=2x = 2x=2.
Infinite Discontinuities
If a function has an infinite discontinuity at some point aaa, then the function has a vertical asymptote at x=ax = ax=a. If any one of the following statements are true, then fff has a vertical asymptote at x=ax = ax=a.
limx→a+f(x)=+∞\lim_{x\to a^+ }f(x) = +\inftylimx→a+f(x)=+∞
limx→a+f(x)=−∞\lim_{x\to a^+ }f(x) = -\inftylimx→a+f(x)=−∞
limx→a−f(x)=+∞\lim_{x\to a^- }f(x) = +\inftylimx→a−f(x)=+∞
limx→a−f(x)=−∞\lim_{x\to a^- }f(x) = -\inftylimx→a−f(x)=−∞
For example, in the function above, there is a vertical asymptote at x=−3x = -3x=−3 and x=0x = 0x=0. Thus, there is an infinite discontinuity at x=−3x = -3x=−3 and x=0x = 0x=0.
If fff and ggg are both continuous at x=cx = cx=c, then the following properties are true:
The sum (f+g)x=f(x)+g(x)(f+g)x = f(x) + g(x)(f+g)x=f(x)+g(x) is continuous at x=cx = cx=c.
The difference (f−g)x=f(x)−g(x)(f-g)x = f(x) - g(x)(f−g)x=f(x)−g(x) is continuous at x=cx = cx=c.
The product (f⋅g)x=f(x)⋅g(x)(f \cdot g)x = f(x) \cdot g(x)(f⋅g)x=f(x)⋅g(x) is continuous at x=cx = cx=c.
The quotient (fg)(x)=f(x)g(x)(\frac{f}{g})(x) = \frac{f(x)}{g(x)}(gf)(x)=g(x)f(x) is continuous, provided g(x)≠0g(x) \neq 0g(x)=0.
The constant multiple k⋅f(x)k \cdot f(x)k⋅f(x) is continuous at x=cx = cx=c for any number kkk.
The composition (f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f∘g)(x)=f(g(x)) is continuous at ccc if fff is continuous at g(c)g(c)g(c).
Extreme Value Theorem
The Extreme Value Theorem states that if a function is continuous on the closed interval [a,b][a,b][a,b], then the function must have both a maximum and a minimum on [a,b][a, b][a,b].
Intermediate Value Theorem
The Intermediate Value Theorem is an extremely useful theorem in math. It's often used to prove that different equations are solvable. It's especially useful for proving that a function has a root on a particular interval. The root of a function is the point at which a function equals zero and crosses the x-axis. The Intermediate Value Theorem states:
Suppose f is a continuous function defined on [a, b], and let s be a number such that f(a) < s < f(b). Then, there must exist some x between a and b such that f(x) = s.
More simply, the Intermediate Value Theorem says that a continuous function must take on every value between f(a)f(a)f(a) and f(b)f(b)f(b) at least once on the interval [a,b][a, b][a,b].
For example, consider the graph of f(x)=2xf(x) = 2xf(x)=2x in the above graph. Let's examine the interval [a,b][a, b][a,b] where a=1a = 1a=1 and b=3b = 3b=3. Since f(a)=2f(a) = 2f(a)=2 and f(b)=6f(b) = 6f(b)=6, we'll choose an in-between value s=4s = 4s=4 for our s-value.
Then, since f(x)=2xf(x) = 2xf(x)=2x is continuous on [1,3][1, 3][1,3], the Intermediate Value Theorem guarantees that there must exist some xxx on [1,3][1, 3][1,3] such that f(x)=4f(x) = 4f(x)=4. By looking at the graph, we can see that x=2x = 2x=2 is the value for which f(x)=s=4f(x) = s = 4f(x)=s=4.
Polynomial Function
Any polynomial function is continuous for all real numbers. A polynomial function is a function consisting of variables and coefficients that involves only non-negative exponents of the variable. Polynomials use only addition, subtraction, and multiplication operations. For example, f(x)=7x3+x2−5f(x) = 7x^3 + x^2 - 5f(x)=7x3+x2−5 is a polynomial function.
Differentiable Function
Every differentiable function is continuous. However, be careful to remember that the converse is not necessarily true. A function could be continuous, but not differentiable. For example, the absolute value function f(x)=∣x∣f(x) = \mid x \midf(x)=∣x∣ below is continuous at x=0x = 0x=0 but not differentiable at x=0x = 0x=0.
Rational, root, trigonometric, exponential, and logarithmic functions are all continuous in their domains. The domain of a function is the set of values that a function can accept as inputs. Many real life examples of continuous functions can be modeled using these function types.
Rational Functions
A rational function is a function that is written as the ratio of two polynomial functions. The domain of rational functions is all numbers except those that make the denominator zero. So, the values where rational functions have vertical asymptotes or removable discontinuities are outside of their domain.
The trigonometric functions sin(x)\sin(x)sin(x) and cos(x)\cos(x)cos(x) have domains that include all real numbers. So, they are continuous for all real numbers.
Other trigonometric functions such as $\tan(x)$ have more selective domains — for example, the domain of tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x) is equal to all real numbers except for where cos(x)=0\cos(x) = 0cos(x)=0. This occurs at every odd multiple of π2\frac{\pi}{2}2π, and so these x-values are outside the domain of tan(x)\tan(x)tan(x).
Exponential functions have the form f(x)=abxf(x) = ab^xf(x)=abx, where a≠0a \neq 0a=0 and bbb is a real number greater than 1. The domain of exponential functions is all real numbers.
Logarithmic functions are only defined for positive inputs. So, the domain of logarithmic functions can be determined by solving the inequality that sets the inside terms to be greater than 0.
The mathematics of change.
Calculating p-Value in Hypothesis Testing
In this article, we'll take a deep dive on p-values, beginning with a description and definition of this key component of statistical hypothesis testing, before moving on to look at how to calculate it for different types of variables.
Derivatives in Math: Definition and Rules
As one of the fundamental operations in calculus, derivatives are an enormously useful tool for measuring rates of change. In this article, we'll first take a high-level view of how derivative rules work, and then dig deeper to examine a number of common rules.
Drew Zemke
What Is the Power Rule?
The Power Rule is one of the fundamental derivative rules in the field of Calculus. In this article, we'll first discuss its definition and how to use it, and then take a deeper dive by looking at its application to a number of specific functions.
How to Find Derivatives in 3 Steps
What Is U-Substitution?
What Is Multivariable Calculus?
What is Partial Derivative? Definition, Rules and Examples
Definite Integrals: What Are They and How to Calculate Them
How To Do Implicit Differentiation? A Step-by-Step Guide With Examples | CommonCrawl |
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.
Pythagorean theorem
TypeTheorem
FieldEuclidean geometry
StatementThe sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).
Symbolic statement$a^{2}+b^{2}=c^{2}$
Generalizations
• Law of cosines
• Solid geometry
• Non-Euclidean geometry
• Differential geometry
Consequences
• Pythagorean triple
• Reciprocal Pythagorean theorem
• Complex number
• Euclidean distance
• Pythagorean trigonometric identity
Geometry
Projecting a sphere to a plane
• Outline
• History (Timeline)
Branches
• Euclidean
• Non-Euclidean
• Elliptic
• Spherical
• Hyperbolic
• Non-Archimedean geometry
• Projective
• Affine
• Synthetic
• Analytic
• Algebraic
• Arithmetic
• Diophantine
• Differential
• Riemannian
• Symplectic
• Discrete differential
• Complex
• Finite
• Discrete/Combinatorial
• Digital
• Convex
• Computational
• Fractal
• Incidence
• Noncommutative geometry
• Noncommutative algebraic geometry
• Concepts
• Features
Dimension
• Straightedge and compass constructions
• Angle
• Curve
• Diagonal
• Orthogonality (Perpendicular)
• Parallel
• Vertex
• Congruence
• Similarity
• Symmetry
Zero-dimensional
• Point
One-dimensional
• Line
• segment
• ray
• Length
Two-dimensional
• Plane
• Area
• Polygon
Triangle
• Altitude
• Hypotenuse
• Pythagorean theorem
Parallelogram
• Square
• Rectangle
• Rhombus
• Rhomboid
Quadrilateral
• Trapezoid
• Kite
Circle
• Diameter
• Circumference
• Area
Three-dimensional
• Volume
• Cube
• cuboid
• Cylinder
• Dodecahedron
• Icosahedron
• Octahedron
• Pyramid
• Platonic Solid
• Sphere
• Tetrahedron
Four- / other-dimensional
• Tesseract
• Hypersphere
Geometers
by name
• Aida
• Aryabhata
• Ahmes
• Alhazen
• Apollonius
• Archimedes
• Atiyah
• Baudhayana
• Bolyai
• Brahmagupta
• Cartan
• Coxeter
• Descartes
• Euclid
• Euler
• Gauss
• Gromov
• Hilbert
• Huygens
• Jyeṣṭhadeva
• Kātyāyana
• Khayyám
• Klein
• Lobachevsky
• Manava
• Minkowski
• Minggatu
• Pascal
• Pythagoras
• Parameshvara
• Poincaré
• Riemann
• Sakabe
• Sijzi
• al-Tusi
• Veblen
• Virasena
• Yang Hui
• al-Yasamin
• Zhang
• List of geometers
by period
BCE
• Ahmes
• Baudhayana
• Manava
• Pythagoras
• Euclid
• Archimedes
• Apollonius
1–1400s
• Zhang
• Kātyāyana
• Aryabhata
• Brahmagupta
• Virasena
• Alhazen
• Sijzi
• Khayyám
• al-Yasamin
• al-Tusi
• Yang Hui
• Parameshvara
1400s–1700s
• Jyeṣṭhadeva
• Descartes
• Pascal
• Huygens
• Minggatu
• Euler
• Sakabe
• Aida
1700s–1900s
• Gauss
• Lobachevsky
• Bolyai
• Riemann
• Klein
• Poincaré
• Hilbert
• Minkowski
• Cartan
• Veblen
• Coxeter
Present day
• Atiyah
• Gromov
The theorem can be written as an equation relating the lengths of the sides a, b and the hypotenuse c, sometimes called the Pythagorean equation:[1]
$a^{2}+b^{2}=c^{2}.$
The theorem is named for the Greek philosopher Pythagoras, born around 570 BC. The theorem has been proved numerous times by many different methods – possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years.
When Euclidean space is represented by a Cartesian coordinate system in analytic geometry, Euclidean distance satisfies the Pythagorean relation: the squared distance between two points equals the sum of squares of the difference in each coordinate between the points.
The theorem can be generalized in various ways: to higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and to objects that are not triangles at all but n-dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power; popular references in literature, plays, musicals, songs, stamps, and cartoons abound.
Proofs using constructed squares
Rearrangement proofs
In one rearrangement proof, two squares are used whose sides have a measure of $a+b$ and which contain four right triangles whose sides are a, b and c, with the hypotenuse being c. In the square on the right side, the triangles are placed such that the corners of the square correspond to the corners of the right angle in the triangles, forming a square in the center whose sides are length c. Each outer square has an area of $(a+b)^{2}$ as well as $2ab+c^{2}$, with $2ab$ representing the total area of the four triangles. Within the big square on the left side, the four triangles are moved to form two similar rectangles with sides of length a and b. These rectangles in their new position have now delineated two new squares, one having side length a is formed in the bottom-left corner, and another square of side length b formed in the top-right corner. In this new position, this left side now has a square of area $(a+b)^{2}$ as well as $2ab+a^{2}+b^{2}$. Since both squares have the area of $(a+b)^{2}$ it follows that the other measure of the square area also equal each other such that $2ab+c^{2}$ = $2ab+a^{2}+b^{2}$. With the area of the four triangles removed from both side of the equation what remains is $a^{2}+b^{2}=c^{2}.$ [2]
In another proof rectangles in the second box can also be placed such that both have one corner that correspond to consecutive corners of the square. In this way they also form two boxes, this time in consecutive corners, with areas $a^{2}$ and $b^{2}$which will again lead to a second square of with the area $2ab+a^{2}+b^{2}$.
English mathematician Sir Thomas Heath gives this proof in his commentary on Proposition I.47 in Euclid's Elements, and mentions the proposals of German mathematicians Carl Anton Bretschneider and Hermann Hankel that Pythagoras may have known this proof. Heath himself favors a different proposal for a Pythagorean proof, but acknowledges from the outset of his discussion "that the Greek literature which we possess belonging to the first five centuries after Pythagoras contains no statement specifying this or any other particular great geometric discovery to him."[3] Recent scholarship has cast increasing doubt on any sort of role for Pythagoras as a creator of mathematics, although debate about this continues.[4]
Algebraic proofs
The theorem can be proved algebraically using four copies of the same triangle arranged symmetrically around a square with side c, as shown in the lower part of the diagram.[5] This results in a larger square, with side a + b and area (a + b)2. The four triangles and the square side c must have the same area as the larger square,
$(b+a)^{2}=c^{2}+4{\frac {ab}{2}}=c^{2}+2ab,$
giving
$c^{2}=(b+a)^{2}-2ab=b^{2}+2ab+a^{2}-2ab=a^{2}+b^{2}.$
A similar proof uses four copies of a right triangle with sides a, b and c, arranged inside a square with side c as in the top half of the diagram.[6] The triangles are similar with area ${\tfrac {1}{2}}ab$, while the small square has side b − a and area (b − a)2. The area of the large square is therefore
$(b-a)^{2}+4{\frac {ab}{2}}=(b-a)^{2}+2ab=b^{2}-2ab+a^{2}+2ab=a^{2}+b^{2}.$
But this is a square with side c and area c2, so
$c^{2}=a^{2}+b^{2}.$
Other proofs of the theorem
This theorem may have more known proofs than any other (the law of quadratic reciprocity being another contender for that distinction); the book The Pythagorean Proposition contains 370 proofs.[7]
Proof using similar triangles
In this section, and as usual in geometry, a "word" of two capital letters, such as AB denotes the length of the line segment defined by the points labeled with the letters, and not a multiplication. So, AB2 denotes the square of the length AB and not the product $A\times B^{2}.$
This proof is based on the proportionality of the sides of three similar triangles, that is, upon the fact that the ratio of any two corresponding sides of similar triangles is the same regardless of the size of the triangles.
Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. Draw the altitude from point C, and call H its intersection with the side AB. Point H divides the length of the hypotenuse c into parts d and e. The new triangle, ACH, is similar to triangle ABC, because they both have a right angle (by definition of the altitude), and they share the angle at A, meaning that the third angle will be the same in both triangles as well, marked as θ in the figure. By a similar reasoning, the triangle CBH is also similar to ABC. The proof of similarity of the triangles requires the triangle postulate: The sum of the angles in a triangle is two right angles, and is equivalent to the parallel postulate. Similarity of the triangles leads to the equality of ratios of corresponding sides:
${\frac {BC}{AB}}={\frac {BH}{BC}}{\text{ and }}{\frac {AC}{AB}}={\frac {AH}{AC}}.$
The first result equates the cosines of the angles θ, whereas the second result equates their sines.
These ratios can be written as
$BC^{2}=AB\times BH{\text{ and }}AC^{2}=AB\times AH.$
Summing these two equalities results in
$BC^{2}+AC^{2}=AB\times BH+AB\times AH=AB(AH+BH)=AB^{2},$
which, after simplification, demonstrates the Pythagorean theorem:
$BC^{2}+AC^{2}=AB^{2}.$
The role of this proof in history is the subject of much speculation. The underlying question is why Euclid did not use this proof, but invented another. One conjecture is that the proof by similar triangles involved a theory of proportions, a topic not discussed until later in the Elements, and that the theory of proportions needed further development at that time.[8]
Einstein's proof by dissection without rearrangement
Albert Einstein gave a proof by dissection in which the pieces do not need to be moved.[9] Instead of using a square on the hypotenuse and two squares on the legs, one can use any other shape that includes the hypotenuse, and two similar shapes that each include one of two legs instead of the hypotenuse (see Similar figures on the three sides). In Einstein's proof, the shape that includes the hypotenuse is the right triangle itself. The dissection consists of dropping a perpendicular from the vertex of the right angle of the triangle to the hypotenuse, thus splitting the whole triangle into two parts. Those two parts have the same shape as the original right triangle, and have the legs of the original triangle as their hypotenuses, and the sum of their areas is that of the original triangle. Because the ratio of the area of a right triangle to the square of its hypotenuse is the same for similar triangles, the relationship between the areas of the three triangles holds for the squares of the sides of the large triangle as well.
Trigonometric proof using Einstein's construction
Both the proof using similar triangles and Einstein's proof rely on constructing the height to the hypotenuse of the right triangle $\triangle ABC$. The same construction provides a trigonometric proof of the Pythagorean theorem using the definition of the sine as a ratio inside a right triangle:
$\sin \alpha ={\frac {a}{c}},$
$\sin \beta ={\frac {b}{c}},$
$c=b\sin \beta +a\sin \alpha ={\frac {b^{2}}{c}}+{\frac {a^{2}}{c}},$
and thus
$c^{2}=a^{2}+b^{2}.$
This proof is essentially the same as the above proof using similar triangles, where some ratios of lengths are replaced by sines.
Euclid's proof
In outline, here is how the proof in Euclid's Elements proceeds. The large square is divided into a left and right rectangle. A triangle is constructed that has half the area of the left rectangle. Then another triangle is constructed that has half the area of the square on the left-most side. These two triangles are shown to be congruent, proving this square has the same area as the left rectangle. This argument is followed by a similar version for the right rectangle and the remaining square. Putting the two rectangles together to reform the square on the hypotenuse, its area is the same as the sum of the area of the other two squares. The details follow.
Let A, B, C be the vertices of a right triangle, with a right angle at A. Drop a perpendicular from A to the side opposite the hypotenuse in the square on the hypotenuse. That line divides the square on the hypotenuse into two rectangles, each having the same area as one of the two squares on the legs.
For the formal proof, we require four elementary lemmata:
1. If two triangles have two sides of the one equal to two sides of the other, each to each, and the angles included by those sides equal, then the triangles are congruent (side-angle-side).
2. The area of a triangle is half the area of any parallelogram on the same base and having the same altitude.
3. The area of a rectangle is equal to the product of two adjacent sides.
4. The area of a square is equal to the product of two of its sides (follows from 3).
Next, each top square is related to a triangle congruent with another triangle related in turn to one of two rectangles making up the lower square.[10]
The proof is as follows:
1. Let ACB be a right-angled triangle with right angle CAB.
2. On each of the sides BC, AB, and CA, squares are drawn, CBDE, BAGF, and ACIH, in that order. The construction of squares requires the immediately preceding theorems in Euclid, and depends upon the parallel postulate.[11]
3. From A, draw a line parallel to BD and CE. It will perpendicularly intersect BC and DE at K and L, respectively.
4. Join CF and AD, to form the triangles BCF and BDA.
5. Angles CAB and BAG are both right angles; therefore C, A, and G are collinear.
6. Angles CBD and FBA are both right angles; therefore angle ABD equals angle FBC, since both are the sum of a right angle and angle ABC.
7. Since AB is equal to FB, BD is equal to BC and angle ABD equals angle FBC, triangle ABD must be congruent to triangle FBC.
8. Since A-K-L is a straight line, parallel to BD, then rectangle BDLK has twice the area of triangle ABD because they share the base BD and have the same altitude BK, i.e., a line normal to their common base, connecting the parallel lines BD and AL. (lemma 2)
9. Since C is collinear with A and G, and this line is parallel to FB, then square BAGF must be twice in area to triangle FBC.
10. Therefore, rectangle BDLK must have the same area as square BAGF = AB2.
11. By applying steps 3 to 10 to the other side of the figure, it can be similarly shown that rectangle CKLE must have the same area as square ACIH = AC2.
12. Adding these two results, AB2 + AC2 = BD × BK + KL × KC
13. Since BD = KL, BD × BK + KL × KC = BD(BK + KC) = BD × BC
14. Therefore, AB2 + AC2 = BC2, since CBDE is a square.
This proof, which appears in Euclid's Elements as that of Proposition 47 in Book 1, demonstrates that the area of the square on the hypotenuse is the sum of the areas of the other two squares.[12][13] This is quite distinct from the proof by similarity of triangles, which is conjectured to be the proof that Pythagoras used.[14][15]
Proofs by dissection and rearrangement
Another by rearrangement is given by the middle animation. A large square is formed with area c2, from four identical right triangles with sides a, b and c, fitted around a small central square. Then two rectangles are formed with sides a and b by moving the triangles. Combining the smaller square with these rectangles produces two squares of areas a2 and b2, which must have the same area as the initial large square.[16]
The third, rightmost image also gives a proof. The upper two squares are divided as shown by the blue and green shading, into pieces that when rearranged can be made to fit in the lower square on the hypotenuse – or conversely the large square can be divided as shown into pieces that fill the other two. This way of cutting one figure into pieces and rearranging them to get another figure is called dissection. This shows the area of the large square equals that of the two smaller ones.[17]
Proof by area-preserving shearing
As shown in the accompanying animation, area-preserving shear mappings and translations can transform the squares on the sides adjacent to the right-angle onto the square on the hypotenuse, together covering it exactly.[18] Each shear leaves the base and height unchanged, thus leaving the area unchanged too. The translations also leave the area unchanged, as they do not alter the shapes at all. Each square is first sheared into a parallelogram, and then into a rectangle which can be translated onto one section of the square on the hypotenuse.
Other algebraic proofs
A related proof was published by future U.S. President James A. Garfield (then a U.S. Representative) (see diagram).[19][20][21] Instead of a square it uses a trapezoid, which can be constructed from the square in the second of the above proofs by bisecting along a diagonal of the inner square, to give the trapezoid as shown in the diagram. The area of the trapezoid can be calculated to be half the area of the square, that is
${\frac {1}{2}}(b+a)^{2}.$
The inner square is similarly halved, and there are only two triangles so the proof proceeds as above except for a factor of ${\frac {1}{2}}$, which is removed by multiplying by two to give the result.
Proof using differentials
One can arrive at the Pythagorean theorem by studying how changes in a side produce a change in the hypotenuse and employing calculus.[22][23][24]
The triangle ABC is a right triangle, as shown in the upper part of the diagram, with BC the hypotenuse. At the same time the triangle lengths are measured as shown, with the hypotenuse of length y, the side AC of length x and the side AB of length a, as seen in the lower diagram part.
If x is increased by a small amount dx by extending the side AC slightly to D, then y also increases by dy. These form two sides of a triangle, CDE, which (with E chosen so CE is perpendicular to the hypotenuse) is a right triangle approximately similar to ABC. Therefore, the ratios of their sides must be the same, that is:
${\frac {dy}{dx}}={\frac {x}{y}}.$
This can be rewritten as $y\,dy=x\,dx$ , which is a differential equation that can be solved by direct integration:
$\int y\,dy=\int x\,dx\,,$
giving
$y^{2}=x^{2}+C.$
The constant can be deduced from x = 0, y = a to give the equation
$y^{2}=x^{2}+a^{2}.$
This is more of an intuitive proof than a formal one: it can be made more rigorous if proper limits are used in place of dx and dy.
Converse
The converse of the theorem is also true:[25]
Given a triangle with sides of length a, b, and c, if a2 + b2 = c2, then the angle between sides a and b is a right angle.
For any three positive real numbers a, b, and c such that a2 + b2 = c2, there exists a triangle with sides a, b and c as a consequence of the converse of the triangle inequality.
This converse appears in Euclid's Elements (Book I, Proposition 48): "If in a triangle the square on one of the sides equals the sum of the squares on the remaining two sides of the triangle, then the angle contained by the remaining two sides of the triangle is right."[26]
It can be proved using the law of cosines or as follows:
Let ABC be a triangle with side lengths a, b, and c, with a2 + b2 = c2. Construct a second triangle with sides of length a and b containing a right angle. By the Pythagorean theorem, it follows that the hypotenuse of this triangle has length c = √a2 + b2, the same as the hypotenuse of the first triangle. Since both triangles' sides are the same lengths a, b and c, the triangles are congruent and must have the same angles. Therefore, the angle between the side of lengths a and b in the original triangle is a right angle.
The above proof of the converse makes use of the Pythagorean theorem itself. The converse can also be proved without assuming the Pythagorean theorem.[27][28]
A corollary of the Pythagorean theorem's converse is a simple means of determining whether a triangle is right, obtuse, or acute, as follows. Let c be chosen to be the longest of the three sides and a + b > c (otherwise there is no triangle according to the triangle inequality). The following statements apply:[29]
• If a2 + b2 = c2, then the triangle is right.
• If a2 + b2 > c2, then the triangle is acute.
• If a2 + b2 < c2, then the triangle is obtuse.
Edsger W. Dijkstra has stated this proposition about acute, right, and obtuse triangles in this language:
sgn(α + β − γ) = sgn(a2 + b2 − c2),
where α is the angle opposite to side a, β is the angle opposite to side b, γ is the angle opposite to side c, and sgn is the sign function.[30]
Consequences and uses of the theorem
Pythagorean triples
Main article: Pythagorean triple
A Pythagorean triple has three positive integers a, b, and c, such that a2 + b2 = c2. In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths.[1] Such a triple is commonly written (a, b, c). Some well-known examples are (3, 4, 5) and (5, 12, 13).
A primitive Pythagorean triple is one in which a, b and c are coprime (the greatest common divisor of a, b and c is 1).
The following is a list of primitive Pythagorean triples with values less than 100:
(3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53), (33, 56, 65), (36, 77, 85), (39, 80, 89), (48, 55, 73), (65, 72, 97)
Inverse Pythagorean theorem
Given a right triangle with sides $a,b,c$ and altitude $d$ (a line from the right angle and perpendicular to the hypotenuse $c$). The Pythagorean theorem has,
$a^{2}+b^{2}=c^{2}$
while the inverse Pythagorean theorem relates the two legs $a,b$ to the altitude $d$,[31]
${\frac {1}{a^{2}}}+{\frac {1}{b^{2}}}={\frac {1}{d^{2}}}$
The equation can be transformed to,
${\frac {1}{(xz)^{2}}}+{\frac {1}{(yz)^{2}}}={\frac {1}{(xy)^{2}}}$
where $x^{2}+y^{2}=z^{2}$ for any non-zero real $x,y,z$. If the $a,b,d$ are to be integers, the smallest solution $a>b>d$ is then
${\frac {1}{20^{2}}}+{\frac {1}{15^{2}}}={\frac {1}{12^{2}}}$
using the smallest Pythagorean triple $3,4,5$. The reciprocal Pythagorean theorem is a special case of the optic equation
${\frac {1}{p}}+{\frac {1}{q}}={\frac {1}{r}}$
where the denominators are squares and also for a heptagonal triangle whose sides $p,q,r$ are square numbers.
Incommensurable lengths
One of the consequences of the Pythagorean theorem is that line segments whose lengths are incommensurable (so the ratio of which is not a rational number) can be constructed using a straightedge and compass. Pythagoras' theorem enables construction of incommensurable lengths because the hypotenuse of a triangle is related to the sides by the square root operation.
The figure on the right shows how to construct line segments whose lengths are in the ratio of the square root of any positive integer.[32] Each triangle has a side (labeled "1") that is the chosen unit for measurement. In each right triangle, Pythagoras' theorem establishes the length of the hypotenuse in terms of this unit. If a hypotenuse is related to the unit by the square root of a positive integer that is not a perfect square, it is a realization of a length incommensurable with the unit, such as √2, √3, √5 . For more detail, see Quadratic irrational.
Incommensurable lengths conflicted with the Pythagorean school's concept of numbers as only whole numbers. The Pythagorean school dealt with proportions by comparison of integer multiples of a common subunit.[33] According to one legend, Hippasus of Metapontum (ca. 470 B.C.) was drowned at sea for making known the existence of the irrational or incommensurable.[34][35]
Complex numbers
For any complex number
$z=x+iy,$
the absolute value or modulus is given by
$r=|z|={\sqrt {x^{2}+y^{2}}}.$
So the three quantities, r, x and y are related by the Pythagorean equation,
$r^{2}=x^{2}+y^{2}.$
Note that r is defined to be a positive number or zero but x and y can be negative as well as positive. Geometrically r is the distance of the z from zero or the origin O in the complex plane.
This can be generalised to find the distance between two points, z1 and z2 say. The required distance is given by
$|z_{1}-z_{2}|={\sqrt {(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}},$
so again they are related by a version of the Pythagorean equation,
$|z_{1}-z_{2}|^{2}=(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}.$
Euclidean distance
Main article: Euclidean distance
The distance formula in Cartesian coordinates is derived from the Pythagorean theorem.[36] If (x1, y1) and (x2, y2) are points in the plane, then the distance between them, also called the Euclidean distance, is given by
${\sqrt {(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}}.$
More generally, in Euclidean n-space, the Euclidean distance between two points, $A\,=\,(a_{1},a_{2},\dots ,a_{n})$ and $B\,=\,(b_{1},b_{2},\dots ,b_{n})$, is defined, by generalization of the Pythagorean theorem, as:
${\sqrt {(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}+\cdots +(a_{n}-b_{n})^{2}}}={\sqrt {\sum _{i=1}^{n}(a_{i}-b_{i})^{2}}}.$
If instead of Euclidean distance, the square of this value (the squared Euclidean distance, or SED) is used, the resulting equation avoids square roots and is simply a sum of the SED of the coordinates:
$(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}+\cdots +(a_{n}-b_{n})^{2}=\sum _{i=1}^{n}(a_{i}-b_{i})^{2}.$
The squared form is a smooth, convex function of both points, and is widely used in optimization theory and statistics, forming the basis of least squares.
Euclidean distance in other coordinate systems
If Cartesian coordinates are not used, for example, if polar coordinates are used in two dimensions or, in more general terms, if curvilinear coordinates are used, the formulas expressing the Euclidean distance are more complicated than the Pythagorean theorem, but can be derived from it. A typical example where the straight-line distance between two points is converted to curvilinear coordinates can be found in the applications of Legendre polynomials in physics. The formulas can be discovered by using Pythagoras' theorem with the equations relating the curvilinear coordinates to Cartesian coordinates. For example, the polar coordinates (r, θ) can be introduced as:
$x=r\cos \theta ,\ y=r\sin \theta .$
Then two points with locations (r1, θ1) and (r2, θ2) are separated by a distance s:
$s^{2}=(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}=(r_{1}\cos \theta _{1}-r_{2}\cos \theta _{2})^{2}+(r_{1}\sin \theta _{1}-r_{2}\sin \theta _{2})^{2}.$
Performing the squares and combining terms, the Pythagorean formula for distance in Cartesian coordinates produces the separation in polar coordinates as:
${\begin{aligned}s^{2}&=r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}\left(\cos \theta _{1}\cos \theta _{2}+\sin \theta _{1}\sin \theta _{2}\right)\\&=r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}\cos \left(\theta _{1}-\theta _{2}\right)\\&=r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}\cos \Delta \theta ,\end{aligned}}$
using the trigonometric product-to-sum formulas. This formula is the law of cosines, sometimes called the generalized Pythagorean theorem.[37] From this result, for the case where the radii to the two locations are at right angles, the enclosed angle Δθ = π/2, and the form corresponding to Pythagoras' theorem is regained: $s^{2}=r_{1}^{2}+r_{2}^{2}.$ The Pythagorean theorem, valid for right triangles, therefore is a special case of the more general law of cosines, valid for arbitrary triangles.
Pythagorean trigonometric identity
Main article: Pythagorean trigonometric identity
In a right triangle with sides a, b and hypotenuse c, trigonometry determines the sine and cosine of the angle θ between side a and the hypotenuse as:
$\sin \theta ={\frac {b}{c}},\quad \cos \theta ={\frac {a}{c}}.$
From that it follows:
${\cos }^{2}\theta +{\sin }^{2}\theta ={\frac {a^{2}+b^{2}}{c^{2}}}=1,$
where the last step applies Pythagoras' theorem. This relation between sine and cosine is sometimes called the fundamental Pythagorean trigonometric identity.[38] In similar triangles, the ratios of the sides are the same regardless of the size of the triangles, and depend upon the angles. Consequently, in the figure, the triangle with hypotenuse of unit size has opposite side of size sin θ and adjacent side of size cos θ in units of the hypotenuse.
Relation to the cross product
The Pythagorean theorem relates the cross product and dot product in a similar way:[39]
$\|\mathbf {a} \times \mathbf {b} \|^{2}+(\mathbf {a} \cdot \mathbf {b} )^{2}=\|\mathbf {a} \|^{2}\|\mathbf {b} \|^{2}.$
This can be seen from the definitions of the cross product and dot product, as
${\begin{aligned}\mathbf {a} \times \mathbf {b} &=ab\mathbf {n} \sin {\theta }\\\mathbf {a} \cdot \mathbf {b} &=ab\cos {\theta },\end{aligned}}$
with n a unit vector normal to both a and b. The relationship follows from these definitions and the Pythagorean trigonometric identity.
This can also be used to define the cross product. By rearranging the following equation is obtained
$\|\mathbf {a} \times \mathbf {b} \|^{2}=\|\mathbf {a} \|^{2}\|\mathbf {b} \|^{2}-(\mathbf {a} \cdot \mathbf {b} )^{2}.$
This can be considered as a condition on the cross product and so part of its definition, for example in seven dimensions.[40][41]
Generalizations
Similar figures on the three sides
The Pythagorean theorem generalizes beyond the areas of squares on the three sides to any similar figures. This was known by Hippocrates of Chios in the 5th century BC,[42] and was included by Euclid in his Elements:[43]
If one erects similar figures (see Euclidean geometry) with corresponding sides on the sides of a right triangle, then the sum of the areas of the ones on the two smaller sides equals the area of the one on the larger side.
This extension assumes that the sides of the original triangle are the corresponding sides of the three congruent figures (so the common ratios of sides between the similar figures are a:b:c).[44] While Euclid's proof only applied to convex polygons, the theorem also applies to concave polygons and even to similar figures that have curved boundaries (but still with part of a figure's boundary being the side of the original triangle).[44]
The basic idea behind this generalization is that the area of a plane figure is proportional to the square of any linear dimension, and in particular is proportional to the square of the length of any side. Thus, if similar figures with areas A, B and C are erected on sides with corresponding lengths a, b and c then:
${\frac {A}{a^{2}}}={\frac {B}{b^{2}}}={\frac {C}{c^{2}}}\,,$
$\Rightarrow A+B={\frac {a^{2}}{c^{2}}}C+{\frac {b^{2}}{c^{2}}}C\,.$
But, by the Pythagorean theorem, a2 + b2 = c2, so A + B = C.
Conversely, if we can prove that A + B = C for three similar figures without using the Pythagorean theorem, then we can work backwards to construct a proof of the theorem. For example, the starting center triangle can be replicated and used as a triangle C on its hypotenuse, and two similar right triangles (A and B ) constructed on the other two sides, formed by dividing the central triangle by its altitude. The sum of the areas of the two smaller triangles therefore is that of the third, thus A + B = C and reversing the above logic leads to the Pythagorean theorem a2 + b2 = c2. (See also Einstein's proof by dissection without rearrangement)
Law of cosines
Main article: Law of cosines
The Pythagorean theorem is a special case of the more general theorem relating the lengths of sides in any triangle, the law of cosines, which states that
$a^{2}+b^{2}-2ab\cos {\theta }=c^{2}$
where $\theta $ is the angle between sides $a$ and $b$.[45]
When $\theta $ is ${\frac {\pi }{2}}$ radians or 90°, then $\cos {\theta }=0$, and the formula reduces to the usual Pythagorean theorem.
Arbitrary triangle
At any selected angle of a general triangle of sides a, b, c, inscribe an isosceles triangle such that the equal angles at its base θ are the same as the selected angle. Suppose the selected angle θ is opposite the side labeled c. Inscribing the isosceles triangle forms triangle CAD with angle θ opposite side b and with side r along c. A second triangle is formed with angle θ opposite side a and a side with length s along c, as shown in the figure. Thābit ibn Qurra stated that the sides of the three triangles were related as:[47][48]
$a^{2}+b^{2}=c(r+s)\ .$
As the angle θ approaches π/2, the base of the isosceles triangle narrows, and lengths r and s overlap less and less. When θ = π/2, ADB becomes a right triangle, r + s = c, and the original Pythagorean theorem is regained.
One proof observes that triangle ABC has the same angles as triangle CAD, but in opposite order. (The two triangles share the angle at vertex A, both contain the angle θ, and so also have the same third angle by the triangle postulate.) Consequently, ABC is similar to the reflection of CAD, the triangle DAC in the lower panel. Taking the ratio of sides opposite and adjacent to θ,
${\frac {c}{b}}={\frac {b}{r}}\ .$
Likewise, for the reflection of the other triangle,
${\frac {c}{a}}={\frac {a}{s}}\ .$
Clearing fractions and adding these two relations:
$cs+cr=a^{2}+b^{2}\ ,$
the required result.
The theorem remains valid if the angle $\theta $ is obtuse so the lengths r and s are non-overlapping.
General triangles using parallelograms
Pappus's area theorem is a further generalization, that applies to triangles that are not right triangles, using parallelograms on the three sides in place of squares (squares are a special case, of course). The upper figure shows that for a scalene triangle, the area of the parallelogram on the longest side is the sum of the areas of the parallelograms on the other two sides, provided the parallelogram on the long side is constructed as indicated (the dimensions labeled with arrows are the same, and determine the sides of the bottom parallelogram). This replacement of squares with parallelograms bears a clear resemblance to the original Pythagoras' theorem, and was considered a generalization by Pappus of Alexandria in 4 AD[49][50]
The lower figure shows the elements of the proof. Focus on the left side of the figure. The left green parallelogram has the same area as the left, blue portion of the bottom parallelogram because both have the same base b and height h. However, the left green parallelogram also has the same area as the left green parallelogram of the upper figure, because they have the same base (the upper left side of the triangle) and the same height normal to that side of the triangle. Repeating the argument for the right side of the figure, the bottom parallelogram has the same area as the sum of the two green parallelograms.
Solid geometry
In terms of solid geometry, Pythagoras' theorem can be applied to three dimensions as follows. Consider a rectangular solid as shown in the figure. The length of diagonal BD is found from Pythagoras' theorem as:
${\overline {BD}}^{\,2}={\overline {BC}}^{\,2}+{\overline {CD}}^{\,2}\ ,$
where these three sides form a right triangle. Using horizontal diagonal BD and the vertical edge AB, the length of diagonal AD then is found by a second application of Pythagoras' theorem as:
${\overline {AD}}^{\,2}={\overline {AB}}^{\,2}+{\overline {BD}}^{\,2}\ ,$
or, doing it all in one step:
${\overline {AD}}^{\,2}={\overline {AB}}^{\,2}+{\overline {BC}}^{\,2}+{\overline {CD}}^{\,2}\ .$
This result is the three-dimensional expression for the magnitude of a vector v (the diagonal AD) in terms of its orthogonal components {vk} (the three mutually perpendicular sides):
$\|\mathbf {v} \|^{2}=\sum _{k=1}^{3}\|\mathbf {v} _{k}\|^{2}.$
This one-step formulation may be viewed as a generalization of Pythagoras' theorem to higher dimensions. However, this result is really just the repeated application of the original Pythagoras' theorem to a succession of right triangles in a sequence of orthogonal planes.
A substantial generalization of the Pythagorean theorem to three dimensions is de Gua's theorem, named for Jean Paul de Gua de Malves: If a tetrahedron has a right angle corner (like a corner of a cube), then the square of the area of the face opposite the right angle corner is the sum of the squares of the areas of the other three faces. This result can be generalized as in the "n-dimensional Pythagorean theorem":[51]
Let $x_{1},x_{2},\ldots ,x_{n}$ be orthogonal vectors in Rn. Consider the n-dimensional simplex S with vertices $0,x_{1},\ldots ,x_{n}$. (Think of the (n − 1)-dimensional simplex with vertices $x_{1},\ldots ,x_{n}$ not including the origin as the "hypotenuse" of S and the remaining (n − 1)-dimensional faces of S as its "legs".) Then the square of the volume of the hypotenuse of S is the sum of the squares of the volumes of the n legs.
This statement is illustrated in three dimensions by the tetrahedron in the figure. The "hypotenuse" is the base of the tetrahedron at the back of the figure, and the "legs" are the three sides emanating from the vertex in the foreground. As the depth of the base from the vertex increases, the area of the "legs" increases, while that of the base is fixed. The theorem suggests that when this depth is at the value creating a right vertex, the generalization of Pythagoras' theorem applies. In a different wording:[52]
Given an n-rectangular n-dimensional simplex, the square of the (n − 1)-content of the facet opposing the right vertex will equal the sum of the squares of the (n − 1)-contents of the remaining facets.
Inner product spaces
The Pythagorean theorem can be generalized to inner product spaces,[53] which are generalizations of the familiar 2-dimensional and 3-dimensional Euclidean spaces. For example, a function may be considered as a vector with infinitely many components in an inner product space, as in functional analysis.[54]
In an inner product space, the concept of perpendicularity is replaced by the concept of orthogonality: two vectors v and w are orthogonal if their inner product $\langle \mathbf {v} ,\mathbf {w} \rangle $ is zero. The inner product is a generalization of the dot product of vectors. The dot product is called the standard inner product or the Euclidean inner product. However, other inner products are possible.[55]
The concept of length is replaced by the concept of the norm ‖v‖ of a vector v, defined as:[56]
$\lVert \mathbf {v} \rVert \equiv {\sqrt {\langle \mathbf {v} ,\mathbf {v} \rangle }}\,.$
In an inner-product space, the Pythagorean theorem states that for any two orthogonal vectors v and w we have
$\left\|\mathbf {v} +\mathbf {w} \right\|^{2}=\left\|\mathbf {v} \right\|^{2}+\left\|\mathbf {w} \right\|^{2}.$
Here the vectors v and w are akin to the sides of a right triangle with hypotenuse given by the vector sum v + w. This form of the Pythagorean theorem is a consequence of the properties of the inner product:
${\begin{aligned}\left\|\mathbf {v} +\mathbf {w} \right\|^{2}&=\langle \mathbf {v+w} ,\ \mathbf {v+w} \rangle \\[3mu]&=\langle \mathbf {v} ,\ \mathbf {v} \rangle +\langle \mathbf {w} ,\ \mathbf {w} \rangle +\langle \mathbf {v,\ w} \rangle +\langle \mathbf {w,\ v} \rangle \\[3mu]&=\left\|\mathbf {v} \right\|^{2}+\left\|\mathbf {w} \right\|^{2},\end{aligned}}$
where $\langle \mathbf {v,\ w} \rangle =\langle \mathbf {w,\ v} \rangle =0$ because of orthogonality.
A further generalization of the Pythagorean theorem in an inner product space to non-orthogonal vectors is the parallelogram law :[56]
$2\|\mathbf {v} \|^{2}+2\|\mathbf {w} \|^{2}=\|\mathbf {v+w} \|^{2}+\|\mathbf {v-w} \|^{2}\ ,$
which says that twice the sum of the squares of the lengths of the sides of a parallelogram is the sum of the squares of the lengths of the diagonals. Any norm that satisfies this equality is ipso facto a norm corresponding to an inner product.[56]
The Pythagorean identity can be extended to sums of more than two orthogonal vectors. If v1, v2, ..., vn are pairwise-orthogonal vectors in an inner-product space, then application of the Pythagorean theorem to successive pairs of these vectors (as described for 3-dimensions in the section on solid geometry) results in the equation[57]
$\left\|\sum _{k=1}^{n}\mathbf {v} _{k}\right\|^{2}=\sum _{k=1}^{n}\|\mathbf {v} _{k}\|^{2}$
Sets of m-dimensional objects in n-dimensional space
Another generalization of the Pythagorean theorem applies to Lebesgue-measurable sets of objects in any number of dimensions. Specifically, the square of the measure of an m-dimensional set of objects in one or more parallel m-dimensional flats in n-dimensional Euclidean space is equal to the sum of the squares of the measures of the orthogonal projections of the object(s) onto all m-dimensional coordinate subspaces.[58]
In mathematical terms:
$\mu _{ms}^{2}=\sum _{i=1}^{x}\mathbf {\mu ^{2}} _{mp_{i}}$
where:
• $\mu _{m}$ is a measure in m-dimensions (a length in one dimension, an area in two dimensions, a volume in three dimensions, etc.).
• $s$ is a set of one or more non-overlapping m-dimensional objects in one or more parallel m-dimensional flats in n-dimensional Euclidean space.
• $\mu _{ms}$ is the total measure (sum) of the set of m-dimensional objects.
• $p$ represents an m-dimensional projection of the original set onto an orthogonal coordinate subspace.
• $\mu _{mp_{i}}$ is the measure of the m-dimensional set projection onto m-dimensional coordinate subspace $i$. Because object projections can overlap on a coordinate subspace, the measure of each object projection in the set must be calculated individually, then measures of all projections added together to provide the total measure for the set of projections on the given coordinate subspace.
• $x$ is the number of orthogonal, m-dimensional coordinate subspaces in n-dimensional space (Rn) onto which the m-dimensional objects are projected (m ≤ n):
$x={\binom {n}{m}}={\frac {n!}{m!(n-m)!}}$
Non-Euclidean geometry
The Pythagorean theorem is derived from the axioms of Euclidean geometry, and in fact, were the Pythagorean theorem to fail for some right triangle, then the plane in which this triangle is contained cannot be Euclidean. More precisely, the Pythagorean theorem implies, and is implied by, Euclid's Parallel (Fifth) Postulate.[59][60] Thus, right triangles in a non-Euclidean geometry[61] do not satisfy the Pythagorean theorem. For example, in spherical geometry, all three sides of the right triangle (say a, b, and c) bounding an octant of the unit sphere have length equal to π/2, and all its angles are right angles, which violates the Pythagorean theorem because $a^{2}+b^{2}=2c^{2}>c^{2}$.
Here two cases of non-Euclidean geometry are considered—spherical geometry and hyperbolic plane geometry; in each case, as in the Euclidean case for non-right triangles, the result replacing the Pythagorean theorem follows from the appropriate law of cosines.
However, the Pythagorean theorem remains true in hyperbolic geometry and elliptic geometry if the condition that the triangle be right is replaced with the condition that two of the angles sum to the third, say A+B = C. The sides are then related as follows: the sum of the areas of the circles with diameters a and b equals the area of the circle with diameter c.[62]
Spherical geometry
For any right triangle on a sphere of radius R (for example, if γ in the figure is a right angle), with sides a, b, c, the relation between the sides takes the form:[63]
$\cos {\frac {c}{R}}=\cos {\frac {a}{R}}\,\cos {\frac {b}{R}}.$
This equation can be derived as a special case of the spherical law of cosines that applies to all spherical triangles:
$\cos {\frac {c}{R}}=\cos {\frac {a}{R}}\,\cos {\frac {b}{R}}+\sin {\frac {a}{R}}\,\sin {\frac {b}{R}}\,\cos {\gamma }.$
For infinitesimal triangles on the sphere (or equivalently, for finite spherical triangles on a sphere of infinite radius), the spherical relation between the sides of a right triangle reduces to the Euclidean form of the Pythagorean theorem. To see how, assume we have a spherical triangle of fixed side lengths a, b, and c on a sphere with expanding radius R. As R approaches infinity the quantities a/R, b/R, and c/R tend to zero and the spherical Pythagorean identity reduces to $1=1,$ so we must look at its asymptotic expansion.
The Maclaurin series for the cosine function can be written as $ \cos x=1-{\tfrac {1}{2}}x^{2}+O{\left(x^{4}\right)}$ with the remainder term in big O notation. Letting $x=c/R$ be a side of the triangle, and treating the expression as an asymptotic expansion in terms of R for a fixed c,
${\begin{aligned}\cos {\frac {c}{R}}=1-{\frac {c^{2}}{2R^{2}}}+O{\left(R^{-4}\right)}\end{aligned}}$
and likewise for a and b. Substituting the asymptotic expansion for each of the cosines into the spherical relation for a right triangle yields
${\begin{aligned}1-{\frac {c^{2}}{2R^{2}}}+O{\left(R^{-4}\right)}&=\left(1-{\frac {a^{2}}{2R^{2}}}+O{\left(R^{-4}\right)}\right)\left(1-{\frac {b^{2}}{2R^{2}}}+O{\left(R^{-4}\right)}\right)\\&=1-{\frac {a^{2}}{2R^{2}}}-{\frac {b^{2}}{2R^{2}}}+O{\left(R^{-4}\right)}.\end{aligned}}$
Subtracting 1 and then negating each side,
${\frac {c^{2}}{2R^{2}}}={\frac {a^{2}}{2R^{2}}}+{\frac {b^{2}}{2R^{2}}}+O{\left(R^{-4}\right)}.$
Multiplying through by 2R2, the asymptotic expansion for c in terms of fixed a, b and variable R is
$c^{2}=a^{2}+b^{2}+O{\left(R^{-2}\right)}.$
The Euclidean Pythagorean relationship $ c^{2}=a^{2}+b^{2}$ is recovered in the limit, as the remainder vanishes when the radius R approaches infinity.
For practical computation in spherical trigonometry with small right triangles, cosines can be replaced with sines using the double-angle identity $\cos {2\theta }=1-2\sin ^{2}{\theta }$ to avoid loss of significance. Then the spherical Pythagorean theorem can alternately be written as
$\sin ^{2}{\frac {c}{2R}}=\sin ^{2}{\frac {a}{2R}}+\sin ^{2}{\frac {b}{2R}}-2\sin ^{2}{\frac {a}{2R}}\,\sin ^{2}{\frac {b}{2R}}.$
Hyperbolic geometry
In a hyperbolic space with uniform Gaussian curvature −1/R2, for a right triangle with legs a, b, and hypotenuse c, the relation between the sides takes the form:[64]
$\cosh {\frac {c}{R}}=\cosh {\frac {a}{R}}\,\cosh {\frac {b}{R}}$
where cosh is the hyperbolic cosine. This formula is a special form of the hyperbolic law of cosines that applies to all hyperbolic triangles:[65]
$\cosh {\frac {c}{R}}=\cosh {\frac {a}{R}}\ \cosh {\frac {b}{R}}-\sinh {\frac {a}{R}}\ \sinh {\frac {b}{R}}\ \cos \gamma \ ,$
with γ the angle at the vertex opposite the side c.
By using the Maclaurin series for the hyperbolic cosine, cosh x ≈ 1 + x2/2, it can be shown that as a hyperbolic triangle becomes very small (that is, as a, b, and c all approach zero), the hyperbolic relation for a right triangle approaches the form of Pythagoras' theorem.
For small right triangles (a, b << R), the hyperbolic cosines can be eliminated to avoid loss of significance, giving
$\sinh ^{2}{\frac {c}{2R}}=\sinh ^{2}{\frac {a}{2R}}+\sinh ^{2}{\frac {b}{2R}}+2\sinh ^{2}{\frac {a}{2R}}\sinh ^{2}{\frac {b}{2R}}\,.$
Very small triangles
For any uniform curvature K (positive, zero, or negative), in very small right triangles (|K|a2, |K|b2 << 1) with hypotenuse c, it can be shown that
$c^{2}=a^{2}+b^{2}-{\frac {K}{3}}a^{2}b^{2}-{\frac {K^{2}}{45}}a^{2}b^{2}(a^{2}+b^{2})-{\frac {2K^{3}}{945}}a^{2}b^{2}(a^{2}-b^{2})^{2}+O(K^{4}c^{10})\,.$
Differential geometry
The Pythagorean theorem applies to infinitesimal triangles seen in differential geometry. In three dimensional space, the distance between two infinitesimally separated points satisfies
$ds^{2}=dx^{2}+dy^{2}+dz^{2},$
with ds the element of distance and (dx, dy, dz) the components of the vector separating the two points. Such a space is called a Euclidean space. However, in Riemannian geometry, a generalization of this expression useful for general coordinates (not just Cartesian) and general spaces (not just Euclidean) takes the form:[66]
$ds^{2}=\sum _{i,j}^{n}g_{ij}\,dx_{i}\,dx_{j}$
which is called the metric tensor. (Sometimes, by abuse of language, the same term is applied to the set of coefficients gij.) It may be a function of position, and often describes curved space. A simple example is Euclidean (flat) space expressed in curvilinear coordinates. For example, in polar coordinates:
$ds^{2}=dr^{2}+r^{2}d\theta ^{2}\ .$
History
There is debate whether the Pythagorean theorem was discovered once, or many times in many places, and the date of first discovery is uncertain, as is the date of the first proof. Historians of Mesopotamian mathematics have concluded that the Pythagorean rule was in widespread use during the Old Babylonian period (20th to 16th centuries BC), over a thousand years before Pythagoras was born.[68][69][70][71] The history of the theorem can be divided into four parts: knowledge of Pythagorean triples, knowledge of the relationship among the sides of a right triangle, knowledge of the relationships among adjacent angles, and proofs of the theorem within some deductive system.
Written c. 1800 BC, the Egyptian Middle Kingdom Berlin Papyrus 6619 includes a problem whose solution is the Pythagorean triple 6:8:10, but the problem does not mention a triangle. The Mesopotamian tablet Plimpton 322, also written c. 1800 BC near Larsa, contains many entries closely related to Pythagorean triples.[72]
In India, the Baudhayana Shulba Sutra, the dates of which are given variously as between the 8th and 5th century BC,[73] contains a list of Pythagorean triples and a statement of the Pythagorean theorem, both in the special case of the isosceles right triangle and in the general case, as does the Apastamba Shulba Sutra (c. 600 BC).[lower-alpha 1]
Byzantine Neoplatonic philosopher and mathematician Proclus, writing in the fifth century AD, states two arithmetic rules, "one of them attributed to Plato, the other to Pythagoras",[75] for generating special Pythagorean triples. The rule attributed to Pythagoras (c. 570 – c. 495 BC) starts from an odd number and produces a triple with leg and hypotenuse differing by one unit; the rule attributed to Plato (428/427 or 424/423 – 348/347 BC) starts from an even number and produces a triple with leg and hypotenuse differing by two units. According to Thomas L. Heath (1861–1940), no specific attribution of the theorem to Pythagoras exists in the surviving Greek literature from the five centuries after Pythagoras lived.[76] However, when authors such as Plutarch and Cicero attributed the theorem to Pythagoras, they did so in a way which suggests that the attribution was widely known and undoubted.[77][78] Classicist Kurt von Fritz wrote, "Whether this formula is rightly attributed to Pythagoras personally, but one can safely assume that it belongs to the very oldest period of Pythagorean mathematics."[35] Around 300 BC, in Euclid's Elements, the oldest extant axiomatic proof of the theorem is presented.[79]
With contents known much earlier, but in surviving texts dating from roughly the 1st century BC, the Chinese text Zhoubi Suanjing (周髀算经), (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven) gives a reasoning for the Pythagorean theorem for the (3, 4, 5) triangle — in China it is called the "Gougu theorem" (勾股定理).[80][81] During the Han Dynasty (202 BC to 220 AD), Pythagorean triples appear in The Nine Chapters on the Mathematical Art,[82] together with a mention of right triangles.[83] Some believe the theorem arose first in China,[84] where it is alternatively known as the "Shang Gao theorem" (商高定理),[85] named after the Duke of Zhou's astronomer and mathematician, whose reasoning composed most of what was in the Zhoubi Suanjing.[86]
See also
• Addition in quadrature
• At Dulcarnon
• British flag theorem
• Fermat's Last Theorem
• Inverse Pythagorean theorem
• Kepler triangle
• Linear algebra
• List of triangle topics
• Lp space
• Nonhypotenuse number
• Parallelogram law
• Parseval's identity
• Ptolemy's theorem
• Pythagoras in popular culture
• Pythagorean expectation
• Pythagorean tiling
• Rational trigonometry in Pythagoras' theorem
• Thales theorem
Notes and references
Notes
1. Van der Waerden believed that this material "was certainly based on earlier traditions". Carl Boyer states that the Pythagorean theorem in the Śulba-sũtram may have been influenced by ancient Mesopotamian math, but there is no conclusive evidence in favor or opposition of this possibility.[74]
References
1. Judith D. Sally; Paul Sally (2007). "Chapter 3: Pythagorean triples". Roots to research: a vertical development of mathematical problems. American Mathematical Society Bookstore. p. 63. ISBN 978-0-8218-4403-8.
2. Benson, Donald. The Moment of Proof : Mathematical Epiphanies, pp. 172–173 (Oxford University Press, 1999).
3. Euclid (1956), pp. 351–352
4. Huffman, Carl (23 February 2005). "Pythagoras". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy (Winter 2018 Edition)., "It should now be clear that decisions about sources are crucial in addressing the question of whether Pythagoras was a mathematician and scientist. The view of Pythagoras's cosmos sketched in the first five paragraphs of this section, according to which he was neither a mathematician nor a scientist, remains the consensus."
5. Alexander Bogomolny. "Cut-the-knot.org: Pythagorean theorem and its many proofs, Proof #4". Cut the Knot. Retrieved 4 November 2010.
6. Alexander Bogomolny. "Cut-the-knot.org: Pythagorean theorem and its many proofs, Proof #3". Cut the Knot. Retrieved 4 November 2010.
7. (Loomis 1940)
8. (Maor 2007, p. 39)
9. Schroeder, Manfred Robert (2012). Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. Courier Corporation. pp. 3–4. ISBN 978-0486134789.
10. See for example Pythagorean theorem by shear mapping Archived 2016-10-14 at the Wayback Machine, Saint Louis University website Java applet
11. Jan Gullberg (1997). Mathematics: from the birth of numbers. W. W. Norton & Company. p. 435. ISBN 0-393-04002-X.
12. Heiberg, J.L. "Euclid's Elements of Geometry" (PDF). pp. 46–47.
13. "Euclid's Elements, Book I, Proposition 47". See also a web page version using Java applets by Prof. David E. Joyce, Clark University.
14. Stephen W. Hawking (2005). God created the integers: the mathematical breakthroughs that changed history. Philadelphia: Running Press Book Publishers. p. 12. ISBN 0-7624-1922-9. This proof first appeared after a computer program was set to check Euclidean proofs.
15. The proof by Pythagoras probably was not a general one, as the theory of proportions was developed only two centuries after Pythagoras; see (Maor 2007, p. 25)
16. Alexander Bogomolny. "Pythagorean theorem, proof number 10". Cut the Knot. Retrieved 27 February 2010.
17. (Loomis 1940, p. 113, Geometric proof 22 and Figure 123)
18. Polster, Burkard (2004). Q.E.D.: Beauty in Mathematical Proof. Walker Publishing Company. p. 49.
19. Published in a weekly mathematics column: James A Garfield (1876). "Pons Asinorum". The New England Journal of Education. 3 (14): 161. as noted in William Dunham (1997). The mathematical universe: An alphabetical journey through the great proofs, problems, and personalities. Wiley. p. 96. ISBN 0-471-17661-3. and in A calendar of mathematical dates: April 1, 1876 Archived July 14, 2010, at the Wayback Machine by V. Frederick Rickey
20. Lantz, David. "Garfield's proof of the Pythagorean Theorem". Math.Colgate.edu. Archived from the original on 2013-08-28. Retrieved 2018-01-14.
21. Maor, Eli, The Pythagorean Theorem, Princeton University Press, 2007: pp. 106-107.
22. Mike Staring (1996). "The Pythagorean proposition: A proof by means of calculus". Mathematics Magazine. Mathematical Association of America. 69 (1): 45–46. doi:10.2307/2691395. JSTOR 2691395.
23. Bogomolny, Alexander. "Pythagorean Theorem". Interactive Mathematics Miscellany and Puzzles. Alexander Bogomolny. Archived from the original on 2010-07-06. Retrieved 2010-05-09.
24. Bruce C. Berndt (1988). "Ramanujan – 100 years old (fashioned) or 100 years new (fangled)?". The Mathematical Intelligencer. 10 (3): 24–31. doi:10.1007/BF03026638. S2CID 123311054.
25. Judith D. Sally; Paul J. Sally Jr. (2007-12-21). "Theorem 2.4 (Converse of the Pythagorean theorem).". Roots to Research. American Mathematical Society. pp. 54–55. ISBN 978-0-8218-4403-8.
26. Euclid's Elements, Book I, Proposition 48 From D.E. Joyce's web page at Clark University
27. Casey, Stephen, "The converse of the theorem of Pythagoras", Mathematical Gazette 92, July 2008, 309–313.
28. Mitchell, Douglas W., "Feedback on 92.47", Mathematical Gazette 93, March 2009, 156.
29. Ernest Julius Wilczynski; Herbert Ellsworth Slaught (1914). "Theorem 1 and Theorem 2". Plane trigonometry and applications. Allyn and Bacon. p. 85.
30. Dijkstra, Edsger W. (September 7, 1986). "On the theorem of Pythagoras". EWD975. E. W. Dijkstra Archive.
31. Alexander Bogomolny, Pythagorean Theorem for the Reciprocals,https://www.cut-the-knot.org/pythagoras/PTForReciprocals.shtml
32. Law, Henry (1853). "Corollary 5 of Proposition XLVII (Pythagoras's Theorem)". The Elements of Euclid: with many additional propositions, and explanatory notes, to which is prefixed an introductory essay on logic. John Weale. p. 49.
33. Shaughan Lavine (1994). Understanding the infinite. Harvard University Press. p. 13. ISBN 0-674-92096-1.
34. (Heath 1921, Vol I, pp. 65); Hippasus was on a voyage at the time, and his fellows cast him overboard. See James R. Choike (1980). "The pentagram and the discovery of an irrational number". The College Mathematics Journal. 11: 312–316.
35. A careful discussion of Hippasus's contributions is found in Kurt Von Fritz (Apr 1945). "The Discovery of Incommensurability by Hippasus of Metapontum". Annals of Mathematics. Second Series. 46 (2): 242–264. doi:10.2307/1969021. JSTOR 1969021.
36. Jon Orwant; Jarkko Hietaniemi; John Macdonald (1999). "Euclidean distance". Mastering algorithms with Perl. O'Reilly Media, Inc. p. 426. ISBN 1-56592-398-7.
37. Wentworth, George (2009). Plane Trigonometry and Tables. BiblioBazaar, LLC. p. 116. ISBN 978-1-103-07998-8., Exercises, page 116
38. Lawrence S. Leff (2005). PreCalculus the Easy Way (7th ed.). Barron's Educational Series. p. 296. ISBN 0-7641-2892-2.
39. WS Massey (Dec 1983). "Cross products of vectors in higher-dimensional Euclidean spaces" (PDF). The American Mathematical Monthly. Mathematical Association of America. 90 (10): 697–701. doi:10.2307/2323537. JSTOR 2323537. S2CID 43318100. Archived from the original (PDF) on 2021-02-26.
40. Pertti Lounesto (2001). "§7.4 Cross product of two vectors". Clifford algebras and spinors (2nd ed.). Cambridge University Press. p. 96. ISBN 0-521-00551-5.
41. Francis Begnaud Hildebrand (1992). Methods of applied mathematics (Reprint of Prentice-Hall 1965 2nd ed.). Courier Dover Publications. p. 24. ISBN 0-486-67002-3.
42. Heath, T. L., A History of Greek Mathematics, Oxford University Press, 1921; reprinted by Dover, 1981.
43. Euclid's Elements: Book VI, Proposition VI 31: "In right-angled triangles the figure on the side subtending the right angle is equal to the similar and similarly described figures on the sides containing the right angle."
44. Putz, John F. and Sipka, Timothy A. "On generalizing the Pythagorean theorem", The College Mathematics Journal 34 (4), September 2003, pp. 291–295.
45. Lawrence S. Leff (2005-05-01). cited work. Barron's Educational Series. p. 326. ISBN 0-7641-2892-2.
46. Howard Whitley Eves (1983). "§4.8:...generalization of Pythagorean theorem". Great moments in mathematics (before 1650). Mathematical Association of America. p. 41. ISBN 0-88385-310-8.
47. Aydin Sayili (Mar 1960). "Thâbit ibn Qurra's Generalization of the Pythagorean Theorem". Isis. 51 (1): 35–37. doi:10.1086/348837. JSTOR 227603. S2CID 119868978.
48. Judith D. Sally; Paul Sally (2007-12-21). "Exercise 2.10 (ii)". Roots to Research: A Vertical Development of Mathematical Problems. American Mathematical Soc. p. 62. ISBN 978-0-8218-4403-8.
49. For the details of such a construction, see Jennings, George (1997). "Figure 1.32: The generalized Pythagorean theorem". Modern geometry with applications: with 150 figures (3rd ed.). Springer. p. 23. ISBN 0-387-94222-X.
50. Claudi Alsina, Roger B. Nelsen: Charming Proofs: A Journey Into Elegant Mathematics. MAA, 2010, ISBN 9780883853481, pp. 77–78 (excerpt, p. 77, at Google Books)
51. Rajendra Bhatia (1997). Matrix analysis. Springer. p. 21. ISBN 0-387-94846-5.
52. For an extended discussion of this generalization, see, for example, Willie W. Wong Archived 2009-12-29 at the Wayback Machine 2002, A generalized n-dimensional Pythagorean theorem.
53. Ferdinand van der Heijden; Dick de Ridder (2004). Classification, parameter estimation, and state estimation. Wiley. p. 357. ISBN 0-470-09013-8.
54. Qun Lin; Jiafu Lin (2006). Finite element methods: accuracy and improvement. Elsevier. p. 23. ISBN 7-03-016656-6.
55. Howard Anton; Chris Rorres (2010). Elementary Linear Algebra: Applications Version (10th ed.). Wiley. p. 336. ISBN 978-0-470-43205-1.
56. Karen Saxe (2002). "Theorem 1.2". Beginning functional analysis. Springer. p. 7. ISBN 0-387-95224-1.
57. Douglas, Ronald G. (1998). Banach Algebra Techniques in Operator Theory (2nd ed.). New York, New York: Springer-Verlag New York, Inc. pp. 60–61. ISBN 978-0-387-98377-6.
58. Donald R Conant & William A Beyer (Mar 1974). "Generalized Pythagorean Theorem". The American Mathematical Monthly. Mathematical Association of America. 81 (3): 262–265. doi:10.2307/2319528. JSTOR 2319528.
59. Eric W. Weisstein (2003). CRC concise encyclopedia of mathematics (2nd ed.). CRC Press. p. 2147. ISBN 1-58488-347-2. The parallel postulate is equivalent to the Equidistance postulate, Playfair axiom, Proclus axiom, the Triangle postulate and the Pythagorean theorem.
60. Alexander R. Pruss (2006). The principle of sufficient reason: a reassessment. Cambridge University Press. p. 11. ISBN 0-521-85959-X. We could include...the parallel postulate and derive the Pythagorean theorem. Or we could instead make the Pythagorean theorem among the other axioms and derive the parallel postulate.
61. Stephen W. Hawking (2005). cited work. Running Press. p. 4. ISBN 0-7624-1922-9.
62. Victor Pambuccian (December 2010). "Maria Teresa Calapso's Hyperbolic Pythagorean Theorem". The Mathematical Intelligencer. 32 (4): 2. doi:10.1007/s00283-010-9169-0.
63. Barrett O'Neill (2006). "Exercise 4". Elementary Differential Geometry (2nd ed.). Academic Press. p. 441. ISBN 0-12-088735-5.
64. Saul Stahl (1993). "Theorem 8.3". The Poincaré half-plane: a gateway to modern geometry. Jones & Bartlett Learning. p. 122. ISBN 0-86720-298-X.
65. Jane Gilman (1995). "Hyperbolic triangles". Two-generator discrete subgroups of PSL(2,R). American Mathematical Society Bookstore. ISBN 0-8218-0361-1.
66. Tai L. Chow (2000). Mathematical methods for physicists: a concise introduction. Cambridge University Press. p. 52. ISBN 0-521-65544-7.
67. Neugebauer 1969, p. 36.
68. Neugebauer 1969: p. 36 "In other words it was known during the whole duration of Babylonian mathematics that the sum of the squares on the lengths of the sides of a right triangle equals the square of the length of the hypotenuse."
69. Friberg, Jöran (1981). "Methods and traditions of Babylonian mathematics: Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations". Historia Mathematica. 8: 277–318. doi:10.1016/0315-0860(81)90069-0.: p. 306 "Although Plimpton 322 is a unique text of its kind, there are several other known texts testifying that the Pythagorean theorem was well known to the mathematicians of the Old Babylonian period."
70. Høyrup, Jens. "Pythagorean 'Rule' and 'Theorem' – Mirror of the Relation Between Babylonian and Greek Mathematics". In Renger, Johannes (ed.). Babylon: Focus mesopotamischer Geschichte, Wiege früher Gelehrsamkeit, Mythos in der Moderne. 2. Internationales Colloquium der Deutschen Orient-Gesellschaft 24.–26. März 1998 in Berlin (PDF). Berlin: Deutsche Orient-Gesellschaft / Saarbrücken: SDV Saarbrücker Druckerei und Verlag. pp. 393–407., p. 406, "To judge from this evidence alone it is therefore likely that the Pythagorean rule was discovered within the lay surveyors’ environment, possibly as a spin-off from the problem treated in Db2-146, somewhere between 2300 and 1825 BC." (Db2-146 is an Old Babylonian clay tablet from Eshnunna concerning the computation of the sides of a rectangle given its area and diagonal.)
71. Robson, E. (2008). Mathematics in Ancient Iraq: A Social History. Princeton University Press.: p. 109 "Many Old Babylonian mathematical practitioners … knew that the square on the diagonal of a right triangle had the same area as the sum of the squares on the length and width: that relationship is used in the worked solutions to word problems on cut-and-paste ‘algebra’ on seven different tablets, from Ešnuna, Sippar, Susa, and an unknown location in southern Babylonia."
72. Robson, Eleanor (2001). "Neither Sherlock Holmes nor Babylon: a reassessment of Plimpton 322". Historia Mathematica. 28 (3): 167–206. doi:10.1006/hmat.2001.2317.
73. Kim Plofker (2009). Mathematics in India. Princeton University Press. pp. 17–18. ISBN 978-0-691-12067-6.
74. Carl Benjamin Boyer; Uta C. Merzbach (2011). "China and India". A history of mathematics (3rd ed.). Wiley. p. 229. ISBN 978-0470525487. Quote: [In Sulba-sutras,] we find rules for the construction of right angles by means of triples of cords the lengths of which form Pythagorean triages, such as 3, 4, and 5, or 5, 12, and 13, or 8, 15, and 17, or 12, 35, and 37. Although Mesopotamian influence in the Sulvasũtras is not unlikely, we know of no conclusive evidence for or against this. Aspastamba knew that the square on the diagonal of a rectangle is equal to the sum of the squares on the two adjacent sides. Less easily explained is another rule given by Apastamba – one that strongly resembles some of the geometric algebra in Book II of Euclid's Elements. (...)
75. Proclus (1970). A Commentary of the First Book of Euclid's Elements. Translated by Morrow, Glenn R. Princeton University Press. 428.6.
76. "Introduction and books 1,2". The University Press. March 25, 1908 – via Google Books.
77. (Heath 1921, Vol I, p. 144): "Though this is the proposition universally associated by tradition with the name of Pythagoras, no really trustworthy evidence exists that it was actually discovered by him. The comparatively late writers who attribute it to him add the story that he sacrificed an ox to celebrate his discovery."
78. An extensive discussion of the historical evidence is provided in (Euclid 1956, p. 351) page=351
79. Asger Aaboe (1997). Episodes from the early history of mathematics. Mathematical Association of America. p. 51. ISBN 0-88385-613-1. ...it is not until Euclid that we find a logical sequence of general theorems with proper proofs.
80. Robert P. Crease (2008). The great equations: breakthroughs in science from Pythagoras to Heisenberg. W W Norton & Co. p. 25. ISBN 978-0-393-06204-5.
81. A rather extensive discussion of the origins of the various texts in the Zhou Bi is provided by Christopher Cullen (2007). Astronomy and Mathematics in Ancient China: The 'Zhou Bi Suan Jing'. Cambridge University Press. pp. 139 ff. ISBN 978-0-521-03537-8.
82. This work is a compilation of 246 problems, some of which survived the book burning of 213 BC, and was put in final form before 100 AD. It was extensively commented upon by Liu Hui in 263 AD. Philip D. Straffin Jr. (2004). "Liu Hui and the first golden age of Chinese mathematics". In Marlow Anderson; Victor J. Katz; Robin J. Wilson (eds.). Sherlock Holmes in Babylon: and other tales of mathematical history. Mathematical Association of America. pp. 69 ff. ISBN 0-88385-546-1. See particularly §3: Nine chapters on the mathematical art, pp. 71 ff.
83. Kangshen Shen; John N. Crossley; Anthony Wah-Cheung Lun (1999). The nine chapters on the mathematical art: companion and commentary. Oxford University Press. p. 488. ISBN 0-19-853936-3.
84. In particular, Li Jimin; see Centaurus, Volume 39. Copenhagen: Munksgaard. 1997. pp. 193, 205.
85. Chen, Cheng-Yih (1996). "§3.3.4 Chén Zǐ's formula and the Chóng-Chã method; Figure 40". Early Chinese work in natural science: a re-examination of the physics of motion, acoustics, astronomy and scientific thoughts. Hong Kong University Press. p. 142. ISBN 962-209-385-X.
86. Wen-tsün Wu (2008). "The Gougu theorem". Selected works of Wen-tsün Wu. World Scientific. p. 158. ISBN 978-981-279-107-8.
Works cited
• Bell, John L. (1999). The Art of the Intelligible: An Elementary Survey of Mathematics in its Conceptual Development. Kluwer. ISBN 0-7923-5972-0.
• Euclid (1956). The Thirteen Books of Euclid's Elements, Translated from the Text of Heiberg, with Introduction and Commentary. Vol. 1 (Books I and II). Translated by Heath, Thomas L. (Reprint of 2nd (1925) ed.). Dover. On-line text at archive.org
• Heath, Sir Thomas (1921). "The 'Theorem of Pythagoras'". A History of Greek Mathematics (2 Vols.) (Dover Publications, Inc. (1981) ed.). Clarendon Press, Oxford. pp. 144 ff. ISBN 0-486-24073-8.
• Libeskind, Shlomo (2008). Euclidean and transformational geometry: a deductive inquiry. Jones & Bartlett Learning. ISBN 978-0-7637-4366-6. This high-school geometry text covers many of the topics in this WP article.
• Loomis, Elisha Scott (1940). The Pythagorean Proposition (2nd ed.). Ann Arbor, Michigan: Edwards Brothers. ISBN 9780873530361. Reissued 1968 by the National Council of Teachers of Mathematics. A lower-quality scan was published online by the Education Resources Information Center, ERIC ED037335.
• Maor, Eli (2007). The Pythagorean Theorem: A 4,000-Year History. Princeton, New Jersey: Princeton University Press. ISBN 978-0-691-12526-8.
• Neugebauer, Otto (1969). The exact sciences in antiquity. pp. 1–191. ISBN 0-486-22332-9. PMID 14884919. {{cite book}}: |journal= ignored (help)
• Robson, Eleanor and Jacqueline Stedall, eds., The Oxford Handbook of the History of Mathematics, Oxford: Oxford University Press, 2009. pp. vii + 918. ISBN 978-0-19-921312-2.
• Stillwell, John (1989). Mathematics and Its History. Springer-Verlag. ISBN 0-387-96981-0. Also ISBN 3-540-96981-0.
• Swetz, Frank; Kao, T. I. (1977). Was Pythagoras Chinese?: An Examination of Right Triangle Theory in Ancient China. Pennsylvania State University Press. ISBN 0-271-01238-2.
• van der Waerden, Bartel Leendert (1983). Geometry and Algebra in Ancient Civilizations. Springer. ISBN 3-540-12159-5. Pythagorean triples Babylonian scribes van der Waerden.
External links
• Pythagorean theorem at ProofWiki
Wikimedia Commons has media related to Pythagorean theorem.
• Euclid (1997) [c. 300 BC]. David E. Joyce (ed.). Elements. Retrieved 2006-08-30. In HTML with Java-based interactive figures.
• "Pythagorean theorem". Encyclopedia of Mathematics. EMS Press. 2001 [1994].
• History topic: Pythagoras's theorem in Babylonian mathematics
• Interactive links:
• Interactive proof in Java of the Pythagorean theorem
• Another interactive proof in Java of the Pythagorean theorem
• Pythagorean theorem with interactive animation
• Animated, non-algebraic, and user-paced Pythagorean theorem
• Pythagorean theorem water demo on YouTube
• Pythagorean theorem (more than 70 proofs from cut-the-knot)
• Weisstein, Eric W. "Pythagorean theorem". MathWorld.
Authority control: National
• Spain
• France
• BnF data
• Germany
• Israel
• United States
• Japan
| Wikipedia |
# ALGEBRAIC CURVES
An Introduction to Algebraic Geometry
WILLIAM FULTON
January 28, 2008
## Third Preface, 2008
This text has been out of print for several years, with the author holding copyrights. Since I continue to hear from young algebraic geometers who used this as their first text, I am glad now to make this edition available without charge to anyone interested. I am most grateful to Kwankyu Lee for making a careful LaTeX version, which was the basis of this edition; thanks also to Eugene Eisenstein for help with the graphics.
As in 1989, I have managed to resist making sweeping changes. I thank all who have sent corrections to earlier versions, especially Grzegorz Bobiński for the most recent and thorough list. It is inevitable that this conversion has introduced some new errors, and I and future readers will be grateful if you will send any errors you find to me at [email protected].
## Second Preface, 1989
When this book first appeared, there were few texts available to a novice in modern algebraic geometry. Since then many introductory treatises have appeared, including excellent texts by Shafarevich, Mumford, Hartshorne, Griffiths-Harris, Kunz, Clemens, Iitaka, Brieskorn-Knörrer, and Arbarello-Cornalba-Griffiths-Harris.
The past two decades have also seen a good deal of growth in our understanding of the topics covered in this text: linear series on curves, intersection theory, and the Riemann-Roch problem. It has been tempting to rewrite the book to reflect this progress, but it does not seem possible to do so without abandoning its elementary character and destroying its original purpose: to introduce students with a little algebra background to a few of the ideas of algebraic geometry and to help them gain some appreciation both for algebraic geometry and for origins and applications of many of the notions of commutative algebra. If working through the book and its exercises helps prepare a reader for any of the texts mentioned above, that will be an added benefit.
## First Preface, 1969
Although algebraic geometry is a highly developed and thriving field of mathematics, it is notoriously difficult for the beginner to make his way into the subject. There are several texts on an undergraduate level that give an excellent treatment of the classical theory of plane curves, but these do not prepare the student adequately for modern algebraic geometry. On the other hand, most books with a modern approach demand considerable background in algebra and topology, often the equivalent of a year or more of graduate study. The aim of these notes is to develop the theory of algebraic curves from the viewpoint of modern algebraic geometry, but without excessive prerequisites.
We have assumed that the reader is familiar with some basic properties of rings, ideals, and polynomials, such as is often covered in a one-semester course in modern algebra; additional commutative algebra is developed in later sections. Chapter 1 begins with a summary of the facts we need from algebra. The rest of the chapter is concerned with basic properties of affine algebraic sets; we have given Zariski's proof of the important Nullstellensatz.
The coordinate ring, function field, and local rings of an affine variety are studied in Chapter 2. As in any modern treatment of algebraic geometry, they play a fundamental role in our preparation. The general study of affine and projective varieties is continued in Chapters 4 and 6, but only as far as necessary for our study of curves.
Chapter 3 considers affine plane curves. The classical definition of the multiplicity of a point on a curve is shown to depend only on the local ring of the curve at the point. The intersection number of two plane curves at a point is characterized by its properties, and a definition in terms of a certain residue class ring of a local ring is shown to have these properties. Bézout's Theorem and Max Noether's Fundamental Theorem are the subject of Chapter 5. (Anyone familiar with the cohomology of projective varieties will recognize that this cohomology is implicit in our proofs.)
In Chapter 7 the nonsingular model of a curve is constructed by means of blowing up points, and the correspondence between algebraic function fields on one variable and nonsingular projective curves is established. In the concluding chapter the algebraic approach of Chevalley is combined with the geometric reasoning of Brill and Noether to prove the Riemann-Roch Theorem.
These notes are from a course taught to Juniors at Brandeis University in 196768. The course was repeated (assuming all the algebra) to a group of graduate students during the intensive week at the end of the Spring semester. We have retained an essential feature of these courses by including several hundred problems. The results of the starred problems are used freely in the text, while the others range from exercises to applications and extensions of the theory.
From Chapter 3 on, $k$ denotes a fixed algebraically closed field. Whenever convenient (including without comment many of the problems) we have assumed $k$ to be of characteristic zero. The minor adjustments necessary to extend the theory to arbitrary characteristic are discussed in an appendix.
Thanks are due to Richard Weiss, a student in the course, for sharing the task of writing the notes. He corrected many errors and improved the clarity of the text. Professor Paul Monsky provided several helpful suggestions as I taught the course. “Je n'ai jamais été assez loin pour bien sentir l'application de l'algèbre à la géométrie. Je n'ai mois point cette manière d'opérer sans voir ce qu'on fait, et il me sembloit que résoudre un probleme de géométrie par les équations, c'étoit jouer un air en tournant une manivelle. La premiere fois que je trouvai par le calcul que le carré d'un binôme étoit composé du carré de chacune de ses parties, et du double produit de l'une par l'autre, malgré la justesse de ma multiplication, je n'en voulus rien croire jusqu'à ce que j'eusse fai la figure. Ce n'étoit pas que je n'eusse un grand goût pour l'algèbre en n'y considérant que la quantité abstraite; mais appliquée a l'étendue, je voulois voir l'opération sur les lignes; autrement je n'y comprenois plus rien."
Les Confessions de J.-J. Rousseau
## Chapter 1
## Affine Algebraic Sets
### Algebraic Preliminaries
This section consists of a summary of some notation and facts from commutative algebra. Anyone familiar with the italicized terms and the statements made here about them should have sufficient background to read the rest of the notes.
When we speak of a ring, we shall always mean a commutative ring with a multiplicative identity. A ring homomorphism from one ring to another must take the multiplicative identity of the first ring to that of the second. A domain, or integral domain, is a ring (with at least two elements) in which the cancellation law holds. A field is a domain in which every nonzero element is a unit, i.e., has a multiplicative inverse.
$\mathbb{Z}$ will denote the domain of integers, while $\mathbb{Q}, \mathbb{R}$, and $\mathbb{C}$ will denote the fields of rational, real, complex numbers, respectively.
Any domain $R$ has a quotient field $K$, which is a field containing $R$ as a subring, and any elements in $K$ may be written (not necessarily uniquely) as a ratio of two elements of $R$. Any one-to-one ring homomorphism from $R$ to a field $L$ extends uniquely to a ring homomorphism from $K$ to $L$. Any ring homomorphism from a field to a nonzero ring is one-to-one.
For any ring $R, R[X]$ denotes the ring of polynomials with coefficients in $R$. The degree of a nonzero polynomial $\sum a_{i} X^{i}$ is the largest integer $d$ such that $a_{d} \neq 0$; the polynomial is monic if $a_{d}=1$.
The ring of polynomials in $n$ variables over $R$ is written $R\left[X_{1}, \ldots, X_{n}\right]$. We often write $R[X, Y]$ or $R[X, Y, Z]$ when $n=2$ or 3 . The monomials in $R\left[X_{1}, \ldots, X_{n}\right]$ are the polynomials $X_{1}^{i_{1}} X_{2}^{i_{2}} \cdots X_{n}^{i_{n}}, i_{j}$ nonnegative integers; the degree of the monomial is $i_{1}+\cdots+i_{n}$. Every $F \in R\left[X_{1}, \ldots, X_{n}\right]$ has a unique expression $F=\sum a_{(i)} X^{(i)}$, where the $X^{(i)}$ are the monomials, $a_{(i)} \in R$. We call $F$ homogeneous, or a form, of degree $d$, if all coefficients $a_{(i)}$ are zero except for monomials of degree $d$. Any polynomial $F$ has a unique expression $F=F_{0}+F_{1}+\cdots+F_{d}$, where $F_{i}$ is a form of degree $i$; if $F_{d} \neq 0, d$ is the degree of $F$, written $\operatorname{deg}(F)$. The terms $F_{0}, F_{1}, F_{2}, \ldots$ are called the constant, linear, quadratic, ... terms of $F ; F$ is constant if $F=F_{0}$. The zero polynomial is allowed to have any degree. If $R$ is a domain, $\operatorname{deg}(F G)=\operatorname{deg}(F)+\operatorname{deg}(G)$. The ring $R$ is a subring of $R\left[X_{1}, \ldots, X_{n}\right]$, and $R\left[X_{1}, \ldots, X_{n}\right]$ is characterized by the following property: if $\varphi$ is a ring homomorphism from $R$ to a ring $S$, and $s_{1}, \ldots, s_{n}$ are elements in $S$, then there is a unique extension of $\varphi$ to a ring homomorphism $\tilde{\varphi}$ from $R\left[X_{1}, \ldots, X_{n}\right]$ to $S$ such that $\tilde{\varphi}\left(X_{i}\right)=s_{i}$, for $1 \leq i \leq n$. The image of $F$ under $\tilde{\varphi}$ is written $F\left(s_{1}, \ldots, s_{n}\right)$. The ring $R\left[X_{1}, \ldots, X_{n}\right]$ is canonically isomorphic to $R\left[X_{1}, \ldots, X_{n-1}\right]\left[X_{n}\right]$.
An element $a$ in a ring $R$ is irreducible if it is not a unit or zero, and for any factorization $a=b c, b, c \in R$, either $b$ or $c$ is a unit. A domain $R$ is a unique factorization domain, written UFD, if every nonzero element in $R$ can be factored uniquely, up to units and the ordering of the factors, into irreducible elements.
If $R$ is a UFD with quotient field $K$, then (by Gauss) any irreducible element $F \in$ $R[X]$ remains irreducible when considered in $K[X]$; it follows that if $F$ and $G$ are polynomials in $R[X]$ with no common factors in $R[X]$, they have no common factors in $K[X]$.
If $R$ is a UFD, then $R[X]$ is also a UFD. Consequently $k\left[X_{1}, \ldots, X_{n}\right]$ is a UFD for any field $k$. The quotient field of $k\left[X_{1}, \ldots, X_{n}\right]$ is written $k\left(X_{1}, \ldots, X_{n}\right)$, and is called the field of rational functions in $n$ variables over $k$.
If $\varphi: R \rightarrow S$ is a ring homomorphism, the set $\varphi^{-1}(0)$ of elements mapped to zero is the kernel of $\varphi$, written $\operatorname{Ker}(\varphi)$. It is an ideal in $R$. And ideal $I$ in a ring $R$ is proper if $I \neq R$. A proper ideal is maximal if it is not contained in any larger proper ideal. A prime ideal is an ideal $I$ such that whenever $a b \in I$, either $a \in I$ or $b \in I$.
A set $S$ of elements of a ring $R$ generates an ideal $I=\left\{\sum a_{i} s_{i} \mid s_{i} \in S, a_{i} \in R\right\}$. An ideal is finitely generated if it is generated by a finite set $S=\left\{f_{1}, \ldots, f_{n}\right\}$; we then write $I=\left(f_{1}, \ldots, f_{n}\right)$. An ideal is principal if it is generated by one element. A domain in which every ideal is principal is called a principal ideal domain, written PID. The ring of integers $Z$ and the ring of polynomials $k[X]$ in one variable over a field $k$ are examples of PID's. Every PID is a UFD. A principal ideal $I=(a)$ in a UFD is prime if and only if $a$ is irreducible (or zero).
Let $I$ be an ideal in a ring $R$. The residue class ring of $R$ modulo $I$ is written $R / I$; it is the set of equivalence classes of elements in $R$ under the equivalence relation: $a \sim b$ if $a-b \in I$. The equivalence class containing $a$ may be called the $I$-residue of $a$; it is often denoted by $\bar{a}$. The classes $R / I$ form a ring in such a way that the mapping $\pi: R \rightarrow R / I$ taking each element to its $I$-residue is a ring homomorphism. The ring $R / I$ is characterized by the following property: if $\varphi: R \rightarrow S$ is a ring homomorphism to a ring $S$, and $\varphi(I)=0$, then there is a unique ring homomorphism $\bar{\varphi}: R / I \rightarrow S$ such that $\varphi=\bar{\varphi} \circ \pi$. A proper ideal $I$ in $R$ is prime if and only if $R / I$ is a domain, and maximal if and only if $R / I$ is a field. Every maximal ideal is prime.
Let $k$ be a field, $I$ a proper ideal in $k\left[X_{1}, \ldots, X_{n}\right]$. The canonical homomorphism $\pi$ from $k\left[X_{1}, \ldots, X_{n}\right]$ to $k\left[X_{1}, \ldots, X_{n}\right] / I$ restricts to a ring homomorphism from $k$ to $k\left[X_{1}, \ldots, X_{n}\right] / I$. We thus regard $k$ as a subring of $k\left[X_{1}, \ldots, X_{n}\right] / I$; in particular, $k\left[X_{1}, \ldots, X_{n}\right] / I$ is a vector space over $k$.
Let $R$ be a domain. The characteristic of $R, \operatorname{char}(R)$, is the smallest integer $p$ such that $1+\cdots+1$ ( $p$ times) $=0$, if such a $p$ exists; otherwise $\operatorname{char}(R)=0$. If $\varphi: \mathbb{Z} \rightarrow R$ is the unique ring homomorphism from $\mathbb{Z}$ to $R$, then $\operatorname{Ker}(\varphi)=(p)$, so char $(R)$ is a prime number or zero.
If $R$ is a ring, $a \in R, F \in R[X]$, and $a$ is a root of $F$, then $F=(X-a) G$ for a unique $G \in R[X]$. A field $k$ is algebraically closed if any non-constant $F \in k[X]$ has a root. It follows that $F=\mu \Pi\left(X-\lambda_{i}\right)^{e_{i}}, \mu, \lambda_{i} \in k$, where the $\lambda_{i}$ are the distinct roots of $F$, and $e_{i}$ is the multiplicity of $\lambda_{i}$. A polynomial of degree $d$ has $d$ roots in $k$, counting multiplicities. The field $\mathbb{C}$ of complex numbers is an algebraically closed field.
Let $R$ be any ring. The derivative of a polynomial $F=\sum a_{i} X^{i} \in R[X]$ is defined to be $\sum i a_{i} X^{i-1}$, and is written either $\frac{\partial F}{\partial X}$ or $F_{X}$. If $F \in R\left[X_{1}, \ldots, X_{n}\right], \frac{\partial F}{\partial X_{i}}=F_{X_{i}}$ is defined by considering $F$ as a polynomial in $X_{i}$ with coefficients in $R\left[X_{1}, \ldots, X_{i-1}, X_{i+1}, \ldots, X_{n}\right]$. The following rules are easily verified:
(1) $(a F+b G)_{X}=a F_{X}+b G_{X}, a, b \in R$.
(2) $F_{X}=0$ if $F$ is a constant.
(3) $(F G)_{X}=F_{X} G+F G_{X}$, and $\left(F^{n}\right)_{X}=n F^{n-1} F_{X}$.
(4) If $G_{1}, \ldots, G_{n} \in R[X]$, and $F \in R\left[X_{1}, \ldots, X_{n}\right]$, then
$$
F\left(G_{1}, \ldots, G_{n}\right)_{X}=\sum_{i=1}^{n} F_{X_{i}}\left(G_{1}, \ldots, G_{n}\right)\left(G_{i}\right)_{X} .
$$
(5) $F_{X_{i} X_{j}}=F_{X_{j} X_{i}}$, where we have written $F_{X_{i} X_{j}}$ for $\left(F_{X_{i}}\right)_{X_{j}}$.
(6) (Euler's Theorem) If $F$ is a form of degree $m$ in $R\left[X_{1}, \ldots, X_{n}\right]$, then
$$
m F=\sum_{i=1}^{n} X_{i} F_{X_{i}}
$$
## Problems
1.1* Let $R$ be a domain. (a) If $F, G$ are forms of degree $r, s$ respectively in $R\left[X_{1}, \ldots, X_{n}\right]$, show that $F G$ is a form of degree $r+s$. (b) Show that any factor of a form in $R\left[X_{1}, \ldots, X_{n}\right]$ is also a form.
1.2* Let $R$ be a UFD, $K$ the quotient field of $R$. Show that every element $z$ of $K$ may be written $z=a / b$, where $a, b \in R$ have no common factors; this representative is unique up to units of $R$.
1.3. Let $R$ be a PID, Let $P$ be a nonzero, proper, prime ideal in $R$. (a) Show that $P$ is generated by an irreducible element. (b) Show that $P$ is maximal.
1.4. Let $k$ be an infinite field, $F \in k\left[X_{1}, \ldots, X_{n}\right]$. Suppose $F\left(a_{1}, \ldots, a_{n}\right)=0$ for all $a_{1}, \ldots, a_{n} \in k$. Show that $F=0$. (Hint: Write $F=\sum F_{i} X_{n}^{i}, F_{i} \in k\left[X_{1}, \ldots, X_{n-1}\right]$. Use induction on $n$, and the fact that $F\left(a_{1}, \ldots, a_{n-1}, X_{n}\right)$ has only a finite number of roots if any $F_{i}\left(a_{1}, \ldots, a_{n-1}\right) \neq 0$.)
1.5. Let $k$ be any field. Show that there are an infinite number of irreducible monic polynomials in $k[X]$. (Hint: Suppose $F_{1}, \ldots, F_{n}$ were all of them, and factor $F_{1} \cdots F_{n}+$ 1 into irreducible factors.)
1.6. Show that any algebraically closed field is infinite. (Hint: The irreducible monic polynomials are $X-a, a \in k$.)
1.7. Let $k$ be a field, $F \in k\left[X_{1}, \ldots, X_{n}\right], a_{1}, \ldots, a_{n} \in k$. (a) Show that
$$
F=\sum \lambda_{(i)}\left(X_{1}-a_{1}\right)^{i_{1}} \ldots\left(X_{n}-a_{n}\right)^{i_{n}}, \quad \lambda_{(i)} \in k .
$$
(b) If $F\left(a_{1}, \ldots, a_{n}\right)=0$, show that $F=\sum_{i=1}^{n}\left(X_{i}-a_{i}\right) G_{i}$ for some (not unique) $G_{i}$ in $k\left[X_{1}, \ldots, X_{n}\right]$.
### Affine Space and Algebraic Sets
Let $k$ be any field. By $\mathbb{A}^{n}(k)$, or simply $\mathbb{A}^{n}$ (if $k$ is understood), we shall mean the cartesian product of $k$ with itself $n$ times: $\mathbb{A}^{n}(k)$ is the set of $n$-tuples of elements of $k$. We call $\mathbb{A}^{n}(k)$ affine $n$-space over $k$; its elements will be called points. In particular, $\mathbb{A}^{1}(k)$ is the affine line, $\mathbb{A}^{2}(k)$ the affine plane.
If $F \in k\left[X_{1}, \ldots, X_{n}\right]$, a point $P=\left(a_{1}, \ldots, a_{n}\right)$ in $\mathbb{A}^{n}(k)$ is called a zero of $F$ if $F(P)=$ $F\left(a_{1}, \ldots, a_{n}\right)=0$. If $F$ is not a constant, the set of zeros of $F$ is called the hypersurface defined by $F$, and is denoted by $V(F)$. A hypersurface in $\mathbb{A}^{2}(k)$ is called an affine plane curve. If $F$ is a polynomial of degree one, $V(F)$ is called a hyperplane in $\mathbb{A}^{n}(k)$; if $n=2$, it is a line.
Examples. Let $k=\mathbb{R}$.
a. $V\left(Y^{2}-X\left(X^{2}-1\right)\right) \subset \mathbb{A}^{2}$
c. $V\left(Z^{2}-\left(X^{2}+Y^{2}\right)\right) \subset \mathbb{A}^{3}$
b. $V\left(Y^{2}-X^{2}(X+1)\right) \subset \mathbb{A}^{2}$
d. $\left.V\left(Y^{2}-X Y-X^{2} Y+X^{3}\right)\right) \subset \mathbb{A}^{2}$
More generally, if $S$ is any set of polynomials in $k\left[X_{1}, \ldots, X_{n}\right]$, we let $V(S)=\{P \in$ $\mathbb{A}^{n} \mid F(P)=0$ for all $\left.F \in S\right\}: V(S)=\bigcap_{F \in S} V(F)$. If $S=\left\{F_{1}, \ldots, F_{r}\right\}$, we usually write $V\left(F_{1}, \ldots, F_{r}\right)$ instead of $V\left(\left\{F_{1}, \ldots, F_{r}\right\}\right)$. A subset $X \subset \mathbb{A}^{n}(k)$ is an affine algebraic set, or simply an algebraic set, if $X=V(S)$ for some $S$. The following properties are easy to verify:
(1) If $I$ is the ideal in $k\left[X_{1}, \ldots, X_{n}\right]$ generated by $S$, then $V(S)=V(I)$; so every algebraic set is equal to $V(I)$ for some ideal $I$.
(2) If $\left\{I_{\alpha}\right\}$ is any collection of ideals, then $V\left(\cup_{\alpha} I_{\alpha}\right)=\bigcap_{\alpha} V\left(I_{\alpha}\right)$; so the intersection of any collection of algebraic sets is an algebraic set.
(3) If $I \subset J$, then $V(I) \supset V(J)$. (4) $V(F G)=V(F) \cup V(G)$ for any polynomials $F, G ; V(I) \cup V(J)=V(\{F G \mid F \in$ $I, G \in J\})$; so any finite union of algebraic sets is an algebraic set.
(5) $V(0)=\mathbb{A}^{n}(k) ; V(1)=\varnothing ; V\left(X_{1}-a_{1}, \ldots, X_{n}-a_{n}\right)=\left\{\left(a_{1}, \ldots, a_{n}\right)\right\}$ for $a_{i} \in k$. So any finite subset of $\mathbb{A}^{n}(k)$ is an algebraic set.
## Problems
1.8. Show that the algebraic subsets of $\mathbb{A}^{1}(k)$ are just the finite subsets, together with $\mathbb{A}^{1}(k)$ itself.
1.9. If $k$ is a finite field, show that every subset of $\mathbb{A}^{n}(k)$ is algebraic.
1.10. Give an example of a countable collection of algebraic sets whose union is not algebraic.
1.11. Show that the following are algebraic sets:
(a) $\left\{\left(t, t^{2}, t^{3}\right) \in \mathbb{A}^{3}(k) \mid t \in k\right\}$;
(b) $\left\{(\cos (t), \sin (t)) \in \mathbb{A}^{2}(\mathbb{R}) \mid t \in \mathbb{R}\right\}$;
(c) the set of points in $\mathbb{A}^{2}(\mathbb{R})$ whose polar coordinates $(r, \theta)$ satisfy the equation $r=\sin (\theta)$.
1.12. Suppose $C$ is an affine plane curve, and $L$ is a line in $\mathbb{A}^{2}(k), L \not \subset C$. Suppose $C=V(F), F \in k[X, Y]$ a polynomial of degree $n$. Show that $L \cap C$ is a finite set of no more than $n$ points. (Hint: Suppose $L=V(Y-(a X+b)$ ), and consider $F(X, a X+b) \in$ $k[X]$.
1.13. Show that each of the following sets is not algebraic:
(a) $\left\{(x, y) \in \mathbb{A}^{2}(\mathbb{R}) \mid y=\sin (x)\right\}$.
(b) $\left\{\left.(z, w) \in \mathbb{A}^{2}(\mathbb{C})|| z\right|^{2}+|w|^{2}=1\right\}$, where $|x+i y|^{2}=x^{2}+y^{2}$ for $x, y \in \mathbb{R}$.
(c) $\left\{(\cos (t), \sin (t), t) \in \mathbb{A}^{3}(\mathbb{R}) \mid t \in \mathbb{R}\right\}$.
1.14. Let $F$ be a nonconstant polynomial in $k\left[X_{1}, \ldots, X_{n}\right], k$ algebraically closed. Show that $\mathbb{A}^{n}(k) \backslash V(F)$ is infinite if $n \geq 1$, and $V(F)$ is infinite if $n \geq 2$. Conclude that the complement of any proper algebraic set is infinite. (Hint: See Problem 1.4.)
1.15. Let $V \subset \mathbb{A}^{n}(k), W \subset \mathbb{A}^{m}(k)$ be algebraic sets. Show that
$$
V \times W=\left\{\left(a_{1}, \ldots, a_{n}, b_{1}, \ldots, b_{m}\right) \mid\left(a_{1}, \ldots, a_{n}\right) \in V,\left(b_{1}, \ldots, b_{m}\right) \in W\right\}
$$
is an algebraic set in $\mathbb{A}^{n+m}(k)$. It is called the product of $V$ and $W$.
### The Ideal of a Set of Points
For any subset $X$ of $\mathbb{A}^{n}(k)$, we consider those polynomials that vanish on $X$; they form an ideal in $k\left[X_{1}, \ldots, X_{n}\right]$, called the ideal of $X$, and written $I(X) . I(X)=\{F \in$ $k\left[X_{1}, \ldots, X_{n}\right] \mid F\left(a_{1}, \ldots, a_{n}\right)=0$ for all $\left.\left(a_{1}, \ldots, a_{n}\right) \in X\right\}$. The following properties show some of the relations between ideals and algebraic sets; the verifications are left to the reader (see Problems 1.4 and 1.7):
(6) If $X \subset Y$, then $I(X) \supset I(Y)$. (7) $I(\varnothing)=k\left[X_{1}, \ldots, X_{n}\right] ; I\left(\mathbb{A}^{n}(k)\right)=(0)$ if $k$ is an infinite field; $I\left(\left\{\left(a_{1}, \ldots, a_{n}\right)\right\}\right)=\left(X_{1}-a_{1}, \ldots, X_{n}-a_{n}\right)$ for $a_{1}, \ldots, a_{n} \in k$.
(8) $I(V(S)) \supset S$ for any set $S$ of polynomials; $V(I(X)) \supset X$ for any set $X$ of points.
(9) $V(I(V(S)))=V(S)$ for any set $S$ of polynomials, and $I(V(I(X)))=I(X)$ for any set $X$ of points. So if $V$ is an algebraic set, $V=V(I(V))$, and if $I$ is the ideal of an algebraic set, $I=I(V(I))$.
An ideal that is the ideal of an algebraic set has a property not shared by all ideals: if $I=I(X)$, and $F^{n} \in I$ for some integer $n>0$, then $F \in I$. If $I$ is any ideal in a ring $R$, we define the radical of $I$, written $\operatorname{Rad}(I)$, to be $\left\{a \in R \mid a^{n} \in I\right.$ for some integer $\left.n>0\right\}$. Then $\operatorname{Rad}(I)$ is an ideal (Problem 1.18 below) containing $I$. An ideal $I$ is called a radical ideal if $I=\operatorname{Rad}(I)$. So we have property
(10) $I(X)$ is a radical ideal for any $X \subset \mathbb{A}^{n}(k)$.
## Problems
1.16. Let $V, W$ be algebraic sets in $\mathbb{A}^{n}(k)$. Show that $V=W$ if and only if $I(V)=$ $I(W)$.
1.17* (a) Let $V$ be an algebraic set in $\mathbb{A}^{n}(k), P \in \mathbb{A}^{n}(k)$ a point not in $V$. Show that there is a polynomial $F \in k\left[X_{1}, \ldots, X_{n}\right]$ such that $F(Q)=0$ for all $Q \in V$, but $F(P)=1$. (Hint: $I(V) \neq I(V \cup\{P\})$.) (b) Let $P_{1}, \ldots, P_{r}$ be distinct points in $\mathbb{A}^{n}(k)$, not in an algebraic set $V$. Show that there are polynomials $F_{1}, \ldots, F_{r} \in I(V)$ such that $F_{i}\left(P_{j}\right)=0$ if $i \neq j$, and $F_{i}\left(P_{i}\right)=1$. (Hint: Apply (a) to the union of $V$ and all but one point.) (c) With $P_{1}, \ldots, P_{r}$ and $V$ as in (b), and $a_{i j} \in k$ for $1 \leq i, j \leq r$, show that there are $G_{i} \in I(V)$ with $G_{i}\left(P_{j}\right)=a_{i j}$ for all $i$ and $j$. (Hint: Consider $\sum_{j} a_{i j} F_{j}$.)
1.18* Let $I$ be an ideal in a ring $R$. If $a^{n} \in I, b^{m} \in I$, show that $(a+b)^{n+m} \in I$. Show that $\operatorname{Rad}(I)$ is an ideal, in fact a radical ideal. Show that any prime ideal is radical.
1.19. Show that $I=\left(X^{2}+1\right) \subset \mathbb{R}[X]$ is a radical (even a prime) ideal, but $I$ is not the ideal of any set in $\mathbb{A}^{1}(\mathbb{R})$.
1.20* Show that for any ideal $I$ in $k\left[X_{1}, \ldots, X_{n}\right], V(I)=V(\operatorname{Rad}(I))$, and $\operatorname{Rad}(I) \subset$ $I(V(I))$.
1.21* Show that $I=\left(X_{1}-a_{1}, \ldots, X_{n}-a_{n}\right) \subset k\left[X_{1}, \ldots, X_{n}\right]$ is a maximal ideal, and that the natural homomorphism from $k$ to $k\left[X_{1}, \ldots, X_{n}\right] / I$ is an isomorphism.
### The Hilbert Basis Theorem
Although we have allowed an algebraic set to be defined by any set of polynomials, in fact a finite number will always do.
Theorem 1. Every algebraic set is the intersection of a finite number of hypersurfaces
Proof. Let the algebraic set be $V(I)$ for some ideal $I \subset k\left[X_{1}, \ldots, X_{n}\right]$. It is enough to show that $I$ is finitely generated, for if $I=\left(F_{1}, \ldots, F_{r}\right)$, then $V(I)=V\left(F_{1}\right) \cap \cdots \cap V\left(F_{r}\right)$. To prove this fact we need some algebra: A ring is said to be Noetherian if every ideal in the ring is finitely generated. Fields and PID's are Noetherian rings. Theorem 1, therefore, is a consequence of the
HILBERT BASIS THEOREM. If $R$ is a Noetherian ring, then $R\left[X_{1}, \ldots, X_{n}\right]$ is a Noetherian ring.
Proof. Since $R\left[X_{1}, \ldots, X_{n}\right]$ is isomorphic to $R\left[X_{1}, \ldots, X_{n-1}\right]\left[X_{n}\right]$, the theorem will follow by induction if we can prove that $R[X]$ is Noetherian whenever $R$ is Noetherian. Let $I$ be an ideal in $R[X]$. We must find a finite set of generators for $I$.
If $F=a_{1}+a_{1} X+\cdots+a_{d} X^{d} \in R[X], a_{d} \neq 0$, we call $a_{d}$ the leading coefficient of $F$. Let $J$ be the set of leading coefficients of all polynomials in $I$. It is easy to check that $J$ is an ideal in $R$, so there are polynomials $F_{1}, \ldots, F_{r} \in I$ whose leading coefficients generate $J$. Take an integer $N$ larger than the degree of each $F_{i}$. For each $m \leq N$, let $J_{m}$ be the ideal in $R$ consisting of all leading coefficients of all polynomials $F \in I$ such that $\operatorname{deg}(F) \leq m$. Let $\left\{F_{m j}\right\}$ be a finite set of polynomials in $I$ of degree $\leq m$ whose leading coefficients generate $J_{m}$. Let $I^{\prime}$ be the ideal generated by the $F_{i}$ 's and all the $F_{m j}$ 's. It suffices to show that $I=I^{\prime}$.
Suppose $I^{\prime}$ were smaller than $I$; let $G$ be an element of $I$ of lowest degree that is not in $I^{\prime}$. If $\operatorname{deg}(G)>N$, we can find polynomials $Q_{i}$ such that $\sum Q_{i} F_{i}$ and $G$ have the same leading term. But then $\operatorname{deg}\left(G-\sum Q_{i} F_{i}\right)<\operatorname{deg} G$, so $G-\sum Q_{i} F_{i} \in I^{\prime}$, so $G \in I^{\prime}$. Similarly if $\operatorname{deg}(G)=m \leq N$, we can lower the degree by subtracting off $\sum Q_{j} F_{m j}$ for some $Q_{j}$. This proves the theorem.
Corollary. $k\left[X_{1}, \ldots, X_{n}\right]$ is Noetherian for any field $k$.
## Problem
1.22* Let $I$ be an ideal in a $\operatorname{ring} R, \pi: R \rightarrow R / I$ the natural homomorphism. (a) Show that for every ideal $J^{\prime}$ of $R / I, \pi^{-1}\left(J^{\prime}\right)=J$ is an ideal of $R$ containing $I$, and for every ideal $J$ of $R$ containing $I, \pi(J)=J^{\prime}$ is an ideal of $R / I$. This sets up a natural one-to-one correspondence between \{ideals of $R / I$ \} and \{ideals of $R$ that contain $I$ \} . (b) Show that $J^{\prime}$ is a radical ideal if and only if $J$ is radical. Similarly for prime and maximal ideals. (c) Show that $J^{\prime}$ is finitely generated if $J$ is. Conclude that $R / I$ is Noetherian if $R$ is Noetherian. Any ring of the form $k\left[X_{1}, \ldots, X_{n}\right] / I$ is Noetherian.
### Irreducible Components of an Algebraic Set
An algebraic set may be the union of several smaller algebraic sets (Section 1.2 Example d). An algebraic set $V \subset \mathbb{A}^{n}$ is reducible if $V=V_{1} \cup V_{2}$, where $V_{1}, V_{2}$ are algebraic sets in $\mathbb{A}^{n}$, and $V_{i} \neq V, i=1,2$. Otherwise $V$ is irreducible.
Proposition 1. An algebraic set $V$ is irreducible if and only if $I(V)$ is prime.
Proof. If $I(V)$ is not prime, suppose $F_{1} F_{2} \in I(V), F_{i} \notin I(V)$. Then $V=\left(V \cap V\left(F_{1}\right)\right) \cup$ $\left(V \cap V\left(F_{2}\right)\right.$ ), and $V \cap V\left(F_{i}\right) \varsubsetneqq V$, so $V$ is reducible.
Conversely if $V=V_{1} \cup V_{2}, V_{i} \varsubsetneqq V$, then $I\left(V_{i}\right) \supsetneqq I(V)$; let $F_{i} \in I\left(V_{i}\right), F_{i} \notin I(V)$. Then $F_{1} F_{2} \in I(V)$, so $I(V)$ is not prime. We want to show that an algebraic set is the union of a finite number of irreducible algebraic sets. If $V$ is reducible, we write $V=V_{1} \cup V_{2}$; if $V_{2}$ is reducible, we write $V_{2}=V_{3} \cup V_{4}$, etc. We need to know that this process stops.
Lemma. Let $\mathscr{S}$ be any nonempty collection of ideals in a Noetherian ring $R$. Then $\mathscr{S}$ has a maximal member, i.e. there is an ideal I in $\mathscr{S}$ that is not contained in any other ideal of $\mathscr{S}$.
Proof. Choose (using the axiom of choice) an ideal from each subset of $\mathscr{S}$. Let $I_{0}$ be the chosen ideal for $\mathscr{S}$ itself. Let $\mathscr{S}_{1}=\left\{I \in \mathscr{S} \mid I \supsetneqq I_{0}\right\}$, and let $I_{1}$ be the chosen ideal of $\mathscr{S}_{1}$. Let $\mathscr{S}_{2}=\left\{I \in \mathscr{S} \mid I \supsetneqq I_{1}\right\}$, etc. It suffices to show that some $\mathscr{S}_{n}$ is empty. If not let $I=\cup_{n=0}^{\infty} I_{n}$, an ideal of $R$. Let $F_{1}, \ldots, F_{r}$ generate $I$; each $F_{i} \in I_{n}$ if $n$ is chosen sufficiently large. But then $I_{n}=I$, so $I_{n+1}=I_{n}$, a contradiction.
It follows immediately from this lemma that any collection of algebraic sets in $\mathbb{A}^{n}(k)$ has a minimal member. For if $\left\{V_{\alpha}\right\}$ is such a collection, take a maximal member $I\left(V_{\alpha_{0}}\right)$ from $\left\{I\left(V_{\alpha}\right)\right\}$. Then $V_{\alpha_{0}}$ is clearly minimal in the collection.
Theorem 2. Let $V$ be an algebraic set in $\mathbb{A}^{n}(k)$. Then there are unique irreducible algebraic sets $V_{1}, \ldots, V_{m}$ such that $V=V_{1} \cup \cdots \cup V_{m}$ and $V_{i} \not \subset V_{j}$ for all $i \neq j$.
Proof. Let $\mathscr{S}=$ algebraic sets $V \subset \mathbb{A}^{n}(k) \mid V$ is not the union of a finite number of irreducible algebraic sets\}. We want to show that $\mathscr{S}$ is empty. If not, let $V$ be a minimal member of $\mathscr{S}$. Since $V \in \mathscr{S}, V$ is not irreducible, so $V=V_{1} \cup V_{2}, V_{i} \varsubsetneqq$ $V$. Then $V_{i} \notin \mathscr{S}$, so $V_{i}=V_{i 1} \cup \cdots \cup V_{i m_{i}}, V_{i j}$ irreducible. But then $V=\bigcup_{i, j} V_{i j}$, a contradiction.
So any algebraic set $V$ may be written as $V=V_{1} \cup \cdots \cup V_{m}, V_{i}$ irreducible. To get the second condition, simply throw away any $V_{i}$ such that $V_{i} \subset V_{j}$ for $i \neq j$. To show uniqueness, let $V=W_{1} \cup \cdots \cup W_{m}$ be another such decomposition. Then $V_{i}=$ $\bigcup_{j}\left(W_{j} \cap V_{i}\right)$, so $V_{i} \subset W_{j(i)}$ for some $j(i)$. Similarly $W_{j(i)} \subset V_{k}$ for some $k$. But $V_{i} \subset V_{k}$ implies $i=k$, so $V_{i}=W_{j(i)}$. Likewise each $W_{j}$ is equal to some $V_{i(j)}$.
The $V_{i}$ are called the irreducible components of $V ; V=V_{1} \cup \cdots \cup V_{m}$ is the decomposition of $V$ into irreducible components.
## Problems
1.23. Give an example of a collection $\mathscr{S}$ of ideals in a Noetherian ring such that no maximal member of $\mathscr{S}$ is a maximal ideal.
1.24. Show that every proper ideal in a Noetherian ring is contained in a maximal ideal. (Hint: If $I$ is the ideal, apply the lemma to \{proper ideals that contain $I\}$.)
1.25. (a) Show that $V\left(Y-X^{2}\right) \subset \mathbb{A}^{2}(\mathbb{C})$ is irreducible; in fact, $I\left(V\left(Y-X^{2}\right)\right)=(Y-$ $\left.X^{2}\right)$. (b) Decompose $V\left(Y^{4}-X^{2}, Y^{4}-X^{2} Y^{2}+X Y^{2}-X^{3}\right) \subset \mathbb{A}^{2}(\mathbb{C})$ into irreducible components.
1.26. Show that $F=Y^{2}+X^{2}(X-1)^{2} \in \mathbb{R}[X, Y]$ is an irreducible polynomial, but $V(F)$ is reducible. 1.27. Let $V, W$ be algebraic sets in $\mathbb{A}^{n}(k)$, with $V \subset W$. Show that each irreducible component of $V$ is contained in some irreducible component of $W$.
1.28. If $V=V_{1} \cup \cdots \cup V_{r}$ is the decomposition of an algebraic set into irreducible components, show that $V_{i} \not \subset \bigcup_{j \neq i} V_{j}$.
1.29. Show that $\mathbb{A}^{n}(k)$ is irreducible if $k$ is infinite,.
### Algebraic Subsets of the Plane
Before developing the general theory further, we will take a closer look at the affine plane $\mathbb{A}^{2}(k)$, and find all its algebraic subsets. By Theorem 2 it is enough to find the irreducible algebraic sets.
Proposition 2. Let $F$ and $G$ be polynomials in $k[X, Y]$ with no common factors. Then $V(F, G)=V(F) \cap V(G)$ is a finite set of points.
Proof. $F$ and $G$ have no common factors in $k[X][Y]$, so they also have no common factors in $k(X)[Y]$ (see Section 1). Since $k(X)[Y]$ is a PID, $(F, G)=(1)$ in $k(X)[Y]$, so $R F+S G=1$ for some $R, S \in k(X)[Y]$. There is a nonzero $D \in k[X]$ such that $D R=A$, $D S=B \in k[X, Y]$. Therefore $A F+B G=D$. If $(a, b) \in V(F, G)$, then $D(a)=0$. But $D$ has only a finite number of zeros. This shows that only a finite number of $X$ coordinates appear among the points of $V(F, G)$. Since the same reasoning applies to the $Y$-coordinates, there can be only a finite number of points.
Corollary 1. If $F$ is an irreducible polynomial in $k[X, Y]$ such that $V(F)$ is infinite, then $I(V(F))=(F)$, and $V(F)$ is irreducible.
Proof. If $G \in I(V(F))$, then $V(F, G)$ is infinite, so $F$ divides $G$ by the proposition, i.e., $G \in(F)$. Therefore $I(V(F)) \supset(F)$, and the fact that $V(F)$ is irreducible follows from Proposition 1.
Corollary 2. Suppose $k$ is infinite. Then the irreducible algebraic subsets of $\mathbb{A}^{2}(k)$ are: $\mathbb{A}^{2}(k), \varnothing$, points, and irreducible plane curves $V(F)$, where $F$ is an irreducible polynomial and $V(F)$ is infinite.
Proof. Let $V$ be an irreducible algebraic set in $A^{2}(k)$. If $V$ is finite or $I(V)=(0), V$ is of the required type. Otherwise $I(V)$ contains a nonconstant polynomial $F$; since $I(V)$ is prime, some irreducible polynomial factor of $F$ belongs to $I(V)$, so we may assume $F$ is irreducible. Then $I(V)=(F)$; for if $G \in I(V), G \notin(F)$, then $V \subset V(F, G)$ is finite.
Corollary 3. Assume $k$ is algebraically closed, $F$ a nonconstant polynomial in $k[X, Y]$. Let $F=F_{1}^{n_{1}} \cdots F_{r}^{n_{r}}$ be the decomposition of $F$ into irreducible factors. Then $V(F)=$ $V\left(F_{1}\right) \cup \cdots \cup V\left(F_{r}\right)$ is the decomposition of $V(F)$ into irreducible components, and $I(V(F))=\left(F_{1} \cdots F_{r}\right)$.
Proof. No $F_{i}$ divides any $F_{j}, j \neq i$, so there are no inclusion relations among the $V\left(F_{i}\right)$. And $I\left(\bigcup_{i} V\left(F_{i}\right)\right)=\bigcap_{i} I\left(V\left(F_{i}\right)\right)=\bigcap_{i}\left(F_{i}\right)$. Since any polynomial divisible by each $F_{i}$ is also divisible by $F_{1} \cdots F_{r}, \bigcap_{i}\left(F_{i}\right)=\left(F_{1} \cdots F_{r}\right)$. Note that the $V\left(F_{i}\right)$ are infinite since $k$ is algebraically closed (Problem 1.14).
## Problems
1.30. Let $k=\mathbb{R}$. (a) Show that $I\left(V\left(X^{2}+Y^{2}+1\right)\right)=(1)$. (b) Show that every algebraic subset of $\mathbb{A}^{2}(\mathbb{R})$ is equal to $V(F)$ for some $F \in \mathbb{R}[X, Y]$.
This indicates why we usually require that $k$ be algebraically closed.
1.31. (a) Find the irreducible components of $V\left(Y^{2}-X Y-X^{2} Y+X^{3}\right)$ in $\mathbb{A}^{2}(\mathbb{R})$, and also in $\mathbb{A}^{2}(\mathbb{C})$. (b) Do the same for $V\left(Y^{2}-X\left(X^{2}-1\right)\right)$, and for $V\left(X^{3}+X-X^{2} Y-Y\right)$.
### Hilbert's Nullstellensatz
If we are given an algebraic set $V$, Proposition 2 gives a criterion for telling whether $V$ is irreducible or not. What is lacking is a way to describe $V$ in terms of a given set of polynomials that define $V$. The preceding paragraph gives a beginning to this problem, but it is the Nullstellensatz, or Zeros-theorem, which tells us the exact relationship between ideals and algebraic sets. We begin with a somewhat weaker theorem, and show how to reduce it to a purely algebraic fact. In the rest of this section we show how to deduce the main result from the weaker theorem, and give a few applications.
We assume throughout this section that $k$ is algebraically closed.
WEAK NULLSTELLENSATZ. If I is a proper ideal in $k\left[X_{1}, \ldots, X_{n}\right]$, then $V(I) \neq \varnothing$.
Proof. We may assume that $I$ is a maximal ideal, for there is a maximal ideal $J$ containing $I$ (Problem 1.24), and $V(J) \subset V(I)$. So $L=k\left[X_{1}, \ldots, X_{n}\right] / I$ is a field, and $k$ may be regarded as a subfield of $L$ (cf. Section 1 ).
Suppose we knew that $k=L$. Then for each $i$ there is an $a_{i} \in k$ such that the $I$ residue of $X_{i}$ is $a_{i}$, or $X_{i}-a_{i} \in I$. But $\left(X_{1}-a_{1}, \ldots, X_{n}-a_{n}\right)$ is a maximal ideal (Problem $1.21)$, so $I=\left(X_{1}-a_{1}, \ldots, X_{n}-a_{n}\right)$, and $V(I)=\left\{\left(a_{1}, \ldots, a_{n}\right)\right\} \neq \varnothing$.
Thus we have reduced the problem to showing:
(*) If an algebraically closed field $k$ is a subfield of a field $L$, and there is a ring homomorphism from $k\left[X_{1}, \ldots, X_{n}\right]$ onto $L$ (that is the identity on $k$ ), then $k=L$.
The algebra needed to prove this will be developed in the next two sections; $(*)$ will be proved in Section 10 .
HILBERT'S NULLSTELLENSATZ. Let I be an ideal in $k\left[X_{1}, \ldots, X_{n}\right]$ ( $k$ algebraically closed). Then $I(V(I))=\operatorname{Rad}(I)$.
Note. In concrete terms, this says the following: if $F_{1}, F_{2}, \ldots, F_{r}$ and $G$ are in $k\left[X_{1}, \ldots, X_{n}\right]$, and $G$ vanishes wherever $F_{1}, F_{2}, \ldots, F_{r}$ vanish, then there is an equation $G^{N}=A_{1} F_{1}+A_{2} F_{2}+\cdots+A_{r} F^{r}$, for some $N>0$ and some $A_{i} \in k\left[X_{1}, \ldots, X_{n}\right]$.
Proof. That $\operatorname{Rad}(I) \subset I(V(I))$ is easy (Problem 1.20). Suppose that $G$ is in the ideal $I\left(V\left(F_{1}, \ldots, F_{r}\right)\right), F_{i} \in k\left[X_{1}, \ldots, X_{n}\right]$. Let $J=\left(F_{1}, \ldots, F_{r}, X_{n+1} G-1\right) \subset k\left[X_{1}, \ldots, X_{n}, X_{n+1}\right]$. Then $V(J) \subset \mathbb{A}^{n+1}(k)$ is empty, since $G$ vanishes wherever all that $F_{i}$ 's are zero. Applying the Weak Nullstellensatz to $J$, we see that $1 \in J$, so there is an equation $1=$ $\sum A_{i}\left(X_{1}, \ldots, X_{n+1}\right) F_{i}+B\left(X_{1}, \ldots, X_{n+1}\right)\left(X_{n+1} G-1\right)$. Let $Y=1 / X_{n+1}$, and multiply the equation by a high power of $Y$, so that an equation $Y^{N}=\sum C_{i}\left(X_{1}, \ldots, X_{n}, Y\right) F_{i}+$ $D\left(X_{1}, \ldots, X_{n}, Y\right)(G-Y)$ in $k\left[X_{1}, \ldots, X_{n}, Y\right]$ results. Substituting $G$ for $Y$ gives the required equation.
The above proof is due to Rabinowitsch. The first three corollaries are immediate consequences of the theorem.
Corollary 1. If $I$ is a radical ideal in $k\left[X_{1}, \ldots, X_{n}\right]$, then $I(V(I))=I$. So there is a one-to-one correspondence between radical ideals and algebraic sets.
Corollary 2. If I is a prime ideal, then $V(I)$ is irreducible. There is a one-to-one correspondence between prime ideals and irreducible algebraic sets. The maximal ideals correspond to points.
Corollary 3. Let $F$ be a nonconstant polynomial in $k\left[X_{1}, \ldots, X_{n}\right], F=F_{1}^{n_{1}} \cdots F_{r}^{n_{r}}$ the decomposition of $F$ into irreducible factors. Then $V(F)=V\left(F_{1}\right) \cup \cdots \cup V\left(F_{r}\right)$ is the decomposition of $V(F)$ into irreducible components, and $I(V(F))=\left(F_{1} \cdots F_{r}\right)$. There is a one-to-one correspondence between irreducible polynomials $F \in k\left[X_{1}, \ldots, X_{n}\right]$ (up to multiplication by a nonzero element of $k$ ) and irreducible hypersurfaces in $\mathbb{A}^{n}(k)$.
Corollary 4. Let I be an ideal in $k\left[X_{1}, \ldots, X_{n}\right]$. Then $V(I)$ is a finite set if and only if $k\left[X_{1}, \ldots, X_{n}\right] / I$ is a finite dimensional vector space over $k$. If this occurs, the number of points in $V(I)$ is at most $\operatorname{dim}_{k}\left(k\left[X_{1}, \ldots, X_{n}\right] / I\right)$.
Proof. Let $P_{1}, \ldots, P_{r} \in V(I)$. Choose $F_{1}, \ldots, F_{r} \in k\left[X_{1}, \ldots, X_{n}\right]$ such that $F_{i}\left(P_{j}\right)=0$ if $i \neq j$, and $F_{i}\left(P_{i}\right)=1$ (Problem 1.17); let $\bar{F}_{i}$ be the $I$-residue of $F_{i}$. If $\sum \lambda_{i} \bar{F}_{i}=0, \lambda_{i} \in k$, then $\sum \lambda_{i} F_{i} \in I$, so $\lambda_{j}=\left(\sum \lambda_{i} F_{i}\right)\left(P_{j}\right)=0$. Thus the $\bar{F}_{i}$ are linearly independent over $k$, so $r \leq \operatorname{dim}_{k}\left(k\left[X_{1}, \ldots, X_{n}\right] / I\right)$.
Conversely, if $V(I)=\left\{P_{1}, \ldots, P_{r}\right\}$ is finite, let $P_{i}=\left(a_{i 1}, \ldots, a_{i n}\right)$, and define $F_{j}$ by $F_{j}=\prod_{i=1}^{r}\left(X_{j}-a_{i j}\right), j=1, \ldots, n$. Then $F_{j} \in I(V(I))$, so $F_{j}^{N} \in I$ for some $N>0$ (Take $N$ large enough to work for all $F_{j}$ ). Taking $I$-residues, $\bar{F}_{j}^{N}=0$, so $\bar{X}_{j}^{r N}$ is a $k$-linear combination of $\overline{1}, \bar{X}_{j}, \ldots, \bar{X}_{j}^{r N-1}$. It follows by induction that $\bar{X}_{j}^{s}$ is a $k$-linear combination of $\overline{1}, \bar{X}_{j}, \ldots, \bar{X}_{j}^{r N-1}$ for all $s$, and hence that the set $\left\{\bar{X}_{1}^{m_{1}}, \cdots \bar{X}_{n}^{m_{n}} \mid m_{i}<r N\right\}$ generates $k\left[X_{1}, \ldots, X_{n}\right] / I$ as a vector space over $k$.
## Problems
1.32. Show that both theorems and all of the corollaries are false if $k$ is not algebraically closed.
1.33. (a) Decompose $V\left(X^{2}+Y^{2}-1, X^{2}-Z^{2}-1\right) \subset \mathbb{A}^{3}(\mathbb{C})$ into irreducible components. (b) Let $V=\left\{\left(t, t^{2}, t^{3}\right) \in \mathbb{A}^{3}(\mathbb{C}) \mid t \in \mathbb{C}\right\}$. Find $I(V)$, and show that $V$ is irreducible. 1.34. Let $R$ be a UFD. (a) Show that a monic polynomial of degree two or three in $R[X]$ is irreducible if and only if it has no roots in $R$. (b) The polynomial $X^{2}-a \in R[X]$ is irreducible if and only if $a$ is not a square in $R$.
1.35. Show that $V\left(Y^{2}-X(X-1)(X-\lambda)\right) \subset \mathbb{A}^{2}(k)$ is an irreducible curve for any algebraically closed field $k$, and any $\lambda \in k$.
1.36. Let $I=\left(Y^{2}-X^{2}, Y^{2}+X^{2}\right) \subset \mathbb{C}[X, Y]$. Find $V(I)$ and $\operatorname{dim}_{\mathbb{C}}(\mathbb{C}[X, Y] / I)$.
1.37* Let $K$ be any field, $F \in K[X]$ a polynomial of degree $n>0$. Show that the residues $\overline{1}, \bar{X}, \ldots, \bar{X}^{n-1}$ form a basis of $K[X] /(F)$ over $K$.
1.38* Let $R=k\left[X_{1}, \ldots, X_{n}\right], k$ algebraically closed, $V=V(I)$. Show that there is a natural one-to-one correspondence between algebraic subsets of $V$ and radical ideals in $k\left[X_{1}, \ldots, X_{n}\right] / I$, and that irreducible algebraic sets (resp. points) correspond to prime ideals (resp. maximal ideals). (See Problem 1.22.)
1.39. (a) Let $R$ be a UFD, and let $P=(t)$ be a principal, proper, prime ideal. Show that there is no prime ideal $Q$ such that $0 \subset Q \subset P, Q \neq 0, Q \neq P$. (b) Let $V=V(F)$ be an irreducible hypersurface in $\mathbb{A}^{n}$. Show that there is no irreducible algebraic set $W$ such that $V \subset W \subset \mathbb{A}^{n}, W \neq V, W \neq \mathbb{A}^{n}$.
1.40. Let $I=\left(X^{2}-Y^{3}, Y^{2}-Z^{3}\right) \subset k[X, Y, Z]$. Define $\alpha: k[X, Y, Z] \rightarrow k[T]$ by $\alpha(X)=$ $T^{9}, \alpha(Y)=T^{6}, \alpha(Z)=T^{4}$. (a) Show that every element of $k[X, Y, Z] / I$ is the residue of an element $A+X B+Y C+X Y D$, for some $A, B, C, D \in k[Z]$. (b) If $F=A+X B+$ $Y C+X Y D, A, B, C, D \in k[Z]$, and $\alpha(F)=0$, compare like powers of $T$ to conclude that $F=0$. (c) Show that $\operatorname{Ker}(\alpha)=I$, so $I$ is prime, $V(I)$ is irreducible, and $I(V(I))=I$.
### Modules; Finiteness Conditions
Let $R$ be a ring. An $R$-module is a commutative group $M$ (the group law on $M$ is written + ; the identity of the group is 0 , or $0_{M}$ ) together with a scalar multiplication, i.e., a mapping from $R \times M$ to $M$ (denote the image of $(a, m)$ by $a \cdot m$ or $a m$ ) satisfying:
(i) $(a+b) m=a m+b m$ for $a, b \in R, m \in M$.
(ii) $a \cdot(m+n)=a m+a n$ for $a \in R, m, n \in M$.
(iii) $(a b) \cdot m=a \cdot(b m)$ for $a, b \in R, m \in M$.
(iv) $1_{R} \cdot m=m$ for $m \in M$, where $1_{R}$ is the multiplicative identity in $R$.
Exercise. Show that $0_{R} \cdot m=0_{M}$ for all $m \in M$.
Examples. (1) AZ्Z-module is just a commutative group, where $( \pm a) m$ is $\pm(m+$ $\cdots+m$ ) ( $a$ times) for $a \in \mathbb{Z}, a \geq 0$.
(2) If $R$ is a field, an $R$-module is the same thing as a vector space over $R$.
(3) The multiplication in $R$ makes any ideal of $R$ into an $R$-module.
(4) If $\varphi: R \rightarrow S$ is a ring homomorphism, we define $r \cdot s$ for $r \in R, s \in S$, by the equation $r \cdot s=\varphi(r) s$. This makes $S$ into an $R$-module. In particular, if a ring $R$ is a subring of a ring $S$, then $S$ is an $R$-module.
A subgroup $N$ of an $R$-module $M$ is called a submodule if $a m \in N$ for all $a \in R$, $m \in N ; N$ is then an $R$-module. If $S$ is a set of elements of an $R$-module $M$, the submodule generated by $S$ is defined to be $\left\{\sum r_{i} s_{i} \mid r_{i} \in R, s_{i} \in S\right\}$; it is the smallest submodule of $M$ that contains $S$. If $S=\left\{s_{1}, \ldots, s_{n}\right\}$ is finite, the submodule generated by $S$ is denoted by $\sum R s_{i}$. The module $M$ is said to be finitely generated if $M=\sum R s_{i}$ for some $s_{1}, \ldots, s_{n} \in M$. Note that this concept agrees with the notions of finitely generated commutative groups and ideals, and with the notion of a finite-dimensional vector space if $R$ is a field.
Let $R$ be a subring of a ring $S$. There are several types of finiteness conditions for $S$ over $R$, depending on whether we consider $S$ as an $R$-module, a ring, or (possibly) a field.
(A) $S$ is said to be module-finite over $R$, if $S$ is finitely generated as an $R$-module. If $R$ and $S$ are fields, and $S$ is module finite over $R$, we denote the dimension of $S$ over $R$ by $[S: R]$.
(B) Let $v_{1}, \ldots, v_{n} \in S$. Let $\varphi: R\left[X_{1}, \ldots, X_{n}\right] \rightarrow S$ be the ring homomorphism taking $X_{i}$ to $v_{i}$. The image of $\varphi$ is written $R\left[v_{1}, \ldots, v_{n}\right]$. It is a subring of $S$ containing $R$ and $v_{1}, \ldots, v_{n}$, and it is the smallest such ring. Explicitly, $R\left[v_{1}, \ldots, v_{n}\right]=\left\{\sum a_{(i)} v_{1}^{i_{1}} \cdots v_{n}^{i_{n}} \mid\right.$ $\left.a_{(i)} \in R\right\}$. The ring $S$ is ring-finite over $R$ if $S=R\left[v_{1}, \ldots, v_{n}\right]$ for some $v_{1}, \ldots, v_{n} \in S$.
(C) Suppose $R=K, S=L$ are fields. If $v_{1}, \ldots, v_{n} \in L$, let $K\left(v_{1}, \ldots, v_{n}\right)$ be the quotient field of $K\left[v_{1}, \ldots, v_{n}\right]$. We regard $K\left(v_{1}, \ldots, v_{n}\right)$ as a subfield of $L$; it is the smallest subfield of $L$ containing $K$ and $v_{1}, \ldots, v_{n}$. The field $L$ is said to be a finitely generated field extension of $K$ if $L=K\left(v_{1}, \ldots, v_{n}\right)$ for some $v_{1}, \ldots, v_{n} \in L$.
## Problems
1.41* If $S$ is module-finite over $R$, then $S$ is ring-finite over $R$.
1.42. Show that $S=R[X]$ (the ring of polynomials in one variable) is ring-finite over $R$, but not module-finite.
1.43. If $L$ is ring-finite over $K(K, L$ fields) then $L$ is a finitely generated field extension of $K$.
1.44* Show that $L=K(X)$ (the field of rational functions in one variable) is a finitely generated field extension of $K$, but $L$ is not ring-finite over $K$. (Hint: If $L$ were ringfinite over $K$, a common denominator of ring generators would be an element $b \in$ $K[X]$ such that for all $z \in L, b^{n} z \in K[X]$ for some $n$; but let $z=1 / c$, where $c$ doesn't divide $b$ (Problem 1.5).)
1.45. Let $R$ be a subring of $S, S$ a subring of $T$.
(a) If $S=\sum R v_{i}, T=\sum S w_{j}$, show that $T=\sum R v_{i} w_{j}$.
(b) If $S=R\left[v_{1}, \ldots, v_{n}\right], T=S\left[w_{1}, \ldots, w_{m}\right]$, show that $T=R\left[v_{1}, \ldots, v_{n}, w_{1}, \ldots, w_{m}\right]$.
(c) If $R, S, T$ are fields, and $S=R\left(v_{1}, \ldots, v_{n}\right), T=S\left(w_{1}, \ldots, w_{m}\right)$, show that $T=$ $R\left(v_{1}, \ldots, v_{n}, w_{1}, \ldots, w_{m}\right)$.
So each of the three finiteness conditions is a transitive relation.
### Integral Elements
Let $R$ be a subring of a ring $S$. An element $v \in S$ is said to be integral over $R$ if there is a monic polynomial $F=X^{n}+a_{1} X^{n-1}+\cdots+a_{n} \in R[X]$ such that $F(v)=0$. If $R$ and $S$ are fields, we usually say that $v$ is algebraic over $R$ if $v$ is integral over $R$.
Proposition 3. Let $R$ be a subring of a domain $S, v \in S$. Then the following are equivalent:
(1) $v$ is integral over $R$.
(2) $R[v]$ is module-finite over $R$.
(3) There is a subring $R^{\prime}$ of $S$ containing $R[v]$ that is module-finite over $R$.
Proof. (1) implies (2): If $v^{n}+a_{1} v^{n-1}+\cdots+a_{n}=0$, then $v^{n} \in \sum_{i=0}^{n-1} R v^{i}$. It follows that $v^{m} \in \sum_{i=0}^{n-1} R v^{i}$ for all $m$, so $R[v]=\sum_{i=0}^{n-1} R v^{i}$.
(2) implies (3): Let $R^{\prime}=R[v]$.
(3) implies (1): If $R^{\prime}=\sum_{i=1}^{n} R w_{i}$, then $v w_{i}=\sum_{j=1}^{n} a_{i j} w_{j}$ for some $a_{i j} \in R$. Then $\sum_{j=1}^{n}\left(\delta_{i j} v-a_{i j}\right) w_{j}=0$ for all $i$, where $\delta_{i j}=0$ if $i \neq j$ and $\delta_{i i}=1$. If we consider these equations in the quotient field of $S$, we see that $\left(w_{1}, \ldots, w_{n}\right)$ is a nontrivial solution, so $\operatorname{det}\left(\delta_{i j} v-a_{i j}\right)=0$. Since $v$ appears only in the diagonal of the matrix, this determinant has the form $v^{n}+a_{1} v^{n-1}+\cdots+a_{n}, a_{i} \in R$. So $v$ is integral over $R$.
Corollary. The set of elements of $S$ that are integral over $R$ is a subring of $S$ containing $R$.
Proof. If $a, b$ are integral over $R$, then $b$ is integral over $R[a] \supset R$, so $R[a, b]$ is modulefinite over $R$ (Problem 1.45(a)). And $a \pm b, a b \in R[a, b]$, so they are integral over $R$ by the proposition.
We say that $S$ is integral over $R$ if every element of $S$ is integral over $R$. If $R$ and $S$ are fields, we say $S$ is an algebraic extension of $R$ if $S$ is integral over $R$. The proposition and corollary extend to the case where $S$ is not a domain, with essentially the same proofs, but we won't need that generality.
## Problems
1.46. Let $R$ be a subring of $S, S$ a subring of (a domain) $T$. If $S$ is integral over $R$, and $T$ is integral over $S$, show that $T$ is integral over $R$. (Hint: Let $z \in T$, so we have $z^{n}+a_{1} z^{n-1}+\cdots+a_{n}=0, a_{i} \in S$. Then $R\left[a_{1}, \ldots, a_{n}, z\right]$ is module-finite over $R$.)
1.47* Suppose (a domain) $S$ is ring-finite over $R$. Show that $S$ is module-finite over $R$ if and only if $S$ is integral over $R$.
1.48. Let $L$ be a field, $k$ an algebraically closed subfield of $L$. (a) Show that any element of $L$ that is algebraic over $k$ is already in $k$. (b) An algebraically closed field has no module-finite field extensions except itself.
1.49. Let $K$ be a field, $L=K(X)$ the field of rational functions in one variable over $K$. (a) Show that any element of $L$ that is integral over $K[X]$ is already in $K[X]$. (Hint: If $z^{n}+a_{1} z^{n-1}+\cdots=0$, write $z=F / G, F, G$ relatively prime. Then $F^{n}+a_{1} F^{n-1} G+\cdots=0$, so $G$ divides $F$.) (b) Show that there is no nonzero element $F \in K[X]$ such that for every $z \in L, F^{n} z$ is integral over $K[X]$ for some $n>0$. (Hint: See Problem 1.44.)
1.50* Let $K$ be a subfield of a field $L$. (a) Show that the set of elements of $L$ that are algebraic over $K$ is a subfield of $L$ containing $K$. (Hint: If $v^{n}+a_{1} v^{n-1}+\cdots+a_{n}=0$, and $a_{n} \neq 0$, then $v\left(v^{n-1}+\cdots\right)=-a_{n}$.) (b) Suppose $L$ is module-finite over $K$, and $K \subset R \subset L$. Show that $R$ is a field.
### Field Extensions
Suppose $K$ is a subfield of a field $L$, and suppose $L=K(v)$ for some $v \in L$. Let $\varphi: K[X] \rightarrow L$ be the homomorphism taking $X$ to $v$. Let $\operatorname{Ker}(\varphi)=(F), F \in K[X]$ (since $K[X]$ is a PID). Then $K[X] /(F)$ is isomorphic to $K[\nu]$, so $(F)$ is prime. Two cases may occur:
Case 1. $F=0$. Then $K[v]$ is isomorphic to $K[X]$, so $K(v)=L$ is isomorphic to $K(X)$. In this case $L$ is not ring-finite (or module-finite) over $K$ (Problem 1.44).
Case 2. $F \neq 0$. We may assume $F$ is monic. Then $(F)$ is prime, so $F$ is irreducible and $(F)$ is maximal (Problem 1.3); therefore $K[v]$ is a field, so $K[v]=K(v)$. And $F(\nu)=0$, so $v$ is algebraic over $K$ and $L=K[v]$ is module-finite over $K$.
To finish the proof of the Nullstellensatz, we must prove the claim $(*)$ of Section 7 ; this says that if a field $L$ is a ring-finite extension of an algebraically closed field $k$, then $L=k$. In view of Problem 1.48, it is enough to show that $L$ is module-finite over $k$. The above discussion indicates that a ring-finite field extension is already module-finite. The next proposition shows that this is always true, and completes the proof of the Nullstellensatz.
Proposition 4 (Zariski). If a field $L$ is ring-finite over a subfield $K$, then $L$ is modulefinite (and hence algebraic) over $K$.
Proof. Suppose $L=K\left[v_{1}, \ldots, v_{n}\right]$. The case $n=1$ is taken care of by the above discussion, so we assume the result for all extensions generated by $n-1$ elements. Let $K_{1}=K\left(v_{1}\right)$. By induction, $L=K_{1}\left[v_{2}, \ldots, v_{n}\right]$ is module-finite over $K_{1}$. We may assume $v_{1}$ is not algebraic over $K$ (otherwise Problem 1.45(a) finishes the proof).
Each $v_{i}$ satisfies an equation $v_{i}^{n_{i}}+a_{i 1} v_{i}^{n_{i}-1}+\cdots=0, a_{i j} \in K_{1}$. If we take $a \in K\left[v_{1}\right]$ that is a multiple of all the denominators of the $a_{i j}$, we get equations $\left(a v_{i}\right)^{n_{i}}+$ $a a_{i 1}\left(a v_{1}\right)^{n_{i}-1}+\cdots=0$. It follows from the Corollary in $\$ 1.9$ that for any $z \in L=$ $K\left[v_{1}, \ldots, v_{n}\right]$, there is an $N$ such that $a^{N} z$ is integral over $K\left[v_{1}\right]$. In particular this must hold for $z \in K\left(v_{1}\right)$. But since $K\left(\nu_{1}\right)$ is isomorphic to the field of rational functions in one variable over $K$, this is impossible (Problem $1.49(b)$ ).
## Problems
1.51. Let $K$ be a field, $F \in K[X]$ an irreducible monic polynomial of degree $n>0$. (a) Show that $L=K[X] /(F)$ is a field, and if $x$ is the residue of $X$ in $L$, then $F(x)=0$. (b) Suppose $L^{\prime}$ is a field extension of $K, y \in L^{\prime}$ such that $F(y)=0$. Show that the homomorphism from $K[X]$ to $L^{\prime}$ that takes $X$ to $Y$ induces an isomorphism of $L$ with $K(y)$. (c) With $L^{\prime}, y$ as in (b), suppose $G \in K[X]$ and $G(y)=0$. Show that $F$ divides $G$. (d) Show that $F=(X-x) F_{1}, F_{1} \in L[X]$.
1.52* Let $K$ be a field, $F \in K[X]$. Show that there is a field $L$ containing $K$ such that $F=\prod_{i=1}^{n}\left(X-x_{i}\right) \in L[X]$. (Hint: Use Problem 1.51(d) and induction on the degree.) $L$ is called a splitting field of $F$.
1.53. Suppose $K$ is a field of characteristic zero, $F$ an irreducible monic polynomial in $K[X]$ of degree $n>0$. Let $L$ be a splitting field of $F$, so $F=\prod_{i=1}^{n}\left(X-x_{i}\right), x_{i} \in L$. Show that the $x_{i}$ are distinct. (Hint: Apply Problem 1.51(c) to $G=F_{X}$; if $(X-x)^{2}$ divides $F$, then $G(x)=0$.)
1.54. Let $R$ be a domain with quotient field $K$, and let $L$ be a finite algebraic extension of $K$. (a) For any $v \in L$, show that there is a nonzero $a \in R$ such that $a v$ is integral over $R$. (b) Show that there is a basis $v_{1}, \ldots, v_{n}$ for $L$ over $K$ (as a vector space) such that each $v_{i}$ is integral over $R$.
## Chapter 2
## Affine Varieties
From now on $k$ will be a fixed algebraically closed field. Affine algebraic sets will be in $\mathbb{A}^{n}=\mathbb{A}^{n}(k)$ for some $n$. An irreducible affine algebraic set is called an affine variety.
All rings and fields will contain $k$ as a subring. By a homomorphism $\varphi: R \rightarrow S$ of such rings, we will mean a ring homomorphism such that $\varphi(\lambda)=\lambda$ for all $\lambda \in k$.
In this chapter we will be concerned only with affine varieties, so we call them simply varieties.
### Coordinate Rings
Let $V \subset \mathbb{A}^{n}$ be a nonempty variety. Then $I(V)$ is a prime ideal in $k\left[X_{1}, \ldots, X_{n}\right]$, so $k\left[X_{1}, \ldots, X_{n}\right] / I(V)$ is a domain. We let $\Gamma(V)=k\left[X_{1}, \ldots, X_{n}\right] / I(V)$, and call it the coordinate ring of $V$.
For any (nonempty) set $V$, we let $\mathscr{F}(V, k)$ be the set of all functions from $V$ to $k$. $\mathscr{F}(V, k)$ is made into a ring in the usual way: if $f, g \in \mathscr{F}(V, k),(f+g)(x)=f(x)+g(x)$, $(f g)(x)=f(x) g(x)$, for all $x \in V$. It is usual to identify $k$ with the subring of $\mathscr{F}(V, k)$ consisting of all constant functions.
If $V \subset \mathbb{A}^{n}$ is a variety, a function $f \in \mathscr{F}(V, k)$ is called a polynomial function if there is a polynomial $F \in k\left[X_{1}, \ldots, X_{n}\right]$ such that $f\left(a_{1}, \ldots, a_{n}\right)=F\left(a_{1}, \ldots, a_{n}\right)$ for all $\left(a_{1}, \ldots, a_{n}\right) \in V$. The polynomial functions form a subring of $\mathscr{F}(V, k)$ containing $k$. Two polynomials $F, G$ determine the same function if and only if $(F-G)\left(a_{1}, \ldots, a_{n}\right)=$ 0 for all $\left(a_{1}, \ldots, a_{n}\right) \in V$, i.e., $F-G \in I(V)$. We may thus identify $\Gamma(V)$ with the subring of $\mathscr{F}(V, k)$ consisting of all polynomial functions on $V$. We have two important ways to view an element of $\Gamma(V)$ - as a function on $V$, or as an equivalence class of polynomials.
## Problems
2.1* Show that the map that associates to each $F \in k\left[X_{1}, \ldots, X_{n}\right]$ a polynomial function in $\mathscr{F}(V, k)$ is a ring homomorphism whose kernel is $I(V)$. 2.2* Let $V \subset \mathbb{A}^{n}$ be a variety. A subvariety of $V$ is a variety $W \subset \mathbb{A}^{n}$ that is contained in $V$. Show that there is a natural one-to-one correspondence between algebraic subsets (resp. subvarieties, resp. points) of $V$ and radical ideals (resp. prime ideals, resp. maximal ideals) of $\Gamma(V)$. (See Problems 1.22, 1.38.)
2.3. Let $W$ be a subvariety of a variety $V$, and let $I_{V}(W)$ be the ideal of $\Gamma(V)$ corresponding to $W$. (a) Show that every polynomial function on $V$ restricts to a polynomial function on $W$. (b) Show that the map from $\Gamma(V)$ to $\Gamma(W)$ defined in part (a) is a surjective homomorphism with kernel $I_{V}(W)$, so that $\Gamma(W)$ is isomorphic to $\Gamma(V) / I_{V}(W)$.
2.4. Let $V \subset \mathbb{A}^{n}$ be a nonempty variety. Show that the following are equivalent: (i) $V$ is a point; (ii) $\Gamma(V)=k$; (iii) $\operatorname{dim}_{k} \Gamma(V)<\infty$.
2.5. Let $F$ be an irreducible polynomial in $k[X, Y]$, and suppose $F$ is monic in $Y$ : $F=Y^{n}+a_{1}(X) Y^{n-1}+\cdots$, with $n>0$. Let $V=V(F) \subset \mathbb{A}^{2}$. Show that the natural homomorphism from $k[X]$ to $\Gamma(V)=k[X, Y] /(F)$ is one-to-one, so that $k[X]$ may be regarded as a subring of $\Gamma(V)$; show that the residues $\overline{1}, \bar{Y}, \ldots, \bar{Y}^{n-1}$ generate $\Gamma(V)$ over $k[X]$ as a module.
### Polynomial Maps
Let $V \subset \mathbb{A}^{n}, W \subset \mathbb{A}^{m}$ be varieties. A mapping $\varphi: V \rightarrow W$ is called a polynomial map if there are polynomials $T_{1}, \ldots, T_{m} \in k\left[X_{1}, \ldots, X_{n}\right]$ such that $\varphi\left(a_{1}, \ldots, a_{n}\right)=$ $\left(T_{1}\left(a_{1}, \ldots, a_{n}\right), \ldots, T_{m}\left(a_{1}, \ldots, a_{n}\right)\right)$ for all $\left(a_{1}, \ldots, a_{n}\right) \in V$.
Any mapping $\varphi: V \rightarrow W$ induces a homomorphisms $\tilde{\varphi}: \mathscr{F}(W, k) \rightarrow \mathscr{F}(V, k)$, by setting $\tilde{\varphi}(f)=f \circ \varphi$. If $\varphi$ is a polynomial map, then $\tilde{\varphi}(\Gamma(W)) \subset \Gamma(V)$, so $\tilde{\varphi}$ restricts to a homomorphism (also denoted by $\tilde{\varphi})$ from $\Gamma(W)$ to $\Gamma(V)$; for if $f \in \Gamma(W)$ is the $I(W)$ residue of a polynomial $F$, then $\tilde{\varphi}(f)=f \circ \varphi$ is the $I(V)$-residue of the polynomial $F\left(T_{1}, \ldots, T_{m}\right)$.
If $V=\mathbb{A}^{n}, W=\mathbb{A}^{m}$, and $T_{1}, \ldots, T_{m} \in k\left[X_{1}, \ldots, X_{n}\right]$ determine a polynomial map $T: \mathbb{A}^{n} \rightarrow \mathbb{A}^{m}$, the $T_{i}$ are uniquely determined by $T$ (see Problem 1.4), so we often write $T=\left(T_{1}, \ldots, T_{m}\right)$.
Proposition 1. Let $V \subset \mathbb{A}^{n}, W \subset \mathbb{A}^{m}$ be affine varieties. There is a natural one-to-one correspondence between the polynomial maps $\varphi: V \rightarrow W$ and the homomorphisms $\tilde{\varphi}: \Gamma(W) \rightarrow \Gamma(V)$. Any such $\varphi$ is the restriction of a polynomial map from $\mathbb{A}^{n}$ to $\mathbb{A}^{m}$.
Proof. Suppose $\alpha: \Gamma(W) \rightarrow \Gamma(V)$ is a homomorphism. Choose $T_{i} \in k\left[X_{1}, \ldots, X_{n}\right]$ such that $\alpha\left(I(W)\right.$-residue of $\left.X_{i}\right)=\left(I(V)\right.$-residue of $\left.T_{i}\right)$, for $i=1, \ldots, m$. Then $T=$ $\left(T_{1}, \ldots, T_{m}\right)$ is a polynomial map from $\mathbb{A}^{n}$ to $\mathbb{A}^{m}$, inducing $\tilde{T}: \Gamma\left(\mathbb{A}^{m}\right)=k\left[X_{1}, \ldots, X_{m}\right] \rightarrow$ $\Gamma\left(\mathbb{A}^{n}\right)=k\left[X_{1}, \ldots, X_{n}\right]$. It is easy to check that $\tilde{T}(I(W)) \subset I(V)$, and hence that $T(V) \subset$ $W$, and so $T$ restricts to a polynomial map $\varphi: V \rightarrow W$. It is also easy to verify that $\tilde{\varphi}=\alpha$. Since we know how to construct $\tilde{\varphi}$ from $\varphi$, this completes the proof.
A polynomial map $\varphi: V \rightarrow W$ is an isomorphism if there is a polynomial map $\psi: W \rightarrow V$ such that $\psi \circ \varphi=$ identity on $V, \varphi \circ \psi=$ identity on $W$. Proposition 1 shows that two affine varieties are isomorphic if and only if their coordinate rings are isomorphic (over $k$ ).
## Problems
2.6. Let $\varphi: V \rightarrow W, \psi: W \rightarrow Z$. Show that $\widetilde{\psi \circ \varphi}=\tilde{\varphi} \circ \tilde{\psi}$. Show that the composition of polynomial maps is a polynomial map.
2.7. If $\varphi: V \rightarrow W$ is a polynomial map, and $X$ is an algebraic subset of $W$, show that $\varphi^{-1}(X)$ is an algebraic subset of $V$. If $\varphi^{-1}(X)$ is irreducible, and $X$ is contained in the image of $\varphi$, show that $X$ is irreducible. This gives a useful test for irreducibility.
2.8. (a) Show that $\left\{\left(t, t^{2}, t^{3}\right) \in \mathbb{A}^{3}(k) \mid t \in k\right\}$ is an affine variety. (b) Show that $V(X Z$ $\left.Y^{2}, Y Z-X^{3}, Z^{2}-X^{2} Y\right) \subset \mathbb{A}^{3}(\mathbb{C})$ is a variety. (Hint:: $Y^{3}-X^{4}, Z^{3}-X^{5}, Z^{4}-Y^{5} \in I(V)$. Find a polynomial map from $\mathbb{A}^{1}(\mathbb{C})$ onto $V$.)
2.9. Let $\varphi: V \rightarrow W$ be a polynomial map of affine varieties, $V^{\prime} \subset V, W^{\prime} \subset W$ subvarieties. Suppose $\varphi\left(V^{\prime}\right) \subset W^{\prime}$. (a) Show that $\tilde{\varphi}\left(I_{W}\left(W^{\prime}\right)\right) \subset I_{V}\left(V^{\prime}\right)$ (see Problems 2.3). (b) Show that the restriction of $\varphi$ gives a polynomial map from $V^{\prime}$ to $W^{\prime}$.
2.10. Show that the projection map pr: $\mathbb{A}^{n} \rightarrow \mathbb{A}^{r}, n \geq r$, defined by $\operatorname{pr}\left(a_{1}, \ldots, a_{n}\right)=$ $\left(a_{1}, \ldots, a_{r}\right)$ is a polynomial map.
2.11. Let $f \in \Gamma(V), V$ a variety $\subset \mathbb{A}^{n}$. Define
$$
G(f)=\left\{\left(a_{1}, \ldots, a_{n}, a_{n+1}\right) \in \mathbb{A}^{n+1} \mid\left(a_{1}, \ldots, a_{n}\right) \in V \text { and } a_{n+1}=f\left(a_{1}, \ldots, a_{n}\right)\right\},
$$
the graph of $f$. Show that $G(f)$ is an affine variety, and that the map $\left(a_{1}, \ldots, a_{n}\right) \mapsto$ $\left(a_{1}, \ldots, a_{n}, f\left(a_{1}, \ldots, a_{n}\right)\right)$ defines an isomorphism of $V$ with $G(f)$. (Projection gives the inverse.)
2.12. (a)] Let $\varphi: \mathbb{A}^{1} \rightarrow V=V\left(Y^{2}-X^{3}\right) \subset \mathbb{A}^{2}$ be defined by $\varphi(t)=\left(t^{2}, t^{3}\right)$. Show that although $\varphi$ is a one-to-one, onto polynomial map, $\varphi$ is not an isomorphism. (Hint:: $\tilde{\varphi}(\Gamma(V))=k\left[T^{2}, T^{3}\right] \subset k[T]=\Gamma\left(\mathbb{A}^{1}\right)$.) (b) Let $\varphi: \mathbb{A}^{1} \rightarrow V=V\left(Y^{2}-X^{2}(X+1)\right)$ be defined by $\varphi(t)=\left(t^{2}-1, t\left(t^{2}-1\right)\right)$. Show that $\varphi$ is one-to-one and onto, except that $\varphi( \pm 1)=(0,0)$.
2.13. Let $V=V\left(X^{2}-Y^{3}, Y^{2}-Z^{3}\right) \subset \mathbb{A}^{3}$ as in Problem $1.40, \bar{\alpha}: \Gamma(V) \rightarrow k[T]$ induced by the homomorphism $\alpha$ of that problem. (a) What is the polynomial map $f$ from $\mathbb{A}^{1}$ to $V$ such that $\tilde{f}=\bar{\alpha}$ ? (b) Show that $f$ is one-to-one and onto, but not an isomorphism.
### Coordinate Changes
If $T=\left(T_{1}, \ldots, T_{m}\right)$ is a polynomial map from $\mathbb{A}^{n}$ to $\mathbb{A}^{m}$, and $F$ is a polynomial in $k\left[X_{1}, \ldots, X_{m}\right]$, we let $F^{T}=\tilde{T}(F)=F\left(T_{1}, \ldots, T_{m}\right)$. For ideals $I$ and algebraic sets $V$ in $\mathbb{A}^{m}, I^{T}$ will denote the ideal in $k\left[X_{1}, \ldots, X_{n}\right]$ generated by $\left\{F^{T} \mid F \in I\right\}$ and $V^{T}$ the algebraic set $T^{-1}(V)=V\left(I^{T}\right)$, where $I=I(V)$. If $V$ is the hypersurface of $F, V^{T}$ is the hypersurface of $F^{T}$ (if $F^{T}$ is not a constant).
An affine change of coordinates on $\mathbb{A}^{n}$ is a polynomial map $T=\left(T_{1}, \ldots, T_{n}\right): \mathbb{A}^{n} \rightarrow$ $\mathbb{A}^{n}$ such that each $T_{i}$ is a polynomial of degree 1 , and such that $T$ is one-to-one and onto. If $T_{i}=\sum a_{i j} X_{j}+a_{i 0}$, then $T=T^{\prime \prime} \circ T^{\prime}$, where $T^{\prime}$ is a linear map $\left(T_{i}^{\prime}=\right.$ $\left.\sum a_{i j} X_{j}\right)$ and $T^{\prime \prime}$ is a translation $\left(T_{i}^{\prime \prime}=X_{i}+a_{i 0}\right)$. Since any translation has an inverse (also a translation), it follows that $T$ will be one-to-one (and onto) if and only if $T^{\prime}$ is invertible. If $T$ and $U$ are affine changes of coordinates on $\mathbb{A}^{n}$, then so are $T \circ U$ and $T^{-1} ; T$ is an isomorphism of the variety $\mathbb{A}^{n}$ with itself.
## Problems
2.14. A set $V \subset \mathbb{A}^{n}(k)$ is called a linear subvariety of $\mathbb{A}^{n}(k)$ if $V=V\left(F_{1}, \ldots, F_{r}\right)$ for some polynomials $F_{i}$ of degree 1. (a) Show that if $T$ is an affine change of coordinates on $\mathbb{A}^{n}$, then $V^{T}$ is also a linear subvariety of $\mathbb{A}^{n}(k)$. (b) If $V \neq \varnothing$, show that there is an affine change of coordinates $T$ of $\mathbb{A}^{n}$ such that $V^{T}=V\left(X_{m+1}, \ldots, X_{n}\right)$. (Hint:: use induction on $r$.) So $V$ is a variety. (c) Show that the $m$ that appears in part (b) is independent of the choice of $T$. It is called the dimension of $V$. Then $V$ is then isomorphic (as a variety) to $\mathbb{A}^{m}(k)$. (Hint:: Suppose there were an affine change of coordinates $T$ such that $V\left(X_{m+1}, \ldots, X_{n}\right)^{T}=V\left(X_{s+1}, \ldots, X_{n}\right), m<s$; show that $T_{m+1}, \ldots, T_{n}$ would be dependent.)
2.15. Let $P=\left(a_{1}, \ldots, a_{n}\right), Q=\left(b_{1}, \ldots, b_{n}\right)$ be distinct points of $\mathbb{A}^{n}$. The line through $P$ and $Q$ is defined to be $\left.\left\{a_{1}+t\left(b_{1}-a_{1}\right), \ldots, a_{n}+t\left(b_{n}-a_{n}\right)\right) \mid t \in k\right\}$. (a) Show that if $L$ is the line through $P$ and $Q$, and $T$ is an affine change of coordinates, then $T(L)$ is the line through $T(P)$ and $T(Q)$. (b) Show that a line is a linear subvariety of dimension 1 , and that a linear subvariety of dimension 1 is the line through any two of its points. (c) Show that, in $\mathbb{A}^{2}$, a line is the same thing as a hyperplane. (d) Let $P, P^{\prime} \in \mathbb{A}^{2}, L_{1}, L_{2}$ two distinct lines through $P, L_{1}^{\prime}, L_{2}^{\prime}$ distinct lines through $P^{\prime}$. Show that there is an affine change of coordinates $T$ of $\mathbb{A}^{2}$ such that $T(P)=P^{\prime}$ and $T\left(L_{i}\right)=L_{i}^{\prime}, i=1,2$.
2.16. Let $k=\mathbb{C}$. Give $\mathbb{A}^{n}(\mathbb{C})=\mathbb{C}^{n}$ the usual topology (obtained by identifying $\mathbb{C}$ with $\mathbb{R}^{2}$, and hence $\mathbb{C}^{n}$ with $\left.\mathbb{R}^{2 n}\right)$. Recall that a topological space $X$ is path-connected if for any $P, Q \in X$, there is a continuous mapping $\gamma:[0,1] \rightarrow X$ such that $\gamma(0)=P, \gamma(1)=Q$. (a) Show that $\mathbb{C} \backslash S$ is path-connected for any finite set $S$. (b) Let $V$ be an algebraic set in $\mathbb{A}^{n}\left(\mathbb{C}\right.$ ). Show that $\mathbb{A}^{n}(\mathbb{C}) \backslash V$ is path-connected. (Hint:: If $P, Q \in \mathbb{A}^{n}(\mathbb{C}) \backslash V$, let $L$ be the line through $P$ and $Q$. Then $L \cap V$ is finite, and $L$ is isomorphic to $\mathbb{A}^{1}(\mathbb{C})$.)
### Rational Functions and Local Rings
Let $V$ be a nonempty variety in $\mathbb{A}^{n}, \Gamma(V)$ its coordinate ring. Since $\Gamma(V)$ is a domain, we may form its quotient field. This field is called the field of rational functions on $V$, and is written $k(V)$. An element of $k(V)$ is a rational function on $V$.
If $f$ is a rational function on $V$, and $P \in V$, we say that $f$ is defined at $P$ if for some $a, b \in \Gamma(V), f=a / b$, and $b(P) \neq 0$. Note that there may be many different ways to write $f$ as a ratio of polynomial functions; $f$ is defined at $P$ if it is possible to find a "denominator" for $f$ that doesn't vanish at $P$. If $\Gamma(V)$ is a UFD, however, there is an essentially unique representation $f=a / b$, where $a$ and $b$ have no common factors (Problem 1.2), and then $f$ is defined at $P$ if and only if $b(P) \neq 0$.
Example. $V=V(X W-Y Z) \subset A^{4}(k) . \Gamma(V)=k[X, Y, Z, W] /(X W-Y Z)$. Let $\bar{X}, \bar{Y}, \bar{Z}, \bar{W}$ be the residues of $X, Y, Z, W$ in $\Gamma(V)$. Then $\bar{X} / \bar{Y}=\bar{Z} / \bar{W}=f \in k(V)$ is defined at $P=(x, y, z, w) \in V$ if $y \neq 0$ or $w \neq 0$ (see Problem 2.20). Let $P \in V$. We define $\mathscr{O}_{P}(V)$ to be the set of rational functions on $V$ that are defined at $P$. It is easy to verify that $\mathscr{O}_{P}(V)$ forms a subring of $k(V)$ containing $\Gamma(V)$ : $k \subset \Gamma(V) \subset \mathscr{O}_{P}(V) \subset k(V)$. The ring $\mathscr{O}_{P}(V)$ is called the local ring of $V$ at $P$.
The set of points $P \in V$ where a rational function $f$ is not defined is called the pole set of $f$.
Proposition 2. (1) The pole set of a rational function is an algebraic subset of $V$.
(2) $\Gamma(V)=\bigcap_{P \in V} \mathscr{O}_{P}(V)$.
Proof. Suppose $V \subset \mathbb{A}^{n}$. For $G \in k\left[X_{1}, \ldots, X_{n}\right]$, denote the residue of $G$ in $\Gamma(V)$ by $\bar{G}$. Let $f \in k(V)$. Let $J_{f}=\left\{G \in k\left[X_{1}, \ldots, X_{n}\right] \mid \bar{G} f \in \Gamma(V)\right\}$. Note that $J_{f}$ is an ideal in $k\left[X_{1}, \ldots, X_{n}\right]$ containing $I(V)$, and the points of $V\left(J_{f}\right)$ are exactly those points where $f$ is not defined. This proves (1). If $f \in \bigcap_{P \in V} \mathscr{O}_{P}(V), V\left(J_{f}\right)=\varnothing$, so $1 \in J_{f}$ (Nullstellensatz!), i.e., $1 \cdot f=f \in \Gamma(V)$, which proves (2).
Suppose $f \in \mathscr{O}_{P}(V)$. We can then define the value of $f$ at $P$, written $f(P)$, as follows: write $f=a / b, a, b \in \Gamma(V), b(P) \neq 0$, and let $f(P)=a(P) / b(P)$ (one checks that this is independent of the choice of $a$ and $b$.) The ideal $\mathfrak{m}_{P}(V)=\left\{f \in \mathscr{O}_{P}(V) \mid f(P)=\right.$ $0\}$ is called the maximal ideal of $V$ at $P$. It is the kernel of the evaluation homomorphism $f \mapsto f(P)$ of $\mathscr{O}_{P}(V)$ onto $k$, so $\mathscr{O}_{P}(V) / \mathfrak{m}_{P}(V)$ is isomorphic to $k$. An element $f \in \mathscr{O}_{P}(V)$ is a unit in $\mathscr{O}_{P}(V)$ if and only if $f(P) \neq 0$, so $\mathfrak{m}_{P}(V)=$ nnon-units of $\left.\mathscr{O}_{P}(V)\right\}$.
Lemma. The following conditions on a ring $R$ are equivalent:
(1) The set of non-units in $R$ forms an ideal.
(2) $R$ has a unique maximal ideal that contains every proper ideal of $R$.
Proof. Let $\mathfrak{m}=$ non-units of $R\}$. Clearly every proper ideal of $R$ is contained in $\mathfrak{m}$; the lemma is an immediate consequence of this.
A ring satisfying the conditions of the lemma is called a local ring; the units are those elements not belonging to the maximal ideal. We have seen that $\mathscr{O}_{P}(V)$ is a local ring, and $\mathfrak{m}_{P}(V)$ is its unique maximal ideal. These local rings play a prominent role in the modern study of algebraic varieties. All the properties of $V$ that depend only on a "neighborhood" of $P$ (the "local" properties) are reflected in the $\operatorname{ring} \mathscr{O}_{P}(V)$. See Problem 2.18 for one indication of this.
Proposition 3. $\mathscr{O}_{P}(V)$ is a Noetherian local domain.
Proof. We must show that any ideal $I$ of $\mathscr{O}_{P}(V)$ is finitely generated. Since $\Gamma(V)$ is Noetherian (Problem 1.22), choose generators $f_{1}, \ldots, f_{r}$ for the ideal $I \cap \Gamma(V)$ of $\Gamma(V)$. We claim that $f_{1}, \ldots, f_{r}$ generate $I$ as an ideal in $\mathscr{O}_{P}(V)$. For if $f \in I \subset \mathscr{O}_{P}(V)$, there is a $b \in \Gamma(V)$ with $b(P) \neq 0$ and $b f \in \Gamma(V)$; then $b f \in \Gamma(V) \cap I$, so $b f=\sum a_{i} f_{i}, a_{i} \in \Gamma(V)$; therefore $f=\sum\left(a_{i} / b\right) f_{i}$, as desired.
## Problems
2.17. Let $V=V\left(Y^{2}-X^{2}(X+1)\right) \subset \mathbb{A}^{2}$, and $\bar{X}, \bar{Y}$ the residues of $X, Y$ in $\Gamma(V)$; let $z=\bar{Y} / \bar{X} \in k(V)$. Find the pole sets of $z$ and of $z^{2}$. 2.18. Let $\mathscr{O}_{P}(V)$ be the local ring of a variety $V$ at a point $P$. Show that there is a natural one-to-one correspondence between the prime ideals in $\mathscr{O}_{P}(V)$ and the subvarieties of $V$ that pass through $P$. (Hint:: If $I$ is prime in $\mathscr{O}_{P}(V), I \cap \Gamma(V)$ is prime in $\Gamma(V)$, and $I$ is generated by $I \cap \Gamma(V)$; use Problem 2.2.)
2.19. Let $f$ be a rational function on a variety $V$. Let $U=\{P \in V \mid f$ is defined at $P$. Then $f$ defines a function from $U$ to $k$. Show that this function determines $f$ uniquely. So a rational function may be considered as a type of function, but only on the complement of an algebraic subset of $V$, not on $V$ itself.
2.20. In the example given in this section, show that it is impossible to write $f=a / b$, where $a, b \in \Gamma(V)$, and $b(P) \neq 0$ for every $P$ where $f$ is defined. Show that the pole set of $f$ is exactly $\{(x, y, z, w) \mid y=0$ and $w=0\}$.
2.21* Let $\varphi: V \rightarrow W$ be a polynomial map of affine varieties, $\tilde{\varphi}: \Gamma(W) \rightarrow \Gamma(V)$ the induced map on coordinate rings. Suppose $P \in V, \varphi(P)=Q$. Show that $\tilde{\varphi}$ extends uniquely to a ring homomorphism (also written $\tilde{\varphi}$ ) from $\mathscr{O}_{Q}(W)$ to $\mathscr{O}_{P}(V)$. (Note that $\tilde{\varphi}$ may not extend to all of $k(W)$.) Show that $\tilde{\varphi}\left(\mathfrak{m}_{Q}(W)\right) \subset \mathfrak{m}_{P}(V)$.
2.22* Let $T: \mathbb{A}^{n} \rightarrow \mathbb{A}^{n}$ be an affine change of coordinates, $T(P)=Q$. Show that $\tilde{T}: \mathscr{O}_{Q}\left(\mathbb{A}^{n}\right) \rightarrow \mathscr{O}_{P}\left(\mathbb{A}^{n}\right)$ is an isomorphism. Show that $\tilde{T}$ induces an isomorphism from $\mathscr{O}_{Q}(V)$ to $\mathscr{O}_{P}\left(V^{T}\right)$ if $P \in V^{T}$, for $V$ a subvariety of $\mathbb{A}^{n}$.
### Discrete Valuation Rings
Our study of plane curves will be made easier if we have at our disposal several concepts and facts of an algebraic nature. They are put into the next few sections to avoid disrupting later geometric arguments.
Proposition 4. Let $R$ be a domain that is not a field. Then the following are equivalent:
(1) $R$ is Noetherian and local, and the maximal ideal is principal.
(2) There is an irreducible element $t \in R$ such that every nonzero $z \in R$ may be written uniquely in the form $z=u t^{n}, u$ a unit in $R, n$ a nonnegative integer.
Proof. (1) implies (2): Let $\mathfrak{m}$ be the maximal ideal, $t$ a generator for $\mathfrak{m}$. Suppose $u t^{n}=v t^{m}, u, v$ units, $n \geq m$. Then $u t^{n-m}=v$ is a unit, so $n=m$ and $u=v$. Thus the expression of any $z=u t^{n}$ is unique. To show that any $z$ has such an expression, we may assume $z$ is not a unit, so $z=z_{1} t$ for some $z_{1} \in R$. If $z_{1}$ is a unit we are finished, so assume $z_{1}=z_{2} t$. Continuing in this way, we find an infinite sequence $z_{1}, z_{2}, \ldots$ with $z_{i}=z_{i+1} t$. Since $R$ is Noetherian, the chain of ideals $\left(z_{1}\right) \subset\left(z_{2}\right) \subset \cdots$ must have a maximal member (Chapter 1, Section 5), so $\left(z_{n}\right)=\left(z_{n+1}\right)$ for some $n$. Then $z_{n+1}=v z_{n}$ for some $v \in R$, so $z_{n}=v t z_{n}$ and $v t=1$. But $t$ is not a unit.
(2) implies (1): (We don't really need this part.) $\mathfrak{m}=(t)$ is clearly the set of nonunits. It is not hard to see that the only ideals in $R$ are the principal ideals $\left(t^{n}\right), n$ a nonnegative integer, so $R$ is a PID.
A ring $R$ satisfying the conditions of Proposition 4 is called a discrete valuation ring, written DVR. An element $t$ as in (2) is called a uniformizing parameter for $R$; any other uniformizing parameter is of the form $u t, u$ a unit in $R$. Let $K$ be the quotient field of $R$. Then (when $t$ is fixed) any nonzero element $z \in K$ has a unique expression $z=u t^{n}, u$ a unit in $R, n \in \mathbb{Z}$ (see Problem 1.2). The exponent $n$ is called the $\operatorname{order}$ of $z$, and is written $n=\operatorname{ord}(z)$; we define $\operatorname{ord}(0)=\infty$. Note that $R=\{z \in K \mid$ $\operatorname{ord}(z) \geq 0\}$, and $\mathfrak{m}=\{z \in K \mid \operatorname{ord}(z)>0\}$ is the maximal ideal in $R$.
## Problems
2.23. Show that the order function on $K$ is independent of the choice of uniformizing parameter.
2.24. Let $V=\mathbb{A}^{1}, \Gamma(V)=k[X], K=k(V)=k(X)$. (a) For each $a \in k=V$, show that $\mathscr{O}_{a}(V)$ is a DVR with uniformizing parameter $t=X-a$. (b) Show that $\mathscr{O}_{\infty}=\{F / G \in$ $k(X) \mid \operatorname{deg}(G) \geq \operatorname{deg}(F)\}$ is also a DVR, with uniformizing parameter $t=1 / X$.
2.25. Let $p \in \mathbb{Z}$ be a prime number. Show that $\{r \in Q \mid r=a / b, a, b \in \mathbb{Z}, p$ doesn't divide $b\}$ is a DVR with quotient field $\mathbb{Q}$.
2.26. Let $R$ be a DVR with quotient field $K$; let $\mathfrak{m}$ be the maximal ideal of $R$. (a) Show that if $z \in K, z \notin R$, then $z^{-1} \in m$. (b) Suppose $R \subset S \subset K$, and $S$ is also a DVR. Suppose the maximal ideal of $S$ contains $\mathfrak{m}$. Show that $S=R$.
2.27. Show that the DVR's of Problem 2.24 are the only DVR's with quotient field $k(X)$ that contain $k$. Show that those of Problem 2.25 are the only DVR's with quotient field $\mathbb{Q}$.
2.28. An order function on a field $K$ is a function $\varphi$ from $K$ onto $\mathbb{Z} \cup\{\infty\}$, satisfying:
(i) $\varphi(a)=\infty$ if and only if $a=0$.
(ii) $\varphi(a b)=\varphi(a)+\varphi(b)$.
(iii) $\varphi(a+b) \geq \min (\varphi(a), \varphi(b))$.
Show that $R=\{z \in K \mid \varphi(z) \geq 0\}$ is a DVR with maximal ideal $\mathfrak{m}=\{z \mid \varphi(z)>0\}$, and quotient field $K$. Conversely, show that if $R$ is a DVR with quotient field $K$, then the function ord: $K \rightarrow \mathbb{Z} \cup\{\infty\}$ is an order function on $K$. Giving a DVR with quotient field $K$ is equivalent to defining an order function on $K$.
2.29. Let $R$ be a DVR with quotient field $K$, ord the order function on $K$. (a) If $\operatorname{ord}(a)<\operatorname{ord}(b), \operatorname{show}$ that $\operatorname{ord}(a+b)=\operatorname{ord}(a)$. (b) If $a_{1}, \ldots, a_{n} \in K$, and for some $i, \operatorname{ord}\left(a_{i}\right)<\operatorname{ord}\left(a_{j}\right)($ all $j \neq i)$, then $a_{1}+\cdots+a_{n} \neq 0$.
2.30. Let $R$ be a DVR with maximal ideal $\mathfrak{m}$, and quotient field $K$, and suppose a field $k$ is a subring of $R$, and that the composition $k \rightarrow R \rightarrow R / \mathfrak{m}$ is an isomorphism of $k$ with $R / m$ (as for example in Problem 2.24). Verify the following assertions:
(a) For any $z \in R$, there is a unique $\lambda \in k$ such that $z-\lambda \in \mathfrak{m}$.
(b) Let $t$ be a uniformizing parameter for $R, z \in R$. Then for any $n \geq 0$ there are unique $\lambda_{0}, \lambda_{1}, \ldots, \lambda_{n} \in k$ and $z_{n} \in R$ such that $z=\lambda_{0}+\lambda_{1} t+\lambda_{2} t^{2}+\cdots+\lambda_{n} t^{n}+z_{n} t^{n+1}$. (Hint:: For uniqueness use Problem 2.29; for existence use (a) and induction.)
2.31. Let $k$ be a field. The ring of formal power series over $k$, written $k[[X]]$, is defined to be $\left\{\sum_{i=0}^{\infty} a_{i} X^{i} \mid a_{i} \in k\right\}$. (As with polynomials, a rigorous definition is best given in terms of sequences $\left(a_{0}, a_{1}, \ldots\right)$ of elements in $k$; here we allow an infinite number of nonzero terms.) Define the sum by $\sum a_{i} X^{i}+\sum b_{i} X^{i}=\sum\left(a_{i}+b_{i}\right) X^{i}$, and the product by $\left(\sum a_{i} X^{i}\right)\left(\sum b_{i} X^{i}\right)=\sum c_{i} X^{i}$, where $c_{i}=\sum_{j+k=i} a_{j} b_{k}$. Show that $k[[X]]$ is a ring containing $k[X]$ as a subring. Show that $k[[X]]$ is a DVR with uniformizing parameter $X$. Its quotient field is denoted $k((X))$.
2.32. Let $R$ be a DVR satisfying the conditions of Problem 2.30. Any $z \in R$ then determines a power series $\lambda_{i} X^{i}$, if $\lambda_{0}, \lambda_{1}, \ldots$ are determined as in Problem 2.30(b). (a) Show that the map $z \mapsto \sum \lambda_{i} X^{i}$ is a one-to-one ring homomorphism of $R$ into $k[[X]]$. We often write $z=\sum \lambda_{i} t^{i}$, and call this the power series expansion of $z$ in terms of $t$. (b) Show that the homomorphism extends to a homomorphism of $K$ into $k((X))$, and that the order function on $k((X))$ restricts to that on $K$. (c) Let $a=0$ in Problem $2.24, t=X$. Find the power series expansion of $z=(1-X)^{-1}$ and of $(1-X)\left(1+X^{2}\right)^{-1}$ in terms of $t$.
### Forms
Let $R$ be a domain. If $F \in R\left[X_{1}, \ldots, X_{n+1}\right]$ is a form, we define $F_{*} \in R\left[X_{1}, \ldots, X_{n}\right]$ by setting $F_{*}=F\left(X_{1}, \ldots, X_{n}, 1\right)$. Conversely, for any polynomial $f \in R\left[X_{1}, \ldots, X_{n}\right]$ of degree $d$, write $f=f_{0}+f_{1}+\cdots+f_{d}$, where $f_{i}$ is a form of degree $i$, and define $f^{*} \in$ $R\left[X_{1}, \ldots, X_{n+1}\right]$ by setting
$$
f^{*}=X_{n+1}^{d} f_{0}+X_{n+1}^{d-1} f_{1}+\cdots+f_{d}=X_{n+1}^{d} f\left(X_{1} / X_{n+1}, \ldots, X_{n} / X_{n+1}\right) ;
$$
$f^{*}$ is a form of degree $d$. (These processes are often described as "dehomogenizing" and "homogenizing" polynomials with respect to $X_{n+1}$.) The proof of the following proposition is left to the reader:
Proposition 5. (1) $(F G)_{*}=F_{*} G_{*} ;(f g)^{*}=f^{*} g^{*}$.
(2) If $F \neq 0$ and $r$ is the highest power of $X_{n+1}$ that divides $F$, then $X_{n+1}^{r}\left(F_{*}\right)^{*}=F$; $\left(f^{*}\right)_{*}=f$.
(3) $(F+G)_{*}=F_{*}+G_{*} ; X_{n+1}^{t}(f+g)^{*}=X_{n+1}^{r} f^{*}+X_{n+1}^{s} g^{*}$, where $r=\operatorname{deg}(g), s=$ $\operatorname{deg}(f)$, and $t=r+s-\operatorname{deg}(f+g)$.
Corollary. Up to powers of $X_{n+1}$, factoring a form $F \in R\left[X_{1}, \ldots, X_{n+1}\right]$ is the same as factoring $F_{*} \in R\left[X_{1}, \ldots, X_{n}\right]$. In particular, if $F \in k[X, Y]$ is a form, $k$ algebraically closed, then $F$ factors into a product of linear factors.
Proof. The first claim follows directly from (1) and (2) of the proposition. For the second, write $F=Y^{r} G$, where $Y$ doesn't divide $G$. Then $F_{*}=G_{*}=\epsilon \prod\left(X-\lambda_{i}\right)$ since $k$ is algebraically closed, so $F=\epsilon Y^{r} \Pi\left(X-\lambda_{i} Y\right)$.
## Problems
2.33. Factor $Y^{3}-2 X Y^{2}+2 X^{2} Y+X^{3}$ into linear factors in $\mathbb{C}[X, Y]$.
2.34. Suppose $F, G \in k\left[X_{1}, \ldots, X_{n}\right]$ are forms of degree $r, r+1$ respectively, with no common factors ( $k$ a field). Show that $F+G$ is irreducible. 2.35* (a) Show that there are $d+1$ monomials of degree $d$ in $R[X, Y]$, and $1+2+$ $\cdots+(d+1)=(d+1)(d+2) / 2$ monomials of degree $d$ in $R[X, Y, Z]$. (b) Let $V(d, n)=$ \{forms of degree $d$ in $\left.k\left[X_{1}, \ldots, X_{n}\right]\right\}, k$ a field. Show that $V(d, n)$ is a vector space over $k$, and that the monomials of degree $d$ form a basis. So $\operatorname{dim} V(d, 1)=1 ; \operatorname{dim} V(d, 2)=$ $d+1 ; \operatorname{dim} V(d, 3)=(d+1)(d+2) / 2$. (c) Let $L_{1}, L_{2}, \ldots$ and $M_{1}, M_{2}, \ldots$ be sequences of nonzero linear forms in $k[X, Y]$, and assume no $L_{i}=\lambda M_{j}, \lambda \in k$. Let $A_{i j}=$ $L_{1} L_{2} \ldots L_{i} M_{1} M_{2} \ldots M_{j}, i, j \geq 0\left(A_{00}=1\right)$. Show that $\left\{A_{i j} \mid i+j=d\right\}$ forms a basis for $V(d, 2)$.
2.36. With the above notation, show that $\operatorname{dim} V(d, n)=\left(\begin{array}{c}d+n-1 \\ n-1\end{array}\right)$, the binomial coefficient.
### Direct Products of Rings
If $R_{1}, \ldots, R_{n}$ are rings, the cartesian product $R_{1} \times \cdots \times R_{n}$ is made into a ring as follows: $\left(a_{1}, \ldots, a_{n}\right)+\left(b_{1}, \ldots, b_{n}\right)=\left(a_{1}+b_{1}, \ldots, a_{n}+b_{n}\right)$, and $\left(a_{1}, \ldots, a_{n}\right)\left(b_{1}, \ldots, b_{n}\right)=$ $\left(a_{1} b_{1}, \ldots, a_{n} b_{n}\right)$. This ring is called the direct product of $R_{1}, \ldots, R_{n}$, and is written $\prod_{i=1}^{n} R_{i}$. The natural projection maps $\pi_{i}: \prod_{j=1}^{n} R_{j} \rightarrow R_{i}$ taking $\left(a_{1}, \ldots, a_{n}\right)$ to $a_{i}$ are ring homomorphisms.
The direct product is characterized by the following property: given any ring $R$, and ring homomorphisms $\varphi_{i}: R \rightarrow R_{i}, i=1, \ldots, n$, there is a unique ring homomorphism $\varphi: R \rightarrow \prod_{i=1}^{n} R_{i}$ such that $\pi_{i} \circ \varphi=\varphi_{i}$. In particular, if a field $k$ is a subring of each $R_{i}, k$ may be regarded as a subring of $\prod_{i=1}^{n} R_{i}$.
## Problems
2.37. What are the additive and multiplicative identities in $\prod R_{i}$ ? Is the map from $R_{i}$ to $\prod R_{j}$ taking $a_{i}$ to $\left(0, \ldots, a_{i}, \ldots, 0\right)$ a ring homomorphism?
2.38. Show that if $k \subset R_{i}$, and each $R_{i}$ is finite-dimensional over $k$, then $\operatorname{dim}\left(\prod R_{i}\right)=$ $\sum \operatorname{dim} R_{i}$.
### Operations with Ideals
Let $I, J$ be ideals in a ring $R$. The ideal generated by $\{a b \mid a \in I, b \in J\}$ is denoted $I J$. Similarly if $I_{1}, \ldots, I_{n}$ are ideals, $I_{1} \cdots I_{n}$ is the ideal generated by $\left\{a_{1} a_{2} \cdots a_{n} \mid a_{i} \in\right.$ $I_{i}$ \}. We define $I^{n}$ to be $I I \cdots I$ ( $n$ times). Note that while $I^{n}$ contains all $n$th powers of elements of $I$, it may not be generated by them. If $I$ is generated by $a_{1}, \ldots, a_{r}$, then $I^{n}$ is generated by $\left\{a_{i}^{i_{1}} \cdots a_{r}^{i_{r}} \mid \sum i_{j}=n\right\}$. And $R=I^{0} \supset I^{1} \supset I^{2} \supset \cdots$.
Example. $R=k\left[X_{1}, \ldots, X_{r}\right], I=\left(X_{1}, \ldots, X_{r}\right)$. Then $I^{n}$ is generated by the monomials of degree $n$, so $I^{n}$ consists of those polynomials with no terms of degree $<n$. It follows that the residues of the monomials of degree $<n$ form a basis of $k\left[X_{1}, \ldots, X_{r}\right] / I^{n}$ over $k$. If $R$ is a subring of a ring $S, I S$ denotes the ideal of $S$ generated by the elements of $I$. It is easy to see that $I^{n} S=(I S)^{n}$.
Let $I, J$ be ideals in a ring $R$. Define $I+J=\{a+b \mid a \in I, b \in J\}$. Then $I+J$ is an ideal; in fact, it is the smallest ideal in $R$ that contains $I$ and $J$.
Two ideals $I, J$ in $R$ are said to be comaximal if $I+J=R$, i.e., if $1=a+b, a \in I$, $b \in J$. For example, two distinct maximal ideals are comaximal.
Lemma. (1) $I J \subset I \cap J$ for any ideals $I$ and $J$.
(2) If I and $J$ are comaximal, $I J=I \cap J$.
Proof. (1) is trivial. If $I+J=R$, then $I \cap J=(I \cap J) R=(I \cap J)(I+J)=(I \cap J) I+(I \cap J) J \subset$ $J I+I J=I J$, proving (2). (See Problem 2.39.)
## Problems
2.39* Prove the following relations among ideals $I_{i}, J$, in a ring $R$ :
(a) $\left(I_{1}+I_{2}\right) J=I_{1} J+I_{2} J$.
(b) $\left(I_{1} \cdots I_{N}\right)^{n}=I_{1}^{n} \cdots I_{N}^{n}$.
2.40* (a) Suppose $I, J$ are comaximal ideals in $R$. Show that $I+J^{2}=R$. Show that $I^{m}$ and $J^{n}$ are comaximal for all $m, n$. (b) Suppose $I_{1}, \ldots, I_{N}$ are ideals in $R$, and $I_{i}$ and $J_{i}=\bigcap_{j \neq i} I_{j}$ are comaximal for all $i$. Show that $I_{1}^{n} \cap \cdots \cap I_{N}^{n}=\left(I_{1} \cdots I_{N}\right)^{n}=$ $\left(I_{1} \cap \cdots \cap I_{N}\right)^{n}$ for all $n$.
2.41* Let $I, J$ be ideals in a ring $R$. Suppose $I$ is finitely generated and $I \subset \operatorname{Rad}(J)$. Show that $I^{n} \subset J$ for some $n$.
2.42* (a) Let $I \subset J$ be ideals in a ring $R$. Show that there is a natural ring homomorphism from $R / I$ onto $R / J$. (b) Let $I$ be an ideal in a ring $R, R$ a subring of a ring $S$. Show that there is a natural ring homomorphism from $R / I$ to $S / I S$.
2.43. Let $P=(0, \ldots, 0) \in \mathbb{A}^{n}, \mathscr{O}=\mathscr{O}_{P}\left(\mathbb{A}^{n}\right), \mathfrak{m}=\mathfrak{m}_{P}\left(\mathbb{A}^{n}\right)$. Let $I \subset k\left[X_{1}, \ldots, X_{n}\right]$ be the ideal generated by $X_{1}, \ldots, X_{n}$. Show that $I \mathscr{O}=m$, so $I^{r} \mathscr{O}=m^{r}$ for all $r$.
2.44* Let $V$ be a variety in $\mathbb{A}^{n}, I=I(V) \subset k\left[X_{1}, \ldots, X_{n}\right], P \in V$, and let $J$ be an ideal of $k\left[X_{1}, \ldots, X_{n}\right]$ that contains $I$. Let $J^{\prime}$ be the image of $J$ in $\Gamma(V)$. Show that there is a natural homomorphism $\varphi$ from $\mathscr{O}_{P}\left(\mathbb{A}^{n}\right) / J \mathscr{O}_{P}\left(\mathbb{A}^{n}\right)$ to $\mathscr{O}_{P}(V) / J^{\prime} \mathscr{O}_{P}(V)$, and that $\varphi$ is an isomorphism. In particular, $\mathscr{O}_{P}\left(\mathbb{A}^{n}\right) / I \mathscr{O}_{P}\left(\mathbb{A}^{n}\right)$ is isomorphic to $\mathscr{O}_{P}(V)$.
2.45* Show that ideals $I, J \subset k\left[X_{1}, \ldots, X_{n}\right]$ ( $k$ algebraically closed) are comaximal if and only if $V(I) \cap V(J)=\varnothing$.
2.46. . Let $I=(X, Y) \subset k[X, Y]$. Show that $\operatorname{dim}_{k}\left(k[X, Y] / I^{n}\right)=1+2+\cdots+n=\frac{n(n+1)}{2}$.
### Ideals with a Finite Number of Zeros
The proposition of this section will be used to relate local questions (in terms of the local rings $\mathscr{O}_{P}(V)$ ) with global ones (in terms of coordinate rings). Proposition 6. Let I be an ideal in $k\left[X_{1}, \ldots, X_{n}\right]$ ( $k$ algebraically closed), and suppose $V(I)=\left\{P_{1}, \ldots, P_{N}\right\}$ is finite. Let $\mathscr{O}_{i}=\mathscr{O}_{P_{i}}\left(\mathbb{A}^{n}\right)$. Then there is a natural isomorphism of $k\left[X_{1}, \ldots, X_{n}\right] / I$ with $\prod_{i=1}^{N} \mathscr{O}_{i} / I \mathscr{O}_{i}$.
Proof. Let $I_{i}=I\left(\left\{P_{i}\right\}\right) \subset k\left[X_{1}, \ldots, X_{n}\right]$ be the distinct maximal ideals that contain $I$. Let $R=k\left[X_{1}, \ldots, X_{n}\right] / I, R_{i}=\mathscr{O}_{i} / I \mathscr{O}_{i}$. The natural homomorphisms (Problem 2.42(b)) $\varphi_{i}$ from $R$ to $R_{i}$ induce a homomorphism $\varphi$ from $R$ to $\prod_{i=1}^{N} R_{i}$.
By the Nullstellensatz, $\operatorname{Rad}(I)=I\left(\left\{P_{1}, \ldots, P_{N}\right\}\right)=\bigcap_{i=1}^{N} I_{i}$, so $\left(\cap I_{i}\right)^{d} \subset I$ for some $d$ (Problem 2.41). Since $\bigcap_{j \neq i} I_{j}$ and $I_{i}$ are comaximal (Problem 2.45), it follows (Problem 2.40) that $\cap I_{j}^{d}=\left(I_{1} \cdots I_{N}\right)^{d}=\left(\cap I_{j}\right)^{d} \subset I$.
Now choose $F_{i} \in k\left[X_{1}, \ldots, X_{n}\right]$ such that $F_{i}\left(P_{j}\right)=0$ if $i \neq j, F_{i}\left(P_{i}\right)=1$ (Problem 1.17). Let $E_{i}=1-\left(1-F_{i}^{d}\right)^{d}$. Note that $E_{i}=F_{i}^{d} D_{i}$ for some $D_{i}$, so $E_{i} \in I_{j}^{d}$ if $i \neq j$, and $1-\sum_{i} E_{i}=\left(1-E_{j}\right)-\sum_{i \neq j} E_{i} \in \cap I_{j}^{d} \subset I$. In addition, $E_{i}-E_{i}^{2}=E_{i}\left(1-F_{i}^{d}\right)^{d}$ is in $\bigcap_{j \neq i} I_{j}^{d} \cdot I_{i}^{d} \subset I$. If we let $e_{i}$ be the residue of $E_{i}$ in $R$, we have $e_{i}^{2}=e_{i}, e_{i} e_{j}=0$ if $i \neq j$, and $\sum e_{i}=1$.
Claim. If $G \in k\left[X_{1}, \ldots, X_{n}\right]$, and $G\left(P_{i}\right) \neq 0$, then there is a $t \in R$ such that $t g=e_{i}$, where $g$ is the $I$-residue of $G$.
Assuming the claim for the moment, we show how to conclude that $\varphi$ is an isomorphism:
$\varphi$ is one-to-one: If $\varphi(f)=0$, then for each $i$ there is a $G_{i}$ with $G_{i}\left(P_{i}\right) \neq 0$ and $G_{i} F \in I(f=I$-residue of $F)$. Let $t_{i} g_{i}=e_{i}$. Then $f=\sum e_{i} f=\sum t_{i} g_{i} f=0$.
$\varphi$ is onto: Since $E_{i}\left(P_{i}\right)=1, \varphi_{i}\left(e_{i}\right)$ is a unit in $R_{i}$; since $\varphi_{i}\left(e_{i}\right) \varphi_{i}\left(e_{j}\right)=\varphi_{i}\left(e_{i} e_{j}\right)=0$ if $i \neq j, \varphi_{i}\left(e_{j}\right)=0$ for $i \neq j$. Therefore $\varphi_{i}\left(e_{i}\right)=\varphi_{i}\left(\sum e_{j}\right)=\varphi_{i}(1)=1$. Now suppose $z=\left(a_{1} / s_{1}, \ldots, a_{N} / s_{N}\right) \in \prod R_{i}$. By the claim, we may find $t_{i}$ so that $t_{i} s_{i}=e_{i}$; then $a_{i} / s_{i}=a_{i} t_{i}$ in $R_{i}$, so $\varphi_{i}\left(\sum t_{j} a_{j} e_{j}\right)=\varphi_{i}\left(t_{i} a_{i}\right)=a_{i} / s_{i}$, and $\varphi\left(\sum t_{j} a_{j} e_{j}\right)=z$.
To prove the claim, we may assume that $G\left(P_{i}\right)=1$. Let $H=1-G$. It follows that $(1-H)\left(E_{i}+H E_{i}+\cdots+H^{d-1} E_{i}\right)=E_{i}-H^{d} E_{i}$. Then $H \in I_{i}$, so $H^{d} E_{i} \in I$. Therefore $g\left(e_{i}+h e_{i}+\cdots+h^{d-1} e_{i}\right)=e_{i}$, as desired.
Corollary 1. $\operatorname{dim}_{k}\left(k\left[X_{1}, \ldots, X_{n}\right] / I\right)=\sum_{i=1}^{N} \operatorname{dim}_{k}\left(\mathscr{O}_{i} / I \mathscr{O}_{i}\right)$.
Corollary 2. If $V(I)=\{P\}$, then $k\left[X_{1}, \ldots, X_{n}\right] / I$ is isomorphic to $\mathscr{O}_{P}\left(\mathbb{A}^{n}\right) / I \mathscr{O}_{P}\left(\mathbb{A}^{n}\right)$.
## Problem
2.47. Suppose $R$ is a ring containing $k$, and $R$ is finite dimensional over $k$. Show that $R$ is isomorphic to a direct product of local rings.
### Quotient Modules and Exact Sequences
Let $R$ be a ring, $M, M^{\prime} R$-modules. A homomorphism $\varphi: M \rightarrow M^{\prime}$ of abelian groups is called an $R$-module homomorphism if $\varphi(a m)=a \varphi(m)$ for all $a \in R, m \in$ $M$. It is an $R$-module isomorphism if it is one-to-one and onto.
If $N$ is a submodule of an $R$-module $M$, the quotient group $M / N$ of cosets of $N$ in $M$ is made into an $R$-module in the following way: if $\bar{m}$ is the coset (or equivalence class) containing $m$, and $a \in R$, define $a \bar{m}=\overline{a m}$. It is easy to verify that this makes $M / N$ into an $R$-module in such a way that the natural map from $M$ to $M / N$ is an $R$-module homomorphism. This $M / N$ is called the quotient module of $M$ by $N$.
Let $\psi: M^{\prime} \rightarrow M, \varphi: M \rightarrow M^{\prime \prime}$ be $R$-module homomorphisms. We say that the sequence (of modules and homomorphisms)
$$
M^{\prime} \stackrel{\psi}{\longrightarrow} M \stackrel{\varphi}{\longrightarrow} M^{\prime \prime}
$$
is exact (or exact at $M$ ) if $\operatorname{Im}(\psi)=\operatorname{Ker}(\varphi)$. Note that there are unique $R$-module homomorphism from the zero-module 0 to any $R$-module $M$, and from $M$ to 0 . Thus $M \stackrel{\varphi}{\longrightarrow} M^{\prime \prime} \longrightarrow 0$ is exact if and only if $\varphi$ is onto, and $0 \longrightarrow M^{\prime} \stackrel{\psi}{\longrightarrow} M$ is exact if and only if $\psi$ is one-to-one.
If $\varphi_{i}: M_{i} \rightarrow M_{i+1}$ are $R$-module homomorphisms, we say that the sequence
$$
M_{1} \stackrel{\varphi_{1}}{\longrightarrow} M_{2} \stackrel{\varphi_{2}}{\longrightarrow} \cdots \stackrel{\varphi_{n}}{\longrightarrow} M_{n+1}
$$
is exact if $\operatorname{Im}\left(\varphi_{i}\right)=\operatorname{Ker}\left(\varphi_{i+1}\right)$ for each $i=1, \ldots, n$. Thus $0 \longrightarrow M^{\prime} \stackrel{\psi}{\longrightarrow} M \stackrel{\varphi}{\longrightarrow} M^{\prime \prime} \longrightarrow 0$ is exact if and only if $\varphi$ is onto, and $\psi$ maps $M^{\prime}$ isomorphically onto the kernel of $\varphi$.
Proposition 7. (1) Let $0 \longrightarrow V^{\prime} \stackrel{\psi}{\longrightarrow} V \stackrel{\varphi}{\longrightarrow} V^{\prime \prime} \longrightarrow 0$ be an exact sequence of finitedimensional vector spaces over a field $k$. Then $\operatorname{dim} V^{\prime}+\operatorname{dim} V^{\prime \prime}=\operatorname{dim} V$.
(2) Let
$$
0 \longrightarrow V_{1} \stackrel{\varphi_{1}}{\longrightarrow} V_{2} \stackrel{\varphi_{2}}{\longrightarrow} V_{3} \stackrel{\varphi_{3}}{\longrightarrow} V_{4} \longrightarrow 0
$$
be an exact sequence of finite-dimensional vector spaces. Then
$$
\operatorname{dim} V_{4}=\operatorname{dim} V_{3}-\operatorname{dim} V_{2}+\operatorname{dim} V_{1} .
$$
Proof. (1) is just an abstract version of the rank-nullity theorem for a linear transformation $\varphi: V \rightarrow V^{\prime \prime}$ of finite-dimensional vector spaces.
(2) follows from (1) by letting $W=\operatorname{Im}\left(\varphi_{2}\right)=\operatorname{Ker}\left(\varphi_{3}\right)$. For then $0 \longrightarrow V_{1} \stackrel{\varphi_{1}}{\longrightarrow} V_{2} \stackrel{\varphi_{2}}{\longrightarrow}$ $W \longrightarrow 0$ and $0 \longrightarrow W \stackrel{\psi}{\longrightarrow} V_{3} \stackrel{\varphi_{3}}{\longrightarrow} V_{4} \longrightarrow 0$ are exact, where $\psi$ is the inclusion, and the result follows by subtraction.
## Problems
2.48. Verify that for any $R$-module homomorphism $\varphi: M \rightarrow M^{\prime}, \operatorname{Ker}(\varphi)$ and $\operatorname{Im}(\varphi)$ are submodules of $M$ and $M^{\prime}$ respectively. Show that
$$
0 \longrightarrow \operatorname{Ker}(\varphi) \longrightarrow M \stackrel{\varphi}{\longrightarrow} \operatorname{Im}(\varphi) \longrightarrow 0
$$
is exact.
2.49. (a) Let $N$ be a submodule of $M, \pi: M \rightarrow M / N$ the natural homomorphism. Suppose $\varphi: M \rightarrow M^{\prime}$ is a homomorphism of $R$-modules, and $\varphi(N)=0$. Show that there is a unique homomorphism $\bar{\varphi}: M / N \rightarrow M^{\prime}$ such that $\bar{\varphi} \circ \pi=\varphi$. (b) If $N$ and $P$ are submodules of a module $M$, with $P \subset N$, then there are natural homomorphisms from $M / P$ onto $M / N$ and from $N / P$ into $M / P$. Show that the resulting sequence
$$
0 \longrightarrow N / P \longrightarrow M / P \longrightarrow M / N \longrightarrow 0
$$
is exact ("Second Noether Isomorphism Theorem"). (c) Let $U \subset W \subset V$ be vector spaces, with $V / U$ finite-dimensional. Then $\operatorname{dim} V / U=\operatorname{dim} V / W+\operatorname{dim} W / U$. (d) If $J \subset I$ are ideals in a ring $R$, there is a natural exact sequence of $R$-modules:
$$
0 \longrightarrow I / J \longrightarrow R / J \longrightarrow R / I \longrightarrow 0
$$
(e) If $\mathscr{O}$ is a local ring with maximal ideal $\mathfrak{m}$, there is a natural exact sequence of $\mathscr{O}$-modules
$$
0 \longrightarrow \mathfrak{m}^{n} / \mathfrak{m}^{n+1} \longrightarrow \mathscr{O} / \mathfrak{m}^{n+1} \longrightarrow \mathscr{O} / \mathfrak{m}^{n} \longrightarrow 0
$$
2.50. Let $R$ be a DVR satisfying the conditions of Problem 2.30. Then $\mathfrak{m}^{n} / \mathfrak{m}^{n+1}$ is an $R$-module, and so also a $k$-module, since $k \subset R$. (a) Show that $\operatorname{dim}_{k}\left(\mathfrak{m}^{n} / \mathfrak{m}^{n+1}\right)=1$ for all $n \geq 0$. (b) Show that $\operatorname{dim}_{k}\left(R / \mathfrak{m}^{n}\right)=n$ for all $n>0$. (c) Let $z \in R$. Show that $\operatorname{ord}(z)=n$ if $(z)=\mathfrak{m}^{n}$, and hence that $\operatorname{ord}(z)=\operatorname{dim}_{k}(R /(z))$.
2.51. Let $0 \longrightarrow V_{1} \longrightarrow \cdots \longrightarrow V_{n} \longrightarrow 0$ be an exact sequence of finite-dimensional vector spaces. Show that $\sum(-1)^{i} \operatorname{dim}\left(V_{i}\right)=0$.
2.52. Let $N, P$ be submodules of a module $M$. Show that the subgroup $N+P=$ $\{n+p \mid n \in N, p \in P\}$ is a submodule of $M$. Show that there is a natural $R$-module isomorphism of $N / N \cap P$ onto $N+P / P$ ("First Noether Isomorphism Theorem").
2.53. Let $V$ be a vector space, $W$ a subspace, $T: V \rightarrow V$ a one-to-one linear map such that $T(W) \subset W$, and assume $V / W$ and $W / T(W)$ are finite-dimensional. (a) Show that $T$ induces an isomorphism of $V / W$ with $T(V) / T(W)$. (b) Construct an isomorphism between $T(V) /(W \cap T(V))$ and $(W+T(V)) / W$, and an isomorphism between $W /(W \cap T(V))$ and $(W+T(V)) / T(V)$. (c) Use Problem 2.49(c) to show that $\operatorname{dim} V /(W+T(V))=\operatorname{dim}(W \cap T(V)) / T(W)$. (d) Conclude finally that $\operatorname{dim} V / T(V)=$ $\operatorname{dim} W / T(W)$.
### Free Modules
Let $R$ be a ring, $X$ any set. Let $M_{X}=$ mapping s $\varphi: X \rightarrow R \mid \varphi(x)=0$ for all but a finite number of $x \in X$. This $M_{X}$ is made into an $R$-module as follows: $(\varphi+\psi)(x)=$ $\varphi(x)+\psi(x)$, and $(a \varphi)(x)=a \varphi(x)$ for $\varphi, \psi \in M_{X}, a \in R, x \in X$. The module $M_{X}$ is called the free $R$-module on the set $X$. If we define $\varphi_{x} \in M_{X}$ by the rules: $\varphi_{x}(y)=0$ if $y \neq x, \varphi_{x}(x)=1$, then every $\varphi \in M_{X}$ has a unique expression $\varphi=\sum a_{x} \varphi_{x}$, where $a_{x} \in R$ (in fact, $a_{x}=\varphi(x)$ ). Usually we write $x$ instead of $\varphi_{x}$, and consider $X$ as a subset of $M_{X}$. We say that $X$ is a basis for $M_{X}$ : the elements of $M_{X}$ are just "formal sums" $\sum a_{x} x$.
$M_{X}$ is characterized by the following property: If $\alpha: X \rightarrow M$ is any mapping from the set $X$ to an $R$-module $M$, then $\alpha$ extends uniquely to a homomorphism from $M_{X}$ to $M$. An $R$-module $M$ is said to be free with basis $m_{1}, \ldots, m_{n} \in M$ if for $X$ the set $\left\{m_{1}, \ldots, m_{n}\right\}$ with $n$ elements, the natural homomorphism from $M_{X}$ to $M$ is an isomorphism.
If $R=\mathbb{Z}$, a free $\mathbb{Z}$-module on $X$ is called the free abelian group on $X$.
## Problems
2.54. What does $M$ being free on $m_{1}, \ldots, m_{n}$ say in terms of the elements of $M$ ?
2.55. Let $F=X^{n}+a_{1} X^{n-1}+\cdots+a_{n}$ be a monic polynomial in $R[X]$. Show that $R[X] /(F)$ is a free $R$-module with basis $\overline{1}, \bar{X}, \ldots, \bar{X}^{n-1}$, where $\bar{X}$ is the residue of $X$.
2.56. Show that a subset $X$ of a module $M$ generates $M$ if and only if the homomorphism $M_{X} \rightarrow M$ is onto. Every module is isomorphic to a quotient of a free module.
## Chapter 3
## Local Properties of Plane Curves
### Multiple Points and Tangent Lines
We have seen that affine plane curves correspond to nonconstant polynomials $F \in k[X, Y]$ without multiple factors, where $F$ is determined up to multiplication by a nonzero constant (Chapter 1, Section 6). For some purposes it is useful to allow $F$ to have multiple factors, so we modify our definition slightly:
We say that two polynomials $F, G \in k[X, Y]$ are equivalent if $F=\lambda G$ for some nonzero $\lambda \in k$. We define an affine plane curve to be an equivalence class of nonconstant polynomials under this equivalence relation. We often slur over this equivalence distinction, and say, e.g., "the plane curve $Y^{2}-X^{3}$ ", or even "the plane curve $Y^{2}=X^{3}$ ". The degree of a curve is the degree of a defining polynomial for the curve. A curve of degree one is a line; so we speak of "the line $a X+b Y+c$ ", or "the line $a X+b Y+c=0$ ". If $F=\prod F_{i}^{e_{i}}$, where the $F_{i}$ are the irreducible factors of $F$, we say that the $F_{i}$ are the components of $F$ and $e_{i}$ is the multiplicity of the component $F_{i} . F_{i}$ is a simple component if $e_{i}=1$, and multiple otherwise. Note that the components $F_{i}$ of $F$ can be recovered (up to equivalence) from $V(F)$, but the multiplicities of the components cannot.
If $F$ is irreducible, $V(F)$ is a variety in $\mathbb{A}^{2}$. We will usually write $\Gamma(F), k(F)$, and $\mathscr{O}_{P}(F)$ instead of $\Gamma(V(F)), k(V(F))$, and $\mathscr{O}_{P}(V(F))$.
Let $F$ be a curve, $P=(a, b) \in F$. The point $P$ is called a simple point of $F$ if either derivative $F_{X}(P) \neq 0$ or $F_{Y}(P) \neq 0$. In this case the line $F_{X}(P)(X-a)+F_{Y}(P)(Y-b)=0$ is called the tangent line to $F$ at $P$. A point that isn't simple is called multiple (or singular). A curve with only simple points is called a nonsingular curve.
We sketch some examples. Since $\mathbb{R} \subset \mathbb{C}, \mathbb{A}^{2}(\mathbb{R}) \subset \mathbb{A}^{2}(\mathbb{C})$. If $F$ is a curve in $\mathbb{A}^{2}(\mathbb{C})$, we can only sketch the real part of $F$, i.e. $F \cap \mathbb{A}^{2}(\mathbb{R})$. While the pictures are an aid to the imagination, they should not be relied upon too heavily.
## Examples.
$A=Y-X^{2}$
$$
C=Y^{2}-X^{3}
$$
$$
E=\left(X^{2}+Y^{2}\right)^{2}+3 X^{2} Y-Y^{3}
$$
$$
B=Y^{2}-X^{3}+X
$$
$$
D=Y^{2}-X^{3}-X^{2}
$$
$$
F=\left(X^{2}+Y^{2}\right)^{3}-4 X^{2} Y^{2}
$$
A calculation with derivatives shows that $A$ and $B$ are nonsingular curves, and that $P=(0,0)$ is the only multiple point on $C, D, E$, and $F$. In the first two examples, the linear term of the equation for the curve is just the tangent line to the curve at $(0,0)$. The lowest terms in $C, D, E$, and $F$ respectively are $Y^{2}, Y^{2}-X^{2}=(Y-X)(Y+X)$, $3 X^{2} Y-Y^{3}=Y(\sqrt{3} X-Y)(\sqrt{3} X+Y)$, and $-4 X^{2} Y^{2}$. In each case, the lowest order form picks out those lines that can best be called tangent to the curve at $(0,0)$.
Let $F$ be any curve, $P=(0,0)$. Write $F=F_{m}+F_{m+1}+\cdots+F_{n}$, where $F_{i}$ is a form in $k[X, Y]$ of degree $i, F_{m} \neq 0$. We define $m$ to be the multiplicity of $F$ at $P=(0,0)$, write $m=m_{P}(F)$. Note that $P \in F$ if and only if $m_{P}(F)>0$. Using the rules for derivatives, it is easy to check that $P$ is a simple point on $F$ if and only if $m_{P}(F)=1$, and in this case $F_{1}$ is exactly the tangent line to $F$ at $P$. If $m=2, P$ is called a double point; if $m=3$, a triple point, etc. Since $F_{m}$ is a form in two variables, we can write $F_{m}=\prod L_{i}^{r_{i}}$ where the $L_{i}$ are distinct lines (Corollary in $\$ 2.6$ ). The $L_{i}$ are called the tangent lines to $F$ at $P=(0,0)$; $r_{i}$ is the multiplicity of the tangent. The line $L_{i}$ is a simple (resp. double, etc.) tangent if $r_{i}=1$ (resp. 2, etc.). If $F$ has $m$ distinct (simple) tangents at $P$, we say that $P$ is an ordinary multiple point of $F$. An ordinary double point is called a node. (The curve $D$ has a node at $(0,0), E$ has an ordinary triple point, while $C$ and $F$ have a nonordinary multiple point at $(0,0)$.) For convenience, we call a line through $P$ a tangent of multiplicity zero if it is not tangent to $F$ at $P$.
Let $F=\prod F_{i}^{e_{i}}$ be the factorization of $F$ into irreducible components. Then $m_{P}(F)=$ $\sum e_{i} m_{P}\left(F_{i}\right)$; and if $L$ is a tangent line to $F_{i}$ with multiplicity $r_{i}$, then $L$ is tangent to $F$ with multiplicity $\sum e_{i} r_{i}$. (This is a consequence of the fact that the lowest degree term of $F$ is the product of the lowest degree terms of its factors.)
In particular, a point $P$ is a simple point of $F$ if and only if $P$ belongs to just one component $F_{i}$ of $F, F_{i}$ is a simple component of $F$, and $P$ is a simple point of $F_{i}$.
To extend these definitions to a point $P=(a, b) \neq(0,0)$, Let $T$ be the translation that takes $(0,0)$ to $P$, i.e., $T(x, y)=(x+a, y+b)$. Then $F^{T}=F(X+a, Y+b)$. Define $m_{P}(F)$ to be $m_{(0,0)}\left(F^{T}\right)$, i.e., write $F^{T}=G_{m}+G_{m+1}+\cdots, G_{i}$ forms, $G_{m} \neq 0$, and let $m=m_{P}(F)$. If $G_{m}=\prod L_{i}^{r_{i}}, L_{i}=\alpha_{i} X+\beta_{i} Y$, the lines $\alpha_{i}(X-a)+\beta_{i}(Y-b)$ are defined to be the tangent lines to $F$ at $P$, and $r_{i}$ is the multiplicity of the tangent, etc. Note that $T$ takes the points of $F^{T}$ to the points of $F$, and the tangents to $F^{T}$ at $(0,0)$ to the tangents to $F$ at $P$. Since $F_{X}(P)=F_{X}^{T}(0,0)$ and $F_{Y}(P)=F_{Y}^{T}(0,0), P$ is a simple point on $F$ if and only if $m_{P}(F)=1$, and the two definitions of tangent line to a simple point coincide.
## Problems
3.1. Prove that in the above examples $P=(0,0)$ is the only multiple point on the curves $C, D, E$, and $F$.
3.2. Find the multiple points, and the tangent lines at the multiple points, for each of the following curves:
(a) $Y^{3}-Y^{2}+X^{3}-X^{2}+3 X Y^{2}+3 X^{2} Y+2 X Y$
(b) $X^{4}+Y^{4}-X^{2} Y^{2}$
(c) $X^{3}+Y^{3}-3 X^{2}-3 Y^{2}+3 X Y+1$
(d) $Y^{2}+\left(X^{2}-5\right)\left(4 X^{4}-20 X^{2}+25\right)$
Sketch the part of the curve in (d) that is contained in $\mathbb{A}^{2}(\mathbb{R}) \subset \mathbb{A}^{2}(\mathbb{C})$.
3.3. If a curve $F$ of degree $n$ has a point $P$ of multiplicity $n$, show that $F$ consists of $n$ lines through $P$ (not necessarily distinct).
3.4. Let $P$ be a double point on a curve $F$. Show that $P$ is a node if and only if $F_{X Y}(P)^{2} \neq F_{X X}(P) F_{Y Y}(P)$.
3.5. $(\operatorname{char}(k)=0)$ Show that $m_{P}(F)$ is the smallest integer $m$ such that for some $i+$ $j=m, \frac{\partial^{m} F}{\partial x^{i} \partial y^{j}}(P) \neq 0$. Find an explicit description for the leading form for $F$ at $P$ in terms of these derivatives. 3.6. Irreducible curves with given tangent lines $L_{i}$ of multiplicity $r_{i}$ may be constructed as follows: if $\sum r_{i}=m$, let $F=\prod L_{i}^{r_{i}}+F_{m+1}$, where $F_{m+1}$ is chosen to make $F$ irreducible (see Problem 2.34).
3.7. (a) Show that the real part of the curve $E$ of the examples is the set of points in $\mathbb{A}^{2}(\mathbb{R})$ whose polar coordinates $(r, \theta)$ satisfy the equation $r=-\sin (3 \theta)$. Find the polar equation for the curve $F$. (b) If $n$ is an odd integer $\geq 1$, show that the equation $r=$ $\sin (n \theta)$ defines the real part of a curve of degree $n+1$ with an ordinary $n$-tuple point at $(0,0)$. (Use the fact that $\sin (n \theta)=\operatorname{Im}\left(e^{i n \theta}\right)$ to get the equation; note that rotation by $\pi / n$ is a linear transformation that takes the curve into itself.) (c) Analyze the singularities that arise by looking at $r^{2}=\sin ^{2}(n \theta), n$ even. (d) Show that the curves constructed in (b) and (c) are all irreducible in $\mathbb{A}^{2}(\mathbb{C}$ ). (Hint:: Make the polynomials homogeneous with respect to a variable $Z$, and use $\S 2.1$.)
3.8. Let $T: \mathbb{A}^{2} \rightarrow \mathbb{A}^{2}$ be a polynomial map, $T(Q)=P$. (a) Show that $m_{Q}\left(F^{T}\right) \geq$ $m_{P}(F)$. (b) Let $T=\left(T_{1}, T_{2}\right)$, and define $J_{Q} T=\left(\partial T_{i} / \partial X_{j}(Q)\right)$ to be the Jacobian matrix of $T$ at $Q$. Show that $m_{Q}\left(F^{T}\right)=m_{P}(F)$ if $J_{Q} T$ is invertible. (c) Show that the converse of (b) is false: let $T=\left(X^{2}, Y\right), F=Y-X^{2}, P=Q=(0,0)$.
3.9. Let $F \in k\left[X_{1}, \ldots, X_{n}\right]$ define a hypersurface $V(F) \subset \mathbb{A}^{n}$. Let $P \in \mathbb{A}^{n}$. (a) Define the multiplicity $m_{P}(F)$ of $F$ at $P$. (b) If $m_{P}(F)=1$, define the tangent hyperplane to $F$ at $P$. (c) Examine $F=X^{2}+Y^{2}-Z^{2}, P=(0,0)$. Is it possible to define tangent hyperplanes at multiple points?
3.10. Show that an irreducible plane curve has only a finite number of multiple points. Is this true for hypersurfaces?
3.11. Let $V \subset \mathbb{A}^{n}$ be an affine variety, $P \in V$. The tangent space $T_{P}(V)$ is defined to be $\left\{\left(\nu_{1}, \ldots, v_{n}\right) \in \mathbb{A}^{n} \mid\right.$ for all $\left.G \in I(V), \sum G_{X_{i}}(P) v_{i}=0\right\}$. If $V=V(F)$ is a hypersurface, $F$ irreducible, show that $T_{P}(V)=\left\{\left(\nu_{1}, \ldots, v_{n}\right) \mid \sum F_{X_{i}}(P) \nu_{i}=0\right\}$. How does the dimension of $T_{P}(V)$ relate to the multiplicity of $F$ at $P$ ?
### Multiplicities and Local Rings
Let $F$ be an irreducible plane curve, $P \in F$. In this section we find the multiplicity of $P$ on $F$ in terms of the local ring $\mathscr{O}_{P}(F)$. The following notation will be useful: for any polynomial $G \in k[X, Y]$, denote its image (residue) in $\Gamma(F)=k[X, Y] /(F)$ by $g$.
Theorem 1. $P$ is a simple point of $F$ if and only if $\mathscr{O}_{P}(F)$ is a discrete valuation ring. In this case, if $L=a X+b Y+c$ is any line through $P$ that is not tangent to $F$ at $P$, then the image $l$ of $L$ in $\mathscr{O}_{P}(F)$ is a uniformizing parameter for $\mathscr{O}_{P}(F)$.
Proof. Suppose $P$ is a simple point on $F$, and $L$ is a line through $P$, not tangent to $F$ at $P$. By making an affine change of coordinates, we may assume that $P=(0,0)$, that $Y$ is the tangent line, and that $L=X$ (Problems 2.15(d) and 2.22). By Proposition 4 of $\S 2.5$, it suffices to show that $\mathfrak{m}_{P}(F)$ is generated by $x$.
First note that $\mathfrak{m}_{P}(F)=(x, y)$, whether $P$ is simple or not (Problems 2.43, 2.44).
Now with the above assumptions, $F=Y+$ higher terms. Grouping together those terms with $Y$, we can write $F=Y G-X^{2} H$, where $G=1+$ higher terms, $H \in k[X]$. Then $y g=x^{2} h \in \Gamma(F)$, so $y=x^{2} h g^{-1} \in(x)$, since $g(P) \neq 0$. Thus $m_{P}(F)=(x, y)=(x)$, as desired.
The converse will follow from Theorem 2.
Suppose $P$ is a simple point on an irreducible curve $F$. We let $\operatorname{ord}_{P}^{F}$ be the order function on $k(F)$ defined by the DVR $\mathscr{O}_{P}(F)$; when $F$ is fixed, we may write simply $\operatorname{ord}_{P}$. If $G \in k[X, Y]$, and $g$ is the image of $G$ in $\Gamma(F)$, we write $\operatorname{ord}_{P}^{F}(G)$ instead of $\operatorname{ord}_{P}^{F}(g)$.
If $P$ is a simple point on a reducible curve $F$, we write $\operatorname{ord}_{P}^{F} \operatorname{instead}$ of $\operatorname{ord}_{P}^{F_{i}}$, where $F_{i}$ is the component of $F$ containing $P$.
Suppose $P$ is a simple point on $F$, and $L$ is any line through $P$. $\operatorname{Then}^{\operatorname{ord}_{P}^{F}}(L)=1$ if $L$ is not tangent to $F$ at $P$, and $\operatorname{ord}_{P}^{F}(L)>1$ if $L$ is tangent to $F$ at $P$. For we may assume the conditions are as in the proof of Theorem $1 ; Y$ is the tangent, $y=x^{2} h g^{-1}$, so $\operatorname{ord}_{P}(y)=\operatorname{ord}_{P}\left(x^{2}\right)+\operatorname{ord}_{P}\left(h g^{-1}\right) \geq 2$.
The proof of the next theorem introduces a technique that will reappear at several places in our study of curves. It allows us to calculate the dimensions of certain vector spaces of the type $\mathscr{O}_{P}(V) / I$, where $I$ is an ideal in $\mathscr{O}_{P}(V)$.
Theorem 2. Let $P$ be a point on an irreducible curve $F$. Then for all sufficiently large $n$,
$$
m_{P}(F)=\operatorname{dim}_{k}\left(\mathfrak{m}_{P}(F)^{n} / \mathfrak{m}_{P}(F)^{n+1}\right) .
$$
In particular, the multiplicity of $F$ at $P$ depends only on the local ring $\mathscr{O}_{P}(F)$.
Proof. Write $\mathscr{O}, \mathfrak{m}$ for $\mathscr{O}_{P}(F), \mathfrak{m}_{P}(F)$ respectively. From the exact sequence
$$
0 \longrightarrow \mathfrak{m}^{n} / \mathfrak{m}^{n+1} \longrightarrow \mathscr{O} / \mathfrak{m}^{n+1} \longrightarrow \mathscr{O} / \mathfrak{m}^{n} \longrightarrow 0
$$
it follows that it is enough to prove that $\operatorname{dim}_{k}\left(\mathscr{O} / \mathfrak{m}^{n}\right)=n m_{P}(F)+s$, for some constant $s$, and all $n \geq m_{P}(F)$ (Problem 2.49(e) and Proposition 7 of $\$ 2.10$ ). We may assume that $P=(0,0)$, so $\mathfrak{m}^{n}=I^{n} \mathscr{O}$, where $I=(X, Y) \subset k[X, Y]$ (Problem 2.43). Since $V\left(I^{n}\right)=$ $\{P\}, k[X, Y] /\left(I^{n}, F\right) \cong \mathscr{O}_{P}\left(\mathbb{A}^{2}\right) /\left(I^{n}, F\right) \mathscr{O}_{P}\left(\mathbb{A}^{2}\right) \cong \mathscr{O}_{P}(F) / I^{n} \mathscr{O}_{P}(F)=\mathscr{O} / \mathfrak{m}^{n}$ (Corollary 2 of $\S 2.9$ and Problem 2.44).
So we are reduced to calculating the dimension of $k[X, Y] /\left(I^{n}, F\right)$. Let $m=m_{P}(F)$. Then $F G \in I^{n}$ whenever $G \in I^{n-m}$. There is a natural ring homomorphism $\varphi$ from $k[X, Y] / I^{n}$ to $k[X, Y] /\left(I^{n}, F\right)$, and a $k$-linear map $\psi$ from $k[X, Y] / I^{n-m}$ to $k[X, Y] / I^{n}$ defined by $\psi(\bar{G})=\overline{F G}$ (where the bars denote residues). It is easy to verify that the sequence
$$
0 \longrightarrow k[X, Y] / I^{n-m} \stackrel{\psi}{\longrightarrow} k[X, Y] / I^{n} \stackrel{\varphi}{\longrightarrow} k[X, Y] /\left(I^{n}, F\right) \longrightarrow 0
$$
is exact. Applying Problem 2.46 and Proposition 7 of $\$ 2.10$ again, we see that
$$
\operatorname{dim}_{k}\left(k[X, Y] /\left(I^{n}, F\right)\right)=n m-\frac{m(m-1)}{2}
$$
for all $n \geq m$, as desired. Note that if $\mathscr{O}_{P}(F)$ is a DVR, then Theorem 2 implies that $m_{P}(F)=1$ (Problem 2.50) so $P$ is simple. This completes the proof of Theorem 1.
It should at least be remarked that the function $\chi(n)=\operatorname{dim}_{k}\left(\mathscr{O} / \mathfrak{m}^{n}\right)$, which is a polynomial in $n$ (for large $n$ ) is called the Hilbert-Samuel polynomial of the local $\operatorname{ring} \mathscr{O}$; it plays an important role in the modern study of the multiplicities of local rings.
## Problems
3.12. A simple point $P$ on a curve $F$ is called a flex if $\operatorname{ord}_{P}^{F}(L) \geq 3$, where $L$ is the tangent to $F$ at $P$. The flex is called ordinary if $\operatorname{ord}_{P}(L)=3$, a higher flex otherwise. (a) Let $F=Y-X^{n}$. For which $n$ does $F$ have a flex at $P=(0,0)$, and what kind of flex? (b) Suppose $P=(0,0), L=Y$ is the tangent line, $F=Y+a X^{2}+\cdots$. Show that $P$ is a flex on $F$ if and only if $a=0$. Give a simple criterion for calculating $\operatorname{ord}_{P}^{F}(Y)$, and therefore for determining if $P$ is a higher flex.
3.13. With the notation of Theorem 2 , and $\mathfrak{m}=\mathfrak{m}_{P}(F)$, show that $\operatorname{dim}_{k}\left(\mathfrak{m}^{n} / \mathfrak{m}^{n+1}\right)=$ $n+1$ for $0 \leq n<m_{P}(F)$. In particular, $P$ is a simple point if and only if $\operatorname{dim}_{k}\left(\mathfrak{m} / \mathfrak{m}^{2}\right)=$ 1 ; otherwise $\operatorname{dim}_{k}\left(\mathfrak{m} / \mathfrak{m}^{2}\right)=2$.
3.14. Let $V=V\left(X^{2}-Y^{3}, Y^{2}-Z^{3}\right) \subset \mathbb{A}^{3}, P=(0,0,0), \mathfrak{m}=\mathfrak{m}_{P}(V)$. Find $\operatorname{dim}_{k}\left(\mathfrak{m} / \mathfrak{m}^{2}\right)$. (See Problem 1.40.)
3.15. (a) Let $\mathscr{O}=\mathscr{O}_{P}\left(\mathbb{A}^{2}\right)$ for some $P \in \mathbb{A}^{2}, \mathfrak{m}=\mathfrak{m}_{P}\left(\mathbb{A}^{2}\right)$. Calculate $\chi(n)=\operatorname{dim}_{k}\left(\mathscr{O} / \mathfrak{m}^{n}\right)$. (b) Let $\mathscr{O}=\mathscr{O}_{P}\left(\mathbb{A}^{r}(k)\right)$. Show that $\chi(n)$ is a polynomial of degree $r$ in $n$, with leading coefficient $1 / r$ ! (see Problem 2.36).
3.16. Let $F \in k\left[X_{1}, \ldots, X_{n}\right]$ define a hypersurface in $\mathbb{A}^{r}$. Write $F=F_{m}+F_{m+1}+\cdots$, and let $m=m_{P}(F)$ where $P=(0,0)$. Suppose $F$ is irreducible, and let $\mathscr{O}=\mathscr{O}_{P}(V(F))$, $\mathfrak{m}$ its maximal ideal. Show that $\chi(n)=\operatorname{dim}_{k}\left(\mathscr{O} / \mathfrak{m}^{n}\right)$ is a polynomial of degree $r-1$ for sufficiently large $n$, and that the leading coefficient of $\chi$ is $m_{P}(F) /(r-1)$ !.
Can you find a definition for the multiplicity of a local ring that makes sense in all the cases you know?
### Intersection Numbers
Let $F$ and $G$ be plane curves, $P \in \mathbb{A}^{2}$. We want to define the intersection number of $F$ and $G$ at $P$; it will be denoted by $I(P, F \cap G)$. Since the definition is rather unintuitive, we shall first list seven properties we want this intersection number to have. We then prove that there is only one possible definition, and at the same time we find a simple procedure for calculating $I(P, F \cap G)$ in a reasonable number of steps.
We say that $F$ and $G$ intersect properly at $P$ if $F$ and $G$ have no common component that passes through $P$. Our first requirements are:
(1) $I(P, F \cap G)$ is a nonnegative integer for any $F, G$, and $P$ such that $F$ and $G$ intersect properly at $P . I(P, F \cap G)=\infty$ if $F$ and $G$ do not intersect properly at $P$. (2) $I(P, F \cap G)=0$ if and only if $P \notin F \cap G . I(P, F \cap G)$ depends only on the components of $F$ and $G$ that pass through $P$. And $I(P, F \cap G)=0$ if $F$ or $G$ is a nonzero constant.
(3) If $T$ is an affine change of coordinates on $A^{2}$, and $T(Q)=P$, then $I(P, F \cap G)=$ $I\left(Q, F^{T} \cap G^{T}\right)$.
(4) $I(P, F \cap G)=I(P, G \cap F)$.
Two curves $F$ and $G$ are said to intersect transversally at $P$ if $P$ is a simple point both on $F$ and on $G$, and if the tangent line to $F$ at $P$ is different from the tangent line to $G$ at $P$. We want the intersection number to be one exactly when $F$ and $G$ meet transversally at $P$. More generally, we require
(5) $I(P, F \cap G) \geq m_{P}(F) m_{P}(G)$, with equality occurring if and only if $F$ and $G$ have not tangent lines in common at $P$.
The intersection numbers should add when we take unions of curves:
(6) If $F=\prod F_{i}^{r_{i}}$, and $G=\prod G_{j}^{s_{j}}$, then $I(P, F \cap G)=\sum_{i, j} r_{i} s_{j} I\left(P, F_{i} \cap G_{j}\right)$.
The last requirement is probably the least intuitive. If $F$ is irreducible, it says that $I(P, F \cap G)$ should depend only on the image of $G$ in $\Gamma(F)$. Or, for arbitrary $F$,
(7) $I(P, F \cap G)=I(P, F \cap(G+A F))$ for any $A \in k[X, Y]$.
Theorem 3. There is a unique intersection number $I(P, F \cap G)$ defined for all plane curves $F, G$, and all points $P \in \mathbb{A}^{2}$, satisfying properties (1)-(7). It is given by the formula
$$
I(P, F \cap G)=\operatorname{dim}_{k}\left(\mathscr{O}_{P}\left(\mathbb{A}^{2}\right) /(F, G)\right)
$$
Proof of Uniqueness. Assume we have a number $I(P, F \cap G)$ defined for all $F, G$, and $P$, satisfying (1)-(7). We will give a constructive procedure for calculating $I(P, F \cap G)$ using only these seven properties, that is stronger than the required uniqueness. We may suppose $P=(0,0)$ (by (3)), and that $I(P, F \cap G)$ is finite (by (1)). The case when $I(P, F \cap G)=0$ is taken care of by (2), so we may proceed by induction; assume $I(P, F \cap G)=n>0$, and $I(P, A \cap B)$ can be calculated whenever $I(P, A \cap B)<n$. Let $F(X, 0), G(X, 0) \in k[X]$ be of degree $r, s$ respectively, where $r$ or $s$ is taken to be zero if the polynomial vanishes. We may suppose $r \leq s$ (by (4)).
Case 1: $r=0$. Then $Y$ divides $F$, so $F=Y H$, and
$$
I(P, F \cap G)=I(P, Y \cap G)+I(P, H \cap G)
$$
(by (6)). If $G(X, 0)=X^{m}\left(a_{0}+a_{1} X+\cdots\right), a_{0} \neq 0$, then $I(P, Y \cap G)=I(P, Y \cap G(X, 0))=$ $I\left(P, Y \cap X^{m}\right)=m$ (by (7), (2), (6), and (5)). Since $P \in G, m>0$, so $I(P, H \cap G)<n$, and we are done by induction.
Case 2: $r>0$. We may multiply $F$ and $G$ by constants to make $F(X, 0)$ and $G(X, 0)$ monic. Let $H=G-X^{s-r} F$. Then $I(P, F \cap G)=I(P, F \cap H)($ by $(7))$, and $\operatorname{deg}(H(X, 0))=$ $t<s$. Repeating this process (interchanging the order of $F$ and $H$ if $t<r$ ) a finite number of times we eventually reach a pair of curves $A, B$ that fall under Case 1 , and with $I(P, F \cap G)=I(P, A \cap B)$. This concludes the proof.
Proof of Existence. Define $I(P, F \cap G)$ to be $\operatorname{dim}_{k}\left(\mathscr{O}_{P}\left(\mathbb{A}^{2}\right) /(F, G)\right)$. We must show that properties (1)-(7) are satisfied. Since $I(P, F \cap G)$ depends only on the ideal in $\mathscr{O}_{P}\left(\mathbb{A}^{2}\right)$ generated by $F$ and $G$, properties (2), (4), and (7) are obvious. Since an affine change of coordinates gives an isomorphism of local rings (Problem 2.22), (3) is also clear. We may thus assume that $P=(0,0)$, and that all the components of $F$ and $G$ pass through $P$. Let $\mathscr{O}=\mathscr{O}_{P}\left(\mathbb{A}^{2}\right)$.
If $F$ and $G$ have no common components, $I(P, F \cap G)$ is finite by Corollary 1 of $\$ 2.9$. If $F$ and $G$ have a common component $H$, then $(F, G) \subset(H)$, so there is a homomorphism from $\mathscr{O} /(F, G)$ onto $\mathscr{O} /(H)$ (Problem 2.42), and $I(P, F \cap G) \geq \operatorname{dim}_{k}(\mathscr{O} /(H)$ ). But $\mathscr{O} /(H)$ is isomorphic to $\mathscr{O}_{P}(H)$ (Problem 2.44), and $\mathscr{O}_{P}(H) \supset \Gamma(H)$, with $\Gamma(H)$ infinite-dimensional by Corollary 4 to the Nullstellensatz. This proves (1).
To prove (6), it is enough to show that $I(P, F \cap G H)=I(P, F \cap G)+I(P, F \cap H)$ for any $F, G, H$. We may assume $F$ and $G H$ have no common components, since the result is clear otherwise. Let $\varphi: \mathscr{O} /(F, G H) \rightarrow \mathscr{O} /(F, G)$ be the natural homomorphism (Problem 2.42), and define a $k$-linear map $\psi: \mathscr{O} /(F, H) \rightarrow \mathscr{O} /(F, G H)$ by letting $\psi(\bar{z})=$ $\overline{G z}, z \in \mathscr{O}$ (the bar denotes residues). By Proposition 7 of $\$ 2.10$, it is enough to show that the sequence
$$
0 \longrightarrow \mathscr{O} /(F, H) \stackrel{\psi}{\longrightarrow} \mathscr{O} /(F, G H) \stackrel{\varphi}{\longrightarrow} \mathscr{O} /(F, G) \longrightarrow 0
$$
is exact.
We will verify that $\psi$ is one-to-one; the rest (which is easier) is left to the reader. If $\psi(\bar{z})=0$, then $G z=u F+v G H$ where $u, v \in \mathscr{O}$. Choose $S \in k[X, Y]$ with $S(P) \neq 0$, and $S u=A, S v=B$, and $S z=C \in k[X, Y]$. Then $G(C-B H)=A F$ in $k[X, Y]$. Since $F$ and $G$ have no common factors, $F$ must divide $C-B H$, so $C-B H=D F$. Then $z=(B / S) H+(D / S) F$, or $\bar{z}=0$, as claimed.
Property (5) is the hardest. Let $m=m_{P}(F), n=m_{P}(G)$. Let $I$ be the ideal in $k[X, Y]$ generated by $X$ and $Y$. Consider the following diagram of vector spaces and linear maps:
where $\varphi, \pi$, and $\alpha$ are the natural ring homomorphisms, and $\psi$ is defined by letting $\psi(\bar{A}, \bar{B})=\overline{A F+B G}$.
Now $\varphi$ and $\pi$ are clearly surjective, and, since $V\left(I^{m+n}, F, G\right) \subset\{P\}, \alpha$ is an isomorphism by Corollary 2 in $\$ 2.9$. It is easy to check that the top row is exact. It follows that
$$
\operatorname{dim}\left(k[X, Y] / I^{n}\right)+\operatorname{dim}\left(k[X, Y] / I^{m}\right) \geq \operatorname{dim}(\operatorname{Ker}(\varphi))
$$
with equality if and only if $\psi$ is one-to-one, and that
$$
\operatorname{dim}\left(k[X, Y] /\left(I^{m+n}, F, G\right)\right)=\operatorname{dim}\left(k[X, Y] / I^{m+n}\right)-\operatorname{dim}(\operatorname{Ker}(\varphi))
$$
Putting all this together, we get the following string of inequalities:
$$
\begin{aligned}
I(P, F \cap G) & =\operatorname{dim}(\mathscr{O} /(F, G)) \geq \operatorname{dim}\left(\mathscr{O} /\left(I^{m+n}, F, G\right)\right) \\
& =\operatorname{dim}\left(k[X, Y] /\left(I^{m+n}, F, G\right)\right) \\
& \geq \operatorname{dim}\left(k[X, Y] / I^{m+n}\right)-\operatorname{dim}\left(k[X, Y] / I^{n}\right)-\operatorname{dim}\left(k[X, Y] / I^{m}\right) \\
& =m n
\end{aligned}
$$
(by Problem 2.46 and arithmetic).
This shows that $I(P, F \cap G) \geq m n$, and that $I(P, F \cap G)=m n$ if and only if both inequalities in the above string are equalities. The first such inequality is an equality if $\pi$ is an isomorphism, i.e., if $I^{m+n} \subset(F, G) \mathscr{O}$. The second is an equality if and only if $\psi$ is one-to-one. Property (5) is therefore a consequence of:
Lemma. (a) If $F$ and $G$ have no common tangents at $P$, then $I^{t} \subset(F, G) \mathscr{O}$ for $t \geq m+n-1$.
(b) $\psi$ is one-to-one if and only if $F$ and $G$ have distinct tangents at $P$.
Proof of (a). Let $L_{1}, \ldots, L_{m}$ be the tangents to $F$ at $P, M_{1}, \ldots, M_{n}$ the tangents to $G$. Let $L_{i}=L_{m}$ if $i>m, M_{j}=M_{n}$ if $j>n$, and let $A_{i j}=L_{1} \cdots L_{i} M_{1} \cdots M_{j}$ for all $i, j \geq 0$ $\left(A_{00}=1\right)$. The set $\left\{A_{i j} \mid i+j=t\right\}$ forms a basis for the vector space of all forms of degree $t$ in $k[X, Y]$ (Problem 2.35(c)).
To prove (a), it therefore suffices to show that $A_{i j} \in(F, G) \mathscr{O}$ for all $i+j \geq m+n-1$. But $i+j \geq m+n-1$ implies that either $i \geq m$ or $j \geq n$. Say $i \geq m$, so $A_{i j}=A_{m 0} B$, where $B$ is a form of degree $t=i+j-m$. Write $F=A_{m 0}+F^{\prime}$, where all terms of $F^{\prime}$ are of degree $\geq m+1$. Then $A_{i j}=B F-B F^{\prime}$, where each term of $B F^{\prime}$ has degree $\geq i+j+1$. We will be finished, then, if we can show that $I^{t} \subset(F, G) \mathscr{O}$ for all sufficiently large $t$.
This fact is surely a consequence of the Nullstellensatz: let $V(F, G)=\left\{P, Q_{1}, \ldots, Q_{s}\right\}$, and choose a polynomial $H$ so that $H\left(Q_{i}\right)=0, H(P) \neq 0$ (Problem 1.17). Then $H X$ and $H Y$ are in $I(V(F, G))$, so $(H X)^{N},(H Y)^{N} \in(F, G) \subset k[X, Y]$ for some $N$. Since $H^{N}$ is a unit in $\mathscr{O}, X^{N}$ and $Y^{N}$ are in $(F, G) \mathscr{O}$, and therefore $I^{2 N} \subset(F, G) \mathscr{O}$, as desired.
Proof of (b). Suppose the tangents are distinct, and that
$$
\psi(\bar{A}, \bar{B})=\overline{A F+B G}=0,
$$
i.e., $A F+B G$ consists entirely of terms of degree $\geq m+n$. Suppose $r<m$ or $s<n$. Write $A=A_{r}+$ higher terms, $B=B_{s}+\cdots$, so $A F+B G=A_{r} F_{m}+B_{s} G_{n}+\cdots$. Then we must have $r+m=s+n$ and $A_{r} F_{m}=-B_{s} G_{n}$. But $F_{m}$ and $G_{n}$ have no common factors, so $F_{m}$ divides $B_{s}$, and $G_{n}$ divides $A_{r}$. Therefore $s \geq m, r \geq n$, so $(\bar{A}, \bar{B})=(0,0)$.
Conversely, if $L$ were a common tangent to $F$ and $G$ at $P$, write $F_{m}=L F_{m-1}^{\prime}$ and $G_{n}=L G_{n-1}^{\prime}$. Then $\psi\left(\overline{G_{n-1}^{\prime}},-\overline{F_{m-1}^{\prime}}\right)=0$, so $\psi$ is not one-to-one. This completes the proof of the lemma, and also of Theorem 3.
Two things should be noticed about the uniqueness part of the above proof. First, it shows that, as axioms, Properties (1)-(7) are exceedingly redundant; for example, the only part of Property (5) that is needed is that $I((0,0), X \cap Y)=1$. (The reader might try to find a minimal set of axioms that characterizes the intersection number.) Second, the proof shows that the calculation of intersection numbers is a very easy matter. Making imaginative use of (5) and (7) can save much time, but the proof shows that nothing more is needed than some arithmetic with polynomials.
Example. Let us calculate $I(P, E \cap F)$, where $E=\left(X^{2}+Y^{2}\right)^{2}+3 X^{2} Y-Y^{3}, F=\left(X^{2}+\right.$ $\left.Y^{2}\right)^{3}-4 X^{2} Y^{2}$, and $P=(0,0)$, as in the examples of Section 1 . We can get rid of the worst part of $F$ by replacing $F$ by $F-\left(X^{2}+Y^{2}\right) E=Y\left(\left(X^{2}+Y^{2}\right)\left(Y^{2}-3 X^{2}\right)-4 X^{2} Y\right)=$ $Y G$. Since no obvious method is available to find $I(P, E \cap G)$, we apply the process of the uniqueness proof to get rid of the $X$-terms: Replace $G$ by $G+3 E$, which is $Y\left(5 X^{2}-3 Y^{2}+4 Y^{3}+4 X^{2} Y\right)=Y H$. Then $I(P, E \cap F)=2 I(P, E \cap Y)+I(P, E \cap H)$. But $I(P, E \cap Y)=I\left(P, X^{4} \cap Y\right)=4$ (by (7), (6)), and $I(P, E \cap H)=m_{P}(E) m_{P}(H)=6$ (by (5)). So $I(P, E \cap F)=14$.
Two more properties of the intersection number will be useful later; the first of these can also be used to simplify calculations.
(8) If $P$ is a simple point on $F$, then $I(P, F \cap G)=\operatorname{ord}_{P}^{F}(G)$.
Proof. We may assume $F$ is irreducible. If $g$ is the image of $G \operatorname{in} \mathscr{O}_{P}(F)$, then $\operatorname{ord}_{P}^{F}(G)=$ $\operatorname{dim}_{k}\left(\mathscr{O}_{P}(F) /(g)\right)$ (Problem 2.50(c)). Since $\mathscr{O}_{P}(F) /(g)$ is isomorphic to $\mathscr{O}_{P}\left(\mathbb{A}^{2}\right) /(F, G)$ (Problem 2.44), this dimension is $I(P, F \cap G)$.
(9) If $F$ and $G$ have no common components, then
$$
\sum_{P} I(P, F \cap G)=\operatorname{dim}_{k}(k[X, Y] /(F, G)) .
$$
Proof. This is a consequence of Corollary 1 in $\$ 2.9$.
## Problems
3.17. Find the intersection numbers of various pairs of curves from the examplse of Section 1, at the point $P=(0,0)$.
3.18. Give a proof of Property (8) that uses only Properties (1)-(7).
3.19. A line $L$ is tangent to a curve $F$ at a point $P$ if and only if $I(P, F \cap L)>m_{P}(F)$.
3.20. If $P$ is a simple point on $F$, then $I(P, F \cap(G+H)) \geq \min (I(P, F \cap G), I(P, F \cap H))$. Give an example to show that this may be false if $P$ is not simple on $F$.
3.21. Let $F$ be an affine plane curve. Let $L$ be a line that is not a component of $F$. Suppose $L=\{(a+t b, c+t d) \mid t \in k\}$. Define $G(T)=F(a+T b, c+T d)$. Factor $G(T)=$ $\epsilon \prod\left(T-\lambda_{i}\right)^{e_{i}}, \lambda_{i}$ distinct. Show that there is a natural one-to-one correspondence between the $\lambda_{i}$ and the points $P_{i} \in L \cap F$. Show that under this correspondence, $I\left(P_{i}, L \cap F\right)=e_{i}$. In particular, $\sum I(P, L \cap F) \leq \operatorname{deg}(F)$.
3.22. Suppose $P$ is a double point on a curve $F$, and suppose $F$ has only one tangent $L$ at $P$. (a) Show that $I(P, F \cap L) \geq 3$. The curve $F$ is said to have an (ordinary) cusp at $P$ if $I(P, F \cap L)=3$. (b) Suppose $P=(0,0)$, and $L=Y$. Show that $P$ is a cusp if and only if $F_{X X X}(P) \neq 0$. Give some examples. (c) Show that if $P$ is a cusp on $F$, then $F$ has only one component passing through $P$. 3.23. A point $P$ on a curve $F$ is called a hypercusp if $m_{P}(F)>1, F$ has only one tangent line $L$ at $P$, and $I(P, L \cap F)=m_{P}(F)+1$. Generalize the results of the preceding problem to this case.
3.24* The object of this problem is to find a property of the local ring $\mathscr{O}_{P}(F)$ that determines whether or not $P$ is an ordinary multiple point on $F$.
Let $F$ be an irreducible plane curve, $P=(0,0), m=m_{P}(F)>1$. Let $\mathfrak{m}=\mathfrak{m}_{P}(F)$. For $G \in k[X, Y]$, denote its residue in $\Gamma(F)$ by $g$; and for $g \in \mathfrak{m}$, denote its residue in $\mathfrak{m} / \mathfrak{m}^{2}$ by $\bar{g}$. (a) Show that the map from \{forms of degree 1 in $k[X, Y]\}$ to $\mathfrak{m} / \mathfrak{m}^{2}$ taking $a X+$ $b Y$ to $\overline{a x+b y}$ is an isomorphism of vector spaces (see Problem 3.13). (b) Suppose $P$ is an ordinary multiple point, with tangents $L_{1}, \ldots, L_{m}$. Show that $I\left(P, F \cap L_{i}\right)>m$ and $\overline{l_{i}} \neq \lambda \overline{l_{j}}$ for all $i \neq j$, all $\lambda \in k$. (c) Suppose there are $G_{1}, \ldots, G_{m} \in k[X, Y]$ such that $I\left(P, F \cap G_{i}\right)>m$ and $\bar{g}_{i} \neq \lambda \bar{g}_{j}$ for all $i \neq j$, and all $\lambda \in k$. Show that $P$ is an ordinary multiple point on $F$. (Hint:: Write $G_{i}=L_{i}+$ higher terms. $\bar{l}_{i}=\bar{g}_{i} \neq 0$, and $L_{i}$ is the tangent to $G_{i}$, so $L_{i}$ is tangent to $F$ by Property (5) of intersection numbers. Thus $F$ has $m$ tangents at $P$.) (d) Show that $P$ is an ordinary multiple point on $F$ if and only if there are $g_{1}, \ldots, g_{m} \in \mathfrak{m}$ such that $\bar{g}_{i} \neq \lambda \bar{g}_{j}$ for all $i \neq j, \lambda \in k$, and $\operatorname{dim} \mathscr{O}_{P}(F) /\left(g_{i}\right)>m$.
## Chapter 4
## Projective Varieties
### Projective Space
Suppose we want to study all the points of intersection of two curves; consider for example the curve $Y^{2}=X^{2}+1$ and the line $Y=\alpha X, \alpha \in k$. If $\alpha \neq \pm 1$, they intersect in two points. When $\alpha= \pm 1$, they no not intersect, but the curve is asymptotic to the line. We want to enlarge the plane in such a way that two such curves intersect "at infinity".
One way to achieve this is to identify each point $(x, y) \in \mathbb{A}^{2}$ with the point $(x, y, 1) \in$ $\mathbb{A}^{3}$. Every point $(x, y, 1)$ determines a line in $\mathbb{A}^{3}$ that passes through $(0,0,0)$ and $(x, y, 1)$. Every line through $(0,0,0)$ except those lying in the plane $z=0$ corresponds to exactly one such point. The lines through $(0,0,0)$ in the plane $z=0$ can be thought of as corresponding to the "points at infinity". This leads to the following definition:
Let $k$ be any field. Projective $n$-space over $k$, written $\mathbb{P}^{n}(k)$, or simply $\mathbb{P}^{n}$, is defined to be the set of all lines through $(0,0, \ldots, 0)$ in $\mathbb{A}^{n+1}(k)$. Any point $(x)=$ $\left(x_{1}, \ldots, x_{n+1}\right) \neq(0,0, \ldots, 0)$ determines a unique such line, namely $\left\{\left(\lambda x_{1}, \ldots, \lambda x_{n+1}\right) \mid\right.$ $\lambda \in k\}$. Two such points $(x)$ and $(y)$ determine the same line if and only if there is a nonzero $\lambda \in k$ such that $y_{i}=\lambda x_{i}$ for $i=1, \ldots, n+1$; let us say that $(x)$ and $(y)$ are equivalent if this is the case. Then $\mathbb{P}^{n}$ may be identified with the set of equivalence classes of points in $\mathbb{A}^{n+1} \backslash\{(0, \ldots, 0)\}$.
Elements of $\mathbb{P}^{n}$ will be called points. If a point $P \in \mathbb{P}^{n}$ is determined as above by some $\left(x_{1}, \ldots, x_{n+1}\right) \in \mathbb{A}^{n+1}$, we say that $\left(x_{1}, \ldots, x_{n+1}\right)$ are homogeneous coordinates for $P$. We often write $P=\left[x_{1}: \ldots: x_{n+1}\right]$ to indicate that $\left(x_{1}, \ldots, x_{n+1}\right)$ are homogeneous coordinates for $P$. Note that the $i$ th coordinate $x_{i}$ is not well-defined, but that it is a well-defined notion to say whether the $i$ th coordinate is zero or nonzero; and if $x_{i} \neq$ 0 , the ratios $x_{j} / x_{i}$ are well-defined (since they are unchanged under equivalence).
We let $U_{i}=\left\{\left[x_{1}: \ldots: x_{n+1}\right] \in \mathbb{P}^{n} \mid x_{i} \neq 0\right\}$. Each $P \in U_{i}$ can be written uniquely in the form
$$
P=\left[x_{1}: \ldots: x_{i-1}: 1: x_{i+1}: \ldots: x_{n+1}\right] .
$$
The coordinates $\left(x_{1}, \ldots, x_{i-1}, x_{i+1}, \ldots, x_{n+1}\right)$ are called the nonhomogeneous coordinates for $P$ with respect to $U_{i}$ (or $X_{i}$, or $i$ ). If we define $\varphi_{i}: \mathbb{A}^{n} \rightarrow U_{i}$ by $\varphi_{i}\left(a_{1}, \ldots, a_{n}\right)=$ $\left[a_{1}: \ldots: a_{i-1}: 1: a_{i}: \ldots: a_{n}\right]$, then $\varphi_{i}$ sets up a one-to-one correspondence between the points of $\mathbb{A}^{n}$ and the points of $U_{i}$. Note that $\mathbb{P}^{n}=\bigcup_{i=1}^{n+1} U_{i}$, so $\mathbb{P}^{n}$ is covered by $n+1$ sets each of which looks just like affine $n$-space.
For convenience we usually concentrate on $U_{n+1}$. Let
$$
H_{\infty}=\mathbb{P}^{n} \backslash U_{n+1}=\left\{\left[x_{1}: \ldots: x_{n+1}\right] \mid x_{n+1}=0\right\}
$$
$H_{\infty}$ is often called the hyperplane at infinity. The correspondence $\left[x_{1}: \ldots: x_{n}: 0\right] \leftrightarrow$ $\left[x_{1}: \ldots: x_{n}\right]$ shows that $H_{\infty}$ may be identified with $\mathbb{P}^{n-1}$. Thus $\mathbb{P}^{n}=U_{n+1} \cup H_{\infty}$ is the union of an affine $n$-space and a set that gives all directions in affine $n$-space.
Examples. $\quad(0) \mathbb{P}^{0}(k)$ is a point.
(1) $\mathbb{P}^{1}(k)=\{[x: 1] \mid x \in k\} \cup\{[1: 0]\} . \mathbb{P}^{1}(k)$ is the affine line plus one point at infinity. $\mathbb{P}^{1}(k)$ is the projective line over $k$.
(2) $\mathbb{P}^{2}(k)=\left\{[x: y: 1] \mid(x, y) \in \mathbb{A}^{2}\right\} \cup\left\{[x: y: 0] \mid[x: y] \in \mathbb{P}^{1}\right\}$. Here $H_{\infty}$ is called the line at infinity. $\mathbb{P}^{2}(k)$ is called the projective plane over $k$.
(3) Consider a line $Y=m X+b$ in $\mathbb{A}^{2}$. If we identify $\mathbb{A}^{2}$ with $U_{3} \subset \mathbb{P}^{2}$, the points on the line correspond to the points $[x: y: z] \in \mathbb{P}^{2}$ with $y=m x+b z$ and $z \neq 0$. (We must make the equation homogeneous so that solutions will be invariant under equivalence). The set $\left\{[x: y: z] \in \mathbb{P}^{2} \mid y=m x+b z\right\} \cap H_{\infty}=\{[1: m: 0]\}$. So all lines with the same slope, when extended in this way, pass through the same point at infinity.
(4) Consider again the curve $Y^{2}=X^{2}+1$. The corresponding set in $\mathbb{P}^{2}$ is given by the homogeneous equation $Y^{2}=X^{2}+Z^{2}, Z \neq 0 .\left\{[x: y: z] \in \mathbb{P}^{2} \mid y^{2}=x^{2}+z^{2}\right\}$ intersects $H_{\infty}$ in the two points $[1: 1: 0]$ and $[1:-1: 0]$. These are the points where the lines $Y=X$ and $Y=-X$ intersect the curve.
## Problems
4.1. What points in $\mathbb{P}^{2}$ do not belong to two of the three sets $U_{1}, U_{2}, U_{3}$ ?
4.2. Let $F \in k\left[X_{1}, \ldots, X_{n+1}\right]$ ( $k$ infinite). Write $F=\sum F_{i}, F_{i}$ a form of degree $i$. Let $P \in \mathbb{P}^{n}(k)$, and suppose $F\left(x_{1}, \ldots, x_{n+1}\right)=0$ for every choice of homogeneous coordinates $\left(x_{1}, \ldots, x_{n+1}\right)$ for $P$. Show that each $F_{i}\left(x_{1}, \ldots, x_{n+1}\right)=0$ for all homogeneous coordinates for $P$. (Hint: consider $G(\lambda)=F\left(\lambda x_{1}, \ldots, \lambda x_{n+1}\right)=\sum \lambda^{i} F_{i}\left(x_{1}, \ldots, x_{n+1}\right)$ for fixed $\left(x_{1}, \ldots, x_{n+1}\right)$.)
4.3. (a) Show that the definitions of this section carry over without change to the case where $k$ is an arbitrary field. (b) If $k_{0}$ is a subfield of $k$, show that $\mathbb{P}^{n}\left(k_{0}\right)$ may be identified with a subset of $\mathbb{P}^{n}(k)$.
### Projective Algebraic Sets
In this section we develop the idea of algebraic sets in $\mathbb{P}^{n}=\mathbb{P}^{n}(k)$. Since the concepts and most of the proofs are entirely similar to those for affine algebraic sets, many details will be left to the reader. A point $P \in \mathbb{P}^{n}$ is said to be a zero of a polynomial $F \in k\left[X_{1}, \ldots, X_{n+1}\right]$ if
$$
F\left(x_{1}, \ldots, x_{n+1}\right)=0
$$
for every choice of homogeneous coordinates $\left(x_{1}, \ldots, x_{n+1}\right)$ for $P$; we then write $F(P)=0$. If $F$ is a form, and $F$ vanishes at one representative of $P$, then it vanishes at every representative. If we write $F$ as a sum of forms in the usual way, then each form vanishes on any set of homogeneous coordinates for $P$ (Problem 4.2).
For any set $S$ of polynomials in $k\left[X_{1}, \ldots, X_{n+1}\right]$, we let
$$
V(S)=\left\{P \in \mathbb{P}^{n} \mid P \text { is a zero of each } F \in S\right\} .
$$
If $I$ is the ideal generated by $S, V(I)=V(S)$. If $I=\left(F^{(1)}, \ldots, F^{(r)}\right)$, where $F^{(i)}=\sum F_{j}^{(i)}$, $F_{j}^{(i)}$ a form of degree $j$, then $V(I)=V\left(\left\{F_{j}^{(i)}\right\}\right)$, so $V(S)=V\left(\left\{F_{j}^{(i)}\right\}\right)$ is the set of zeros of a finite number of forms. Such a set is called an algebraic set in $\mathbb{P}^{n}$, or a projective algebraic set.
For any set $X \subset \mathbb{P}^{n}$, we let $I(X)=\left\{F \in k\left[X_{1}, \ldots, X_{n+1}\right] \mid\right.$ every $P \in X$ is a zero of $\left.F\right\}$. The ideal $I(X)$ is called the ideal of $X$.
An ideal $I \subset k\left[X_{1}, \ldots, X_{n+1}\right]$ is called homogeneous if for every $F=\sum_{i=0}^{m} F_{i} \in I, F_{i}$ a form of degree $i$, we have also $F_{i} \in I$. For any set $X \subset \mathbb{P}^{n}, I(X)$ is a homogeneous ideal.
Proposition 1. An ideal $I \subset k\left[X_{1}, \ldots, X_{n+1}\right]$ is homogeneous if and only if it is generated by a (finite) set of forms.
Proof. If $I=\left(F^{(1)}, \ldots, F^{(r)}\right)$ is homogeneous, then $I$ is generated by $\left\{F_{j}^{(i)}\right\}$. Conversely, let $S=\left\{F^{(\alpha)}\right\}$ be a set of forms generating an ideal $I$, with $\operatorname{deg}\left(F^{(\alpha)}\right)=d_{\alpha}$, and suppose $F=F_{m}+\cdots+F_{r} \in I, \operatorname{deg}\left(F_{i}\right)=i$. It suffices to show that $F_{m} \in I$, for then $F-F_{m} \in I$, and an inductive argument finishes the proof. Write $F=\sum A^{(\alpha)} F^{(\alpha)}$. Comparing terms of the same degree, we conclude that $F_{m}=\sum A_{m-d_{\alpha}}^{(\alpha)} F^{(\alpha)}$, so $F_{m} \in I$.
An algebraic set $V \subset \mathbb{P}^{n}$ is irreducible if it is not the union of two smaller algebraic sets. The same proof as in the affine case, but using Problem 4.4 below, shows that $V$ is irreducible if and only if $I(V)$ is prime. An irreducible algebraic set in $\mathbb{P}^{n}$ is called a projective variety. Any projective algebraic set can be written uniquely as a union of projective varieties, its irreducible components.
The operations
$$
\left\{\text { homogeneous ideals in } k\left[X_{1}, \ldots, X_{n+1}\right]\right\} \underset{I}{\stackrel{V}{\rightleftarrows}}\left\{\text { algebraic sets in } \mathbb{P}^{n}(k)\right\}
$$
satisfy most of the properties we found in the corresponding affine situation (see Problem 4.6). We have used the same notation in these two situations. In practice it should always be clear which is meant; if there is any danger of confusion, we will write $V_{p}, I_{p}$ for the projective operations, $V_{a}, I_{a}$ for the affine ones.
If $V$ is an algebraic set in $\mathbb{P}^{n}$, we define
$$
C(V)=\left\{\left(x_{1}, \ldots, x_{n+1} \in \mathbb{A}^{n+1} \mid\left[x_{1}: \ldots: x_{n+1}\right] \in V \text { or }\left(x_{1}, \ldots, x_{n+1}\right)=(0, \ldots, 0)\right\}\right.
$$
to be the cone over $V$. If $V \neq \varnothing$, then $I_{a}(C(V))=I_{p}(V)$; and if $I$ is a homogeneous ideal in $k\left[X_{1}, \ldots, X_{n+1}\right]$ such that $V_{p}(I) \neq \varnothing$, then $C\left(V_{p}(I)\right)=V_{a}(I)$. This reduces many questions about $\mathbb{P}^{n}$ to questions about $\mathbb{A}^{n+1}$. For example
PROJECTIVE NULLSTELLENSATZ. Let I be a homogeneous ideal in $k\left[X_{1}, \ldots, X_{n+1}\right]$. Then
(1) $V_{p}(I)=\varnothing$ if and only if there is an integer $N$ such that I contains all forms of degree $\geq N$.
(2) If $V_{p}(I) \neq \varnothing$, then $I_{p}\left(V_{p}(I)\right)=\operatorname{Rad}(I)$.
Proof. (1) The following four conditions are equivalent: (i) $V_{p}(I)=\varnothing$; (ii) $V_{a}(I) \subset$ $\{(0, \ldots, 0)\}$; (iii) $\operatorname{Rad}(I)=I_{a}\left(V_{a}(I)\right) \supset\left(X_{1}, \ldots, X_{n+1}\right)$ (by the affine Nullstellensatz); and (iv) $\left(X_{1}, \ldots, X_{n+1}\right)^{N} \subset I$ (by Problem 2.41).
(2) $I_{p}\left(V_{p}(I)\right)=I_{a}\left(C\left(V_{p}(I)\right)\right)=I_{a}\left(V_{a}(I)\right)=\operatorname{Rad}(I)$.
The usual corollaries of the Nullstellensatz go through, except that we must always make an exception with the ideal $\left(X_{1}, \ldots, X_{n+1}\right)$. In particular, there is a oneto-one correspondence between projective hypersurfaces $V=V(F)$ and the (nonconstant) forms $F$ that define $V$ provided $F$ has no multiple factors ( $F$ is determined up to multiplication by a nonzero $\lambda \in k$ ). Irreducible hypersurfaces correspond to irreducible forms. A hyperplane is a hypersurface defined by a form of degree one. The hyperplanes $V\left(X_{i}\right), i=1, \ldots, n+1$, may be called the coordinate hyperplanes, or the hyperplanes at infinity with respect to $U_{i}$. If $n=2$, the $V\left(X_{i}\right)$ are the three coordinate axes.
Let $V$ be a nonempty projective variety in $\mathbb{P}^{n}$. Then $I(V)$ is a prime ideal, so the residue ring $\Gamma_{h}(V)=k\left[X_{1}, \ldots, X_{n+1}\right] / I(V)$ is a domain. It is called the homogeneous coordinate ring of $V$.
More generally, let $I$ be any homogeneous ideal in $k\left[X_{1}, \ldots, X_{n+1}\right]$, and let $\Gamma=$ $k\left[X_{1}, \ldots, X_{n+1}\right] / I$. An element $f \in \Gamma$ will be called a form of degree $d$ if there is a form $F$ of degree $d$ in $k\left[X_{1}, \ldots, X_{n+1}\right]$ whose residue is $f$.
Proposition 2. Every element $f \in \Gamma$ may be written uniquely as $f=f_{0}+\cdots+f_{m}$, with $f_{i}$ a form of degree $i$.
Proof. If $f$ is the residue of $F \in k\left[X_{1}, \ldots, X_{n+1}\right]$, write $F=\sum F_{i}$, and then $f=\sum f_{i}$, where $f_{i}$ is the residue of $F_{i}$. To show the uniqueness, suppose also $f=\sum g_{i}, g_{i}=$ residue of $G_{i}$. Then $F-\sum G_{i}=\sum\left(F_{i}-G_{i}\right) \in I$, and since $I$ is homogeneous, each $F_{i}-G_{i} \in I$, so $f_{i}=g_{i}$.
Let $k_{h}(V)$ be the quotient field of $\Gamma_{h}(V)$; it is called the homogeneous function field of $V$. In contrast with the case of affine varieties, no elements of $\Gamma_{h}(V)$ except the constants determine functions on $V$; likewise most elements of $k_{h}(V)$ cannot be regarded as functions. However, if $f, g$ are both forms in $\Gamma_{h}(V)$ of the same degree $d$, then $f / g$ does define a function, at least where $g$ is not zero: in fact, $f(\lambda x) / g(\lambda x)=\lambda^{d} f(x) / \lambda^{d} g(x)=f(x) / g(x)$, so the value of $f / g$ is independent of the choice of homogeneous coordinates.
The function field of $V$, written $k(V)$, is defined to be $\left\{z \in k_{h}(V) \mid\right.$ for some forms $f, g \in \Gamma_{h}(V)$ of the same degree, $\left.z=f / g\right\}$. It is not difficult to verify that $k(V)$ is a subfield of $k_{h}(V) . k \subset k(V) \subset k_{h}(V)$, but $\Gamma_{h}(V) \not \subset k(V)$. Elements of $k(V)$ are called rational functions on $V$.
Let $P \in V, z \in k(V)$. We say that $z$ is defined at $P$ if $z$ can be written as $z=f / g$, $f, g$ forms of the same degree, and $g(P) \neq 0$. We let
$$
\mathscr{O}_{P}(V)=\{z \in k(V) \mid z \text { is defined at } P\} ;
$$
$\mathscr{O}_{P}(V)$ is a subring of $k(V)$; it is a local ring, with maximal ideal
$$
\mathfrak{m}_{P}(V)=\{z \mid z=f / g, g(P) \neq 0, f(P)=0\} .
$$
It is called the local ring of $V$ at $P$. The value $z(P)$ of a function $z \in \mathscr{O}_{P}(V)$ is welldefined.
If $T: \mathbb{A}^{n+1} \rightarrow \mathbb{A}^{n+1}$ is a linear change of coordinates, then $T$ takes lines through the origin into lines through the origin (Problem 2.15). So $T$ determines a map from $\mathbb{P}^{n}$ to $\mathbb{P}^{n}$, called a projective change of coordinates. If $V$ is an algebraic set in $\mathbb{P}^{n}$, then $T^{-1}(V)$ is also an algebraic set in $\mathbb{P}^{n}$; we write $V^{T}$ for $T^{-1}(V)$. If $V=V\left(F_{1}, \ldots, F_{r}\right)$, and $T=\left(T_{1}, \ldots, T_{n+1}\right), T_{i}$ forms of degree 1 , then $V^{T}=V\left(F_{1}^{T}, \ldots, F_{r}^{T}\right)$, where $F_{i}^{T}=$ $F_{i}\left(T_{1}, \ldots, T_{n+1}\right)$. Then $V$ is a variety if and only if $V^{T}$ is a variety, and $T$ induces isomorphisms $\tilde{T}: \Gamma_{h}(V) \rightarrow \Gamma_{h}\left(V^{T}\right), k(V) \rightarrow k\left(V^{T}\right)$, and $\mathscr{O}_{P}(V) \rightarrow \mathscr{O}_{Q}\left(V^{T}\right)$ if $T(Q)=$ $P$.
## Problems
4.4. Let $I$ be a homogeneous ideal in $k\left[X_{1}, \ldots, X_{n+1}\right]$. Show that $I$ is prime if and only if the following condition is satisfied; for any forms $F, G \in k\left[X_{1}, \ldots, X_{n+1}\right]$, if $F G \in$ $I$, then $F \in I$ or $G \in I$.
4.5. If $I$ is a homogeneous ideal, show that $\operatorname{Rad}(I)$ is also homogeneous.
4.6. State and prove the projective analogues of properties (1)-(10) of Chapter 1, Sections 2 and 3.
4.7. Show that each irreducible component of a cone is also a cone.
4.8. Let $V=\mathbb{P}^{1}, \Gamma_{h}(V)=k[X, Y]$. Let $t=X / Y \in k(V)$, and show that $k(V)=k(t)$. Show that there is a natural one-to-one correspondence between the points of $\mathbb{P}^{1}$ and the DVR's with quotient field $k(V)$ that contain $k$ (see Problem 2.27); which DVR corresponds to the point at infinity?
4.9. Let $I$ be a homogeneous ideal in $k\left[X_{1}, \ldots, X_{n+1}\right]$, and
$$
\Gamma=k\left[X_{1}, \ldots, X_{n+1}\right] / I .
$$
Show that the forms of degree $d$ in $\Gamma$ form a finite-dimensional vector space over $k$. 4.10. Let $R=k[X, Y, Z], F \in R$ an irreducible form of degree $n, V=V(F) \subset \mathbb{P}^{2}$, and $\Gamma=\Gamma_{h}(V)$. (a) Construct an exact sequence $0 \longrightarrow R \stackrel{\psi}{\longrightarrow} R \stackrel{\varphi}{\longrightarrow} \Gamma \longrightarrow 0$, where $\psi$ is multiplication by $F$. (b) Show that
$$
\operatorname{dim}_{k}\{\text { forms of degree } d \text { in } \Gamma\}=d n-\frac{n(n-3)}{2}
$$
if $d>n$. 4.11. A set $V \subset \mathbb{P}^{n}(k)$ is called a linear subvariety of $\mathbb{P}^{n}(k)$ if $V=V\left(H_{1}, \ldots, H_{r}\right)$, where each $H_{i}$ is a form of degree 1. (a) Show that if $T$ is a projective change of coordinates, then $V^{T}=T^{-1}(V)$ is also a linear subvariety. (b) Show that there is a projective change of coordinates $T$ of $\mathbb{P}^{n}$ such that $V^{T}=V\left(X_{m+2}, \ldots, X_{n+1}\right)$, so $V$ is a variety. (c) Show that the $m$ that appears in part (b) is independent of the choice of $T$. It is called the dimension of $V(m=-1$ if $V=\varnothing)$.
4.12. Let $H_{1}, \ldots, H_{m}$ be hyperplanes in $\mathbb{P}^{n}, m \leq n$. Show that $H_{1} \cap H_{2} \cap \cdots \cap H_{m} \neq \varnothing$.
4.13* Let $P=\left[a_{1}: \ldots: a_{n+1}\right], Q=\left[b_{1}: \ldots: b_{n+1}\right]$ be distinct points of $\mathbb{P}^{n}$. The line $L$ through $P$ and $Q$ is defined by
$$
L=\left\{\left[\lambda a_{1}+\mu b_{1}: \ldots: \lambda a_{n+1}+\mu b_{n+1}\right] \mid \lambda, \mu \in k, \lambda \neq 0 \text { or } \mu \neq 0\right\} .
$$
Prove the projective analogue of Problem 2.15.
4.14. Let $P_{1}, P_{2}, P_{3}$ (resp. $Q_{1}, Q_{2}, Q_{3}$ ) be three points in $\mathbb{P}^{2}$ not lying on a line. Show that there is a projective change of coordinates $T: \mathbb{P}^{2} \rightarrow \mathbb{P}^{2}$ such that $T\left(P_{i}\right)=Q_{i}$, $i=1,2,3$. Extend this to $n+1$ points in $\mathbb{P}^{n}$, not lying on a hyperplane.
4.15. Show that any two distinct lines in $\mathbb{P}^{2}$ intersect in one point.
4.16. Let $L_{1}, L_{2}, L_{3}$ (resp. $M_{1}, M_{2}, M_{3}$ ) be lines in $\mathbb{P}^{2}(k)$ that do not all pass through a point. Show that there is a projective change of coordinates: $T: \mathbb{P}^{2} \rightarrow \mathbb{P}^{2}$ such that $T\left(L_{i}\right)=M_{i}$. (Hint:: Let $P_{i}=L_{j} \cap L_{k}, Q_{i}=M_{j} \cap M_{k}, i, j, k$ distinct, and apply Problem 4.14.) Extend this to $n+1$ hyperplanes in $\mathbb{P}^{n}$, not passing through a point.
4.17* Let $z$ be a rational function on a projective variety $V$. Show that the pole set of $z$ is an algebraic subset of $V$.
4.18. Let $H=V\left(\sum a_{i} X_{i}\right)$ be a hyperplane in $\mathbb{P}^{n}$. Note that $\left(a_{1}, \ldots, a_{n+1}\right)$ is determined by $H$ up to a constant. (a) Show that assigning $\left[a_{1}: \ldots: a_{n+1}\right] \in \mathbb{P}^{n}$ to $H$ sets up a natural one-to-one correspondence between $\left\{\right.$ hyperplanes in $\left.\mathbb{P}^{n}\right\}$ and $\mathbb{P}^{n}$. If $P \in \mathbb{P}^{n}$, let $P^{*}$ be the corresponding hyperplane; if $H$ is a hyperplane, $H^{*}$ denotes the corresponding point. (b) Show that $P^{* *}=P, H^{* *}=H$. Show that $P \in H$ if and only if $H^{*} \in P^{*}$.
This is the well-known duality of the projective space.
### Affine and Projective Varieties
We consider $\mathbb{A}^{n}$ as a subset of $\mathbb{P}^{n}$ by means of the map $\varphi_{n+1}: \mathbb{A}^{n} \rightarrow U_{n+1} \subset \mathbb{P}^{n}$. In this section we study the relations between the algebraic sets in $\mathbb{A}^{n}$ and those in $\mathbb{P}^{n}$.
Let $V$ be an algebraic set in $\mathbb{A}^{n}, I=I(V) \subset k\left[X_{1}, \ldots, X_{n}\right]$. Let $I^{*}$ be the ideal in $k\left[X_{1}, \ldots, X_{n+1}\right]$ generated by $\left\{F^{*} \mid F \in I\right\}$ (see Chapter 2, Section 6 for notation). This $I^{*}$ is a homogeneous ideal; we define $V^{*}$ to be $V\left(I^{*}\right) \subset \mathbb{P}^{n}$.
Conversely, let $V$ be an algebraic set in $\mathbb{P}^{n}, I=I(V) \subset k\left[X_{1}, \ldots, X_{n}\right]$. Let $I_{*}$ be the ideal in $k\left[X_{1}, \ldots, X_{n}\right]$ generated by $\left\{F_{*} \mid F \in I\right\}$. We define $V_{*}$ to be $V\left(I_{*}\right) \subset \mathbb{A}^{n}$.
Proposition 3. (1) If $V \subset \mathbb{A}^{n}$, then $\varphi_{n+1}(V)=V^{*} \cap U_{n+1}$, and $\left(V^{*}\right)_{*}=V$.
(2) If $V \subset W \subset \mathbb{A}^{n}$, then $V^{*} \subset W^{*} \subset \mathbb{P}^{n}$. If $V \subset W \subset \mathbb{P}^{n}$, then $V_{*} \subset W_{*} \subset \mathbb{A}^{n}$. (3) If $V$ is irreducible in $\mathbb{A}^{n}$, then $V^{*}$ is irreducible in $\mathbb{P}^{n}$.
(4) If $V=\bigcup_{i} V_{i}$ is the irreducible decomposition of $V$ in $\mathbb{A}^{n}$, then $V^{*}=\bigcup_{i} V_{i}^{*}$ is the irreducible decomposition of $V^{*}$ in $\mathbb{P}^{n}$.
(5) If $V \subset \mathbb{A}^{n}$, then $V^{*}$ is the smallest algebraic set in $\mathbb{P}^{n}$ that contains $\varphi_{n+1}(V)$.
(6) If $V \varsubsetneqq \mathbb{A}^{n}$ is not empty, then no component of $V^{*}$ lies in or contains $H_{\infty}=$ $\mathbb{P}^{n} \backslash U_{n+1}$.
(7) If $V \subset \mathbb{P}^{n}$, and no component of $V$ lies in or contains $H_{\infty}$, then $V_{*} \varsubsetneqq \mathbb{A}^{n}$ and $\left(V_{*}\right)^{*}=V$.
Proof. (1) follows from Proposition 5 of $\S 2.6$. (2) is obvious. If $V \subset \mathbb{A}^{n}, I=I(V)$, then a form $F$ belongs to $I^{*}$ if and only if $F_{*} \in I$. If $I$ is prime, it follows readily that $I^{*}$ is also prime, which proves (3).
To prove (5), suppose $W$ is an algebraic set in $\mathbb{P}^{n}$ that contains $\varphi_{n+1}(V)$. If $F \in$ $I(W)$, then $F_{*} \in I(V)$, so $F=X_{n+1}^{r}\left(F_{*}\right)^{*} \in I(V)^{*}$. Therefore $I(W) \subset I(V)^{*}$, so $W \supset V^{*}$, as desired.
(4) follows from (2), (3), and (5). To prove (6), we may assume $V$ is irreducible. $V^{*} \not \subset H_{\infty}$ by (1). If $V^{*} \supset H_{\infty}$, then $I(V)^{*} \subset I\left(V^{*}\right) \subset I\left(H_{\infty}\right)=\left(X_{n+1}\right)$. But if $0 \neq F \in$ $I(V)$, then $F^{*} \in I(V)^{*}$, with $F^{*} \notin\left(X_{n+1}\right)$. So $V^{*} \not \supset H_{\infty}$.
(7): We may assume $V \subset \mathbb{P}^{n}$ is irreducible. Since $\varphi_{n+1}\left(V_{*}\right) \subset V$, it suffices to show that $V \subset\left(V_{*}\right)^{*}$, or that $I\left(V_{*}\right)^{*} \subset I(V)$. Let $F \in I\left(V_{*}\right)$. Then $F^{N} \in I(V)_{*}$ for some $N$ (Nullstellensatz), so $X_{n+1}^{t}\left(F^{N}\right)^{*} \in I(V)$ for some $t$ (Proposition 5 (3) of $\$ 2.6$ ). But $I(V)$ is prime, and $X_{n+1} \notin I(V)$ since $V \not \subset H_{\infty}$, so $F^{*} \in I(V)$, as desired.
If $V$ is an algebraic set in $\mathbb{A}^{n}, V^{*} \subset \mathbb{P}^{n}$ is called the projective closure of $V$. If $V=V(F)$ is an affine hypersurface, then $V^{*}=V\left(F^{*}\right)$ (see Problem 4.19). Except for projective varieties lying in $H_{\infty}$, there is a natural one-to-one correspondence between nonempty affine and projective varieties (see Problem 4.22).
Let $V$ be an affine variety, $V^{*} \subset \mathbb{P}^{n}$ its projective closure. If $f \in \Gamma_{h}\left(V^{*}\right)$ is a form of degree $d$, we may define $f_{*} \in \Gamma(V)$ as follows: take a form $F \in k\left[X_{1}, \ldots, X_{n+1}\right]$ whose $I_{p}\left(V^{*}\right)$-residue is $f$, and let $f_{*}=I(V)$-residue of $F_{*}$ (one checks that this is independent of the choice of $F$ ). We then define a natural isomorphism $\alpha: k\left(V^{*}\right) \rightarrow k(V)$ as follows: $\alpha(f / g)=f_{*} / g_{*}$, where $f, g$ are forms of the same degree on $V^{*}$. If $P \in V$, we may consider $P \in V^{*}$ (by means of $\varphi_{n+1}$ ) and then $\alpha$ induces an isomorphism of $\mathscr{O}_{P}\left(V^{*}\right)$ with $\mathscr{O}_{P}(V)$. We usually use $\alpha$ to identify $k(V)$ with $k\left(V^{*}\right)$, and $\mathscr{O}_{P}(V)$ with $\mathscr{O}_{P}\left(V^{*}\right)$.
Any projective variety $V \subset \mathbb{P}^{n}$ is covered by the $n+1$ sets $V \cap U_{i}$. If we form $V_{*}$ with respect to $U_{i}$ (as with $U_{n+1}$ ), the points on $V \cap U_{i}$ correspond to points on $V_{*}$, and the local rings are isomorphic. Thus questions about $V$ near a point $P$ can be reduced to questions about an affine variety $V_{*}$ (at least if the question can be answered by looking at $\left.\mathscr{O}_{P}(V)\right)$.
## Problems
4.19. If $I=(F)$ is the ideal of an affine hypersurface, show that $I^{*}=\left(F^{*}\right)$.
4.20. Let $V=V\left(Y-X^{2}, Z-X^{3}\right) \subset \mathbb{A}^{3}$. Prove: (a) $I(V)=\left(Y-X^{2}, Z-X^{3}\right)$.
(b) $Z W-X Y \in I(V)^{*} \subset k[X, Y, Z, W]$, but $Z W-X Y \notin\left(\left(Y-X^{2}\right)^{*},\left(Z-X^{3}\right)^{*}\right)$.
So if $I(V)=\left(F_{1}, \ldots, F_{r}\right)$, it does not follow that $I(V)^{*}=\left(F_{1}^{*}, \ldots, F_{r}^{*}\right)$.
4.21. Show that if $V \subset W \subset \mathbb{P}^{n}$ are varieties, and $V$ is a hypersurface, then $W=V$ or $W=\mathbb{P}^{n}$ (see Problem 1.30).
4.22. Suppose $V$ is a variety in $\mathbb{P}^{n}$ and $V \supset H_{\infty}$. Show that $V=\mathbb{P}^{n}$ or $V=H_{\infty}$. If $V=\mathbb{P}^{n}, V_{*}=\mathbb{A}^{n}$, while if $V=H_{\infty}, V_{*}=\varnothing$.
4.23. Describe all subvarieties in $\mathbb{P}^{1}$ and in $\mathbb{P}^{2}$.
4.24* Let $P=[0: 1: 0] \in \mathbb{P}^{2}(k)$. Show that the lines through $P$ consist of the following:
(a) The "vertical" lines $L_{\lambda}=V(X-\lambda Z)=\{[\lambda: t: 1] \mid t \in k\} \cup\{P\}$.
(b) The line at infinity $L_{\infty}=V(Z)=\{[x: y: 0] \mid x, y \in k\}$.
4.25. Let $P=[x: y: z] \in \mathbb{P}^{2}$. (a) Show that $\left\{(a, b, c) \in \mathbb{A}^{3} \mid a x+b y+c z=0\right\}$ is a hyperplane in $\mathbb{A}^{3}$. (b) Show that for any finite set of points in $\mathbb{P}^{2}$, there is a line not passing through any of them.
### Multiprojective Space
We want to make the cartesian product of two varieties into a variety. Since $\mathbb{A}^{n} \times$ $\mathbb{A}^{m}$ may be identified with $\mathbb{A}^{n+m}$, this is not difficult for affine varieties. The product $\mathbb{P}^{n} \times \mathbb{P}^{m}$ requires some discussion, however.
Write $k[X, Y]$ for $k\left[X_{1}, \ldots, X_{n+1}, Y_{1}, \ldots, Y_{m+1}\right]$. A polynomial $F \in k[X, Y]$ is called a biform of bidegree $(p, q)$ if $F$ is a form of degree $p$ (resp. $q$ ) when considered as a polynomial in $X_{1}, \ldots, X_{n+1}$ (resp. $Y_{1}, \ldots, Y_{m+1}$ ) with coefficients in $k\left[Y_{1}, \ldots, Y_{m+1}\right]$ (resp. $k\left[X_{1}, \ldots, X_{n+1}\right]$ ). Every $F \in k[X, Y]$ may be written uniquely as $F=\sum_{p, q} F_{p, q}$, where $F_{p, q}$ is a biform of bidegree $(p, q)$.
If $S$ is any set of biforms in $k\left[X_{1}, \ldots, X_{n+1}, Y_{1}, \ldots, Y_{m+1}\right]$, we let $V(S)$ or $V_{b}(S)$ be $\left\{(x, y)\left|\mathbb{P}^{n} \times \mathbb{P}^{m}\right| F(x, y)=0\right.$ for all $\left.F \in S\right\}$. A subset $V$ of $\mathbb{P}^{n} \times \mathbb{P}^{m}$ will be called algebraic if $V=V(S)$ for some $S$. For any $V \subset \mathbb{P}^{n} \times \mathbb{P}^{m}$, define $I(V)$, or $I_{b}(V)$, to be $\{F \in k[X, Y] \mid F(x, y)=0$ for all $(x, y) \in V\}$.
We leave it to the reader to define a bihomogeneous ideal, show that $I_{b}(V)$ is bihomogeneous, and likewise to carry out the entire development for algebraic sets and varieties in $\mathbb{P}^{n} \times \mathbb{P}^{m}$ as was done for $\mathbb{P}^{n}$ in Section 2. If $V \subset \mathbb{P}^{n} \times \mathbb{P}^{m}$ is a variety (i.e., irreducible), $\Gamma_{b}(V)=k[X, Y] / I_{b}(V)$ is the bihomogeneous coordinate ring, $k_{b}(V)$ its quotient field, and
$$
k(V)=\left\{z \in k_{h}(V) \mid z=f / g, f, g \text { biforms of the same bidegree in } \Gamma_{b}(V)\right\}
$$
is the function field of $V$. The local rings $\mathscr{O}_{P}(V)$ are defined as before.
We likewise leave it to the reader to develop the theory of multiprojective varieties in $\mathbb{P}^{n_{1}} \times \mathbb{P}^{n_{2}} \times \cdots \times \mathbb{P}^{n_{r}}$.
If, finally, the reader develops the theory of algebraic subsets and varieties in mixed, or "multispaces" $\mathbb{P}^{n_{1}} \times \mathbb{P}^{n_{2}} \times \cdots \times \mathbb{P}^{n_{r}} \times \mathbb{A}^{m}$ (here a polynomial should be homogeneous in each set of variables that correspond to a projective space $\mathbb{P}^{n_{i}}$, but there is no restriction on those corresponding to $\mathbb{A}^{m}$ ), he or she will have the most general theory needed for the rest of this text. If we define $A^{0}$ to be a point, then all projective, multiprojective, and affine varieties are special cases of varieties in $\mathbb{P}^{n_{1}} \times \cdots \times \mathbb{P}^{n_{r}} \times \mathbb{A}^{m}$
## Problems
4.26. (a) Define maps $\varphi_{i, j}: \mathbb{A}^{n+m} \rightarrow U_{i} \times U_{j} \subset \mathbb{P}^{n} \times \mathbb{P}^{m}$. Using $\varphi_{n+1, m+1}$, define the "biprojective closure" of an algebraic set in $\mathbb{A}^{n+m}$. Prove an analogue of Proposition 3 of $\S 4$.3. (b) Generalize part (a) to maps $\varphi: \mathbb{A}^{n_{1}} \times \mathbb{A}^{n_{r}} \times \mathbb{A}^{m} \rightarrow \mathbb{P}^{n_{1}} \times \mathbb{P}^{n_{r}} \times \mathbb{A}^{m}$. Show that this sets up a correspondence between \{nonempty affine varieties in $\mathbb{A}^{n_{1}+\cdots+m}$ \} and \{varieties in $\mathbb{P}^{n_{1}} \times \cdots \times \mathbb{A}^{m}$ that intersect $U_{n_{1}+1} \times \cdots \times \mathbb{A}^{m}$ \}. Show that this correspondence preserves function fields and local rings.
4.27. Show that the pole set of a rational function on a variety in any multispace is an algebraic subset.
4.28. For simplicity of notation, in this problem we let $X_{0}, \ldots, X_{n}$ be coordinates for $\mathbb{P}^{n}, Y_{0}, \ldots, Y_{m}$ coordinates for $\mathbb{P}^{m}$, and $T_{00}, T_{01}, \ldots, T_{0 m}, T_{10}, \ldots, T_{n m}$ coordinates for $\mathbb{P}^{N}$, where $N=(n+1)(m+1)-1=n+m+n m$.
Define $S: \mathbb{P}^{n} \times \mathbb{P}^{m} \rightarrow \mathbb{P}^{N}$ by the formula:
$$
S\left(\left[x_{0}: \ldots: x_{n}\right],\left[y_{0}: \ldots: y_{m}\right]\right)=\left[x_{0} y_{0}: x_{0} y_{1}: \ldots: x_{n} y_{m}\right] .
$$
$S$ is called the Segre embedding of $\mathbb{P}^{n} \times \mathbb{P}^{m}$ in $\mathbb{P}^{n+m+n m}$.
(a) Show that $S$ is a well-defined, one-to-one mapping. (b) Show that if $W$ is an algebraic subset of $\mathbb{P}^{N}$, then $S^{-1}(W)$ is an algebraic subset of $\mathbb{P}^{n} \times \mathbb{P}^{m}$. (c) Let $V=V\left(\left\{T_{i j} T_{k l}-T_{i l} T_{k j} \mid i, k=0, \ldots, n ; j, l=0, \ldots, m\right\}\right) \subset \mathbb{P}^{N}$. Show that $S\left(\mathbb{P}^{n} \times \mathbb{P}^{m}\right)=V$. In fact, $S\left(U_{i} \times U_{j}\right)=V \cap U_{i j}$, where $U_{i j}=\left\{[t] \mid t_{i j} \neq 0\right\}$. (d) Show that $V$ is a variety.
## Chapter 5
## Projective Plane Curves
### Definitions
A projective plane curve is a hypersurface in $\mathbb{P}^{2}$, except that, as with affine curves, we want to allow multiple components: We say that two nonconstant forms $F, G \in$ $k[X, Y, Z]$ are equivalent if there is a nonzero $\lambda \in k$ such that $G=\lambda F$. A projective plane curve is an equivalence class of forms. The degree of a curve is the degree of a defining form. Curves of degree 1, 2, 3 and 4 are called lines, conics, cubic, and quartics respectively. The notations and conventions regarding affine curves carry over to projective curves (see $\S 3.1$ ): thus we speak of simple and multiple components, and we write $\mathscr{O}_{P}(F)$ instead of $\mathscr{O}_{P}(V(F))$ for an irreducible $F$, etc. Note that when $P=[x: y: 1]$, then $\mathscr{O}_{P}(F)$ is canonically isomorphic to $\mathscr{O}_{(x, y)}\left(F_{*}\right)$, where $F_{*}=F(X, Y, 1)$ is the corresponding affine curve.
The results of Chapter 3 assure us that the multiplicity of a point on an affine curve depends only on the local ring of the curve at that point. So if $F$ is a projective plane curve, $P \in U_{i}\left(i=1,2\right.$ or 3 ), we can dehomogenize $F$ with respect to $X_{i}$, and define the multiplicity of $F$ at $P, m_{P}(F)$, to be $m_{P}\left(F_{*}\right)$. The multiplicity is independent of the choice of $U_{i}$, and invariant under projective change of coordinates (Theorem 2 of $\S 3.2$ ).
The following notation will be useful. If we are considering a finite set of points $P_{1}, \ldots, P_{n} \in \mathbb{P}^{2}$, we can always find a line $L$ that doesn't pass through any of the points (Problem 4.25). If $F$ is a curve of degree $d$, we let $F_{*}=F / L^{d} \in k\left(\mathbb{P}^{2}\right)$. This $F_{*}$ depends on $L$, but if $L^{\prime}$ were another choice, then $F /\left(L^{\prime}\right)^{d}=\left(L / L^{\prime}\right)^{d} F_{*}$ and $L / L^{\prime}$ is a unit in each $\mathscr{O}_{P_{i}}\left(\mathbb{P}^{2}\right)$. Note also that we may always find a projective change of coordinates so that the line $L$ becomes the line $Z$ at infinity: then, under the natural identification of $k\left(\mathbb{A}^{2}\right)$ with $k\left(\mathbb{P}^{2}\right)(\S 4.3)$, this $F_{*}$ is the same as the old $F_{*}=F(X, Y, 1)$.
If $P$ is a simple point on $F$ (i.e., $m_{P}(F)=1$ ), and $F$ is irreducible, then $\mathscr{O}_{P}(F)$ is a DVR. We let $\operatorname{ord}_{P}^{F}$ denote the corresponding order function on $k(F)$. If $G$ is a form in $k[X, Y, Z]$, and $G_{*} \in \mathscr{O}_{P}\left(\mathbb{P}^{2}\right)$ is determined as in the preceding paragraph, and $\bar{G}_{*}$ is the residue of $G_{*}$ in $\mathscr{O}_{P}(F)$, we define $\operatorname{ord}_{P}^{F}(G)$ to $\operatorname{be} \operatorname{ord}_{P}^{F}\left(\bar{G}_{*}\right)$. Equivalently, $\operatorname{ord}_{P}^{F}(G)$ is the order at $P$ of $G / H$, where $H$ is any form of the same degree as $G$ with $H(P) \neq 0$. Let $F, G$ be projective plane curves, $P \in \mathbb{P}^{2}$. We define the intersection number $I(P, F \cap G)$ to be $\operatorname{dim}_{k}\left(\mathscr{O}_{P}\left(\mathbb{P}^{2}\right) /\left(F_{*}, G_{*}\right)\right)$. This is independent of the way $F_{*}$ and $G_{*}$ are formed, and it satisfies Properties (1)-(8) of Section 3 of Chapter 3: in (3), however, $T$ should be a projective change of coordinates, and in (7), $A$ should be a form with $\operatorname{deg}(A)=\operatorname{deg}(G)-\operatorname{deg}(F)$.
We can define a line $L$ to be tangent to a curve $F$ at $P$ if $I(P, F \cap L)>m_{P}(F)$ (see Problem 3.19). A point $P$ in $F$ is an ordinary multiple point of $F$ if $F$ has $m_{P}(F)$ distinct tangents at $P$.
Two curves $F$ and $G$ are said to be projectively equivalent if there is a projective change of coordinates $T$ such that $G=F^{T}$. Everything we will say about curves will be the same for two projectively equivalent curves.
## Problems
5.1. Let $F$ be a projective plane curve. Show that a point $P$ is a multiple point of $F$ if and only if $F(P)=F_{X}(P)=F_{Y}(P)=F_{Z}(P)=0$.
5.2. Show that the following curves are irreducible; find their multiple points, and the multiplicities and tangents at the multiple points.
(a) $X Y^{4}+Y Z^{4}+X Z^{4}$.
(b) $X^{2} Y^{3}+X^{2} Z^{3}+Y^{2} Z^{3}$.
(c) $Y^{2} Z-X(X-Z)(X-\lambda Z), \lambda \in k$.
(d) $X^{n}+Y^{n}+Z^{n}, n>0$.
5.3. Find all points of intersection of the following pairs of curves, and the intersection numbers at these points:
(a) $Y^{2} Z-X(X-2 Z)(X+Z)$ and $Y^{2}+X^{2}-2 X Z$.
(b) $\left(X^{2}+Y^{2}\right) Z+X^{3}+Y^{3}$ and $X^{3}+Y^{3}-2 X Y Z$.
(c) $Y^{5}-X\left(Y^{2}-X Z\right)^{2}$ and $Y^{4}+Y^{3} Z-X^{2} Z^{2}$.
(d) $\left(X^{2}+Y^{2}\right)^{2}+3 X^{2} Y Z-Y^{3} Z$ and $\left(X^{2}+Y^{2}\right)^{3}-4 X^{2} Y^{2} Z^{2}$.
5.4. Let $P$ be a simple point on $F$. Show that the tangent line to $F$ at $P$ has the equation $F_{X}(P) X+F_{Y}(P) Y+F_{Z}(P) Z=0$.
5.5. Let $P=[0: 1: 0], F$ a curve of degree $n, F=\sum F_{i}(X, Z) Y^{n-i}, F_{i}$ a form of degree $i$. Show that $m_{P}(F)$ is the smallest $m$ such that $F_{m} \neq 0$, and the factors of $F_{m}$ determine the tangents to $F$ at $P$.
5.6. For any $F, P \in F$, show that $m_{P}\left(F_{X}\right) \geq m_{P}(F)-1$.
5.7* Show that two plane curves with no common components intersect in a finite number of points.
5.8. Let $F$ be an irreducible curve. (a) Show that $F_{X}, F_{Y}$, or $F_{Z} \neq 0$. (b) Show that $F$ has only a finite number of multiple points.
5.9. (a) Let $F$ be an irreducible conic, $P=[0: 1: 0]$ a simple point on $F$, and $Z=0$ the tangent line to $F$ at $P$. Show that $F=a Y Z-b X^{2}-c X Z-d Z^{2}, a, b \neq 0$. Find a projective change of coordinates $T$ so that $F^{T}=Y Z-X^{2}-c^{\prime} X Z-d^{\prime} Z^{2}$. Find $T^{\prime}$ so that $\left(F^{T}\right)^{T^{\prime}}=Y Z-X^{2}$. $\left(T^{\prime}=\left(X, Y+c^{\prime} X+d^{\prime} Z, Z\right)\right.$.) (b) Show that, up to projective equivalence, there is only one irreducible conic: $Y Z=X^{2}$. Any irreducible conic is nonsingular.
5.10. Let $F$ be an irreducible cubic, $P=[0: 0: 1]$ a cusp on $F, Y=0$ the tangent line to $F$ at $P$. Show that $F=a Y^{2} Z-b X^{3}-c X^{2} Y-d X Y^{2}-e Y^{3}$. Find projective changes of coordinates (i) to make $a=b=1$; (ii) to make $c=0$ (change $X$ to $X-\frac{c}{3} Y$ ); (iii) to make $d=e=0(Z$ to $Z+d X+e Y)$.
Up to projective equivalence, there is only one irreducible cubic with a cusp: $Y^{2} Z=X^{3}$. It has no other singularities.
5.11. Up to projective equivalence, there is only one irreducible cubic with a node: $X Y Z=X^{3}+Y^{3}$. It has no other singularities.
5.12. (a) Assume $P=[0: 1: 0] \in F, F$ a curve of degree $n$. Show that $\sum_{P} I(P, F \cap X)=$ $n$. (b) Show that if $F$ is a curve of degree $n, L$ a line not contained in $F$, then
$$
\sum I(P, F \cap L)=n .
$$
5.13. Prove that an irreducible cubic is either nonsingular or has at most one double point (a node or a cusp). (Hint: Use Problem 5.12, where $L$ is a line through two multiple points; or use Problems 5.10 and 5.11.)
5.14. Let $P_{1}, \ldots, P_{n} \in \mathbb{P}^{2}$. Show that there are an infinite number of lines passing through $P_{1}$, but not through $P_{2}, \ldots, P_{n}$. If $P_{1}$ is a simple point on $F$, we may take these lines transversal to $F$ at $P_{1}$.
5.15. Let $C$ be an irreducible projective plane curve, $P_{1}, \ldots, P_{n}$ simple points on $C$, $m_{1}, \ldots, m_{n}$ integers. Show that there is a $z \in k(C)$ with $\operatorname{ord}_{P_{i}}(z)=m_{i}$ for $i=1, \ldots, n$. (Hint: Take lines $L_{i}$ as in Problem 5.14 for $P_{i}$, and a line $L_{0}$ not through any $P_{j}$, and let $z=\prod L_{i}^{m_{i}} L_{0}-\sum m_{i}$.)
5.16. Let $F$ be an irreducible curve in $\mathbb{P}^{2}$. Suppose $I(P, F \cap Z)=1$, and $P \neq[1: 0: 0]$. Show that $F_{X}(P) \neq 0$. (Hint: If not, use Euler's Theorem to show that $F_{Y}(P)=0$; but $Z$ is not tangent to $F$ at $P$.)
### Linear Systems of Curves
We often wish to study all curves of a given degree $d \geq 1$. Let $M_{1}, \ldots, M_{N}$ be a fixed ordering of the set of monomials in $X, Y, Z$ of degree $d$, where $N$ is $\frac{1}{2}(d+1)(d+2)$ (Problem 2.35). Giving a curve $F$ of degree $d$ is the same thing as choosing $a_{1}, \ldots, a_{N} \in k$, not all zero, and letting $F=\sum a_{i} M_{i}$, except that $\left(a_{1}, \ldots, a_{N}\right)$ and $\left(\lambda a_{1}, \ldots, \lambda a_{N}\right)$ determine the same curve. In other words, each curve $F$ of degree $d$ corresponds to a unique point in $\mathbb{P}^{N-1}=\mathbb{P}^{d(d+3) / 2}$ and each point of $\mathbb{P}^{d(d+3) / 2}$ represents a unique curve. We often identify $F$ with its corresponding point in $\mathbb{P}^{d(d+3) / 2}$, and say e.g. "the curves of degree $d$ form a projective space of dimension $d(d+3) / 2$ ".
Examples. (1) $d=1$. Each line $a X+b Y+c Z$ corresponds to the point $[a: b: c] \in$ $\mathbb{P}^{2}$. The lines in $\mathbb{P}^{2}$ form a $\mathbb{P}^{2}$ (see Problem 4.18).
(2) $d=2$. The conic $a X^{2}+b X Y+c X Z+d Y^{2}+e Y Z+f Z^{2}$ corresponds the point $[a: b: c: d: e: f] \in \mathbb{P}^{5}$. The conics form a $\mathbb{P}^{5}$. (3) The cubics form a $\mathbb{P}^{9}$, the quartics a $\mathbb{P}^{14}$, etc.
If we put conditions on the set of all curves of degree $d$, the curves that satisfy the conditions form a subset of $\mathbb{P}^{d(d+3) / 2}$. If this subset is a linear subvariety (Problem $4.11)$, it is called a linear system of plane curves.
Lemma. (1) Let $P \in \mathbb{P}^{2}$ be a fixed point. The set of curves of degree $d$ that contain $P$ forms a hyperplane in $\mathbb{P}^{d(d+3) / 2}$.
(2) If $T: \mathbb{P}^{2} \rightarrow \mathbb{P}^{2}$ is a projective change of coordinates, then the map $F \mapsto F^{T}$ from \{curves of degree $d$ \} to \{curves of degree $d$ \} is a projective change of coordinates on $\mathbb{P}^{d(d+3) / 2}$.
Proof. If $P=[x: y: z]$, then the curve corresponding to $\left(a_{1}, \ldots, a_{N}\right) \in \mathbb{P}^{d(d+3) / 2}$ passes through $P$ if and only if $\sum a_{i} M_{i}(x, y, z)=0$. Since not all $M_{i}(x, y, z)$ are zero, those $\left[a_{1}: \ldots: a_{N}\right]$ satisfying this equation form a hyperplane. The proof that $F \mapsto F^{T}$ is linear is similar; it is invertible since $F \mapsto F^{T^{-1}}$ is its inverse.
It follows that for any set of points, the curves of degree $d$ that contain them form a linear subvariety of $\mathbb{P}^{d(d+3) / 2}$. Since the intersection of $n$ hyperplanes of $\mathbb{P}^{n}$ is not empty (Problem 4.12), there is a curve of degree $d$ passing through any given $d(d+3) / 2$ points.
Suppose now we fix a point $P$ and an integer $r \leq d+1$. We claim that the curves $F$ of degree $d$ such that $m_{P}(F) \geq r$ form a linear subvariety of dimension $\frac{d(d+3)}{2}-\frac{r(r+1)}{2}$. By (2) of the Lemma, we may assume $P=[0: 0: 1]$. Write $F=\sum F_{i}(X, Y) Z^{d-i}, F_{i}$ a form of degree $i$. Then $m_{P}(F) \geq r$ if and only if $F_{0}=F_{1}=\cdots=F_{r-1}=0$, i.e., the coefficients of all monomials $X^{i} Y^{j} Z^{k}$ with $i+j<r$ are zero (Problem 5.5). And there are $1+2+\cdots+r=\frac{r(r+1)}{2}$ such coefficients.
Let $P_{1}, \ldots, P_{n}$ be distinct points in $\mathbb{P}^{2}, r_{1}, \ldots, r_{n}$ nonnegative integers. We set
$V\left(d ; r_{1} P_{1}, \ldots, r_{n} P_{n}\right)=\left\{\right.$ curves $F$ of degree $\left.d \mid m_{P_{i}}(F) \geq r_{i}, 1 \leq i \leq n\right\}$.
Theorem 1. (1) $V\left(d ; r_{1} P_{1}, \ldots, r_{n} P_{n}\right)$ is a linear subvariety of $\mathbb{P}^{d(d+3) / 2}$ of dimension $\geq \frac{d(d+3)}{2}-\sum \frac{r_{i}\left(r_{i}+1\right)}{2}$.
(2) If $d \geq\left(\sum r_{i}\right)-1$, then $\operatorname{dim} V\left(d ; r_{1} P_{1}, \ldots, r_{n} P_{n}\right)=\frac{d(d+3)}{2}-\sum \frac{r_{i}\left(r_{i}+1\right)}{2}$.
Proof. (1) follows from the above discussion. We prove (2) by induction on $m=$ $\left(\sum r_{i}\right)-1$. We may assume that $m>1, d>1$, since otherwise it is trivial.
Case 1: Each $r_{i}=1$ : Let $V_{i}=V\left(d ; P_{1}, \ldots, P_{i}\right)$. By induction it is enough to show that $V_{n} \neq V_{n-1}$. Choose lines $L_{i}$ passing through $P_{i}$ but not through $P_{j}, j \neq i$ (Problem 5.14), and a line $L_{0}$ not passing through any $P_{i}$. Then $F=L_{1} \cdots L_{n-1} L_{0}^{d-n+1} \in$ $V_{n-1}, F \notin V_{n}$.
Case 2: Some $r_{i}>1$ : Say $r=r_{1}>1$, and $P=P_{1}=[0: 0: 1]$. Let
$$
V_{0}=V\left(d ;(r-1) P, r_{2} P_{2}, \ldots, r_{n} P_{n}\right) .
$$
For $F \in V_{0}$ let $F_{*}=\sum_{i=0}^{r-1} a_{i} X^{i} Y^{r-1-i}+$ higher terms. Let $V_{i}=\left\{F \in V_{0} \mid a_{j}=0\right.$ for $\left.j<i\right\}$. Then $V_{0} \supset V_{1} \supset \cdots \supset V_{r}=V\left(d ; r_{1} P_{1}, r_{2} P_{2}, \ldots, r_{n} P_{n}\right)$, so it is enough to show that $V_{i} \neq V_{i+1}, i=0,1, \ldots, r-1$. Let $W_{0}=V\left(d-1 ;(r-2) P, r_{2} P_{2}, \ldots, r_{n} P_{n}\right)$. For $F \in W_{0}, F_{*}=a_{i} X^{i} Y^{r-2-i}+\cdots$. Set $W_{i}=\left\{F \in W_{0} \mid a_{j}=0\right.$ for $\left.j<i\right\}$. By induction,
$$
W_{0} \supsetneqq W_{1} \supsetneqq \cdots \supsetneqq W_{r-1}=V\left(d-1 ;(r-1) P, r_{2} P_{2}, \ldots, r_{n} P_{n}\right) .
$$
If $F_{i} \in W_{i}, F_{i} \notin W_{i+1}$, then $Y F_{i} \in V_{i}, Y F_{i} \notin V_{i+1}$, and $X F_{r-2} \in V_{r-1}, X F_{r-2} \notin V_{r}$. Thus $V_{i} \neq V_{i+1}$ for $i=0, \ldots, r-1$, and this completes the proof.
## Problems
5.17. Let $P_{1}, P_{2}, P_{3}, P_{4} \in \mathbb{P}^{2}$. Let $V$ be the linear system of conics passing through these points. Show that $\operatorname{dim}(V)=2$ if $P_{1}, \ldots, P_{4}$ lie on a line, and $\operatorname{dim}(V)=1$ otherwise.
5.18. Show that there is only one conic passing through the five points $[0: 0: 1]$, $[0: 1: 0],[1: 0: 0],[1: 1: 1]$, and $[1: 2: 3]$; show that it is nonsingular.
5.19. Consider the nine points $[0: 0: 1],[0: 1: 1]$, $[1: 0: 1],[1: 1: 1],[0: 2: 1]$, $[2: 0: 1],[1: 2: 1],[2: 1: 1]$, and $[2: 2: 1] \in \mathbb{P}^{2}$ (Sketch). Show that there are an infinite number of cubics passing through these points.
### Bézout's Theorem
The projective plane was constructed so that any two distinct lines would intersect at one point. The famous theorem of Bézout tells us that much more is true:
BÉZOUT'S THEOREM. Let $F$ and $G$ be projective plane curves of degree $m$ and $n$ respectively. Assume $F$ and $G$ have no common component. Then
$$
\sum_{P} I(P, F \cap G)=m n
$$
Proof. Since $F \cap G$ is finite (Problem 5.7), we may assume, by a projective change of coordinates if necessary, that none of the points in $F \cap G$ is on the line at infinity $Z=0$.
Then $\sum_{P} I(P, F \cap G)=\sum_{P} I\left(P, F_{*} \cap G_{*}\right)=\operatorname{dim}_{k} k[X, Y] /\left(F_{*}, G_{*}\right)$, by Property (9) for intersection numbers. Let
$$
\Gamma_{*}=k[X, Y] /\left(F_{*}, G_{*}\right), \quad \Gamma=k[X, Y, Z] /(F, G), \quad R=k[X, Y, Z],
$$
and let $\Gamma_{d}$ (resp. $R_{d}$ ) be the vector space of forms of degree $d$ in $\Gamma$ (resp. $R$ ). The theorem will be proved if we can show that $\operatorname{dim} \Gamma_{*}=\operatorname{dim} \Gamma_{d}$ and $\operatorname{dim} \Gamma_{d}=m n$ for some large $d$.
Step 1: $\operatorname{dim} \Gamma_{d}=m n$ for all $d \geq m+n:$ Let $\pi: R \rightarrow \Gamma$ be the natural map, let $\varphi: R \times R \rightarrow R$ be defined by $\varphi(A, B)=A F+B G$, and let $\psi: R \rightarrow R \times R$ be defined by $\psi(C)=(G C,-F C)$. Using the fact that $F$ and $G$ have no common factors, it is not difficult to check the exactness of the following sequence:
$$
0 \longrightarrow R \stackrel{\psi}{\longrightarrow} R \times R \stackrel{\varphi}{\longrightarrow} R \stackrel{\pi}{\longrightarrow} \Gamma \longrightarrow 0 .
$$
If we restrict these maps to the forms of various degrees, we get the following exact sequences:
$$
0 \longrightarrow R_{d-m-n} \stackrel{\psi}{\longrightarrow} R_{d-m} \times R_{d-n} \stackrel{\varphi}{\longrightarrow} R_{d} \stackrel{\pi}{\longrightarrow} \Gamma_{d} \longrightarrow 0 .
$$
Since $\operatorname{dim} R_{d}=\frac{(d+1)(d+2)}{2}$, it follows from Proposition 7 of $\S 2.10$ (with a calculation) that $\operatorname{dim} \Gamma_{d}=m n$ if $d \geq m+n$.
Step 2: The map $\alpha: \Gamma \rightarrow \Gamma$ defined by $\alpha(\bar{H})=\overline{Z H}$ (where the bar denotes the residue modulo $(F, G))$ is one-to-one:
We must show that if $Z H=A F+B G$, then $H=A^{\prime} F+B^{\prime} G$ for some $A^{\prime}, B^{\prime}$. For any $J \in k[X, Y, Z]$, denote (temporarily) $J(X, Y, 0)$ by $J_{0}$. Since $F, G$, and $Z$ have no common zeros, $F_{0}$ and $G_{0}$ are relatively prime forms in $k[X, Y]$.
If $Z H=A F+B G$, then $A_{0} F_{0}=-B_{0} G_{0}$, so $B_{0}=F_{0} C$ and $A_{0}=-G_{0} C$ for some $C \in k[X, Y]$. Let $A_{1}=A+C G, B_{1}=B-C F$. Since $\left(A_{1}\right)_{0}=\left(B_{1}\right)_{0}=0$, we have $A_{1}=Z A^{\prime}$, $B_{1}=Z B^{\prime}$ for some $A^{\prime}, B^{\prime}$; and since $Z H=A_{1} F+B_{1} G$, it follows that $H=A^{\prime} F+B^{\prime} G$, as claimed.
Step 3: Let $d \geq m+n$, and choose $A_{1}, \ldots, A_{m n} \in R_{d}$ whose residues in $\Gamma_{d}$ form a basis for $\Gamma_{d}$. Let $A_{i *}=A_{i}(X, Y, 1) \in k[X, Y]$, and let $a_{i}$ be the residue of $A_{i *}$ in $\Gamma_{*}$. Then $a_{1}, \ldots, a_{m n}$ form a basis for $\Gamma_{*}$ :
First notice that the map $\alpha$ of Step 2 restricts to an isomorphism from $\Gamma_{d}$ onto $\Gamma_{d+1}$, if $d \geq m+n$, since a one-to-one linear map of vector spaces of the same dimension is an isomorphism. It follows that the residues of $Z^{r} A_{1}, \ldots, Z^{r} A_{m n}$ form a basis for $\Gamma_{d+r}$ for all $r \geq 0$.
The $a_{i}$ generate $\Gamma_{*}$ : if $h=\bar{H} \in \Gamma_{*}, H \in k[X, Y]$, some $Z^{N} H^{*}$ is a form of degree $d+r$, so $Z^{N} H^{*}=\sum_{i=1}^{m n} \lambda_{i} Z^{r} A_{i}+B F+C G$ for some $\lambda_{i} \in k, B, C \in k[X, Y, Z]$. Then $H=\left(Z^{N} H^{*}\right)_{*}=\sum \lambda_{i} A_{i *}+B_{*} F_{*}+C_{*} G_{*}$, so $h=\sum \lambda_{i} a_{i}$, as desired.
The $a_{i}$ are independent: For if $\sum \lambda_{i} a_{i}=0$, then $\sum \lambda_{i} A_{i *}=B F_{*}+C G_{*}$. Therefore (by Proposition 5 of \\$2.6) $Z^{r} \sum \lambda_{i} A_{i}=Z^{s} B^{*} F+Z^{t} C^{*} G$ for some $r, s, t$. But then $\sum \lambda_{i} \overline{Z^{r} A_{i}}=0$ in $\Gamma_{d+r}$, and the $\overline{Z^{r} A_{i}}$ form a basis, so each $\lambda_{i}=0$. This finishes the proof.
Combining Property (5) of the intersection number (\\$3.3) with Bézout's Theorem, we deduce
Corollary 1. If $F$ and $G$ have no common component, then
$$
\sum_{P} m_{P}(F) m_{P}(G) \leq \operatorname{deg}(F) \cdot \operatorname{deg}(G) .
$$
Corollary 2. If $F$ and $G$ meet in $m n$ distinct points, $m=\operatorname{deg}(F), n=\operatorname{deg}(G)$, then theses points are all simple points on $F$ and on $G$.
Corollary 3. If two curves of degrees $m$ and $n$ have more than mn points in common, then they have a common component.
## Problems
5.20. Check your answers of Problem 5.3 with Bézout's Theorem. 5.21. Show that every nonsingular projective plane curve is irreducible. Is this true for affine curves?
5.22. Let $F$ be an irreducible curve of degree $n$. Assume $F_{X} \neq 0$. Apply Corollary 1 to $F$ and $F_{X}$, and conclude that $\sum m_{P}(F)\left(m_{P}(F)-1\right) \leq n(n-1)$. In particular, $F$ has at most $\frac{1}{2} n(n-1)$ multiple points. (See Problems 5.6, 5.8.)
5.23. A problem about flexes (see Problem 3.12): Let $F$ be a projective plane curve of degree $n$, and assume $F$ contains no lines.
Let $F_{i}=F_{X_{i}}$ and $F_{i j}=F_{X_{i} X_{j}}$, forms of degree $n-1$ and $n-2$ respectively. Form a $3 \times 3$ matrix with the entry in the $(i, j)$ th place being $F_{i j}$. Let $H$ be the determinant of this matrix, a form of degree $3(n-2)$. This $H$ is called the Hessian of $F$. Problems 5.22 and 6.47 show that $H \neq 0$, for $F$ irreducible. The following theorem shows the relationship between flexes and the Hessian.
Theorem. $(\operatorname{char}(k)=0)$ (1) $P \in H \cap F$ if and only if $P$ is either a flex or a multiple point of $F$. (2) $I(P, H \cap F)=1$ if and only if $P$ is an ordinary flex.
Outline of proof. (a) Let $T$ be a projective change of coordinates. Then the Hessian of $F^{T}=(\operatorname{det}(T))^{2}\left(H^{T}\right)$. So we can assume $P=[0: 0: 1]$; write $f(X, Y)=F(X, Y, 1)$ and $h(X, Y)=H(X, Y, 1)$.
(b) $(n-1) F_{j}=\sum_{i} X_{i} F_{i j}$. (Use Euler's Theorem.)
(c) $I(P, f \cap h)=I(P, f \cap g)$ where $g=f_{y}^{2} f_{x x}+f_{x}^{2} f_{y y}-2 f_{x} f_{y} f_{x y}$. (Hint: Perform row and column operations on the matrix for $h$. Add $x$ times the first row plus $y$ times the second row to the third row, then apply part (b). Do the same with the columns. Then calculate the determinant.)
(d) If $P$ is a multiple point on $F$, then $I(P, f \cap g)>1$.
(e) Suppose $P$ is a simple point, $Y=0$ is the tangent line to $F$ at $P$, so $f=y+$ $a x^{2}+b x y+c y^{2}+d x^{3}+e x^{2} y+\ldots$. Then $P$ is a flex if and only if $a=0$, and $P$ is an ordinary flex if and only if $a=0$ and $d \neq 0$. A short calculation shows that $g=$ $2 a+6 d x+\left(8 a c-2 b^{2}+2 e\right) y+$ higher terms, which concludes the proof.
Corollary. (1) A nonsingular curve of degree $>2$ always has a flex. (2) A nonsingular cubic has nine flexes, all ordinary.
5.24. $(\operatorname{char}(k)=0$ ) (a) Let $[0: 1: 0]$ be a flex on an irreducible cubic $F, Z=0$ the tangent line to $F$ at $[0: 1: 0]$. Show that $F=Z Y^{2}+b Y Z^{2}+c Y X Z+$ terms in $X, Z$. Find a projective change of coordinates (using $Y \mapsto Y-\frac{b}{2} Z-\frac{c}{2} X$ ) to get $F$ to the form $Z Y^{2}=$ cubic in $X, Z$. (b) Show that any irreducible cubic is projectively equivalent to one of the following: $Y^{2} Z=X^{3}, Y^{2} Z=X^{2}(X+Z)$, or $Y^{2} Z=X(X-Z)(X-\lambda Z)$, $\lambda \in k, \lambda \neq 0$, 1. (See Problems 5.10, 5.11.)
### Multiple Points
In Problem 5.22 of the previous section we saw one easy application of Bézout's Theorem: If $F$ is an irreducible curve of degree $n$, and $m_{P}$ denotes the multiplicity of $F$ at $P$, then $\sum \frac{m_{P}\left(m_{P}-1\right)}{2} \leq \frac{n(n-1)}{2}$.
An examination of the cases $n=2,3$ however, indicates that this is not the best possible result (Problems 5.9, 5.13). In fact: Theorem 2. If $F$ is an irreducible curve of degree $n$, then $\sum \frac{m_{P}\left(m_{P}-1\right)}{2} \leq \frac{(n-1)(n-2)}{2}$.
Proof. Since $r:=\frac{(n-1)(n-1+3)}{2}-\sum \frac{\left(m_{P}-1\right)\left(m_{P}\right)}{2} \geq \frac{(n-1) n}{2}-\sum \frac{\left(m_{P}-1\right) m_{P}}{2} \geq 0$, we may choose simple points $Q_{1}, \ldots, Q_{r} \in F$. Then Theorem 1 of $\$ 5.2$ (for $d=n-1$ ) guarantees the existence of a curve $G$ of degree $n-1$ such that $m_{P}(G) \geq m_{P}-1$ for all $P$, and $m_{Q_{i}}(G) \geq 1$.
Now apply Corollary 1 of Bézout's Theorem to $F$ and $G$ (since $F$ is irreducible, there are no common components): $n(n-1) \geq \sum m_{P}\left(m_{P}-1\right)+r$. The theorem follows by substituting the value for $r$ into this inequality.
For small $n$ this gives us some results we have seen in the problems: lines and irreducible conics are nonsingular, and an irreducible cubic can have at most one double point. Letting $n=4$ we see that an irreducible quartic has at most three double points or one triple point, etc. Note that the curve $X^{n}+Y^{n-1} Z$ has the point $[0: 0: 1]$ of multiplicity $n-1$, so the result cannot be strengthened.
## Problems
5.25. Let $F$ be a projective plane curve of degree $n$ with no multiple components, and $c$ simple components. Show that
$$
\sum \frac{m_{P}\left(m_{P}-1\right)}{2} \leq \frac{(n-1)(n-2)}{2}+c-1 \leq \frac{n(n-1)}{2}
$$
(Hint: Let $F=F_{1} F_{2}$; consider separately the points on one $F_{i}$ or on both.)
5.26. (char $(k)=0$ ) Let $F$ be an irreducible curve of degree $n$ in $\mathbb{P}^{2}$. Suppose $P \in \mathbb{P}^{2}$, with $m_{P}(F)=r \geq 0$. Then for all but a finite number of lines $L$ through $P, L$ intersects $F$ in $n-r$ distinct points other than $P$. We outline a proof:
(a) We may assume $P=[0: 1: 0]$. If $L_{\lambda}=\{[\lambda: t: 1] \mid t \in k\} \cup\{P\}$, we need only consider the $L_{\lambda}$. Then $F=A_{r}(X, Z) Y^{n-r}+\cdots+A_{n}(X, Z), A_{r} \neq 0$. (See Problems 4.24, 5.5).
(b) Let $G_{\lambda}(t)=F(\lambda, t, 1)$. It is enough to show that for all but a finite number of $\lambda, G_{\lambda}$ has $n-r$ distinct points.
(c) Show that $G_{\lambda}$ has $n-r$ distinct roots if $A_{r}(\lambda, 1) \neq 0$, and $F \cap F_{Y} \cap L_{\lambda}=\{P\}$ (see Problem 1.53).
5.27. Show that Problem 5.26 remains true if $F$ is reducible, provided it has no multiple components.
5.28. $(\operatorname{char}(k)=p>0) F=X^{p+1}-Y^{p} Z, P=[0: 1: 0]$. Find $L \cap F$ for all lines $L$ passing through $P$. Show that every line that is tangent to $F$ at a simple point passes through $P$ !
### Max Noether's Fundamental Theorem
A zero-cycle on $\mathbb{P}^{2}$ is a formal sum $\sum_{P \in \mathbb{P}^{2}} n_{P} P$, where $n_{P}$ 's are integers, and all but a finite number of $n_{P}$ 's are zero. The set of all zero-cycles on $\mathbb{P}^{2}$ form an abelian group - in fact, it is the free abelian group with basis $X=\mathbb{P}^{2}$, as defined in Chapter 2, Section 11.
The degree of a zero cycle $\sum n_{P} P$ is defined to be $\sum n_{P}$. The zero cycle is positive if each $n_{P} \geq 0$. We say that $\sum n_{P} P$ is bigger than $\sum m_{P} P$, and write $\sum n_{P} P \geq \sum m_{P} P$, if each $n_{P} \geq m_{P}$.
Let $F, G$ be projective plane curves of degrees $m, n$ respectively, with no common components. We define the intersection cycle $F \cdot G$ by
$$
F \cdot G=\sum_{P \in \mathbb{P}^{2}} I(P, F \cap G) P
$$
Bézout's Theorem says that $F \cdot G$ is a positive zero-cycle of degree $m n$.
Several properties of intersection numbers translate nicely into properties of the intersection cycle. For example: $F \cdot G=G \cdot F ; F \cdot G H=F \cdot G+F \cdot H$; and $F \cdot(G+A F)=$ $F \cdot G$ if $A$ is a form and $\operatorname{deg}(A)=\operatorname{deg}(G)-\operatorname{deg}(F)$.
Max Noether's Theorem is concerned with the following situation: Suppose $F, G$, and $H$ are curves, and $H \cdot F \geq G \cdot F$, i.e., $H$ intersects $F$ in a bigger cycle than $G$ does. When is there a curve $B$ so that $B \cdot F=H \cdot F-G \cdot F$ ? Note that necessarily $\operatorname{deg}(B)=$ $\operatorname{deg}(H)-\operatorname{deg}(G)$.
To find such a $B$, it suffices to find forms $A, B$ such that $H=A F+B G$. For then $H \cdot F=B G \cdot F=B \cdot F+G \cdot F$.
Let $P \in \mathbb{P}^{2}, F, G$ curves with no common component through $P, H$ another curve. We say that Noether's Conditions are satisfied at $P$ (with respect to $F, G$, and $H)$, if $H_{*} \in\left(F_{*}, G_{*}\right) \subset \mathscr{O}_{P}\left(\mathbb{P}^{2}\right)$, i.e., if there are $a, b \in \mathscr{O}_{P}\left(\mathbb{P}^{2}\right)$ such that $H_{*}=a F_{*}+b G_{*}$ (see $\$ 5.1$ for notation). Noether's Theorem relates the local and global conditions.
MAX NOETHER'S FUNDAMENTAL THEOREM. Let $F, G, H$ be projective plane curves. Assume $F$ and $G$ have no common components. Then there is an equation $H=A F+$ $B G$ (with $A, B$ forms of degree $\operatorname{deg}(H)-\operatorname{deg}(F), \operatorname{deg}(H)-\operatorname{deg}(G)$ respectively) if and only if Noether's conditions are satisfied at every $P \in F \cap G$.
Proof. If $H=A F+B G$, then $H_{*}=A_{*} F_{*}+B_{*} G_{*}$ at any $P$. To prove the converse, we assume, as in the proof of Bézout's Theorem, that $V(F, G, Z)=\varnothing$. We may take $F_{*}=F(X, Y, 1), G_{*}=G(X, Y, 1), H_{*}=H(X, Y, 1)$. Noether's conditions say that the residue of $H_{*}$ in $\mathscr{O}_{P}\left(\mathbb{P}^{2}\right) /\left(F_{*}, G_{*}\right)$ is zero for each $P \in F \cap G$. It follows from Proposition 6 of $\$ 2.9$ that the residue of $H_{*}$ in $k[X, Y] /\left(F_{*}, G_{*}\right)$ is zero, i.e., $H_{*}=a F_{*}+$ $b G_{*}, a, b \in k[X, Y]$. Then $Z^{r} H=A F+B G$ for some $r, A, B$ (Proposition 7 of $\$ 2.10$ ). But in the proof of Step 2 of Bézout's Theorem we saw that multiplication by $Z$ on $k[X, Y, Z] /(F, G)$ is one-to-one, so $H=A^{\prime} F+B^{\prime} G$ for some $A^{\prime}, B^{\prime}$. If $A^{\prime}=\sum A_{i}^{\prime}$, $B^{\prime}=\sum B_{i}^{\prime}, A_{i}^{\prime}, B_{i}^{\prime}$ forms of degree $i$, then $H=A_{s}^{\prime} F+B_{t}^{\prime} G, s=\operatorname{deg}(H)-\operatorname{deg}(F), t=$ $\operatorname{deg}(H)-\operatorname{deg}(G)$.
Of course, the usefulness of this theorem depends on finding criteria that assure that Noether's conditions hold at $P$ :
Proposition 1. Let $F, G, H$ be plane curves, $P \in F \cap G$. Then Noether's conditions are satisfied at $P$ if any of the following are true:
(1) $F$ and $G$ meet transversally at $P$, and $P \in H$. (2) $P$ is a simple point on $F$, and $I(P, H \cap F) \geq I(P, G \cap F)$.
(3) $F$ and $G$ have distinct tangents at $P$, and $m_{P}(H) \geq m_{P}(F)+m_{P}(G)-1$.
Proof. (2): $I(P, H \cap F) \geq I(P, G \cap F)$ implies that $\operatorname{ord}_{P}^{F}(H) \geq \operatorname{ord}_{P}^{F}(G)$, so $\bar{H}_{*} \in\left(\bar{G}_{*}\right) \subset$ $\mathscr{O}_{P}(F)$. Since $\mathscr{O}_{P}(F) /\left(\bar{G}_{*}\right) \cong \mathscr{O}_{P}\left(\mathbb{P}^{2}\right) /\left(F_{*}, G_{*}\right)$ (Problem 2.44), the residue of $H_{*}$ in $\mathscr{O}_{P}\left(\mathbb{P}^{2}\right) /\left(F_{*}, G_{*}\right)$ is zero, as desired.
(3): We may assume $P=[0: 0: 1]$, and $m_{P}\left(H_{*}\right) \geq m_{P}\left(F_{*}\right)+m_{P}\left(G_{*}\right)-1$. In the notation of the lemma used to prove Property (5) of the intersection number (\\$3.3), this says that $H_{*} \in I^{t}, t \geq m+n-1$. And in that lemma, we showed precisely that $I^{t} \subset\left(F_{*}, G_{*}\right) \subset \mathscr{O}_{P}\left(\mathbb{P}^{2}\right)$ if $t \geq m_{P}(F)+m_{P}(G)-1$.
(1) is a special case both of (2) and of (3) (and is easy by itself).
Corollary. If either
(1) $F$ and $G$ meet in $\operatorname{deg}(F) \operatorname{deg}(G)$ distinct points, and $H$ passes through these points, or
(2) All the points of $F \cap G$ are simple points of $F$, and $H \cdot F \geq G \cdot F$,
then there is a curve $B$ such that $B \cdot F=H \cdot F-G \cdot F$.
In $\S 7.5$ we will find a criterion that works at all ordinary multiple points of $F$.
## Problems
5.29. Fix $F, G$, and $P$. Show that in cases (1) and (2) - but not (3) - of Proposition 1 the conditions on $H$ are equivalent to Noether's conditions.
5.30. Let $F$ be an irreducible projective plane curve. Suppose $z \in k(F)$ is defined at every $P \in F$. Show that $z \in k$. (Hint: Write $z=H / G$, and use Noether's Theorem).
### Applications of Noether's Theorem
We indicate in this section a few of the many interesting consequences of Noether's Theorem. Since they will not be needed in later Chapters, the proofs will be brief.
Proposition 2. Let $C, C^{\prime}$ be cubics, $C^{\prime} \cdot C=\sum_{i=1}^{9} P_{i}$; suppose $Q$ is a conic, and $Q \cdot C=$ $\sum_{i=1}^{6} P_{i}$. Assume $P_{1}, \ldots, P_{6}$ are simple points on $C$. Then $P_{7}, P_{8}$, and $P_{9}$ lie on a straight line.
Proof. Let $F=C, G=Q, H=C^{\prime}$ in (2) of the Corollary to Proposition 1.
Corollary 1 (Pascal). If a hexagon is inscribed in an irreducible conic, then the opposite sides meet in collinear points.
Proof. Let $C$ be three sides, $C^{\prime}$ the three opposite sides, $Q$ the conic, and apply Proposition 2.
Corollary 2 (Pappus). Let $L_{1}, L_{2}$ be two lines; $P_{1}, P_{2}, P_{3} \in L_{1}, Q_{1}, Q_{2}, Q_{3} \in L_{2}$ (none of these points in $L_{1} \cap L_{2}$ ). Let $L_{i j}$ be the line between $P_{i}$ and $Q_{j}$. For each $i, j, k$ with $\{i, j, k\}=\{1,2,3\}$, let $R_{k}=L_{i j} . L_{j i}$. Then $R_{1}, R_{2}$, and $R_{3}$ are collinear. Proof. The two lines form a conic, and the proof is the same as in Corollary 1.
Pascal's Theorem
Pappus' Theorem
Proposition 3. Let $C$ be an irreducible cubic, $C^{\prime}, C^{\prime \prime}$ cubics. Suppose $C^{\prime} \cdot C=\sum_{i=1}^{9} P_{i}$, where the $P_{i}$ are simple (not necessarily distinct) points on $C$, and suppose $C^{\prime \prime} \cdot C=$ $\sum_{i=1}^{8} P_{i}+Q$. Then $Q=P_{9}$.
Proof. Let $L$ be a line through $P_{9}$ that doesn't pass through $Q ; L \cdot C=P_{9}+R+S$. Then $L C^{\prime \prime} \cdot C=C^{\prime} \cdot C+Q+R+S$, so there is a line $L^{\prime}$ such that $L^{\prime} \cdot C=Q+R+S$. But then $L^{\prime}=L$ and so $P_{9}=Q$.
Addition on a cubic. Let $C$ be a nonsingular cubic. For any two points $P, Q \in C$, there is a unique line $L$ such that $L \cdot C=P+Q+R$, for some $R \in C$. (If $P=Q, L$ is the tangent to $C$ at $P$ ). Define $\varphi: C \times C \rightarrow C$ by setting $\varphi(P, Q)=R$. This $\varphi$ is like an addition on $C$, but there is no identity. To remedy this, choose a point $O$ on $C$. Then define an addition $\oplus$ on $C$ as follows: $P \oplus Q=\varphi(O, \varphi(P, Q))$.
Proposition 4. $C$, with the operation $\oplus$, forms an abelian group, with the point $O$ being the identity.
Proof. Only the associativity is difficult: Suppose $P, Q, R \in C$. Let $L_{1} \cdot C=P+Q+S^{\prime}$, $M_{1} \cdot C=O+S^{\prime}+S, L_{2} \cdot C=S+R+T^{\prime}$. Let $M_{2} \cdot C=Q+R+U^{\prime}, L_{3} \cdot C=O+U^{\prime}+U, M_{3} \cdot C=P+U+T^{\prime \prime}$. Since $(P \oplus Q) \oplus R=$ $\varphi\left(O, T^{\prime}\right)$, and $P \oplus(Q \oplus R)=\varphi\left(O, T^{\prime \prime}\right)$, it suffices to show that $T^{\prime}=T^{\prime \prime}$.
Let $C^{\prime}=L_{1} L_{2} L_{3}, C^{\prime \prime}=M_{1} M_{2} M_{3}$, and apply Proposition 3 .
## Problems
5.31. If in Pascal's Theorem we let some adjacent vertices coincide the side being a tangent), we get many new theorems:
(a) State and sketch what happens if $P_{1}=P_{2}, P_{3}=P_{4}, P_{5}=P_{6}$.
(b) Let $P_{1}=P_{2}$, the other four distinct.
(c) From (b) deduce a rule for constructing the tangent to a given conic at a given point, using only a straight-edge.
5.32. Suppose the intersections of the opposite sides of a hexagon lie on a straight line. Show that the vertices lie on a conic.
5.33. Let $C$ be an irreducible cubic, $L$ a line such that $L \cdot C=P_{1}+P_{2}+P_{3}, P_{i}$ distinct. Let $L_{i}$ be the tangent line to $C$ at $P_{i}: L_{i} \bullet C=2 P_{i}+Q_{i}$ for some $Q_{i}$. Show that $Q_{1}, Q_{2}, Q_{3}$ lie on a line. ( $L^{2}$ is a conic!)
5.34. Show that a line through two flexes on a cubic passes through a third flex.
5.35. Let $C$ be any irreducible cubic, or any cubic without multiple components, $C^{\circ}$ the set of simple points of $C, O \in C^{\circ}$. Show that the same definition as in the nonsingular case makes $C^{\circ}$ into an abelian group.
5.36. Let $C$ be an irreducible cubic, $O$ a simple point on $C$ giving rise to the addition $\oplus$ on the set $C^{\circ}$ of simple points. Suppose another $O^{\prime}$ gives rise to an addition $\oplus^{\prime}$. Let $Q=\varphi\left(O, O^{\prime}\right)$, and define $\alpha:(C, O, \oplus) \rightarrow\left(C, O^{\prime}, \oplus^{\prime}\right)$ by $\alpha(P)=\varphi(Q, P)$. Show that $\alpha$ is a group isomorphism. So the structure of the group is independent of the choice of O.
5.37. In Proposition 4, suppose $O$ is a flex on $C$. (a) Show that the flexes form a subgroup of $C$; as an abelian group, this subgroup is isomorphic to $\mathbb{Z} /(3) \times \mathbb{Z} /(3)$. (b) Show that the flexes are exactly the elements of order three in the group. (i.e., exactly those elements $P$ such that $P \oplus P \oplus P=O$ ). (c) Show that a point $P$ is of order two in the group if and only if the tangent to $C$ at $P$ passes through $O$. (d) Let $C=Y^{2} Z-X(X-Z)(X-\lambda Z), \lambda \neq 0,1, O=[0: 1: 0]$. Find the points of order two. (e) Show that the points of order two on a nonsingular cubic form a group isomorphic to $\mathbb{Z} /(2) \times \mathbb{Z} /(2)$. (f) Let $C$ be a nonsingular cubic, $P \in C$. How many lines through $P$ are tangent to $C$ at some point $Q \neq P$ ? (The answer depends on whether $P$ is a flex.)
5.38. Let $C$ be a nonsingular cubic given by the equation $Y^{2} Z=X^{3}+a X^{2} Z+b X Z^{2}+$ $c Z^{3}, O=[0: 1: 0]$. Let $P_{i}=\left[x_{i}: y_{i}: 1\right], i=1,2,3$, and suppose $P_{1} \oplus P_{2}=P_{3}$. If $x_{1} \neq x_{2}$, let $\lambda=\left(y_{1}-y_{2}\right) /\left(x_{1}-x_{2}\right)$; if $P_{1}=P_{2}$ and $y_{1} \neq 0$, let $\lambda=\left(3 x_{1}^{2}+2 a x_{1}+b\right) /\left(2 y_{1}\right)$. Let $\mu=y_{i}-\lambda x_{i}, i=1,2$. Show that $x_{3}=\lambda^{2}-a-x_{1}-x_{2}$, and $y_{3}=-\lambda x_{3}-\mu$. This gives an explicit method for calculating in the group.
5.39. (a) Let $C=Y^{2} Z-X^{3}-4 X Z^{2}, O=[0: 1: 0], A=[0: 0: 1], B=[2: 4: 1]$, and $C=[2:-4: 1]$. Show that $\{0, A, B, C\}$ form a subgroup of $C$ that is cyclic of order 4 . (b) Let $C=Y^{2} Z-X^{3}-43 X Z^{2}-166 Z^{3}$. Let $O=[0: 1: 0], P=[3: 8: 1]$. Show that $P$ is an element of order 7 in $C$.
5.40. Let $k_{0}$ be a subfield of $k$. If $V$ is an affine variety, $V \subset \mathbb{A}^{n}(k)$, a point $P=$ $\left(a_{1}, \ldots, a_{n}\right) \in V$ is rational over $k_{0}$, if each $a_{i} \in k_{0}$. If $V \subset \mathbb{P}^{n}(k)$ is projective, a point $P \in V$ is rational over $k_{0}$ if for some homogeneous coordinates $\left(a_{1}, \ldots, a_{n+1}\right)$ for $P$, each $a_{i} \in k_{0}$.
A curve $F$ of degree $d$ is said to be emphrational over $k_{0}$ if the corresponding point in $\mathbb{P}^{d(d+3) / 2}$ is rational over $k_{0}$.
Suppose a nonsingular cubic $C$ is rational over $k_{0}$. Let $C\left(k_{0}\right)$ be the set of points of $C$ that are rational over $k_{0}$. (a) If $P, Q \in C\left(k_{0}\right)$, show that $\varphi(P, Q)$ is in $C\left(k_{0}\right)$. (b) If $O \in C\left(k_{0}\right)$, show that $C\left(k_{0}\right)$ forms a subgroup of $C$. (If $k_{0}=\mathbb{Q}, k=\mathbb{C}$, this has important applications to number theory.)
5.41. Let $C$ be a nonsingular cubic, $O$ a flex on $C$. Let $P_{1}, \ldots, P_{3 m} \in C$. Show that $P_{1} \oplus \cdots \oplus P_{3 m}=O$ if and only if there is a curve $F$ of degree $m$ such that $F \cdot C=\sum_{i=1}^{3 m} P_{i}$. (Hint: Use induction on $m$. Let $L \cdot C=P_{1}+P_{2}+Q, L^{\prime} \cdot C=P_{3}+P_{4}+R, L^{\prime \prime} \cdot C=Q+R+S$, and apply induction to $S, P_{5}, \ldots, P_{3 m}$; use Noether's Theorem.)
5.42. Let $C$ be a nonsingular cubic, $F, F^{\prime}$ curves of degree $m$ such that $F \cdot C=\sum_{i=1}^{3 m} P_{i}$, $F^{\prime} \cdot C=\sum_{i=1}^{3 m-1} P_{i}+Q$. Show that $P_{3 m}=Q$.
5.43. For which points $P$ on a nonsingular cubic $C$ does there exist a nonsingular conic that intersects $C$ only at $P$ ?
## Chapter 6
## Varieties, Morphisms, and Rational Maps
This chapter begins the study of intrinsic properties of a variety - properties that do not depend on its embedding in affine or projective spaces (or products of these). Making this transition from extrinsic to intrinsic geometry has not been easy historically; the abstract language required demands some fortitude from the reader.
### The Zariski Topology
One of the purposes of considering a topology on a set is to be able to restrict attention to a "neighborhood" of a point in the set. Often this means simply that we throw away a set (not containing the point) on which something we don't like happens. For example, if $z$ is a rational function on a variety $V$, and $z$ is defined at $P \in V$, there should be a neighborhood of $P$ where $z$ is a function - we must throw away the pole set of $z$. We want to be able to discard an algebraic subset from an affine or projective variety, and still think of what is left as some kind of variety.
We first recall some notions from topology. A topology on a set $X$ is a collection of subsets of $X$, called the open subsets of $X$, satisfying:
(1) $X$ and the empty set $\varnothing$ are open.
(2) The union of any family of open subsets of $X$ is open.
(3) The intersection of any finite number of open sets is open.
A topological space is a set $X$ together with a topology on $X$. A set $C$ in $X$ is closed if $X \backslash C$ is open. If $Y \subset X$, any open set of $X$ that contains $Y$ will be called a neighborhood of $Y$. (Sometimes any set containing an open set containing $Y$ is called a neighborhood of $Y$, but we will consider only open neighborhoods.)
If $Y$ is a subset of a topological space $X$, the induced topology on $Y$ is defined as follows: a set $W \subset Y$ is open in $Y$ if there is an open subset $U$ of $X$ such that $W=Y \cap U$. For any subset $Y$ of a topological space $X$, the closure of $Y$ in $X$ is the intersection of all closed subsets of $X$ that contain $Y$. The set $Y$ is said to be dense in $X$ if $X$ is the closure of $Y$ in $X$; equivalently, for every nonempty open subset $U$ of $X, U \cap Y \neq \varnothing$.
If $X$ and $X^{\prime}$ are topological spaces, a mapping $f: X^{\prime} \rightarrow X$ is called continuous if for every open set $U$ of $X, f^{-1}(U)=\left\{x \in X^{\prime} \mid f(x) \in U\right\}$ is an open subset of $X^{\prime}$; equivalently, for every closed subset $C$ of $X, f^{-1}(C)$ is closed in $X^{\prime}$. If, in addition, $f$ is one-to-one and onto $X$, and $f^{-1}$ is continuous, $f$ is said to be homeomorphism.
Let $X=\mathbb{P}^{n_{1}} \times \cdots \times \mathbb{P}^{n_{r}} \times \mathbb{A}^{m}$. The Zariski topology on $X$ is defined as follows: a set $U \subset X$ is open if $X \backslash U$ is an algebraic subset of $X$. That this is a topology follows from the properties of algebraic sets proved in Chapter 1 (see also $\$ 4.4$ ). Any subset $V$ of $X$ is given the induced topology. In particular, if $V$ is a variety in $X$, a subset of $V$ is closed if and only if it is algebraic.
If $X=\mathbb{A}^{1}$ or $\mathbb{P}^{1}$, the proper closed subsets of $X$ are just the finite subsets. If $X=\mathbb{A}^{2}$ or $\mathbb{P}^{2}$, proper closed subsets are finite unions of points and curves.
Note that for any two nonempty open sets $U_{1}, U_{2}$ in a variety $V, U_{1} \cap U_{2} \neq \varnothing$ (for otherwise $V=\left(V \backslash U_{1}\right) \cup\left(V \backslash U_{2}\right)$ would be reducible). So if $P$ and $Q$ are distinct points of $V$, there are never disjoint neighborhoods containing them. And every nonempty open subset of a variety $V$ is dense in $V$.
## Problems
6.1. Let $Z \subset Y \subset X, X$ a topological space. Give $Y$ the induced topology. Show that the topology induced by $Y$ on $Z$ is the same as that induced by $X$ on $Z$.
6.2* (a) Let $X$ be a topological space, $X=\bigcup_{\alpha \in \mathscr{A}} U_{\alpha}, U_{\alpha}$ open in $X$. Show that a subset $W$ of $X$ is closed if and only if each $W \cap U_{\alpha}$ is closed (in the induced topology) in $U_{\alpha}$. (b) Suppose similarly $Y=\bigcup_{\alpha \in \mathscr{A}} V_{\alpha}, V_{\alpha}$ open in $Y$, and suppose $f: X \rightarrow Y$ is a mapping such that $f\left(U_{\alpha}\right) \subset V_{\alpha}$. Show that $f$ is continuous if and only if the restriction of $f$ to each $U_{\alpha}$ is a continuous mapping from $U_{\alpha}$ to $V_{\alpha}$.
6.3. (a) Let $V$ be an affine variety, $f \in \Gamma(V)$. Considering $f$ as a mapping from $V$ to $k=\mathbb{A}^{1}$, show that $f$ is continuous. (b) Show that any polynomial map of affine varieties is continuous.
6.4. Let $U_{i} \subset \mathbb{P}^{n}, \varphi_{i}: \mathbb{A}^{n} \rightarrow U_{i}$ as in Chapter 4. Give $U_{i}$ the topology induced from $\mathbb{P}^{n}$. (a) Show that $\varphi_{i}$ is a homeomorphism. (b) Show that a set $W \subset \mathbb{P}^{n}$ is closed if and only if each $\varphi_{i}^{-1}(W)$ is closed in $\mathbb{A}^{n}, i=1, \ldots, n+1$. (c) Show that if $V \subset \mathbb{A}^{n}$ is an affine variety, then the projective closure $V^{*}$ of $V$ is the closure of $\varphi_{n+1}(V)$ in $\mathbb{P}^{n}$.
6.5. Any infinite subset of a plane curve $V$ is dense in $V$. Any one-to-one mapping from one irreducible plane curve onto another is a homeomorphism.
6.6. Let $X$ be a topological space, $f: X \rightarrow \mathbb{A}^{n}$ a mapping. Then $f$ is continuous if and only if for each hypersurface $V=V(F)$ of $\mathbb{A}^{n}, f^{-1}(V)$ is closed in $X$. A mapping $f: X \rightarrow k=\mathbb{A}^{1}$ is continuous if and only if $f^{-1}(\lambda)$ is closed for any $\lambda \in k$.
6.7. Let $V$ be an affine variety, $f \in \Gamma(V)$. (a)] Show that $V(f)=\{P \in V \mid f(P)=0\}$ is a closed subset of $V$, and $V(f) \neq V$ unless $f=0$. (b) Suppose $U$ is a dense subset of $V$ and $f(P)=0$ for all $P \in U$. Then $f=0$. 6.8* Let $U$ be an open subset of a variety $V, z \in k(V)$. Suppose $z \in \mathscr{O}_{P}(V)$ for all $P \in U$. Show that $U_{z}=\{P \in U \mid z(P) \neq 0\}$ is open, and that the mapping from $U$ to $k=\mathbb{A}^{1}$ defined by $P \mapsto z(P)$ is continuous.
### Varieties
Let $V$ be a nonempty irreducible algebraic set in $\mathbb{P}^{n_{1}} \times \cdots \times \mathbb{A}^{m}$. Any open subset $X$ of $V$ will be called a variety. It is given the topology induced from $V$; this topology is called the Zariski topology on $X$.
We define $k(X)=k(V)$ to be the field of rational functions on $X$, and if $P \in X$, we define $\mathscr{O}_{P}(X)$ to be $\mathscr{O}_{P}(V)$, the local ring of $X$ at $P$.
If $U$ is an open subset of $X$, then $U$ is also open in $V$, so $U$ is also a variety. We say that $U$ is an open subvariety of $X$.
If $Y$ is a closed subset of $X$, we say that $Y$ is irreducible if $Y$ is not the union of two proper closed subsets. Then $Y$ is then also a variety, for if $\bar{Y}$ is the closure of $Y$ in $V$, it is easy to verify that $\bar{Y}$ is irreducible in $V$ and that $Y=\bar{Y} \cap X$, so $Y$ is open in $\bar{Y}$ (see Problem 6.10). Such a $Y$ is called a closed subvariety of $X$.
Let $X$ be a variety, $U$ a nonempty open subset of $X$. We define $\Gamma\left(U, \mathscr{O}_{X}\right)$, or simply $\Gamma(U)$, to be the set of rational functions on $X$ that are defined at each $P \in U$ : $\Gamma(U)=\bigcap_{P \in U} \mathscr{O}_{P}(X)$. The ring $\Gamma(U)$ is a subring of $k(X)$, and if $U^{\prime} \subset U$, then $\Gamma\left(U^{\prime}\right) \supset$ $\Gamma(U)$. Note that if $U=X$ is an affine variety, then $\Gamma(X)$ is the coordinate ring of $X$ (Proposition 2 of $\S 2.4$ ), so this notation is consistent.
If $z \in \Gamma(U), z$ determines a $k$-valued function on $U$ : for if $P \in U, z \in \mathscr{O}_{P}(X)$, and $z(P)$ is well-defined. Let $\mathscr{F}(U, k)$ be the ring of all $k$-valued functions on $U$. The map that associates a function to each $z \in \Gamma(U)$ is a ring homomorphism from $\Gamma(U)$ into $\mathscr{F}(U, k)$. As in $\S 2.1$ we want to identify $\Gamma(U)$ with its image in $\mathscr{F}(U, k)$, so that we may consider $\Gamma(U)$ as a ring of functions on $U$. For this we need the map from $\Gamma(U)$ to $\mathscr{F}(U, k)$ to be one-to-one, i.e.,
Proposition 1. Let $U$ be an open subset of a variety $X$. Suppose $z \in \Gamma(U)$, and $z(P)=0$ for all $P \in U$. Then $z=0$.
Proof. Note first that we may replace $U$ by any nonempty open subset $U^{\prime}$ of $U$, since $\Gamma(U) \subset \Gamma\left(U^{\prime}\right)$.
If $X \subset \mathbb{P}^{n} \times \cdots \times \mathbb{A}^{m}$, we may replace $X$ by its closure, so assume $X$ is closed. Then if $X \cap\left(U_{i_{1}} \times U_{i_{2}} \times \cdots \times \mathbb{A}^{m}\right) \neq \varnothing$, we may replace $X$ and $U$ by the corresponding affine variety $\varphi^{-1}(X)$ and the open set $\varphi^{-1}(U)$ in $\mathbb{A}^{n} \times \cdots \times \mathbb{A}^{m}$, where $\varphi: \mathbb{A}^{n} \times \cdots \times \mathbb{A}^{m} \rightarrow$ $U_{i_{1}} \times \cdots \times \mathbb{A}^{m}$ is as in Problem 4.26.
Thus we may assume $U$ is open in an affine variety $X \subset \mathbb{A}^{N}$. Write $z=f / g$, $f, g \in \Gamma(X)$. Replacing $U$ by $\{P \in U \mid g(P) \neq 0\}$, we may assume $g(P) \neq 0$ for all $P \in U$ (Problem 6.8). Then $f(P)=0$ for all $P \in U$, so $f=0$ (Problem 6.7), and $z=0$.
## Problems
6.9. Let $X=\mathbb{A}^{2} \backslash\{(0,0)\}$, an open subvariety of $\mathbb{A}^{2}$. Show that $\Gamma(X)=\Gamma\left(\mathbb{A}^{2}\right)=k[X, Y]$. 6.10* Let $U$ be an open subvariety of a variety $X, Y$ a closed subvariety of $U$. Let $Z$ be the closure of $Y$ in $X$. Show that
(a) $Z$ is a closed subvariety of $X$.
(b) $Y$ is an open subvariety of $Z$.
6.11. (a) Show that every family of closed subsets of a variety has a minimal member. (b) Show that if a variety is a union of a collection of open subsets, it is a union of a finite number of theses subsets. (All varieties are "quasi-compact".)
6.12* Let $X$ be a variety, $z \in k(X)$. Show that the pole set of $z$ is closed. If $z \in \mathscr{O}_{P}(X)$, there is a neighborhood $U$ of $z$ such that $z \in \Gamma(U)$; so $\mathscr{O}_{P}(X)$ is the union of all $\Gamma(U)$, where $U$ runs through all neighborhoods of $P$.
### Morphisms of Varieties
If $\varphi: X \rightarrow Y$ is any mapping between sets, composition with $\varphi$ gives a homomorphism of rings $\tilde{\varphi}: \mathscr{F}(Y, k) \rightarrow \mathscr{F}(X, k)$; i.e., $\tilde{\varphi}(f)=f \circ \varphi$.
Let $X$ and $Y$ be varieties. A morphism from $X$ to $Y$ is a mapping $\varphi: X \rightarrow Y$ such that
(1) $\varphi$ is continuous;
(2) For every open set $U$ of $Y$, if $f \in \Gamma\left(U, \mathscr{O}_{Y}\right)$, then $\tilde{\varphi}(f)=f \circ \varphi$ is in $\Gamma\left(\varphi^{-1}(U), \mathscr{O}_{X}\right)$.
An isomorphism of $X$ with $Y$ is a one-to-one morphism $\varphi$ from $X$ onto $Y$ such that $\varphi^{-1}$ is a morphism.
A variety that is isomorphic to a closed subvariety of some $\mathbb{A}^{n}$ (resp. $\left.\mathbb{P}^{n}\right)$ is called an affine variety (resp. a projective variety). When we write " $X \subset \mathbb{A}^{n}$ is an affine variety", we mean that $X$ is a closed subvariety of $\mathbb{A}^{n}$ (as in Chapter 2), while if we say only " $X$ is an affine variety" we mean that $X$ is a variety in the general sense of Section 2, but that there exists an isomorphism of $X$ with a closed subvariety of some $\mathbb{A}^{n}$. A similar nomenclature is used for projective varieties.
Proposition 2. Let $X$ and $Y$ be affine varieties. There is a natural one-to-one correspondence between morphisms $\varphi: X \rightarrow Y$ and homomorphisms $\tilde{\varphi}: \Gamma(Y) \rightarrow \Gamma(X)$. If $X \subset \mathbb{A}^{n}, Y \subset \mathbb{A}^{m}$, a morphism from $X$ to $Y$ is the same thing as a polynomial map from $X$ to $Y$.
Proof. We may assume $X \subset \mathbb{A}^{n}, Y \subset \mathbb{A}^{m}$ are closed subvarieties of affine spaces. The proposition follows from the following facts: (i) a polynomial map is a morphism; (ii) a morphism $\varphi$ induces a homomorphism $\tilde{\varphi}: \Gamma(Y) \rightarrow \Gamma(X)$; (iii) any $\tilde{\varphi}: \Gamma(Y) \rightarrow \Gamma(X)$ is induced by a unique polynomial map from $X$ to $Y$ (Proposition 1 of $\$ 2.2$ ); and (iv) all these operations are compatible. The details are left to the reader.
Proposition 3. Let $V$ be a closed subvariety of $\mathbb{P}^{n}, \varphi_{i}: \mathbb{A}^{n} \rightarrow U_{i} \subset \mathbb{P}^{n}$ as in Chapter 4, Section 1. Then $V_{i}=\varphi_{i}^{-1}(V)$ is a closed subvariety of $\mathbb{A}^{n}$, and $\varphi_{i}$ restricts to an isomorphism of $V_{i}$ with $V \cap U_{i}$. A projective variety is a union of a finite number of open affine varieties.
Proof. The proof, together with the natural generalization to multispace (see Problem 4.26), is left to the reader. Proposition 4. Any closed subvariety of $\mathbb{P}^{n_{1}} \times \cdots \times \mathbb{P}^{n_{r}}$ is a projective variety. Any variety is isomorphic to an open subvariety of a projective variety.
Proof. The second statement follows from the first, since $\mathbb{P}^{n_{1}} \times \cdots \times \mathbb{P}^{n_{r}} \times \mathbb{A}^{m}$ is isomorphic to an open subvariety of $\mathbb{P}^{n_{1}} \times \cdots \times \mathbb{P}^{n_{r}} \times \mathbb{P}^{m}$. By induction, it is enough to prove that $\mathbb{P}^{n} \times \mathbb{P}^{m}$ is a projective variety.
In Problem 4.28, we defined the Segre imbedding $S: \mathbb{P}^{n} \times \mathbb{P}^{m} \rightarrow \mathbb{P}^{n+m+n m}$, which mapped $\mathbb{P}^{n} \times \mathbb{P}^{m}$ one-to-one onto a projective variety $V$. We use the notations of that problem.
It suffices to show that the restriction of $S$ to $U_{0} \times U_{0} \rightarrow V \cap U_{00}$ is an isomorphism. These are affine varieties, so it is enough to show that the induced map on coordinate rings is an isomorphism. We identify $\Gamma\left(U_{0} \times U_{0}\right)$ with $k\left[X_{1}, \ldots, X_{n}, Y_{1}, \ldots, Y_{m}\right]$, and $\Gamma\left(V \cap U_{00}\right)$ may be identified with $k\left[T_{10}, \ldots, T_{n m}\right] /\left(\left\{T_{j k}-T_{j 0} T_{0 k} \mid j, k>0\right\}\right)$. The homomorphism from $k\left[X_{1}, \ldots, X_{n}, Y_{1}, \ldots, Y_{m}\right]$ to this ring that takes $X_{i}$ to the residue of $T_{i 0}, Y_{j}$ to that of $T_{0 j}$, is easily checked to be an isomorphism. Since this isomorphism is the one induced by $S^{-1}$, the proof is complete.
Note. It is possible to define more general varieties than those we have considered here. If this were done, the varieties we have defined would be called the "quasiprojective" varieties.
A closed subvariety of an affine variety is also an affine variety. What is more surprising is that an open subvariety of an affine variety may also be affine.
Proposition 5. Let $V$ be an affine variety, and let $f \in \Gamma(V), f \neq 0$. Let $V_{f}=\{P \in V \mid$ $f(P) \neq 0\}$, an open subvariety of $V$. Then
(1) $\Gamma\left(V_{f}\right)=\Gamma(V)[1 / f]=\left\{a / f^{n} \in k(V) \mid a \in \Gamma(V), n \in \mathbb{Z}\right\}$.
(2) $V_{f}$ is an affine variety.
Proof. We may assume $V \subset \mathbb{A}^{n}$; let $I=I(V)$, so $\Gamma(V)=k\left[X_{1}, \ldots, X_{n}\right] / I$. Choose $F \in$ $k\left[X_{1}, \ldots, X_{n}\right]$ whose $I$-residue $\bar{F}$ is $f$.
(1): Let $z \in \Gamma\left(V_{f}\right)$. The pole set of $z$ is $V(J)$, where $J=\left\{G \in k\left[X_{1}, \ldots, X_{n}\right] \mid \bar{G} z \in\right.$ $\Gamma(V)\}$ (proof of Proposition 2 of $\$ 2.4$ ). Since $V(J) \subset V(F), F^{N} \in J$ for some $N$, by the Nullstellensatz. Then $f^{N} z=a \in \Gamma(V)$, so $z=a / f^{N} \in \Gamma(V)[1 / f]$. The other inclusion is obvious.
(2): We must "push the zeros of $F$ off to infinity" (compare with the proof of the Nullstellensatz). Let $I^{\prime}$ be the ideal in $k\left[X_{1}, \ldots, X_{n+1}\right]$ generated by $I$ and by $X_{n+1} F-1$, $V^{\prime}=V\left(I^{\prime}\right) \subset \mathbb{A}^{n+1}$.
Let $\alpha: k\left[X_{1}, \ldots, X_{n+1}\right] \rightarrow \Gamma\left(V_{f}\right)$ be defined by letting $\alpha\left(X_{i}\right)=\bar{X}_{i}$ if $i \leq n, \alpha\left(X_{n+1}\right)=$ $1 / f$. Then $\alpha$ is onto by (1), and it is left to the reader to check that $\operatorname{Ker}(\alpha)=I^{\prime}$. (See Problem 6.13.) In particular, $I^{\prime}$ is prime, so $V^{\prime}$ is a variety, and $\alpha$ induces an isomor$\operatorname{phism} \bar{\alpha}: \Gamma\left(V^{\prime}\right) \rightarrow \Gamma\left(V_{f}\right)$.
The projection $\left(X_{1}, \ldots, X_{n+1}\right) \mapsto\left(X_{1}, \ldots, X_{n}\right)$ from $\mathbb{A}^{n+1}$ to $\mathbb{A}^{n}$ induces a morphism $\varphi: V^{\prime} \rightarrow V_{f}$ (Problem 6.16). This $\varphi$ is one-to-one and onto, and $\tilde{\varphi}=(\bar{\alpha})^{-1}$. If $W$ is closed in $V$, defined by the vanishing of polynomials $G_{\beta}\left(X_{1}, \ldots, X_{n+1}\right)$, then $\varphi(W)$ is closed in $V_{f}$, defined by polynomials $F^{N} G_{\beta}\left(X_{1}, \ldots, X_{n}, 1 / F\right)$, with $N \geq \operatorname{deg}\left(G_{\beta}\right)$; from this it follows that $\varphi^{-1}$ is continuous. We leave it to the reader to complete the proof that $\varphi^{-1}$ is a morphism, and hence $\varphi$ is an isomorphism. Corollary. Let $X$ be a variety, $U$ a neighborhood of a point $P$ in $X$. Then there is a neighborhood $V$ of $P, V \subset U$, such that $V$ is an affine variety.
Proof. If $X$ is open in a projective variety $X^{\prime} \subset \mathbb{P}^{n}$, and $P \in U_{i}$, we may replace $X$ by $X^{\prime} \cap U_{i}, U$ by $U \cap U_{i}$. So we may assume $X \subset \mathbb{A}^{n}$ is affine.
Since $X \backslash U$ is an algebraic subset of $\mathbb{A}^{n}$, there is a polynomial $F \in k\left[X_{1}, \ldots, X_{n}\right]$ such that $F(P) \neq 0$, and $F(Q)=0$ for all $Q \in X \backslash U$ (Problem 1.17). Let $f$ be the image of $F$ in $\Gamma(X)$. Then $P \in X_{f} \subset U$, and $X_{f}$ is affine by the proposition.
## Problems
6.13. Let $R$ be a domain with quotient field $K, f \neq 0$ in $R$. Let $R[1 / f]=\left\{a / f^{n}\right.$ | $a \in R, n \in \mathbb{Z}$ \}, a subring of $K$. (a) Show that if $\varphi: R \rightarrow S$ is any ring homomorphism such that $\varphi(f)$ is a unit in $S$, then $\varphi$ extends uniquely to a ring homomorphism from $R[1 / f]$ to $S$. (b) Show that the ring homomorphism from $R[X] /(X f-1)$ to $R[1 / f]$ that takes $X$ to $1 / f$ is an isomorphism.
6.14. Let $X, Y$ be varieties, $f: X \rightarrow Y$ a mapping. Let $X=\bigcup_{\alpha} U_{\alpha}, Y=\bigcup_{\alpha} V_{\alpha}$, with $U_{\alpha}, V_{\alpha}$ open subvarieties, and suppose $f\left(U_{\alpha}\right) \subset V_{\alpha}$ for all $\alpha$. (a) Show that $f$ is a morphism if and only if each restriction $f_{\alpha}: U_{\alpha} \rightarrow V_{\alpha}$ of $f$ is a morphism. (b) If each $U_{\alpha}, V_{\alpha}$ is affine, $f$ is a morphism if and only if each $\tilde{f}\left(\Gamma\left(V_{\alpha}\right)\right) \subset \Gamma\left(U_{\alpha}\right)$.
6.15. (a) If $Y$ is an open or closed subvariety of $X$, the inclusion $i: Y \rightarrow X$ is a morphism. (b) The composition of morphisms is a morphism.
6.16. Let $f: X \rightarrow Y$ be a morphism of varieties, $X^{\prime} \subset X, Y^{\prime} \subset Y$ subvarieties (open or closed). Assume $f\left(X^{\prime}\right) \subset Y^{\prime}$. Then the restriction of $f$ to $X^{\prime}$ is a morphism from $X^{\prime}$ to $Y^{\prime}$. (Use Problems 6.14 and 2.9.)
6.17. (a) Show that $\mathbb{A}^{2} \backslash\{(0,0)\}$ is not an affine variety (see Problem 6.9). (b) The union of two open affine subvarieties of a variety may not be affine.
6.18. Show that the natural map $\pi$ from $\mathbb{A}^{n+1} \backslash\{(0, \ldots, 0)\}$ to $\mathbb{P}^{n}$ is a morphism of varieties, and that a subset $U$ of $\mathbb{P}^{n}$ is open if and only if $\pi^{-1}(U)$ is open.
6.19* Let $X$ be a variety, $f \in \Gamma(X)$. Let $\varphi: X \rightarrow \mathbb{A}^{1}$ be the mapping defined by $\varphi(P)=$ $f(P)$ for $P \in X$. (a) Show that for $\lambda \in k, \varphi^{-1}(\lambda)$ is the pole set of $z=1 /(f-\lambda)$. (b) Show that $\varphi$ is a morphism of varieties.
6.20* Let $A=\mathbb{P}^{n_{1}} \times \cdots \times \mathbb{A}^{n}, B=\mathbb{P}^{m_{1}} \times \cdots \times \mathbb{A}^{m}$. Let $y \in B, V$ a closed subvariety of $A$. Show that $V \times\{y\}=\{(x, y) \in A \times B \mid x \in V\}$ is a closed subvariety of $A \times B$, and that the map $V \rightarrow V \times\{y\}$ taking $x$ to $(x, y)$ is an isomorphism.
6.21. Any variety is the union of a finite number of open affine subvarieties.
6.22. Let $X$ be a projective variety in $\mathbb{P}^{n}$, and let $H$ be a hyperplane in $\mathbb{P}^{n}$ that doesn't contain $X$. (a) Show that $X \backslash(H \cap X)$ is isomorphic to an affine variety $X_{*} \subset$ $\mathbb{A}^{n}$. (b) If $L$ is the linear form defining $H$, and $l$ is its image in $\Gamma_{h}(X)=k\left[x_{1}, \ldots, x_{n+1}\right]$, then $\Gamma\left(X_{*}\right)$ may be identified with $k\left[x_{1} / l, \ldots, x_{n+1} / l\right]$. (Hint: Change coordinates so $L=X_{n+1}$.)
6.23. Let $P, Q \in X, X$ a variety. Show that there is an affine open set $V$ on $X$ that contains $P$ and $Q$. (Hint: See the proof of the Corollary to Proposition 5, and use Problem 1.17(c).)
6.24* Let $X$ be a variety, $P, Q$ two distinct points of $X$. Show that there is an $f \in k(X)$ that is defined at $P$ and at $Q$, with $f(P)=0, f(Q) \neq 0$ (Problems 6.23, 1.17). So $f \in$ $\mathfrak{m}_{P}(X), 1 / f \in \mathscr{O}_{Q}(X)$. The local rings $\mathscr{O}_{P}(X)$, as $P$ varies in $X$, are distinct.
6.25. Show that $\left[x_{1}: \ldots: x_{n}\right] \mapsto\left[x_{1}: \ldots: x_{n}: 0\right]$ gives an isomorphism of $\mathbb{P}^{n-1}$ with $H_{\infty} \subset \mathbb{P}^{n}$. If a variety $V$ in $\mathbb{P}^{n}$ is contained in $H_{\infty}, V$ is isomorphic to a variety in $\mathbb{P}^{n-1}$. Any projective variety is isomorphic to a closed subvariety $V \subset \mathbb{P}^{n}$ (for some $n$ ) such that $V$ is not contained in any hyperplane in $\mathbb{P}^{n}$.
### Products and Graphs
Let $A=\mathbb{P}^{n_{1}} \times \cdots \times \mathbb{A}^{n}, B=\mathbb{P}^{m_{1}} \times \cdots \times \mathbb{A}^{m}$ be mixed spaces, as in Chapter 4 , Section 4. Then $A \times B=\mathbb{P}^{n_{1}} \times \cdots \times \mathbb{P}^{m_{1}} \times \cdots \times \mathbb{A}^{n+m}$ is also a mixed space. If $U_{i 1} \times \cdots \times \mathbb{A}^{n}$ and $U_{j 1} \times \cdots \times \mathbb{A}^{m}$ are the usual affine open subvarieties that cover $A$ and $B$, then $U_{i 1} \times \cdots \times \mathbb{A}^{n+m}$ are affine open subvarieties that cover $A \times B$.
Proposition 6. Let $V \subset A, W \subset B$ be closed subvarieties. Then $V \times W$ is a closed subvariety of $A \times B$.
Proof. The only difficulty is in showing that $V \times W$ is irreducible. Suppose $V \times W=$ $Z_{1} \cup Z_{2}, Z_{i}$ closed in $A \times B$. Let $U_{i}=\left\{y \in W \mid V \times\{y\} \not \subset Z_{i}\right\}$. Since $V \times\{y\}$ is irreducible (Problem 6.20), $U_{1} \cap U_{2}=\varnothing$. It suffices to show that each $U_{i}$ is open, for then, since $W$ is a variety, one of the $U_{i}$ (say $U_{1}$ ) must be empty, and then $V \times W \subset Z_{1}$, as desired.
Let $F_{\alpha}(X, Y)$ be the "multiforms" defining $Z_{1}$. If $y \in U_{1}$, then for some $\alpha$ and some $x \in V, F_{\alpha}(x, y) \neq 0$. Let $G_{\alpha}(Y)=F_{\alpha}(x, Y)$. Then $\left\{y^{\prime} \in W \mid G_{\alpha}\left(y^{\prime}\right) \neq 0\right\}$ is an open neighborhood of $y$ in $U_{1}$. A set that contains a neighborhood of each of its points is open, so $U_{1}$ (and likewise $U_{2}$ ) is open.
If $X$ and $Y$ are any varieties, say $X$ is open in $V \subset A, Y$ open in $W \subset B, V, W, A, B$ as above. Then $X \times Y$ is open in $V \times W$, so $X \times Y$ is a variety. Note that the product of two affine varieties is an affine variety, and the product of two projective varieties is a projective variety.
Proposition 7. (1) The projections $\mathrm{pr}_{1}: X \times Y \rightarrow X$ and $\mathrm{pr}_{2}: X \times Y \rightarrow Y$ are morphisms.
(2) If $f: Z \rightarrow X, g: Z \rightarrow Y$ are morphisms, then $(f, g): Z \rightarrow X \times Y$ defined by $(f, g)(z)=(f(z), g(z))$ is a morphism.
(3) If $f: X^{\prime} \rightarrow X, g: Y^{\prime} \rightarrow Y$ are morphisms, then $f \times g: X^{\prime} \times Y^{\prime} \rightarrow X \times Y$ defined by $(f \times g)\left(x^{\prime}, y^{\prime}\right)=\left(f\left(x^{\prime}\right), g\left(y^{\prime}\right)\right)$ is a morphism.
(4) The diagonal $\Delta_{X}=\{(x, y) \in X \times X \mid y=x\}$ is a closed subvariety of $X \times X$, and the diagonal map $\delta_{X}: X \rightarrow \Delta_{X}$ defined by $\delta_{X}(x)=(x, x)$ is an isomorphism.
Proof. (1) is left to the reader.
(2): We may reduce first to the case where $X=A, Y=B$ (Problem 6.16). Since being a morphism is local (Problem 6.14), we may cover $A$ and $B$ by the open affine spaces $U_{i 1} \times \cdots \times \mathbb{A}^{r}$. This reduces it to the case where $X=\mathbb{A}^{n}, Y=\mathbb{A}^{m}$. We may also assume $Z$ is affine, since $Z$ is a union of open affine subvarieties. But this case is trivial, since the product of polynomial maps is certainly a polynomial map.
(3): Apply (2) to the morphism $\left(f \circ \mathrm{pr}_{1}, g \circ \mathrm{pr}_{2}\right)$.
(4): The diagonal in $\mathbb{P}^{n} \times \mathbb{P}^{n}$ is clearly an algebraic subset, so $\Delta_{X}$ is closed in any $X$ (Proposition 4 of $\$ 6.3$ ). The restriction of $\mathrm{pr}_{1}: X \times X \rightarrow X$ is inverse to $\delta_{X}$, so $\delta_{X}$ is an isomorphism.
Corollary. If $f, g: X \rightarrow Y$ are morphisms of varieties, then $\{x \in X \mid f(x)=g(x)\}$ is closed in $X$. If $f$ and $g$ agree on a dense set of $X$, then $f=g$.
Proof. $\{x \mid f(x)=g(x)\}=(f, g)^{-1}\left(\Delta_{Y}\right)$.
If $f: X \rightarrow Y$ is a morphism of varieties, the graph of $f, G(f)$, is defined to be $\{(x, y) \in X \times Y \mid y=f(x)\}$.
Proposition 8. $G(f)$ is a closed subvariety of $X \times Y$. The projection of $X \times Y$ onto $X$ restricts to an isomorphism of $G(f)$ with $X$.
Proof. We have $G(f)=(f \times i)^{-1}\left(\Delta_{Y}\right), i=$ identity on $Y$. Now $(j, f): X \rightarrow X \times Y$, where $j=$ identity on $X$, maps $X$ onto $G(f)$, and this is inverse to the projection.
## Problems
6.26. (a) Let $f: X \rightarrow Y$ be a morphism of varieties such that $f(X)$ is dense in $Y$. Show that the homomorphism $\tilde{f}: \Gamma(Y) \rightarrow \Gamma(X)$ is one-to-one. (b) If $X$ and $Y$ are affine, show that $f(X)$ is dense in $Y$ if and only if $\tilde{f}: \Gamma(Y) \rightarrow \Gamma(X)$ is one-to-one. Is this true if $Y$ is not affine?
6.27. Let $U, V$ be open subvarieties of a variety $X$. (a) Show that $U \cap V$ is isomorphic to $(U \times V) \cap \Delta_{X}$. (b) If $U$ and $V$ are affine, show that $U \cap V$ is affine. (Compare Problem 6.17.)
6.28. Let $d \geq 1, N=\frac{(d+1)(d+2)}{2}$, and let $M_{1}, \ldots, M_{N}$ be the monomials of degree $d$ in $X, Y, Z$ (in some order). Let $T_{1}, \ldots, T_{N}$ be homogeneous coordinates for $\mathbb{P}^{N-1}$. Let $V=V\left(\sum_{i=1}^{N} M_{i}(X, Y, Z) T_{i}\right) \subset \mathbb{P}^{2} \times \mathbb{P}^{N-1}$, and let $\pi: V \rightarrow \mathbb{P}^{N-1}$ be the restriction of the projection map. (a) Show that $V$ is an irreducible closed subvariety of $\mathbb{P}^{2} \times \mathbb{P}^{N-1}$, and $\pi$ is a morphism. (b) For each $t=\left(t_{1}, \ldots, t_{N}\right) \in \mathbb{P}^{N-1}$, let $C_{t}$ be the corresponding curve (\\$5.2). Show that $\pi^{-1}(t)=C_{t} \times\{t\}$.
We may thus think of $\pi: V \rightarrow \mathbb{P}^{N-1}$ as a "universal family" of curves of degree $d$. Every curve appears as a fibre $\pi^{-1}(t)$ over some $t \in \mathbb{P}^{N-1}$.
6.29. Let $V$ be a variety, and suppose $V$ is also a group, i.e., there are mappings $\varphi: V \times V \rightarrow V$ (multiplication or addition), and $\psi: V \rightarrow V$ (inverse) satisfying the group axioms. If $\varphi$ and $\psi$ are morphisms, $V$ is said to be an algebraic group. Show that each of the following is an algebraic group:
(a) $\mathbb{A}^{1}=k$, with the usual addition on $k$; this group is often denoted $\mathbb{G}_{a}$.
(b) $\mathbb{A}^{1} \backslash\{(0)\}=k \backslash\{(0)\}$, with the usual multiplication on $k$ : this is denoted $\mathbb{G}_{m}$.
(c) $\mathbb{A}^{n}(k)$ with addition: likewise $M_{n}(k)=\{n$ by $n$ matrices $\}$ under addition may be identified with $\mathbb{A}^{n^{2}}(k)$. (d) $\mathrm{GL}_{n}(k)=$ \{invertible $n \times n$ matrices $\}$ is an affine open subvariety of $M_{n}(k)$, and a group under multiplication.
(e) $C$ a nonsingular plane cubic, $O \in C, \oplus$ the resulting addition (see Problem 5.38).
6.30. (a) Let $C=V\left(Y^{2} Z-X^{3}\right)$ be a cubic with a cusp, $C^{\circ}=C \backslash\{[0: 0: 1]\}$ the simple points, a group with $O=[0: 1: 0]$. Show that the map $\varphi: \mathbb{G}_{a} \rightarrow C^{\circ}$ given by $\varphi(t)=$ $\left[t: 1: t^{3}\right]$ is an isomorphism of algebraic groups. (b) Let $C=V\left(X^{3}+Y^{3}-X Y Z\right)$ be a cubic with a node, $C^{\circ}=C \backslash\{[0: 0: 1]\}, O=[1: 1: 0]$. Show that $\varphi: \mathbb{G}_{m} \rightarrow C^{\circ}$ defined by $\varphi(t)=\left[t: t^{2}: 1-t^{3}\right]$ is an isomorphism of algebraic groups.
### Algebraic Function Fields and Dimension of Vari- eties
Let $K$ be a finitely generated field extension of $k$. The transcendence degree of $K$ over $k$, written $\operatorname{tr}$. $\operatorname{deg}_{k} K$ is defined to be the smallest integer $n$ such that for some $x_{1}, \ldots, x_{n} \in K, K$ is algebraic over $k\left(x_{1}, \ldots, x_{n}\right)$. We say then that $K$ is an algebraic function field in $n$ variables over $k$.
Proposition 9. Let $K$ be an algebraic function field in one variable over $k$, and let $x \in K, x \notin k$. Then
(1) $K$ is algebraic over $k(x)$.
(2) $(\operatorname{char}(k)=0)$ There is an element $y \in K$ such that $K=k(x, y)$.
(3) If $R$ is a domain with quotient field $K, k \subset R$, and $\mathfrak{p}$ is a prime ideal in $R$, $0 \neq \mathfrak{p} \neq R$, then the natural homomorphism from $k$ to $R / \mathfrak{p}$ is an isomorphism.
Proof. (1) Take any $t \in K$ so that $K$ is algebraic over $k(t)$. Since $x$ is algebraic over $k(t)$, there is a polynomial $F \in k[T, X]$ such that $F(t, x)=0$ (clear denominators if necessary). Since $x$ is not algebraic over $k$ (Problem 1.48), $T$ must appear in $F$, so $t$ is algebraic over $k(x)$. Then $k(x, t)$ is algebraic over $k(x)$ (Problem 1.50), so $K$ is algebraic over $k(x)$ (Problem 1.46).
(2) Since $\operatorname{char}(k(x))=0$, this is an immediate consequence of the "Theorem of the Primitive Element." We have outlined a proof of this algebraic fact in Problem 6.31 below.
(3) Suppose there is an $x \in R$ whose residue $\bar{x}$ in $R / \mathfrak{p}$ is not in $k$, and let $y \in \mathfrak{p}, y \neq$ 0 . Choose $F=\sum a_{i}(X) Y^{i} \in k[X, Y]$ so that $F(x, y)=0$. If we choose $F$ of lowest possible degree, then $a_{0}(X) \neq 0$. But then $a_{0}(x) \in P$, so $a_{0}(\bar{x})=0$. But $\bar{x}$ is not algebraic over $k$ (Problem 1.48), so there is no such $x$.
If $X$ is a variety, $k(X)$ is a finitely generated extension of $k$. Define the dimension of $X, \operatorname{dim}(X)$, to be tr. $\operatorname{deg}_{k} k(X)$. A variety of dimension one is called a curve, of dimension two a surface, etc. Part (5) of the next proposition shows that, for subvarieties of $A^{2}$ or $\mathbb{P}^{2}$, this definition agrees with the one given in Chapters 3 and 5 . Note, however, that a "curve" is assumed to be a variety, while a "plane curve" is allowed to have several (even multiple) components.
Proposition 10. (1) If $U$ is an open subvariety of $X$, then $\operatorname{dim} U=\operatorname{dim} X$. (2) If $V^{*}$ is the projective closure of an affine variety $V$, then $\operatorname{dim} V=\operatorname{dim} V^{*}$.
(3) A variety has dimension zero if and only if it is a point.
(4) Every proper closed subvariety of a curve is a point.
(5) A closed subvariety of $\mathbb{A}^{2}$ (resp. $\left.\mathbb{P}^{2}\right)$ has dimension one if and only if it is an affine (resp. projective) plane curve.
Proof. (1) and (2) follow from the fact that the varieties have the same function fields.
(3): Suppose $\operatorname{dim} V=0$. We may suppose $V$ is affine by (1) and (2). Then $k(V)$ is algebraic over $k$, so $k(V)=k$, so $\Gamma(V)=k$. Then Problem 2.4 gives the result.
(4): Again we may assume $V$ is affine. If $W$ is a closed subvariety of $V$, let $R=$ $\Gamma(V), \mathfrak{p}$ the prime ideal of $R$ corresponding to $W$; then $\Gamma(W)=R / P \mathfrak{p}$ (Problem 2.3). Apply Proposition 9 (3).
(5): Assume $V \subset \mathbb{A}^{2}$. Since $k(V)=k(x, y)$, $\operatorname{dim} V$ must be 0,1 , or 2 . So $V$ is either a point, a plane curve $V(F)$, or $V=\mathbb{A}^{2}(\$ 1.6)$. If $F(x, y)=0$, tr. $\operatorname{deg}_{k} k(x, y) \leq 1$. Then the result follows from (3) and (4). Use (2) if $V \subset \mathbb{P}^{2}$.
## Problems
6.31* (Theorem of the Primitive Element) Let $K$ be a field of a characteristic zero, $L$ a finite (algebraic) extension of $K$. Then there is a $z \in L$ such that $L=K(z)$.
Outline of Proof. Step (i): Suppose $L=K(x, y)$. Let $F$ and $G$ be monic irreducible polynomials in $K[T]$ such that $F(x)=0, G(y)=0$. Let $L^{\prime}$ be a field in which $F=$ $\prod_{i=1}^{n}\left(T-x_{i}\right), G=\prod_{j=1}^{m}\left(T-y_{i}\right), x=x_{1}, y=y_{1}, L^{\prime} \supset L$ (see Problems 1.52, 1.53). Choose $\lambda \neq 0$ in $K$ so that $\lambda x+y \neq \lambda x_{i}+y_{j}$ for all $i \neq 1, j \neq 1$. Let $z=\lambda x+y, K^{\prime}=K(z)$. Set $H(T)=G(z-\lambda T) \in K^{\prime}[T]$. Then $H(x)=0, H\left(x_{i}\right) \neq 0$ if $i>0$. Therefore $(H, F)=$ $(T-x) \in K^{\prime}[T]$. Then $x \in K^{\prime}$, so $y \in K^{\prime}$, so $L=K^{\prime}$.
Step (ii): If $L=K\left(x_{1}, \ldots, x_{n}\right)$, use induction on $n$ to find $\lambda_{1}, \ldots, \lambda_{n} \in k$ such that $L=K\left(\sum \lambda_{i} x_{i}\right)$.
6.32* Let $L=K\left(x_{1}, \ldots, x_{n}\right)$ as in Problem 6.31. Suppose $k \subset K$ is an algebraically closed subfield, and $V \varsubsetneqq \mathbb{A}^{n}(k)$ is an algebraic set. Show that $L=K\left(\sum \lambda_{i} x_{i}\right)$ for some $\left(\lambda_{1}, \ldots, \lambda_{n}\right) \in \mathbb{A}^{n} \backslash V$.
6.33. The notion of transcendence degree is analogous to the idea of the dimension of a vector space. If $k \subset K$, we say that $x_{1}, \ldots, x_{n} \in K$ are algebraically independent if there is no nonzero polynomial $F \in k\left[X_{1}, \ldots, X_{n}\right]$ such that $F\left(x_{1}, \ldots, x_{n}\right)=0$. By methods entirely analogous to those for bases of vector spaces, one can prove:
(a) Let $x_{1}, \ldots, x_{n} \in K, K$ a finitely generated extension of $k$. Then $x_{1}, \ldots, x_{n}$ is a minimal set such that $K$ is algebraic over $k\left(x_{1}, \ldots, x_{n}\right)$ if and only if $x_{1}, \ldots, x_{n}$ is a maximal set of algebraically independent elements of $K$. Such $\left\{x_{1}, \ldots, x_{n}\right\}$ is called a transcendence basis of $K$ over $k$.
(b) Any algebraically independent set may be completed to a transcendence basis. Any set $\left\{x_{1}, \ldots, x_{n}\right\}$ such that $K$ is algebraic over $k\left(x_{1}, \ldots, x_{n}\right)$ contains a transcendence basis.
(c) $\operatorname{tr} \operatorname{deg}_{k} K$ is the number of elements in any transcendence basis of $K$ over $k$. 6.34. Show that $\operatorname{dim} \mathbb{A}^{n}=\operatorname{dim} \mathbb{P}^{n}=n$.
6.35. Let $Y$ be a closed subvariety of a variety $X$. Then $\operatorname{dim} Y \leq \operatorname{dim} X$, with equality if and only if $Y=X$.
6.36. Let $K=k\left(x_{1}, \ldots, x_{n}\right)$ be a function field in $r$ variables over $k$. (a) Show that there is an affine variety $V \subset \mathbb{A}^{n}$ with $k(V)=K$. (b) Show that we may find $V \subset \mathbb{A}^{r+1}$ with $k(V)=K, r=\operatorname{dim} V$. (Assume $\operatorname{char}(k)=0$ if you wish.)
### Rational Maps
Let $X, Y$ be varieties. Two morphisms $f_{i}: U_{i} \rightarrow Y$ from open subvarieties $U_{i}$ of $X$ to $Y$ are said to be equivalent if their restrictions to $U_{1} \cap U_{2}$ are the same. Since $U_{1} \cap U_{2}$ is dense in $X$, each $f_{i}$ is determined by its restriction to $U_{1} \cap U_{2}$ (Corollary to Proposition 7 in $\$ 6.4$ ). An equivalence class of such morphisms is called a rational map from $X$ to $Y$.
The domain of a rational map is the union of all open subvarieties $U_{\alpha}$ of $X$ such that some $f_{\alpha}: U_{\alpha} \rightarrow Y$ belongs to the equivalence class of the rational map. If $U$ is the domain of a rational map, the mapping $f: U \rightarrow Y$ defined by $f_{\mid U_{\alpha}}=f_{\alpha}$ is a morphism belonging to the equivalence class of the map; every equivalent morphism is a restriction of $f$. Thus a rational map from $X$ to $Y$ may also be defined as a morphism $f$ from an open subvariety $U$ of $X$ to $Y$ such that $f$ cannot be extended to a morphism from any larger open subset of $X$ to $Y$. For any point $P$ in the domain of $f$, the value $f(P)$ is well-defined in $k$.
A rational map from $X$ to $Y$ is said to be dominating if $f(U)$ is dense in $Y$, where $f: U \rightarrow Y$ is any morphism representing the map (it is easy to see that this is independent of $U)$.
If $A$ and $B$ are local rings, and $A$ is a subring of $B$, we say that $B$ dominates $A$ if the maximal ideal of $B$ contains the maximal ideal of $A$.
Proposition 11. (1) Let $F$ be a dominating rational map from $X$ to $Y$. Let $U \subset X, V \subset$ $Y$ be affine open sets, $f: U \rightarrow V$ a morphism that represents $F$. Then the induced map $\tilde{f}: \Gamma(V) \rightarrow \Gamma(U)$ is one-to-one, so $\tilde{f}$ extends to a one-to-one homomorphism from $k(Y)=k(V)$ into $k(X)=k(U)$. This homomorphism is independent of the choice of $f$, and is denoted by $\tilde{F}$.
(2) If $P$ belongs to the domain of $F$, and $F(P)=Q$, then $\mathscr{O}_{P}(X)$ dominates $\tilde{F}\left(\mathscr{O}_{Q}(Y)\right)$. Conversely, if $P \in X, Q \in Y$, and $\mathscr{O}_{P}(X)$ dominates $\tilde{F}\left(\mathscr{O}_{Q}(Y)\right)$, then $P$ belongs to the domain of $F$, and $F(P)=Q$.
(3) Any homomorphism from $k(Y)$ into $k(X)$ is induced by a unique dominating rational map from $X$ to $Y$.
Proof. (1) is left to the reader (Problem 6.26), as is the first part of (2).
If $\mathscr{O}_{P}(X)$ dominates $\tilde{F}\left(\mathscr{O}_{Q}(Y)\right)$, take affine neighborhoods $V$ of $P, W$ of $Q$. Let $\Gamma(W)=k\left[y_{1}, \ldots, y_{n}\right]$. Then $\tilde{F}\left(y_{i}\right)=a_{i} / b_{i}, a_{i}, b_{i} \in \Gamma(V)$, and $b_{i}(P) \neq 0$. If we let $b=$ $b_{1} \cdots b_{n}$, then $\tilde{F}(\Gamma(W)) \subset \Gamma\left(V_{b}\right)$ (Proposition 5 of $\S 6.3$ ) so $\tilde{F}: \Gamma(W) \rightarrow \Gamma\left(V_{b}\right)$ is induced by a unique morphism $f: V_{b} \rightarrow W$ (Proposition 2 of $\$ 6.3$ ). If $g \in \Gamma(W)$ vanishes at $Q$, then $\tilde{F}(g)$ vanishes at $P$, from which it follows easily $f(P)=Q$. (3) We may assume $X$ and $Y$ are affine. Then, as in (2), if $\varphi: k(Y) \rightarrow k(X)$, $\varphi(\Gamma(Y)) \subset \Gamma\left(X_{b}\right)$ for some $b \in \Gamma(X)$, so $\varphi$ is induced by a morphism $f: X_{b} \rightarrow Y$. Therefore $f\left(X_{b}\right)$ is dense in $Y$ since $\tilde{f}$ is one-to-one (Problem 6.26).
A rational map $F$ from $X$ to $Y$ is said to be birational if there are open sets $U \subset X$, $V \subset Y$, and an isomorphism $f: U \rightarrow V$ that represents $F$. We say that $X$ and $Y$ are birationally equivalent if there is a birational map from $X$ to $Y$ (This is easily seen to be an equivalence relation). A variety is birationally equivalent to any open subvariety of itself. The varieties $\mathbb{A}^{n}$ and $\mathbb{P}^{n}$ are birationally equivalent.
Proposition 12. Two varieties are birationally equivalent if and only if their function fields are isomorphic.
Proof. Since $k(U)=k(X)$ for any open subvariety $U$ of $X$, birationally equivalent varieties have isomorphic function fields.
Conversely, suppose $\varphi: k(X) \rightarrow k(Y)$ is an isomorphism. We may assume $X$ and $Y$ are affine. Then $\varphi(\Gamma(X)) \subset \Gamma\left(Y_{b}\right)$ for some $b \in \Gamma(Y)$, and $\varphi^{-1}(\Gamma(Y)) \subset \Gamma\left(X_{d}\right)$ for some $d \in \Gamma(X)$, as in the proof of Proposition 11. Then $\varphi$ restricts to an isomorphism of $\Gamma\left(\left(X_{d}\right)_{\varphi^{-1}(b)}\right)$ onto $\Gamma\left(\left(Y_{b}\right)_{\varphi(d)}\right)$, so $\left(X_{d}\right)_{\varphi^{-1}(b)}$ is isomorphic to $\left(Y_{b}\right)_{\varphi(d)}$, as desired.
Corollary. Every curve is birationally equivalent to a plane curve.
Proof. If $V$ is a curve, $k(V)=k(x, y)$ for some $x, y \in k(V)$ (Proposition 9 (2) of §6.5). Let $I$ be the kernel of the natural homomorphism from $k[X, Y]$ onto $k[x, y] \subset k(V)$. Then $I$ is prime, so $V^{\prime}=V(I) \subset \mathbb{A}^{2}$ is a variety. Since $\Gamma\left(V^{\prime}\right)=k[X, Y] / I$ is isomorphic to $k[x, y]$, it follows that $k\left(V^{\prime}\right)$ is isomorphic to $k(x, y)=k(V)$. So $\operatorname{dim} V^{\prime}=1$, and $V^{\prime}$ is a plane curve (Proposition 10 (5) of $\$ 6.5$ ). (See Appendix A for the case when $\operatorname{char}(k)=p$.
A variety is said to be rational if it is birationally equivalent to $\mathbb{A}^{n}$ (or $\mathbb{P}^{n}$ ) for some $n$.
## Problems
6.37. Let $C=V\left(X^{2}+Y^{2}-Z^{2}\right) \subset \mathbb{P}^{2}$. For each $t \in k$, let $L_{t}$ be the line between $P_{0}=$ [-1:0:1] and $P_{t}=[0: t: 1]$. (Sketch this.) (a) If $t \neq \pm 1$, show that $L_{t} \bullet C=P_{0}+Q_{t}$, where $Q_{t}=\left[1-t^{2}: 2 t: 1+t^{2}\right]$. (b) Show that the map $\varphi: \mathbb{A}^{1} \backslash\{ \pm 1\} \rightarrow C$ taking $t$ to $Q_{t}$ extends to an isomorphism of $\mathbb{P}^{1}$ with $C$. (c) Any irreducible conic in $\mathbb{P}^{2}$ is rational; in fact, a conic is isomorphic to $\mathbb{P}^{1}$. (d) Give a prescription for finding all integer solutions $(x, y, z)$ to the Pythagorean equation $X^{2}+Y^{2}=Z^{2}$.
6.38. An irreducible cubic with a multiple point is rational (Problems 6.30, 5.10, 5.11).
6.39. $\mathbb{P}^{n} \times \mathbb{P}^{m}$ is birationally equivalent to $\mathbb{P}^{n+m}$. Show that $\mathbb{P}^{1} \times \mathbb{P}^{1}$ is not isomorphic to $\mathbb{P}^{2}$. (Hint: $\mathbb{P}^{1} \times \mathbb{P}^{1}$ has closed subvarieties of dimension one that do not intersect.)
6.40. If there is a dominating rational map from $X$ to $Y$, then $\operatorname{dim}(Y) \leq \operatorname{dim}(X)$. 6.41. Every $n$-dimensional variety is birationally equivalent to a hypersurface in $\mathbb{A}^{n+1}\left(\right.$ or $\left.\mathbb{P}^{n+1}\right)$.
6.42. Suppose $X, Y$ varieties, $P \in X, Q \in Y$, with $\mathscr{O}_{P}(X)$ isomorphic (over $k$ ) to $\mathscr{O}_{Q}(Y)$. Then there are neighborhoods $U$ of $P$ on $X, V$ of $Q$ on $Y$, such that $U$ is isomorphic to $V$. This is another justification for the assertion that properties of $X$ near $P$ should be determined by the local ring $\mathscr{O}_{P}(X)$.
6.43* Let $C$ be a projective curve, $P \in C$. Then there is a birational morphism $f: C \rightarrow$ $C^{\prime}, C^{\prime}$ a projective plane curve, such that $f^{-1}(f(P))=\{P\}$. We outline a proof:
(a) We can assume: $C \subset \mathbb{P}^{n+1}$ Let $T, X_{1}, \ldots, X_{n}, Z$ be coordinates for $\mathbb{P}^{n+1}$; Then $C \cap V(T)$ is finite; $C \cap V(T, Z)=\varnothing ; P=[0: \ldots: 0: 1]$; and $k(C)$ is algebraic over $k(u)$, where $u=\bar{T} / \bar{Z} \in k(C)$.
(b) For each $\lambda=\left(\lambda_{1}, \ldots, \lambda_{n}\right) \in k^{n}$, let $\varphi_{\lambda}: C \rightarrow \mathbb{P}^{2}$ be defined by the formula $\varphi([t$ : $\left.\left.x_{1}: \ldots: x_{n}: z\right]\right)=\left[t: \sum \lambda_{i} x_{i}: z\right]$. Then $\varphi_{\lambda}$ is a well-defined morphism, and $\varphi_{\lambda}(P)=$ $[0: 0: 1]$. Let $C^{\prime}$ be the closure of $\varphi_{\lambda}(C)$.
(c) The variable $\lambda$ can be chosen so $\varphi_{\lambda}$ is a birational morphism from $C$ to $C^{\prime}$, and $\varphi_{\lambda}^{-1}([0: 0: 1])=\{P\}$. (Use Problem 6.32 and the fact that $C \cap V(T)$ is finite).
6.44. Let $V=V\left(X^{2}-Y^{3}, Y^{2}-Z^{3}\right) \subset \mathbb{A}^{3}, f: \mathbb{A}^{1} \rightarrow V$ as in Problem 2.13 . (a) Show that $f$ is birational, so $V$ is a rational curve. (b) Show that there is no neighborhood of $(0,0,0)$ on $V$ that is isomorphic to an open subvariety of a plane curve. (See Problem 3.14.)
6.45. Let $C, C^{\prime}$ be curves, $F$ a rational map from $C^{\prime}$ to $C$. Prove: (a) Either $F$ is dominating, or $F$ is constant (i.e., for some $P \in C, F(Q)=P$, all $Q \in C^{\prime}$ ). (b) If $F$ is dominating, then $k\left(C^{\prime}\right)$ is a finite algebraic extension of $\tilde{F}(k(C))$.
6.46. Let $k\left(\mathbb{P}^{1}\right)=k(T), T=X / Y$ (Problem 4.8). For any variety $V$, and $f \in k(V)$, $f \notin k$, the subfield $k(f)$ generated by $f$ is naturally isomorphic to $k(T)$. Thus a nonconstant $f \in k(V)$ corresponds a homomorphism from $k(T)$ to $k(V)$, and hence to the a dominating rational map from $V$ to $\mathbb{P}^{1}$. The corresponding map is usually denoted also by $f$. If this rational map is a morphism, show that the pole set of $f$ is $f^{-1}([1: 0])$.
6.47. (The dual curve) Let $F$ be an irreducible projective plane curve of degree $n>$ 1. Let $\Gamma_{h}(F)=k[X, Y, Z] /(F)=k[x, y, z]$, and let $u, v, w \in \Gamma_{h}(F)$ be the residues of $F_{X}, F_{Y}, F_{Z}$, respectively. Define $\alpha: k[U, V, W] \rightarrow \Gamma_{h}(F)$ by setting $\alpha(U)=u, \alpha(V)=v$, $\alpha(W)=w$. Let $I$ be the kernel of $\alpha$. (a) Show that $I$ is a homogeneous prime ideal in $k[U, V, W]$, so $V(I)$ is a closed subvariety of $\mathbb{P}^{2}$. (b) Show that for any simple point $P$ on $F,\left[F_{X}(P): F_{Y}(P): F_{Z}(P)\right]$ is in $V(I)$, so $V(I)$ contains the points corresponding to tangent lines to $F$ at simple points. (c) If $V(I) \subset\{[a: b: c]\}$, use Euler's Theorem to show that $F$ divides $a X+b Y+c Z$, which is impossible. Conclude that $V(I)$ is a curve. It is called the dual curve of $F$. (d) Show that the dual curve is the only irreducible curve containing all the points of (b). (See Walker's "Algebraic Curves" for more about dual curves when $\operatorname{char}(k)=0$.)
## Chapter 7
## Resolution of Singularities
### Rational Maps of Curves
A point $P$ on an arbitrary curve $C$ is called a simple point if $\mathscr{O}_{P}(C)$ is a discrete valuation ring. If $C$ is a plane curve, this agrees with our original definition (Theorem1 of \\$3.2). We let $\operatorname{ord}_{P}^{C}$, or $\operatorname{ord}_{P}$ denote the order function on $k(C)$ defined by $\mathscr{O}_{P}(C)$. The curve $C$ is said to be nonsingular if every point on $C$ is simple.
Let $K$ be a field containing $k$. We say that a local ring $A$ is a local ring of $K$ if $A$ is a subring of $K, K$ is the quotient field of $A$, and $A$ contains $k$. For example, if $V$ is any variety, $P \in V$, then $\mathscr{O}_{P}(V)$ is a local ring of $k(V)$. Similarly, a discrete valuation ring of $K$ is a DVR that is a local ring of $K$.
Theorem 1. Let $C$ be a projective curve, $K=k(C)$. Suppose $L$ is a field containing $K$, and $R$ is a discrete valuation ring of $L$. Assume that $R \not \supset K$. Then there is a unique point $P \in C$ such that $R$ dominates $\mathscr{O}_{P}(C)$.
Proof. Uniqueness: If $R$ dominates $\mathscr{O}_{P}(C)$ and $\mathscr{O}_{Q}(C)$, choose $f \in \mathfrak{m}_{P}(C), 1 / f \in \mathscr{O}_{Q}(C)$ (Problem 6.24). Then $\operatorname{ord}(f)>0$ and $\operatorname{ord}(1 / f) \geq 0$, a contradiction.
Existence: We may assume $C$ is a closed subvariety of $\mathbb{P}^{n}$, and that $C \cap U_{i} \neq \varnothing$, $i=1, \ldots, n+1$ (Problem 6.25). Then in $\Gamma_{h}(C)=k\left[X_{1}, \ldots, X_{n+1}\right] / I(C)=k\left[x_{1}, \ldots, x_{n+1}\right]$, each $x_{i} \neq 0$. Let $N=\max _{i, j} \operatorname{ord}\left(x_{i} / x_{j}\right)$. Assume that $\operatorname{ord}\left(x_{j} / x_{n+1}\right)=N$ for some $j$ (changing coordinates if necessary). Then for all $i$,
$$
\operatorname{ord}\left(x_{i} / x_{n+1}\right)=\operatorname{ord}\left(\left(x_{j} / x_{n+1}\right)\left(x_{i} / x_{j}\right)\right)=N-\operatorname{ord}\left(x_{j} / x_{i}\right) \geq 0 .
$$
If $C_{*}$ is the affine curve corresponding to $C \cap U_{n+1}$, then $\Gamma\left(C_{*}\right)$ may be identified with $k\left[x_{1} / x_{n+1}, \ldots, x_{n} / x_{n+1}\right]$, so $R \supset \Gamma\left(C_{*}\right)$.
Let $\mathfrak{m}$ be the maximal ideal of $R, J=\mathfrak{m} \cap \Gamma\left(C_{*}\right)$. Then $J$ is a prime ideal, so $J$ corresponds to a closed subvariety $W$ of $C_{*}$ (Problem 2.2). If $W=C_{*}$, then $J=0$, and every nonzero element of $\Gamma\left(C_{*}\right)$ is a unit in $R$; but then $K \subset R$, which is contrary to our assumption. So $W=\{P\}$ is a point (Proposition 10 of $\S 6.5$ ). It is then easy to check that $R$ dominates $\mathscr{O}_{P}\left(C_{*}\right)=\mathscr{O}_{P}(C)$. Corollary 1. Let $f$ be a rational map from a curve $C^{\prime}$ to a projective curve $C$. Then the domain of $f$ includes every simple point of $C^{\prime}$. If $C^{\prime}$ is nonsingular, $f$ is a morphism.
Proof. If $F$ is not dominating, it is constant (Problem 6.45), and hence its domain is all of $C^{\prime}$. So we may assume $\tilde{F}$ imbeds $K=k(C)$ as a subfield of $L=k\left(C^{\prime}\right)$. Let $P$ be a simple point of $C^{\prime}, R=\mathscr{O}_{P}\left(C^{\prime}\right)$. By Proposition 11 of $\$ 6.5$ and the above Theorem 1 , it is enough to show that $R \not \supset K$. Suppose $K \subset R \subset L$; then $L$ is a finite algebraic extension of $K$ (Problem 6.45), so $R$ is a field (Problem 1.50). But a DVR is not a field.
Corollary 2. If $C$ is a projective curve, $C^{\prime}$ a nonsingular curve, then there is a natural one-to-one correspondence between dominant morphisms $f: C^{\prime} \rightarrow C$ and homomorphisms $\tilde{f}: k(C) \rightarrow k\left(C^{\prime}\right)$.
Corollary 3. Two nonsingular projective curves are isomorphic if and only if their function fields are isomorphic.
Corollary 4. Let $C$ be a nonsingular projective curve, $K=k(C)$. Then there is a natural one-to-one correspondence between the points of $C$ and the discrete valuation rings of $K$. If $P \in C, \mathscr{O}_{P}(C)$ is the corresponding $D V R$.
Proof. Each $\mathscr{O}_{P}(C)$ is certainly a DVR of $K$. If $R$ is any such DVR, then $R$ dominates a unique $\mathscr{O}_{P}(C)$. Since $R$ and $\mathscr{O}_{P}(C)$ are both DVR's of $K$, it follows that $R=\mathscr{O}_{P}(C)$ (Problem 2.26).
Let $C, K$ be as in Corollary 4. Let $X$ be the set of all discrete valuation rings of $K$ over $k$. Give a topology to $X$ as follows: a nonempty set $U$ of $X$ is open if $X \backslash U$ is finite. Then the correspondence $P \mapsto \mathscr{O}_{P}(C)$ from $C$ to $X$ is a homeomorphism. And if $U$ is open in $C, \Gamma\left(U, \mathscr{O}_{C}\right)=\bigcap_{P \in U} \mathscr{O}_{P}(C)$, so all the rings of functions on $C$ may be recovered from $X$. Since $X$ is determined by $K$ alone, this means that $C$ is determined up to isomorphism by $K$ alone (proving Corollary 3 again). In Chevalley's "Algebraic Functions of One Variable", the reader may find a treatment of these functions fields that avoids the concept of a curve entirely.
## Problem
7.1. Show that any curve has only a finite number of multiple points.
### Blowing up a Point in $\mathbb{A}^{2}$
To "resolve the singularities" of a projective curve $C$ means to construct a nonsingular projective curve $X$ and a birational morphism $f: X \rightarrow C$. A rough idea of the procedure we will follow is this:
If $C \subset \mathbb{P}^{2}$, and $P$ is a multiple point on $C$, we will remove the point $P$ from $\mathbb{P}^{2}$ and replace it by a projective line $L$. The points of $L$ will correspond to the tangent directions at $P$. This can be done in such a way that the resulting "blown up" plane $B=\left(\mathbb{P}^{2} \backslash\{P\}\right) \cup L$ is still a variety, and, in fact, a variety covered by open sets isomorphic to $\mathbb{A}^{2}$. The curve $C$ will be birationally equivalent to a curve $C^{\prime}$ on $B$, with $C^{\prime} \backslash\left(C^{\prime} \cap L\right)$ isomorphic to $C \backslash\{P\}$; but $C^{\prime}$ will have "better" multiple points on $L$ than $C$ has at $P$.
In this section we blow up a point in the affine plane, replacing it by an affine line $L$. In this case the equations are quite simple, and easy to relate to the geometry. In the following two sections we consider projective situations; the equations become more involved, but we will see that, locally, everything looks like what is done in this section. Throughout, many mappings between varieties will be defined by explicit formulas; we will leave it to the reader to verify that they are morphisms, using the general techniques of Chapter 6 .
Let $P=(0,0) \in \mathbb{A}^{2}$. Let $U=\left\{(x, y) \in \mathbb{A}^{2} \mid x \neq 0\right\}$. Define a morphism $f: U \rightarrow \mathbb{A}^{1}=k$ by $f(x, y)=y / x$. Let $G \subset U \times \mathbb{A}^{1} \subset \mathbb{A}^{2} \times \mathbb{A}^{1}=\mathbb{A}^{3}$ be the graph of $f$, so $G=\{(x, y, z) \in$ $\left.\mathbb{A}^{3} \mid y=x z, x \neq 0\right\}$.
Let $B=\left\{(x, y, z) \in \mathbb{A}^{3} \mid y=x z\right\}$. Since $Y-X Z$ is irreducible, $B$ is a variety. Let $\pi: B \rightarrow \mathbb{A}^{2}$ be the restriction of the projection from $\mathbb{A}^{3}$ to $\mathbb{A}^{2}: \pi(x, y, z)=(x, y)$. Then $\pi(B)=U \cup\{P\}$. Let $L=\pi^{-1}(P)=\{(0,0, z) \mid z \in k\}$. Since $\pi^{-1}(U)=G, \pi$ restricts to an isomorphism of $\pi^{-1}(U)$ onto $U$. We see that $B$ is the closure of $G$ in $A^{3}, G$ is an open subvariety of $B$, while $L$ is a closed subvariety of $B$.
For $k=\mathbb{C}$, the real part of this can be visualized in $\mathbb{R}^{3}$. The next figure is an attempt. The curve $C=V\left(Y^{2}-X^{2}(X+1)\right)$ is sketched. It appears that if we remove $P$ from $C$, take $\pi^{-1}(C \backslash\{P\})$, and take the closure of this in $B$, we arrive at a nonsingular curve $C^{\prime}$ with two points lying over the double point of $C$ - we have "resolved the singularity".
If we take the side view of $B$, (projecting $B$ onto the $(x, z)$-plane) we see that $B$ is isomorphic to an affine plane, and that $C^{\prime}$ becomes a parabola.
Let $\varphi: \mathbb{A}^{2} \rightarrow B$ be defined by $\varphi(x, z)=(x, x z, z)$. This $\varphi$ is an isomorphism of $\mathbb{A}^{2}$ onto $B$ (projection to $(x, z)$-plane gives the inverse). Let $\psi=\pi \circ \varphi: \mathbb{A}^{2} \rightarrow \mathbb{A}^{2}$; $\psi(x, z)=(x, x z)$. Let $E=\psi^{-1}(P)=\varphi^{-1}(L)=\left\{(x, z) \in \mathbb{A}^{2} \mid x=0\right\}$. Then $\psi: \mathbb{A}^{2} \backslash E \rightarrow U$ is an isomorphism; $\psi$ is a birational morphism of the plane to itself.
Let $C \neq V(X)$ be a curve in $\mathbb{A}^{2}$. Write $C_{0}=C \cap U$, an open subvariety of $C$; let $C_{0}^{\prime}=\psi^{-1}\left(C_{0}\right)$, and let $C^{\prime}$ be the closure of $C_{0}^{\prime}$ in $\mathbb{A}^{2}$. Let $f: C^{\prime} \rightarrow C$ be the restriction of $\psi$ to $C^{\prime}$. Then $f$ is a birational morphism of $C^{\prime}$ to $C$. By means of $\tilde{f}$ we may identify $k(C)=k(x, y)$ with $k\left(C^{\prime}\right)=k(x, z) ; y=x z$.
(1). Let $C=V(F), F=F_{r}+F_{r+1}+\cdots+F_{n}, F_{i}$ a form of degree $i$ in $k[X, Y], r=m_{P}(C)$, $n=\operatorname{deg}(C)$. Then $C^{\prime}=V\left(F^{\prime}\right)$, where $F^{\prime}=F_{r}(1, Z)+X F_{r+1}(1, Z)+\cdots+X^{n-r} F_{n}(1, Z)$.
Proof. $F(X, X Z)=X^{r} F_{r}(1, Z)+X^{r+1} F_{r+1}(1, Z)+\cdots=X^{r} F^{\prime}$. Since $F_{r}(1, Z) \neq 0, X$ doesn't divide $F^{\prime}$.
If $F^{\prime}=G H$, then $F=X^{r} G(X, Y / X) H(X, Y / X)$ would be reducible. Thus $F^{\prime}$ is irreducible, and since $V\left(F^{\prime}\right) \supset C_{0}^{\prime}, V\left(F^{\prime}\right)=C^{\prime}$.
Assumption. $X$ is not tangent to $C$ at $P$. By multiplying $F$ by a constant, we may assume that $F_{r}=\prod_{i=1}^{s}\left(Y-\alpha_{i} X\right)^{r_{i}}$, where $Y-\alpha_{i} X$ are the tangents to $F$ at $P$.
(2). With $F$ as above, $f^{-1}(P)=\left\{P_{1}, \ldots, P_{s}\right\}$. where $P_{i}=\left(0, \alpha_{i}\right)$, and
$$
m_{P_{i}}\left(C^{\prime}\right) \leq I\left(P_{i}, C^{\prime} \cap E\right)=r_{i} .
$$
If $P$ is an ordinary multiple point on $C$, then each $P_{i}$ is a simple point on $C^{\prime}$, and $\operatorname{ord}_{P_{i}}^{C^{\prime}}(x)=1$.
Proof. $f^{-1}(P)=C^{\prime} \cap E=\left\{(0, \alpha) \mid F_{r}(1, \alpha)=0\right\}$. And
$$
m_{P_{i}}\left(C^{\prime}\right) \leq I\left(P_{i}, F^{\prime} \cap X\right)=I\left(P_{i}, \prod_{i=1}^{s}\left(Z-\alpha_{i}\right)^{r_{i}} \cap X\right)=r_{i}
$$
by properties of the intersection number. (3). There is an affine neighborhood $W$ of $P$ on $C$ such that $W^{\prime}=f^{-1}(W)$ is an affine open subvariety on $C^{\prime}, f\left(W^{\prime}\right)=W, \Gamma\left(W^{\prime}\right)$ is module finite over $\Gamma(W)$, and $x^{r-1} \Gamma\left(W^{\prime}\right) \subset \Gamma(W)$.
Proof. Let $F=\sum_{i+j \geq r} a_{i j} X^{i} Y^{j}$. Let $H=\sum_{j \geq r} a_{0 j} Y^{j-r}$, and let $h$ be the image of $H$ in $\Gamma(C)$. Since $H(0,0)=1, W=C_{h}$ is an affine neighborhood of $P$ in $C$. Then $W^{\prime}=f^{-1}(W)=C_{h}^{\prime}$ is also an affine open subvariety of $C^{\prime}$.
To prove the last two claims it suffices to find an equation $z^{r}+b_{1} z^{r-1}+\cdots+b_{r}=0$, $b_{i} \in \Gamma(W)$. In fact, $\Gamma\left(W^{\prime}\right)=\Gamma(W)[z]$, so it will follow that $1, z, \ldots, z^{r-1}$ generate $\Gamma\left(W^{\prime}\right)$ as a module over $\Gamma(W)$ (see Proposition 3 of $\$ 1.9$ ); and $x^{r-1} z^{i} \in \Gamma(W)$ if $i \leq r-1$.
To find the equation, notice that
$$
F^{\prime}(x, z)=\sum a_{i j} x^{i+j-r} z^{j}=\sum a_{i j} y^{i+j-r} z^{r-i},
$$
so we have an equation $z^{r}+b_{1} z^{r-1}+\cdots+b_{r}=0$, where $b_{i}=\left(\sum_{j} a_{i j} y^{i+j-r}\right) / h$ for $i<r$, and $b_{r}=\sum_{i \geq r, j} a_{i j} x^{i-r} y^{j} / h$.
Remarks. (1) We can take the neighborhoods $W$ and $W^{\prime}$ arbitrarily small; i.e., if $P \in U, U^{\prime} \supset\left\{P_{1}, \ldots, P_{s}\right\}$ are any open sets on $C$ and $C^{\prime}$, we may take $W \subset U, W^{\prime} \subset U^{\prime}$. Starting with $W$ as in (3), we may choose $g \in \Gamma(W)$ such that $g(P) \neq 0$, but $g(Q)=$ 0 for all $Q \in(W \backslash U) \cup f\left(W^{\prime} \backslash U^{\prime}\right)$ (Problem 1.17). Then $W_{g}, W_{g}^{\prime}$ are the required neighborhoods.
(2) By taking a linear change of coordinates if necessary, we may also assume that $W$ includes any finite set of points on $C$ we wish. For the points on the $Y$-axis can be moved into $W$ by a change of coordinates $(X, Y) \mapsto(X+\alpha Y, Y)$. And the zeros of $H$ can be moved by $(X, Y) \mapsto(X, Y+\beta X)$.
## Problems
7.2. (a) For each of the curves $F$ in $\$ 3.1$, find $F^{\prime}$; show that $F^{\prime}$ is nonsingular in the first five examples, but not in the sixth. (b) Let $F=Y^{2}-X^{5}$. What is $F^{\prime}$ ? What is $\left(F^{\prime}\right)^{\prime}$ ? What must be done to resolve the singularity of the curve $Y^{2}=X^{2 n+1}$ ?
7.3. Let $F$ be any plane curve with no multiple components. Generalize the results of this section to $F$.
7.4. Suppose $P$ is an ordinary multiple point on $C, f^{-1}(P)=\left\{P_{1}, \ldots, P_{r}\right\}$. With the notation of Step (2), show that $F_{Y}=\sum_{i} \prod_{j \neq i}\left(Y-\alpha_{j} X\right)+\left(F_{r+1}\right)_{Y}+\ldots$, so $F_{Y}(x, y)=$ $x^{r-1}\left(\sum_{j \neq i}\left(z-\alpha_{j}\right)+x+\ldots\right)$. Conclude that $\operatorname{ord}_{P_{i}}^{C^{\prime}}\left(F_{Y}(x, y)\right)=r-1$ for $i=1, \ldots, r$.
7.5. Let $P$ be an ordinary multiple point on $C, f^{-1}(P)=\left\{P_{1}, \ldots, P_{r}\right\}, L_{i}=Y-\alpha_{i} X$ the tangent line corresponding to $P_{i}=\left(0, \alpha_{i}\right)$. Let $G$ be a plane curve with image $g$ in $\Gamma(C) \subset \Gamma\left(C^{\prime}\right)$. (a) Show that $\operatorname{ord}_{P_{i}}^{C^{\prime}}(g) \geq m_{P}(G)$, with equality if $L_{i}$ is not tangent to $G$ at $P$. (b) If $s \leq r$, and $\operatorname{ord}_{R_{i}}^{C^{\prime}}(g) \geq s$ for each $i=1, \ldots, r$, show that $m_{P}(G) \geq s$. (Hint: How many tangents would $G$ have otherwise?)
7.6. If $P$ is an ordinary cusp on $C$, show that $f^{-1}(P)=\left\{P_{1}\right\}$, where $P_{1}$ is a simple point on $C^{\prime}$.
### Blowing up Points in $\mathbb{P}^{2}$
Let $P_{1}, \ldots, P_{t} \in \mathbb{P}^{2}$. We are going to blow up all of these points, replacing each by a projective line. We assume for simplicity that $P_{i}=\left[a_{i 1}: a_{i 2}: 1\right]$, leaving the reader to make the necessary changes if $P_{i} \notin U_{3}$.
Let $U=\mathbb{P}^{2} \backslash\left\{P_{1}, \ldots, P_{t}\right\}$. Define morphisms $f_{i}: U \rightarrow \mathbb{P}^{1}$ by the formula
$$
f_{i}\left[x_{1}: x_{2}: x_{3}\right]=\left[x_{1}-a_{i 1} x_{3}: x_{2}-a_{i 2} x_{3}\right] .
$$
Let $f=\left(f_{1}, \ldots, f_{t}\right): U \rightarrow \mathbb{P}^{1} \times \cdots \times \mathbb{P}^{1}(t$ times) be the product (Proposition 7 of $\S 6.4)$, and let $G \subset U \times \mathbb{P}^{1} \times \cdots \times \mathbb{P}^{1}$ be the graph of $f$.
Let $X_{1}, X_{2}, X_{3}$ be homogeneous coordinates for $\mathbb{P}^{2}, Y_{i 1}, Y_{i 2}$ homogeneous coordinates for the $i^{\text {th }}$ copy of $\mathbb{P}^{1}$. Let
$$
B=V\left(\left\{Y_{i 1}\left(X_{2}-a_{i 2} X_{3}\right)-Y_{i 2}\left(X_{1}-a_{i 1} X_{3} \mid i=1, \ldots, t\right\}\right) \subset \mathbb{P}^{2} \times \mathbb{P}^{1} \times \cdots \times \mathbb{P}^{1} .\right.
$$
Then $B \supset G$, and we will soon see that $B$ is the closure of $G$ in $\mathbb{P}^{2} \times \cdots \times \mathbb{P}^{1}$, so $B$ is a variety. Let $\pi: B \rightarrow \mathbb{P}^{2}$ be the restriction of the projection from $\mathbb{P}^{2} \times \cdots \times \mathbb{P}^{1}$ to $\mathbb{P}^{2}$. Let $E_{i}=\pi^{-1}\left(P_{i}\right)$.
(1). $E_{i}=\left\{P_{i}\right\} \times\left\{f_{1}\left(P_{i}\right)\right\} \times \cdots \times \mathbb{P}^{1} \times \cdots \times\left\{f_{t}\left(P_{i}\right)\right\}$, where $\mathbb{P}^{1}$ appears in the $i$ th place. So each $E_{i}$ is canonically isomorphic to $\mathbb{P}^{1}$.
(2). $B \backslash \bigcup_{i=1}^{t} E_{i}=B \cap\left(U \times \mathbb{P}^{1} \times \cdots \times \mathbb{P}^{1}\right)=G$, so $\pi$ restricts to an isomorphism of $B \backslash \bigcup_{i=1}^{t} E_{i}$ with $U$.
(3). If $T$ is any projective change of coordinates of $\mathbb{P}^{2}$, with $T\left(P_{i}\right)=P_{i}^{\prime}$, and $f_{i}^{\prime}: \mathbb{P}^{2} \backslash\left\{P_{1}^{\prime}, \ldots, P_{t}^{\prime}\right\} \rightarrow \mathbb{P}^{1}$ are defined using $P_{i}^{\prime}$ instead of the $P_{i}$, then there are unique projective changes of coordinates $T_{i}$ of $\mathbb{P}^{1}$ such that $T_{i} \circ f_{i}=f_{i}^{\prime} \circ T$ (see Problem 7.7 below). If $f^{\prime}=\left(f_{i}^{\prime}, \ldots, f_{t}^{\prime}\right)$, then $\left(T_{1} \times \cdots \times T_{t}\right) \circ f=f^{\prime} \circ T$, and $T \times T_{1} \times \cdots \times T_{t}$ maps $G$, $B$ and $E_{i}$ isomorphically onto the corresponding $G^{\prime}, B^{\prime}$ and $E_{i}^{\prime}$ constructed from $f^{\prime}$.
(4). If $T_{i}$ is a projective change of coordinates of $\mathbb{P}^{1}$ (for one $i$ ), then there is a projective change of coordinates $T$ of $\mathbb{P}^{2}$ such that $T\left(P_{i}\right)=P_{i}$ and $f_{i} \circ T=T_{i} \circ f_{i}$ (see Problem 7.8 below).
(5). We want to study $\pi$ in a neighborhood of a point $Q$ in some $E_{i}$. We may assume $i=1$, and by (3) and (4) we may assume that $P_{1}=[0: 0: 1]$, and that $Q$ corresponds to $[\lambda: 1] \in \mathbb{P}^{1}, \lambda \in k$ (even $\lambda=0$ if desired).
Let $\varphi_{3}: \mathbb{A}^{2} \rightarrow U_{3} \subset \mathbb{P}^{2}$ be the usual morphism: $\varphi_{3}(x, y)=[x: y: 1]$. Let $V=$ $U_{3} \backslash\left\{P_{2}, \ldots, P_{t}\right\}, W=\varphi_{3}^{-1}(V)$. Let $\psi: \mathbb{A}^{2} \rightarrow \mathbb{A}^{2}$ be as in Section 2: $\psi(x, z)=(x, x z)$; and let $W^{\prime}=\psi^{-1}(W)$.
Define $\varphi: W^{\prime} \rightarrow \mathbb{P}^{2} \times \mathbb{P}^{1} \times \cdots \times \mathbb{P}^{1}$ by setting
$$
\varphi(x, z)=[x: x z: 1] \times[1: z] \times f_{2}([x: x z: 1]) \times \cdots \times f_{t}([x: x z: 1]) .
$$
Then $\varphi$ is a morphism, and $\pi \circ \varphi=\varphi_{3} \circ \psi$. Let $V^{\prime}=\varphi\left(W^{\prime}\right)=B \backslash\left(\bigcup_{i>1} E_{i} \cup V\left(X_{3}\right) \cup\right.$ $V\left(Y_{12}\right)$ ). This $V^{\prime}$ is a neighborhood of $Q$ on $B$.
(6). $B$ is the closure of $G$ in $\mathbb{P}^{2} \times \cdots \times \mathbb{P}^{1}$, and hence $B$ is a variety. For if $S$ is any closed set in $\mathbb{P}^{2} \times \cdots \times \mathbb{P}^{1}$ that contains $G$, then $\varphi^{-1}(S)$ is closed in $W^{\prime}$ and contains $\varphi^{-1}(G)=W^{\prime} \backslash V(X)$. Since $W^{\prime} \backslash V(X)$ is open in $W^{\prime}$, it is dense, so $\varphi^{-1}(S)=W^{\prime}$. Therefore $Q \in S$, and since $Q$ was an arbitrary point of $B \backslash G, S \supset B$.
(7). The morphism from $\mathbb{P}^{2} \times \cdots \times \mathbb{P}^{1} \backslash V\left(X_{3} Y_{12}\right)$ to $\mathbb{A}^{2}$ taking $\left[x_{1}: x_{2}: x_{3}\right] \times\left[y_{11}\right.$ : $\left.y_{12}\right] \times \cdots$ to $\left[x_{1} / x_{3}: y_{11} / y_{12}\right]$, when restricted to $V^{\prime}$, is the inverse morphism to $\varphi$. Thus we have the following diagram:
Locally, $\pi: B \rightarrow \mathbb{P}^{2}$ looks just like the map $\psi: \mathbb{A}^{2} \rightarrow \mathbb{A}^{2}$ of Section 2 .
(8). Let $C$ be an irreducible curve in $\mathbb{P}^{2}$. Let $C_{0}=C \cap U, C_{0}^{\prime}=\pi^{-1}\left(C_{0}\right) \subset G$, and let $C^{\prime}$ be the closure of $C_{0}^{\prime}$ in $B$. then $\pi$ restricts to a birational morphism $f: C^{\prime} \rightarrow C$, which is an isomorphism from $C_{0}^{\prime}$ to $C_{0}$. By (7) we know that, locally, $f$ looks just like the corresponding affine map of Section 2.
Proposition 1. Let $C$ be an irreducible projective plane curve, and suppose all the multiple points of $C$ are ordinary. Then there is a nonsingular projective curve $C^{\prime}$ and a birational morphism $f$ from $C^{\prime}$ onto $C$.
Proof. Let $P_{1}, \ldots, P_{t}$ be the multiple points of $C$, and apply the above process. Step (2) of Section 2, together with (8) above, guarantees that $C^{\prime}$ is nonsingular.
## Problems
7.7. Suppose $P_{1}=[0: 0: 1], P_{1}^{\prime}=\left[a_{11}: a_{12}: 1\right]$, and
$$
T=\left(a X+b Y+a_{11} z, c X+d Y+a_{12} Z, e X+f Y+Z\right) .
$$
Show that $T_{1}=\left(\left(a-a_{11} e\right) X+\left(b-a_{11} f\right) Y,\left(c-a_{12} e\right) X+\left(d-a_{12} f\right) Y\right)$ satisfies $T_{1} \circ f_{1}=$ $f_{1}^{\prime} \circ T$. Use this to prove Step (3) above.
7.8. Let $P_{1}=[0: 0: 1], T_{1}=(a X+b Y, c X+d Y)$. Show that $T=(a X+b Y, c X+d Y, Z)$ satisfies $f_{1} \circ T=T_{1} \circ f_{1}$. Use this to prove Step (4).
7.9. Let $C=V\left(X^{4}+Y^{4}-X Y Z^{2}\right)$. Write down equations for a nonsingular curve $X$ in some $\mathbb{P}^{N}$ that is birationally equivalent to $C$. (Use the Segre imbedding.)
### Quadratic Transformations
A disadvantage of the procedure in Section 7.3 is that the new curve $C^{\prime}$, although having better singularities than $C$, is no longer a plane curve. The facts we have learned about plane curves don't apply to $C^{\prime}$, and it is difficult to repeat the process to $C^{\prime}$, getting a better curve $C^{\prime \prime}$. (The latter can be done, but it requires more technique than we have developed here.) If we want $C^{\prime}$ to be a plane curve, we must allow it acquire some new singularities. These new singularities can be taken to be ordinary multiple points, while the old singularities of $C$ become better on $C^{\prime}$.
Let $P=[0: 0: 1], P^{\prime}=[0: 1: 0], P^{\prime \prime}=[1: 0: 0]$ in $\mathbb{P}^{2}$; call these three points the fundamental points. Let $L=V(Z), L^{\prime}=V(Y), L^{\prime \prime}=V(X)$; call these the exceptional lines. Note that the lines $L^{\prime}$ and $L^{\prime \prime}$ intersect in $P$, and $L$ is the line through $P^{\prime}$ and $P^{\prime \prime}$. Let $U=\mathbb{P}^{2} \backslash V(X Y Z)$.
Define $Q: \mathbb{P}^{2} \backslash\left\{P, P^{\prime}, P^{\prime \prime}\right\} \rightarrow \mathbb{P}^{2}$ by the formula $Q([x: y: z]=[y z: x z: x y]$. This $Q$ is a morphism from $\mathbb{P}^{2} \backslash\left\{P, P^{\prime}, P^{\prime \prime}\right\}$ onto $U \cup\left\{P, P^{\prime}, P^{\prime \prime}\right\}$. And $Q^{-1}(P)=L-\left\{P^{\prime}, P^{\prime \prime}\right\}$. (By the symmetry of $Q$, it is enough to write one such equality; the results for the other fundamental points and exceptional lines are then clear - and we usually omit writing them.)
If $[x: y: z] \in U$, then $Q(Q([x: y: z]))=[x z x y: y z x y: y z x z]=[x: y: z]$. So $Q$ maps $U$ one-to-one onto itself, and $Q=Q^{-1}$ on $U$, so $Q$ is an isomorphism of $U$ with itself. In particular, $Q$ is a birational map of $\mathbb{P}^{2}$ with itself. It is called the standard quadratic transformation, or sometimes the standard Cremona transformation (a Cremona transformation is any birational map of $\mathbb{P}^{2}$ with itself).
Let $C$ be an irreducible curve in $\mathbb{P}^{2}$. Assume $C$ is not an exceptional line. Then $C \cap U$ is open in $C$, and closed in $U$. Therefore $Q^{-1}(C \cap U)=Q(C \cap U)$ is a closed curve in $U$. Let $C^{\prime}$ be the closure of $Q^{-1}(C \cap U)$ in $\mathbb{P}^{2}$. Then $Q$ restricts to a birational morphism from $C^{\prime} \backslash\left\{P, P^{\prime}, P^{\prime \prime}\right\}$ to $C$. Note that $\left(C^{\prime}\right)^{\prime}=C$, since $Q \circ Q$ is the identity on $U$.
Let $F \in k[X, Y, Z]$ be the equation of $C, n=\operatorname{deg}(F)$. Let $F^{Q}=F(Y Z, X Z, X Y)$, called the algebraic transform of $F$. So $F^{Q}$ is a form of degree $2 n$.
(1). If $m_{P}(C)=r$, then $Z^{r}$ is the largest power of $Z$ that divides $F^{Q}$.
Proof. Write $F=F_{r}(X, Y) Z^{n-r}+\cdots+F_{n}(X, Y), F_{i}$ a form of degree $i$ (Problem 5.5). Then
$F^{Q}=F_{r}(Y Z, X Z)(X Y)^{n-r}+\cdots=Z^{r}\left(F_{r}(Y, X)(X Y)^{n-r}+Z F_{r+1}(Y, X)(X Y)^{n-r-1}+\cdots\right)$,
from which the result follows.
Let $m_{P}(C)=r, m_{P^{\prime}}(C)=r^{\prime}, m_{P^{\prime \prime}}(C)=r^{\prime \prime}$. Then $F^{Q}=Z^{r} Y^{r^{\prime}} X^{r^{\prime \prime}} F^{\prime}$, where $X, Y$, and $Z$ do not divide $F^{\prime}$. This $F^{\prime}$ is called the proper transform of $F$.
(2). $\operatorname{deg}\left(F^{\prime}\right)=2 n-r-r^{\prime}-r^{\prime \prime},\left(F^{\prime}\right)^{\prime}=F, F^{\prime}$ is irreducible, and $V\left(F^{\prime}\right)=C^{\prime}$.
Proof. From $\left(F^{Q}\right)^{Q}=(X Y Z)^{n} F$, it follows that $F^{\prime}$ is irreducible and $\left(F^{\prime}\right)^{\prime}=F$. Since $V\left(F^{\prime}\right) \supset Q^{-1}(C \cap U), V\left(F^{\prime}\right)$ must be $C^{\prime}$.
(3). $m_{P}\left(F^{\prime}\right)=n-r^{\prime}-r^{\prime \prime}$. (Similarly for $P^{\prime}$ and $P^{\prime \prime}$.)
Proof. $F^{\prime}=\sum_{i=0}^{n-r} F_{r+i}(Y, X) X^{n-r-r^{\prime \prime}-i} Y^{n-r-r^{\prime}-i} Z^{i}$, so the leading form of $F^{\prime}$ at $P=$ $[0: 0: 1]$ is $F_{n}(Y, X) X^{-r^{\prime \prime}} Y^{-r^{\prime}}$.
Let us say that $C$ is in good position if no exceptional line is tangent to $C$ at a fundamental point.
(4). If $C$ is in good position, so is $C^{\prime}$. Proof. The line $L$ is tangent to $C^{\prime}$ at $P^{\prime}$ if and only if $I\left(P^{\prime}, F^{\prime} \cap Z\right)>m_{P^{\prime}}\left(C^{\prime}\right)$. Equivalently, $I\left(P^{\prime}, F_{r}(Y, X) X^{n-r-r^{\prime \prime}} Y^{n-r-r^{\prime}} \cap Z\right)>n-r-r^{\prime \prime}$, or $I\left(P^{\prime}, F_{r}(Y, X) \cap Z\right)>0$, or $F_{r}(1,0)=0$. But if $Y$ is not tangent to $F$ at $P$, then $F_{r}(1,0) \neq 0$. By symmetry, the same holds for the other lines and points.
Assume that $C$ is in good position, and that $P \in C$. Let $C_{0}=(C \cap U) \cup\{P\}$, a neighborhood of $P$ on $C$. Let $C_{0}^{\prime}=C^{\prime} \backslash V(X Y)$. Let $f: C_{0}^{\prime} \rightarrow C_{0}$ be the restriction of $Q$ to $C_{0}^{\prime}$.
Let $F_{*}=F(X, Y, 1), C_{*}=V\left(F_{*}\right) \subset \mathbb{A}^{2}$. Define $\left(F_{*}\right)^{\prime}=F(X, X Z, 1) X^{-r}, C_{*}^{\prime}=V\left(F_{*}^{\prime}\right) \subset$ $\mathbb{A}^{2}$ and $f_{*}: C_{*}^{\prime} \rightarrow C_{*}$ by $f_{*}(x, z)=(x, x z)$, all as in Section 2 .
(5). There is a neighborhood $W$ of $(0,0)$ in $C_{*}$, and there are isomorphisms $\varphi: W \rightarrow$ $C_{0}$ and $\varphi^{\prime}: W^{\prime}=f_{*}^{-1}(W) \rightarrow C_{0}^{\prime}$ such that $\varphi f_{*}=f \varphi^{\prime}$, i.e., the following diagram commutes:
Proof. Take $W=\left(C_{*} \backslash V(X Y)\right) \cup\{(0,0)\} ; \varphi(x, y)=[x: y: 1]$, and $\varphi^{\prime}(x, z)=[z: 1: x z]$. The inverse of $\varphi^{\prime}$ is given by $[x: y: z] \mapsto(z / x, x / y)$. We leave the rest to the reader.
(6). If $C$ is in good position, and $P_{1}, \ldots, P_{s}$ are the non-fundamental points on $C^{\prime} \cap L$, then $m_{P_{i}}\left(C^{\prime}\right) \leq I\left(P_{i}, C^{\prime} \cap L\right)$, and $\sum_{i=1}^{s} I\left(P_{i}, C^{\prime} \cap Z\right)=r$.
Proof. As in the proof of (4), $\sum I\left(P_{i}, F^{\prime} \cap Z\right)=\sum I\left(P_{i}, F_{r}(Y, X) \cap Z\right)=r$.
Remark. If $P \notin C$, the same argument shows that there are no non-fundamental points on $C^{\prime} \cap L$.
Example. (See Problem 7.10)
Let us say that $C$ is in excellent position if $C$ is in good position, and, in addition, $L$ intersects $C$ (transversally) in $n$ distinct non-fundamental points, and $L^{\prime}$ and $L^{\prime \prime}$ each intersect $C$ (transversally) in $n-r$ distinct non-fundamental points. (This condition is no longer symmetric in $P, P^{\prime}, P^{\prime \prime}$.)
(7). If $C$ is in excellent position, then $C^{\prime}$ has the following multiple points:
(a) Those on $C^{\prime} \cap U$ correspond to multiple points on $C \cap U$, the correspondence preserving multiplicity and ordinary multiple points.
(b) $P, P^{\prime}$ and $P^{\prime \prime}$ are ordinary multiple points on $C^{\prime}$ with multiplicities $n, n-r$, and $n-r$ respectively.
(c) There are no non-fundamental points on $C^{\prime} \cap L^{\prime}$ or on $C^{\prime} \cap L^{\prime \prime}$. Let $P_{1}, \ldots, P_{s}$ be the non-fundamental points on $C^{\prime} \cap L$. Then $m_{P_{i}}\left(C^{\prime}\right) \leq I\left(P_{i}, C^{\prime} \cap L\right)$ and $\sum I\left(P_{i}, C^{\prime} \cap\right.$ $L)=r$.
Proof. (a) follows from the fact that $C^{\prime} \cap U$ and $C \cap U$ are isomorphic, and from Theorem 2 of $\$ 3.2$ and Problem 3.24. (c) follows from (6), and (b) follows by applying (6) to the curves $C^{\prime}$ and $\left(C^{\prime}\right)^{\prime}=C$ (observing by (4) that $C^{\prime}$ is in good position).
For any irreducible projective plane curve $C$ of degree $n$, with multiple points of multiplicity $r_{P}=m_{P}(C)$, let
$$
g^{*}(C)=\frac{(n-1)(n-2)}{2}-\sum \frac{r_{P}\left(r_{P}-1\right)}{2} .
$$
We know that $g^{*}(C)$ is a nonnegative integer. (Theorem 2 of $\$ 5.4$ ).
(8). If $C$ is in excellent position, then $g^{*}\left(C^{\prime}\right)=g^{*}(C)-\sum_{i=1}^{s} \frac{r_{i}\left(r_{i}-1\right)}{2}$, where $r_{i}=$ $m_{P_{i}}\left(C^{\prime}\right)$, and $P_{1}, \ldots, P_{s}$ are the non-fundamental points on $C^{\prime} \cap L$.
Proof. A calculation, using (2) and (7).
The special notation used in (1)-(8) regarding fundamental points and exceptional lines has served its purpose; from now on $P, P^{\prime}$, etc. may be any points, $L, L^{\prime}$, etc. any lines.
Lemma 1. $(\operatorname{char}(k)=0)$ Let $F$ be an irreducible projective plane curve, $P$ a point of $F$. Then there is a projective change of coordinates $T$ such that $F^{T}$ is in excellent position, and $T([0: 0: 1])=P$.
Proof. Let $\operatorname{deg}(F)=n, m_{P}(F)=r$. By Problem 4.16, it is enough to find lines $L, L^{\prime}, L^{\prime \prime}$ such that $L^{\prime} \cdot C=r P+P_{r+1}^{\prime}+\cdots+P_{n}^{\prime}, L^{\prime \prime} \cdot C=r P+P_{r+1}^{\prime \prime}+\cdots+P_{n}^{\prime \prime}$, and $L \cdot C=P_{1}+\cdots+P_{n}$, with all these points distinct; then there is a change of coordinates $T$ with $L^{T}=Z$, $L^{\prime T}=Y, L^{\prime \prime T}=X$. The existence of such lines follows from Problem 5.26 (Take $L^{\prime}, L^{\prime \prime}$ first, then $L$ ).
If $T$ is any projective change of coordinates, $Q \circ T$ is called a quadratic transformation, and $\left(F^{T}\right)^{\prime}$ is called a quadratic transformation of $F$. If $F^{T}$ is in excellent position, and $T([0: 0: 1])=P$, we say that the quadratic transformation is centered at $P$. If $F=F_{1}, F_{2}, \ldots, F_{m}=G$ are curves, and each $F_{i}$ is a quadratic transformation of $F_{i-1}$, we say that $F$ is transformed into $G$ by a finite sequence of quadratic transformations. Theorem 2. By a finite sequence of quadratic transformations, any irreducible projective plane curve may be transformed into a curve whose only singularities are ordinary multiple points.
Proof. Take successive quadratic transformations, centering each one at a nonordinary multiple point.
From (7) and (8) we see that at each step, either $C^{\prime}$ has one less non-ordinary multiple point than $C$, or $g^{*}\left(C^{\prime}\right)=g^{*}(C)-\sum \frac{r_{i}\left(r_{i}-1\right)}{2}<g^{*}(C)$. If the original curve $C$ has $N$ non-ordinary multiple points, we reach the desired curve after at most $N+$ $g^{*}(C)$ steps. (See the Appendix for a proof in characteristic $p$.)
## Problems
7.10. Let $F=8 X^{3} Y+8 X^{3} Z+4 X^{2} Y Z-10 X Y^{3}-10 X Y^{2} Z-3 Y^{3} Z$. Show that $F$ is in good position, and that $F^{\prime}=8 Y^{2} Z+8 Y^{3}+4 X Y^{2}-10 X^{2} Z-10 X^{2} Y-3 X^{3}$. Show that $F$ and $F^{\prime}$ have singularities as in the example sketched, and find the multiple points of $F$ and $F^{\prime}$.
7.11. Find a change of coordinates $T$ so that $\left(Y^{2} Z-X^{3}\right)^{T}$ is in excellent position, and $T([0: 0: 1])=[0: 0: 1]$. Calculate the quadratic transformation.
7.12. Find a quadratic transformation of $Y^{2} Z^{2}-X^{4}-Y^{4}$ with only ordinary multiple points. Do the same with $Y^{4}+Z^{4}-2 X^{2}(Y-Z)^{2}$.
7.13. (a) Show that in the lemma, we may choose $T$ in such a way that for a given finite set $S$ of points of $F$, with $P \notin S, T^{-1}(S) \cap V(X Y Z)=\varnothing$. Then there is a neighborhood of $S$ on $F$ that is isomorphic to an open set on $\left(F^{T}\right)^{\prime}$. (b) If $S$ is a finite set of simple points on a plane curve $F$, there is a curve $F^{\prime}$ with only ordinary multiple points, and a neighborhood $U$ of $S$ on $F$, and an open set $U^{\prime}$ on $F^{\prime}$ consisting entirely of simple points, such that $U$ is isomorphic to $U^{\prime}$.
7.14. (a) What happens to the degree, and to $g^{*}(F)$, when a quadratic transformation is centered at: (i) an ordinary multiple point; (ii) a simple point; (iii) a point not on $F$ ? (b) Show that the curve $F^{\prime}$ of Problem 7.13(b) may be assumed to have arbitrarily large degree.
7.15. Let $F=F_{1}, \ldots, F_{m}$ be a sequence of quadratic transformations of $F$, such that $F_{m}$ has only ordinary multiple points. Let $P_{i 1}, P_{i 2}, \ldots$ be the points on $F_{i}$ introduced, as in (7) (c), in going from $F_{i-1}$ to $F_{i}$. (These are called "neighboring singularities"; see Walker’s "Algebraic Curves", Chap. III, §7.6, §7.7). If $n=\operatorname{deg}(F)$, show that
$$
(n-1)(n-2) \geq \sum_{P \in F} m_{P}(F)\left(m_{P}(F)-1\right)+\sum_{i>1, j} m_{P_{i j}}\left(F_{i}\right)\left(m_{P_{i j}}\left(F_{i}\right)-1\right) .
$$
7.16. (a) Show that everything in this section, including Theorem2, goes through for any plane curve with no multiple components. (b) If $F$ and $G$ are two curves with no common components, and no multiple components, apply (a) to the curve $F G$. Deduce that there are sequences of quadratic transformations $F=F_{1}, \ldots, F_{S}=F^{\prime}$ and $G=G_{1}, \ldots, G_{s}=G^{\prime}$, where $F^{\prime}$ and $G^{\prime}$ have only ordinary multiple points, and no tangents in common at points of intersection. Show that
$$
\operatorname{deg}(F) \operatorname{deg}(G)=\sum m_{P}(F) m_{P}(G)+\sum_{i>1, j} m_{P_{i j}}\left(F_{i}\right) m_{P_{i j}}\left(G_{i}\right),
$$
where the $P_{i j}$ are the neighboring singularities of $F G$, determined as in Problem 7.15.
### Nonsingular Models of Curves
Theorem 3. Let $C$ be a projective curve. Then there is a nonsingular projective curve $X$ and a birational morphism $f$ from $X$ onto $C$. If $f^{\prime}: X^{\prime} \rightarrow C$ is another, then there is a unique isomorphism $g: X \rightarrow X^{\prime}$ such that $f^{\prime} g=f$.
Proof. The uniqueness follows from Corollary 2 of $\$ 7$.1. For the existence, the Corollary in $\S 6.6$ says $C$ is birationally equivalent to a plane curve. By Theorem 2 of $\S 7.4$, this plane curve can be taken to have only ordinary multiple points as singularities, and Proposition 1 of $\$ 7.3$ replaces this curve by a nonsingular curve $X$. That the birational map from $X$ to $C$ is a morphism follows from Corollary 1 of $\S 7.1$.
If $C$ is a plane curve, the fact that $f$ maps $X$ onto $C$ follows from the construction of $X$ from $C$; indeed, if $P \in C$, we may find $C^{\prime}$ with ordinary multiple points so that the rational map from $C^{\prime}$ to $C$ is defined at some point $P^{\prime}$ and maps $P^{\prime}$ to $P$ (see Problem 7.13); and the map from $X$ to $C^{\prime}$ is onto (Proposition 1 of $\$ 7.3$ ).
If $C \subset \mathbb{P}^{n}$, and $P \in C$, choose a morphism $g: C \rightarrow C_{1}$ from $C$ to a plane curve $C_{1}$ such that $\{P\}=g^{-1}(g(P))$ (Problem 6.43). Then if $g f(x)=g(P)$, it follows that $f(x)=P$.
Corollary. There is a natural one-to-one correspondence between nonsingular projective curves $X$ and algebraic function fields in one variable $K$ over $k: K=k(X)$. If $X$ and $X^{\prime}$ are two such curves, dominant morphisms from $X^{\prime}$ to $X$ correspond to homomorphisms from $k(X)$ into $k\left(X^{\prime}\right)$.
We will see later that a dominant morphism between projective curves must be surjective (Problem 8.18).
Let $C$ be any projective curve, $f: X \rightarrow C$ as in Theorem 3 . We say that $X$ is the nonsingular model of $C$, or of $K=k(C)$. We identify $k(X)$ with $K$ by means of $\tilde{f}$.
The points $Q$ of $X$ are in one-to-one correspondence with the discrete valuation rings $\mathscr{O}_{Q}(X)$ of $K$ (Corollary 4 of $\S 7.1$ ). Then $f(Q)=P$ exactly when $\mathscr{O}_{Q}(X)$ dominates $\mathscr{O}_{P}(C)$. The points of $X$ will be called places of $C$, or of $K$. A place $Q$ is centered at $P$ if $f(Q)=P$.
Lemma 2. Let $C$ be a projective plane curve, $P \in C$. Then there is an affine neighborhood $U$ of $P$ such that:
(1) $f^{-1}(U)=U^{\prime}$ is an affine open subvariety of $X$.
(2) $\Gamma\left(U^{\prime}\right)$ is module-finite over $\Gamma(U)$.
(3) For some $0 \neq t \in \Gamma(U), t \Gamma\left(U^{\prime}\right) \subset \Gamma(U)$.
(4) The vector space $\Gamma\left(U^{\prime}\right) / \Gamma(U)$ is finite dimensional over $k$. The neighborhood $U$ may be taken to exclude any finite set $S$ of points in $C$, if $P \notin S$.
Proof. We will choose successive quadratic transformations $C=C_{1}, \ldots, C_{n}$ so that $C_{n}$ has only ordinary multiple points, and open affine sets $W_{i} \subset C_{i}$, so that (i) $P \in W_{1}$ and $S \cap W_{1}=\varnothing$; (ii) the birational map from $C_{i+1}$ to $C_{i}$ is represented by a morphism $f_{i}: W_{i+1} \rightarrow W_{i}$; (iii) $f_{i}: W_{i+1} \rightarrow W_{i}$ satisfies all the conditions of Section 2, Step (3). These quadratic transformations and neighborhoods are chosen inductively. At each stage it may be necessary to shrink the previous neighborhoods; the remarks in Section 2, together with Problem 7.13, show that there is no difficulty in doing this.
Likewise let $f_{n}: X \rightarrow C_{n}$ be the nonsingular model of $C_{n}$ (Proposition 1 of $\$ 7.3$ ), and let $W_{n+1}=f_{n}^{-1}\left(W_{n}\right)$ (shrinking again if necessary); this time Section 3, Step (7) guarantees that the same conditions hold.
Let $U=W_{1}, U^{\prime}=W_{n+1}$. That $\Gamma\left(U^{\prime}\right)$ is module-finite over $\Gamma(U)$ follows from Problem 1.45. Suppose $\Gamma\left(U^{\prime}\right)=\sum_{i=1}^{m} \Gamma(U) z_{i}$. Since $\Gamma(U)$ and $\Gamma\left(U^{\prime}\right)$ have the same quotient field, there is a $t \in \Gamma(U)$ with $t z_{i} \in \Gamma(U), t \neq 0$. This $t$ satisfies (3).
Define $\varphi: \Gamma\left(U^{\prime}\right) / \Gamma(U) \rightarrow \Gamma(U) /(t)$ by $\varphi(\bar{z})=\overline{t z}$. Since $\varphi$ is a one-to-one $k$-linear map, it is enough to show that $\Gamma(U) /(t)$ is finite-dimensional. Since $t$ has only finitely many zeros in $U$, this follows from Corollary 4 to the Nullstellensatz in $\S 1.7$.
Notation. Let $f: X \rightarrow C$ as above $Q \in X, f(Q)=P \in C$. Suppose $C$ is a plane curve. For any plane curve $G$ (possibly reducible), form $G_{*} \in \mathscr{O}_{P}\left(\mathbb{P}^{2}\right)$ as in Chapter 5 , Section 1; let $g$ be the image of $G_{*}$ in $\mathscr{O}_{P}(C) \subset k(C)=k(X)$. Define $\operatorname{ord}_{Q}(G)$ to be $\operatorname{ord}_{Q}(g)$. As usual, this is independent of the choice of $G_{*}$.
Proposition 2. Let $C$ be an irreducible projective plane curve, $P \in C, f: X \rightarrow C$ as above. Let $G$ be a (possibly reducible) plane curve. Then $I(P, C \cap G)=\sum_{Q \in f^{-1}(P)} \operatorname{ord}_{Q}(G)$.
Proof. Let $g$ be the image of $G_{*}$ in $\mathscr{O}_{P}(C)$. Choose $U$ as in Lemma 2, and so small that $g$ is a unit in all $\mathscr{O}_{P^{\prime}}(C), P^{\prime} \in U, P^{\prime} \neq P$. Then $I(P, C \cap G)=\operatorname{dim}_{k}\left(\mathscr{O}_{P}\left(\mathbb{P}^{2}\right) /\left(F_{*}, G_{*}\right)\right)=$ $\operatorname{dim}_{k}\left(\mathscr{O}_{P}(C) /(g)\right.$ ) (Problem 2.44) $=\operatorname{dim}_{k}(\Gamma(U) /(g)$ ) (Corollary 1 of $\$ 2.9$ ). Let $V=$ $\Gamma(U), V^{\prime}=\Gamma\left(U^{\prime}\right)$, and let $T: V \rightarrow V^{\prime}$ be defined by $T(z)=g z$. Since $V^{\prime} / V$ is finite dimensional, Problem 2.53 applies: $\operatorname{dim} V / T(V)=\operatorname{dim} V^{\prime} / T\left(V^{\prime}\right)$, so $\operatorname{dim}_{k}(\Gamma(U) /(g))=$ $\operatorname{dim}\left(\Gamma\left(U^{\prime}\right) /(g)\right)$. By Corollary 1 of $\$ 2.9$ again,
$$
\operatorname{dim} \Gamma\left(U^{\prime}\right) /(g)=\sum_{Q \in f^{-1}(P)} \operatorname{dim}\left(\mathscr{O}_{Q}(X) /(g)\right)=\sum \operatorname{ord}_{Q}(g),
$$
as desired (see Problem 2.50).
Lemma 3. Suppose $P$ is an ordinary multiple point on $C$ of multiplicity $r$. Let $f^{-1}(P)=$ $\left\{P_{1}, \ldots, P_{r}\right\}$. If $z \in k(C)$, and $\operatorname{ord}_{P_{i}}(z) \geq r-1$, then $z \in \mathscr{O}_{P}(C)$.
Proof. Take a small neighborhood $U$ of $P$ so that $z \in \mathscr{O}_{P^{\prime}}(C)$ for all $P^{\prime} \in U, P^{\prime} \neq P$ (The pole set is an algebraic subset, hence finite), and so that $f: U^{\prime}=f^{-1}(U) \rightarrow U$ looks like Section 2 Step (3). By Step (2) of Section 2, we know that $\operatorname{ord}_{P_{i}}(x)=1$. Therefore $z x^{1-r} \in \Gamma\left(U^{\prime}\right)$. But $x^{r-1} \Gamma\left(U^{\prime}\right) \subset \Gamma(U)$, so $z=x^{r-1}\left(z x^{1-r}\right) \in \Gamma(U) \subset \mathscr{O}_{P}(C)$, as desired. Proposition 3. Let $F$ be an irreducible projective plane curve, $P$ an ordinary multiple point of multiplicity $r$ on $F$. Let $P_{1}, \ldots, P_{r}$ be the places centered at $P$. Let $G, H$ be plane curves, possibly reducible. Then Noether's conditions are satisfied at $P$ (with respect to $F, G, H)$ if $\operatorname{ord}_{P_{i}}(H) \geq \operatorname{ord}_{P_{i}}(G)+r-1$ for $i=1, \ldots, r$.
Proof. $H_{*} \in\left(F_{*}, G_{*}\right) \subset \mathscr{O}_{P}\left(\mathbb{P}^{2}\right)$ is equivalent with $\bar{H}_{*} \in\left(\bar{G}_{*}\right) \subset \mathscr{O}_{P}(F)$, or with $z=$ $\bar{H}_{*} / \bar{G}_{*} \in \mathscr{O}_{P}(F)$. Applying Lemma 3 to $z$ gives the result.
## Problems
7.17. (a) Show that for any irreducible curve $C$ (projective or not) there is a nonsingular curve $X$ and a birational morphism $f$ from $X$ onto $C$. What conditions on $X$ will make it unique? (b) Let $f: X \rightarrow C$ as in (a), and let $C^{\circ}$ be the set of simple points of $C$. Show that the restriction of $f$ to $f^{-1}\left(C^{\circ}\right)$ gives an isomorphism of $f^{-1}\left(C^{\circ}\right)$ with $C^{\circ}$.
7.18. Show that for any place $P$ of a curve $C$, and choice $t$ of uniformizing parameter for $\mathscr{O}_{P}(X)$, there is a homomorphism $\varphi: k(X) \rightarrow k((T))$ taking $t$ to $T$ (see Problem 2.32). Show how to recover the place from $\varphi$. (In many treatments of curves, a place is defined to be a suitable equivalence class of "power series expansions".)
7.19* Let $f: X \rightarrow C$ as above, $C$ a projective plane curve. Suppose $P$ is an ordinary multiple point of multiplicity $r$ on $C, Q_{1}, \ldots, Q_{r}$ the places on $X$ centered at $P$. Let $G$ be any projective plane curve, and let $s \leq r$. Show that $m_{P}(G) \geq s$ if and only if $\operatorname{ord}_{Q_{i}}(G) \geq s$ for $i=1, \ldots, r$. (See Problem 7.5.)
7.20. Let $R$ be a domain with quotient field $K$. The integral closure $R^{\prime}$ of $R$ is $\{z \in k \mid$ $z$ is integral over $R\}$. Prove:
(a) If $R$ is a DVR, then $R^{\prime}=R$.
(b) If $R_{\alpha}^{\prime}=R_{\alpha}$, then $\left(\cap R_{\alpha}\right)^{\prime}=\left(\cap R_{\alpha}\right)$.
(c) With $f: X \rightarrow C$ as in Lemma 2, show that $\Gamma\left(f^{-1}(U)\right)=\Gamma(U)^{\prime}$ for all open sets $U$ of $C$. This gives another algebraic characterization of $X$.
7.21* Let $X$ be a nonsingular projective curve, $P_{1}, \ldots, P_{s} \in X$. (a) Show that there is projective plane curve $C$ with only ordinary multiple points, and a birational morphism $f: X \rightarrow C$ such that $f\left(P_{i}\right)$ is simple on $C$ for each $i$. (Hint: if $f\left(P_{i}\right)$ is multiple, do a quadratic transform centered at $f\left(P_{i}\right)$.) (b) For any $m_{1}, \ldots, m_{r} \in \mathbb{Z}$, show that there is a $z \in k(X)$ such that $\operatorname{ord}_{P_{i}}(z)=m_{i}$ (Problem 5.15). (c) Show that the curve $C$ of Part (a) may be found with arbitrarily large degree (Problem 7.14).
7.22. Let $P$ be a node on an irreducible plane curve $F$, and let $L_{1}, L_{2}$ be the tangents to $F$ at $P$. $F$ is called a simple node if $I\left(P, L_{i} \cap F\right)=3$ for $i=1,2$. Let $H$ be the Hessian of $F$. (a) If $P$ is a simple node on $F$, show that $I(P, F \cap H)=6$. (Hint: We may take $P=[0: 0: 1], F_{*}=x y+\cdots$, and use Proposition 2 to show that all monomials of degree $\geq 4$ may be ignored - see Problem 5.23). (b) If $P$ is an ordinary cusp on $F$, show that $I(P, F \cap H)=8$ (see Problem 7.6). (c) Use (a) and (b) to show that every cubic has one, three, or nine flexes; then Problem 5.24 gives another proof that every cubic is projectively equivalent to one of the type $Y^{2} Z=$ cubic in $X$ and $Z$. (d) If the curve $F$ has degree $n$, and $i$ flexes (all ordinary), and $\delta$ simple nodes, and $k$ cusps, and no other singularities, then
$$
i+6 \delta+8 k=3 n(n-2)
$$
This is one of "Plücker's formulas" (see Walker's "Algebraic Curves" for the others).
## Chapter 8
## Riemann-Roch Theorem
Throughout this chapter, $C$ will be an irreducible projective curve, $f: X \rightarrow C$ the birational morphism from the nonsingular model $X$ onto $C, K=k(C)=k(X)$ the function field, as in Chapter 7, Section 5. The points $P \in C$ will be identified with the places of $K$; $\operatorname{ord}_{P}$ denotes the corresponding order function on $K$.
### Divisors
A divisor on $X$ is a formal sum $D=\sum_{P \in X} n_{P} P, n_{P} \in \mathbb{Z}$, and $n_{P}=0$ for all but a finite number of $P$. The divisors on $X$ form an abelian group - it is the free abelian group on the set $X$ (Chapter 2, Section 11).
The degree of a divisor is the sum of its coefficients: $\operatorname{deg}\left(\sum n_{P} P\right)=\sum n_{P}$. Clearly $\operatorname{deg}\left(D+D^{\prime}\right)=\operatorname{deg}(D)+\operatorname{deg}\left(D^{\prime}\right)$. A divisor $D=\sum n_{P} P$ is said to be effective (or positive) if each $n_{P} \geq 0$, and we write $\sum n_{P} P \geq \sum m_{P} P$ if each $n_{P} \geq m_{P}$.
Suppose $C$ is a plane curve of degree $n$, and $G$ is a plane curve not containing $C$ as a component. Define the divisor of $G, \operatorname{div}(G)$, to be $\sum_{P \in X} \operatorname{ord}_{P}(G) P$, where $\operatorname{ord}_{P}(G)$ is defined as in Chapter 7, Section 5. By Proposition 2 of $\$ 7.5, \sum_{P \in X} \operatorname{ord}_{P}(G)=$ $\sum_{Q \in C} I(Q, C \cap G)$. By Bézout's theorem, $\operatorname{div}(G)$ is a divisor of degree $m n$, where $m$ is the degree of $G$. Note that $\operatorname{div}(G)$ contains more information than the intersection cycle $G \cdot C$.
For any nonzero $z \in K$, define the divisor of $z$, $\operatorname{div}(z)$, to be $\sum_{P \in X} \operatorname{ord}_{P}(z) P$. Since $z$ has only a finite number of poles and zeros (Problem 4.17), $\operatorname{div}(z)$ is a well-defined divisor. If we let $(z)_{0}=\sum_{\operatorname{ord}_{p}(z)>0} \operatorname{ord}_{P}(z) P$, the divisor of zeros of $z$, and $(z)_{\infty}=$ $\sum_{\operatorname{ord}_{P}(z)<0}-\operatorname{ord}_{P}(z) P$, the divisor of poles of $z$, then $\operatorname{div}(z)=(z)_{0}-(z)_{\infty}$. Note that $\operatorname{div}\left(z z^{\prime}\right)=\operatorname{div}(z)+\operatorname{div}\left(z^{\prime}\right)$, and $\operatorname{div}\left(z^{-1}\right)=-\operatorname{div}(z)$.
Proposition 1. For any nonzero $z \in K, \operatorname{div}(z)$ is a divisor of degree zero. A rational function has the same number of zeros as poles, if they are counted properly.
Proof. Take $C$ to be a plane curve of degree $n$. Let $z=g / h, g, h$ forms of the same degree in $\Gamma_{h}(C)$; say $g, h$ are residues of forms $G, H$ of degree $m$ in $k[X, Y, Z]$. Then $\operatorname{div}(z)=\operatorname{div}(G)-\operatorname{div}(H)$, and we have seen that $\operatorname{div}(G)$ and $\operatorname{div}(H)$ have same degree $m n$. Corollary 1. Let $0 \neq z \in K$. Then the following are equivalent: (i) $\operatorname{div}(z) \geq 0$; (ii) $z \in k$; (iii) $\operatorname{div}(z)=0$.
Proof. If $\operatorname{div}(z) \geq 0, z \in \mathscr{O}_{P}(X)$ for all $P \in X$. If $z\left(P_{0}\right)=\lambda_{0}$ for some $P_{0}$, then $\operatorname{div}(z-$ $\left.\lambda_{0}\right) \geq 0$ and $\operatorname{deg}\left(\operatorname{div}\left(z-\lambda_{0}\right)\right)>0$, a contradiction, unless $z-\lambda_{0}=0$, i.e., $z \in k$.
Corollary 2. Let $z, z^{\prime} \in K$, both nonzero. Then $\operatorname{div}(z)=\operatorname{div}\left(z^{\prime}\right)$ if and only if $z^{\prime}=\lambda z$ for some $\lambda \in k$.
Two divisors $D, D^{\prime}$ are said to be linearly equivalent if $D^{\prime}=D+\operatorname{div}(z)$ for some $z \in K$, in which case we write $D^{\prime} \equiv D$.
Proposition 2. (1) The relation $\equiv$ is an equivalence relation.
(2) $D \equiv 0$ if and only if $D=\operatorname{div}(z), z \in K$.
(3) If $D \equiv D^{\prime}$, then $\operatorname{deg}(D)=\operatorname{deg}\left(D^{\prime}\right)$.
(4) If $D \equiv D^{\prime}$, and $D_{1} \equiv D_{1}^{\prime}$, then $D+D_{1} \equiv D^{\prime}+D_{1}^{\prime}$.
(5) Let $C$ be a plane curve. Then $D \equiv D^{\prime}$ if and only if there are two curves $G, G^{\prime}$ of the same degree with $D+\operatorname{div}(G)=D^{\prime}+\operatorname{div}\left(G^{\prime}\right)$.
Proof. (1)-(4) are left to the reader. For (5) it suffices to write $z=G / G^{\prime}, \operatorname{since} \operatorname{div}(z)=$ $\operatorname{div}(G)-\operatorname{div}\left(G^{\prime}\right)$ in this case.
The criterion proved in Chapter 7, Section 5 for Noether's conditions to hold translates nicely into the language of divisors:
Assume $C$ is a plane curve with only ordinary multiple points. For each $Q \in X$, let $r_{Q}=m_{f(Q)}(C)$. Define the divisor $E=\sum_{Q \in X}\left(r_{Q}-1\right) Q$. This $E$ is effective; its degree is $\sum m_{P}(C)\left(m_{P}(C)-1\right)$. Any plane curve $G$ such that $\operatorname{div}(G) \geq E$ is called an adjoint of $C$. From Problem 7.19 it follows that a curve $G$ is an adjoint to $C$ if and only if $m_{P}(G) \geq m_{P}(C)-1$ for every (multiple) point $P \in C$. If $C$ is nonsingular, every curve is an adjoint.
RESIDUE THEOREM. Let $C, E$ be as above. Suppose $D$ and $D^{\prime}$ are effective divisors on $X$, with $D^{\prime} \equiv D$. Suppose $G$ is an adjoint of degree $m$ such that $\operatorname{div}(G)=D+E+A$, for some effective divisor $A$. Then there is an adjoint $G^{\prime}$ of degree $m$ such that $\operatorname{div}\left(G^{\prime}\right)=$ $D^{\prime}+E+A$.
Proof. Let $H, H^{\prime}$ be curves of the same degree such that $D+\operatorname{div}(H)=D^{\prime}+\operatorname{div}\left(H^{\prime}\right)$. Then $\operatorname{div}(G H)=\operatorname{div}\left(H^{\prime}\right)+D^{\prime}+E+A \geq \operatorname{div}\left(H^{\prime}\right)+E$. Let $F$ be the form defining $C$. Applying the criterion of Proposition 3 of $\$ 7.5$ to $F, H^{\prime}$, and $G H$, we see that Noether's conditions are satisfied at all $P \in C$. By Noether's theorem, $G H=F^{\prime} F+G^{\prime} H^{\prime}$ for some $F^{\prime}, G^{\prime}$, where $\operatorname{deg}\left(G^{\prime}\right)=m$. Then $\operatorname{div}\left(G^{\prime}\right)=\operatorname{div}(G H)-\operatorname{div}\left(H^{\prime}\right)=D^{\prime}+E+A$, as desired.
## Problems
8.1. Let $X=C=\mathbb{P}^{1}, k(X)=k(t)$, where $t=X_{1} / X_{2}, X_{1}, X_{2}$ homogeneous coordinates on $\mathbb{P}^{1}$. (a) Calculate $\operatorname{div}(t)$. (b) Calculate $\operatorname{div}(f / g), f, g$ relatively prime in $k[t]$. (c) Prove Proposition 1 directly in this case. 8.2. Let $X=C=V\left(Y^{2} Z-X(X-Z)(X-\lambda Z)\right) \subset \mathbb{P}^{2}, \lambda \in k, \lambda \neq 0,1$. Let $x=X / Z$, $y=Y / Z \in K ; K=k(x, y)$. Calculate $\operatorname{div}(x)$ and $\operatorname{div}(y)$.
8.3. Let $C=X$ be a nonsingular cubic. (a) Let $P, Q \in C$. Show that $P \equiv Q$ if and only if $P=Q$. (Hint: Lines are adjoints of degree 1.) (b) Let $P, Q, R, S \in C$. Show that $P+Q \equiv R+S$ if and only if the line through $P$ and $Q$ intersects the line through $R$ and $S$ in a point on $C$ (if $P=Q$ use the tangent line). (c) Let $P_{0}$ be a fixed point on $C$, thus defining an addition $\oplus$ on $C$ (Chapter 5, Section 6). Show that $P \oplus Q=R$ if and only if $P+Q=R+P_{0}$. Use this to give another proof of Proposition 4 of $\S 5.6$.
8.4. Let $C$ be a cubic with a node. Show that for any two simple points $P, Q$ on $C$, $P \equiv Q$.
8.5. Let $C$ be a nonsingular quartic, $P_{1}, P_{2}, P_{3} \in C$. Let $D=P_{1}+P_{2}+P_{3}$. Let $L$ and $L^{\prime}$ be lines such that $L \cdot \cdot C=P_{1}+P_{2}+P_{4}+P_{5}, L^{\prime} \cdot C=P_{1}+P_{3}+P_{6}+P_{7}$. Suppose these seven points are distinct. Show that $D$ is not linearly equivalent to any other effective divisor. (Hint: Apply the residue theorem to the conic $L L^{\prime}$.) Investigate in a similar way other divisors of small degree on quartics with various types of multiple points.
8.6. Let $D(X)$ be the group of divisors on $X, D_{0}(X)$ the subgroup consisting of divisors of degree zero, and $P(X)$ the subgroup of $D_{0}(X)$ consisting of divisors of rational functions. Let $C_{0}(X)=D_{0}(X) / P(X)$ be the quotient group. It is the divisor class group on $X$. (a) If $X=\mathbb{P}^{1}$, then $C_{0}(X)=0$. (b) Let $X=C$ be a nonsingular cubic. Pick $P_{0} \in C$, defining $\oplus$ on $C$. Show that the map from $C$ to $C_{0}(X)$ that sends $P$ to the residue class of the divisor $P-P_{0}$ is an isomorphism from $(C, \oplus)$ onto $C_{0}(X)$.
8.7. When do two curves $G, H$ have the same divisor ( $C$ and $X$ are fixed)?
### The Vector Spaces $L(D)$
Let $D=\sum n_{P} P$ be a divisor on $X$. Each $D$ picks out a finite number of points, and assigns integers to them. We want to determine when there is a rational function with poles only at the chosen points, and with poles no "worse" than order $n_{P}$ at $P$; if so, how many such functions are there?
Define $L(D)$ to be $\left\{f \in K \mid \operatorname{ord}_{P}(f) \geq-n_{P}\right.$ for all $\left.P \in X\right\}$, where $D=\sum n_{P} P$. Thus a rational function $f$ belongs to $L(D)$ if $\operatorname{div}(f)+D \geq 0$, or if $f=0 . L(D)$ forms a vector space over $k$. Denote the dimension of $L(D)$ by $l(D)$; the next proposition shows that $l(D)$ is finite.
Proposition 3. (1) If $D \leq D^{\prime}$, then $L(D) \subset L\left(D^{\prime}\right)$, and
$$
\operatorname{dim}_{k}\left(L\left(D^{\prime}\right) / L(D)\right) \leq \operatorname{deg}\left(D^{\prime}-D\right)
$$
(2) $L(0)=k ; L(D)=0$ if $\operatorname{deg}(D)<0$.
(3) $L(D)$ is finite dimensional for all $D$. If $\operatorname{deg}(D) \geq 0$, then $l(D) \leq \operatorname{deg}(D)+1$.
(4) If $D \equiv D^{\prime}$, then $l(D)=l\left(D^{\prime}\right)$.
Proof. (1): $D^{\prime}=D+P_{1}+\cdots+P_{s}$, and $L(D) \subset L\left(D+P_{1}\right) \subset \cdots \subset L\left(D+P_{1}+\cdots+P_{s}\right)$, so it suffices to show that $\operatorname{dim}(L(D+P) / L(D)) \leq 1$ (Problem 2.49). To prove this, let $t$ be a uniformizing parameter in $\mathscr{O}_{P}(X)$, and let $r=n_{P}$ be the coefficient of $P$ in $D$. Define $\varphi: L(D+P) \rightarrow k$ by letting $\varphi(f)=\left(t^{r+1} f\right)(P)$; since $\operatorname{ord}_{P}(f) \geq-r-1$, this is well-defined; $\varphi$ is a linear map, and $\operatorname{Ker}(\varphi)=L(D)$, so $\varphi$ induces a one-to-one linear $\operatorname{map} \bar{\varphi}: L(D+P) / L(D) \rightarrow k$, which gives the result.
(2): This follows from Corollary 1 and Proposition 2 (3) of $\$ 8.1$.
(3): If $\operatorname{deg}(D)=n \geq 0$, choose $P \in X$, and let $D^{\prime}=D-(n+1) P$. Then $L\left(D^{\prime}\right)=0$, and by (1), $\operatorname{dim}\left(L(D) / L\left(D^{\prime}\right)\right) \leq n+1$, so $l(D) \leq n+1$.
(4): Suppose $D^{\prime}=D+\operatorname{div}(g)$. Define $\psi: L(D) \rightarrow L\left(D^{\prime}\right)$ by setting $\psi(f)=f g$. Since $\psi$ is an isomorphism of vector spaces, $l(D)=l\left(D^{\prime}\right)$.
More generally, for any subset $S$ of $X$, and any divisor $D=\sum n_{P} P$ on $X$, define $\operatorname{deg}^{S}(D)=\sum_{P \in S} n_{P}$, and $L^{S}(D)=\left\{f \in K \mid \operatorname{ord}_{P}(f) \geq-n_{P}\right.$ for all $\left.P \in S\right\}$.
Lemma 1. If $D \leq D^{\prime}$, then $L^{S}(D) \subset L^{S}\left(D^{\prime}\right)$. If $S$ is finite, then $\operatorname{dim}\left(L^{S}\left(D^{\prime}\right) / L^{S}(D)\right)=$ $\operatorname{deg}^{S}\left(D^{\prime}-D\right)$.
Proof. Proceeding as in Proposition 3, we assume $D^{\prime}=D+P$, and define $\varphi: L^{S}(D+$ $P) \rightarrow k$ the same way. We must show that $\varphi$ maps $L^{S}(D+P)$ onto $k$, i.e., $\varphi \neq 0$, for then $\bar{\varphi}$ is an isomorphism. Thus we need to find an $f \in K$ with $\operatorname{ord}_{P}(f)=-r-1$, and with $\operatorname{ord}_{Q}(f) \geq-n_{Q}$ for all $Q \in S$. But this is easy, since $S$ is finite (Problem 7.21(b)).
The next proposition is an important first step in calculating the dimension $l(D)$. The proof (see Chevalley's "Algebraic Functions of One Variable”, Chap. I.) involves only the field of rational functions.
Proposition 4. Let $x \in K, x \notin k$. Let $Z=(x)_{0}$ be the divisor of zeros of $x$, and let $n=[K: k(x)]$. Then
(1) $Z$ is an effective divisor of degree $n$.
(2) There is a constant $\tau$ such that $l(r Z) \geq r n-\tau$ for all $r$.
Proof. Let $Z=(x)_{0}=\sum n_{P} P$, and let $m=\operatorname{deg}(Z)$. We show first that $m \leq n$.
Let $S=\left\{P \in X \mid n_{P}>0\right\}$. Choose $v_{1}, \ldots, v_{m} \in L^{S}(0)$ so that the residues $\bar{v}_{1}, \ldots, \bar{v}_{m} \in$ $L^{S}(0) / L^{S}(-Z)$ form a basis for this vector space (Lemma 1). We will show that $v_{1}, \ldots, v_{m}$ are linearly independent over $k(x)$. If not (by clearing denominators and multiplying by a power of $x$ ), there would be polynomials $g_{i}=\lambda_{i}+x h_{i} \in k[x]$ with $\lambda_{i} \in k$, $\sum g_{i} v_{i}=0$, not all $\lambda_{i}=0$. But then $\sum \lambda_{i} v_{i}=-x \sum h_{i} v_{i} \in L^{S}(-Z)$, so $\sum \lambda_{i} \bar{v}_{i}=0$, a contradiction. So $m \leq n$. Next we prove (2).
Let $w_{1}, \ldots, w_{n}$ be a basis of $K$ over $k(x)$ (Proposition 9 of $\$ 6.5$ ). We may assume that each $w_{i}$ satisfies an equation $w_{i}^{n_{i}}+a_{i 1} w_{i}^{n_{i}-1}+\cdots=0, a_{i j} \in k\left[x^{-1}\right]$ (Problem 1.54). Then $\operatorname{ord}_{P}\left(a_{i j}\right) \geq 0$ if $P \notin S$. If $\operatorname{ord}_{P}\left(w_{i}\right)<0, P \notin S$, then $\operatorname{ord}_{P}\left(w_{i}^{n_{i}}\right)<$ $\operatorname{ord}_{P}\left(a_{i j} w_{i}^{n_{i}-j}\right)$, which is impossible (Problem 2.29). It follows that for some $t>0$, $\operatorname{div}\left(w_{i}\right)+t Z \geq 0, i=1, \ldots, n$. Then $w_{i} x^{-j} \in L((r+t) Z)$ for $i=1, \ldots, n, j=0,1, \ldots, r$. Since the $w_{i}$ are independent over $k(x)$, and $1, x^{-1}, \ldots, x^{-r}$ are independent over $k$, $\left\{w_{i} x^{-j} \mid i=1, \ldots, n, j=0, \ldots, r\right\}$ are independent over $k$. So $l((r+t) Z) \geq n(r+1)$. But $l((r+t) Z)=l(r Z)+\operatorname{dim}(L((r+t) Z) / L(r Z)) \leq l(r Z)+t m$ by Proposition $3(1)$. Therefore $l(r Z) \geq n(r+1)-t m=r n-\tau$, as desired.
Lastly, since $r n-\tau \leq l(r Z) \leq r m+1$ (Proposition 3 (3)), if we let $r$ get large, we see that $n \leq m$. Corollary. The following are equivalent:
(1) $C$ is rational.
(2) $X$ is isomorphic to $\mathbb{P}^{1}$.
(3) There is an $x \in K$ with $\operatorname{deg}\left(\left(x_{0}\right)\right)=1$.
(4) For some $P \in X, l(P)>1$.
Proof. (4) says that there is nonconstant $x \in L(P)$, so $(x)_{\infty}=P$. Then $\operatorname{deg}\left((x)_{0}\right)=$ $\operatorname{deg}\left((x)_{\infty}\right)=1$, so $[K: k(x)]=1$, i.e., $K=k(x)$ is rational. The rest is easy (see Problem 8.1).
## Problems
8.8. If $D \leq D^{\prime}$, then $l\left(D^{\prime}\right) \leq l(D)+\operatorname{deg}\left(D^{\prime}-D\right)$, i.e., $\operatorname{deg}(D)-l(D) \leq \operatorname{deg}\left(D^{\prime}\right)-l\left(D^{\prime}\right)$.
8.9. Let $X=\mathbb{P}^{1}, t$ as in Problem 8.1. Calculate $L\left(r(t)_{0}\right)$ explicitly, and show that $l\left(r(t)_{0}\right)=r+1$.
8.10. Let $X=C$ be a cubic, $x, y$ as in Problem 8.2. Let $z=x^{-1}$. Show that $L\left(r(z)_{0}\right) \subset$ $k[x, y]$, and show that $l\left(r(z)_{0}\right)=2 r$ if $r>0$.
8.11. Let $D$ be a divisor. Show that $l(D)>0$ if and only if $D$ is linearly equivalent to an effective divisor.
8.12. Show that $\operatorname{deg}(D)=0$ and $l(D)>0$ are true if and only if $D \equiv 0$.
8.13. Suppose $l(D)>0$, and let $f \neq 0, f \in L(D)$. Show that $f \notin L(D-P)$ for all but a finite number of $P$. So $l(D-P)=l(D)-1$ for all but a finite number of $P$.
### Riemann's Theorem
If $D$ is a large divisor, $L(D)$ should be large. Proposition 4 shows this for divisors of a special form.
RIEMANN'S THEOREM. There is an integer $g$ such that $l(D) \geq \operatorname{deg}(D)+1-g$ for all divisors D. The smallest such $g$ is called the genus of $X$ (or of $K$, or $C$ ). The genus is a nonnegative integer.
Proof. For each $D$, let $s(D)=\operatorname{deg}(D)+1-l(D)$. We want to find $g$ so that $s(D) \leq g$ for all $D$.
(1) $s(0)=0$, so $g \geq 0$ if it exists.
(2) If $D \equiv D^{\prime}$, then $s(D)=s\left(D^{\prime}\right)$ (Propositions 2 and 3 of $\S 8.2$ ).
(3) If $D \leq D^{\prime}$, then $s(D) \leq s\left(D^{\prime}\right)$ (Problem 8.8). Let $x \in K, x \notin k$, let $Z=(x)_{0}$, and let $\tau$ be the smallest integer that works for Proposition 4 (2). Since $s(r Z) \leq \tau+1$ for all $r$, and since $r Z \leq(r+1) Z$, we deduce from (3) that
(4) $s(r Z)=\tau+1$ for all large $r>0$. Let $g=\tau+1$. To finish the proof, it suffices (by (2) and (3)) to show:
(5) For any divisor $D$, there is a divisor $D^{\prime} \equiv D$, and an integer $r \geq 0$ such that $D^{\prime} \leq$ $r Z$. To prove this, let $Z=\sum n_{P} P, D=\sum m_{P} P$. We want $D^{\prime}=D-\operatorname{div}(f)$, so we need $m_{P}-\operatorname{ord}_{P}(f) \leq r n_{P}$ for all $P$. Let $y=x^{-1}$, and let $T=\left\{P \in X \mid m_{P}>0\right.$ and $\left.\operatorname{ord}_{P}(y) \geq 0\right\}$. Let $f=\prod_{P \in T}(y-y(P))^{m_{P}}$. Then $m_{P}-\operatorname{ord}_{P}(f) \leq 0$ whenever $\operatorname{ord}_{P}(y) \geq 0$. If $\operatorname{ord}(y)<$ 0 , then $n_{P}>0$, so a large $r$ will take care of this.
Corollary 1. If $l\left(D_{0}\right)=\operatorname{deg}\left(D_{0}\right)+1-g$, and $D \equiv D^{\prime} \geq D_{0}$, then $l(D)=\operatorname{deg}(D)+1-g$.
Corollary 2. If $x \in K, x \notin k$, then $g=\operatorname{deg}\left(r(x)_{0}\right)-l\left(r(x)_{0}\right)+1$ for all sufficiently large $r$.
Corollary 3. There is an integer $N$ such that for all divisors $D$ of degree $>N, l(D)=$ $\operatorname{deg}(D)+1-g$.
Proofs. The first two corollaries were proved on the way to proving Riemann's Theorem. For the third, choose $D_{0}$ such that $l\left(D_{0}\right)=\operatorname{deg}\left(D_{0}\right)+1-g$, and let $N=\operatorname{deg}\left(D_{0}\right)+$ $g$. Then if $\operatorname{deg}(D) \geq N, \operatorname{deg}\left(D-D_{0}\right)+1-g>0$, so by Riemann's Theorem, $l\left(D-D_{0}\right)>$ 0 . Then $D-D_{0}+\operatorname{div}(f) \geq 0$ for some $f$, i.e., $D \equiv D+\operatorname{div}(f) \geq D_{0}$, and the result follows from Corollary 1.
Examples. (1) $g=0$ if and only if $C$ is rational. If $C$ is rational, $g=0$ by Corollary 2 and Problem 8.9 (or Proposition 5 below). Conversely, if $g=0, l(P)>1$ for any $P \in X$, and the result follows from the Corollary to Proposition 4 of $\S 8.2$.
(2) $(\operatorname{char}(k) \neq 2) g=1$ if and only if $C$ is birationally equivalent to a nonsingular cubic.
For if $X$ is a nonsingular cubic, the result follows from Corollary 2, Problems 8.10 and 5.24 (or Proposition 5 below). Conversely, if $g=1$, then $l(P) \geq 1$ for all $P$. By the Corollary of $\$ 8.2, l(P)=1$, and by the above Corollary $1, l(r P)=r$ for all $r>0$.
Let $1, x$ be a basis for $L(2 P)$. Then $(x)_{\infty}=2 P$ since if $(x)_{\infty}=P, C$ would be rational. So $[K: k(x)]=2$. Let $1, x, y$ be a basis for $L(3 P)$. Then $(y)_{\infty}=3 P$, so $y \notin k(x)$, so $K=k(x, y)$. Since $1, x, y, x^{2}, x y, y^{2} \in L(6 P)$, there is a relation of the form $a y^{2}+(b x+c) y=Q(x), Q$ a polynomial of degree $\leq 3$. By calculating $\operatorname{ord}_{P}$ of both sides, we see that $a \neq 0$ and $\operatorname{deg} Q=3$, so we may assume $a=1$. Replacing $y$ by $y+\frac{1}{2}(b x+c)$, we may assume $y^{2}=\prod_{i=1}^{3}\left(x-\alpha_{i}\right)$. If $\alpha_{1}=\alpha_{2}$, then $\left(y /\left(x-\alpha_{1}\right)\right)^{2}=x-\alpha_{3}$, so $x, y \in k\left(y /\left(x-\alpha_{1}\right)\right)$; but then $X$ would be rational, which contradicts the first example. So the $\alpha_{i}$ are distinct.
It follows that $K=k(C)$, where $C=V\left(Y^{2} Z-\prod_{i=1}^{3}\left(X-\alpha_{i} Z\right)\right)$ is a nonsingular cubic.
The usefulness of Riemann's Theorem depends on being able to calculate the genus of a curve. By its definition the genus depends only on the nonsingular model, or the function field, so two birationally equivalent curves have the same genus. Since we have a method for finding a plane curve with only ordinary multiple points that is birationally equivalent to a given curve, the following proposition is all that we need:
Proposition 5. Let $C$ be a plane curve with only ordinary multiple points. Let $n$ be the degree of $C, r_{P}=m_{P}(C)$. Then the genus $g$ of $C$ is given by the formula
$$
g=\frac{(n-1)(n-2)}{2}-\sum_{P \in C} \frac{r_{P}\left(r_{P}-1\right)}{2} .
$$
Proof. By the above Corollary 3, we need to find some "large" divisors $D$ for which we can calculate $l(D)$. The Residue Theorem allows us to find all effective divisors linearly equivalent to certain divisors $D$. These two observations lead to the calculation of $g$.
We may assume that the line $Z=0$ intersect $C$ in $n$ distinct points $P_{1}, \ldots, P_{n}$. Let $F$ be the form defining $C$.
Let $E=\sum_{Q \in X}\left(r_{Q}-1\right) Q, r_{Q}=r_{f(Q)}=m_{f(Q)}(C)$ as in Section 1. Let
$$
E_{m}=m \sum_{i=1}^{n} P_{i}-E
$$
So $E_{m}$ is a divisor of degree $m n-\sum_{P \in C} r_{P}\left(r_{P}-1\right)$.
Let $V_{m}=$ fforms $G$ of degree $m$ such that $G$ is adjoint to $C$ \}. Since $G$ is adjoint if and only if $m_{P}(G) \geq r_{P}-1$ for all $P \in C$, we may apply Theorem 1 of $\S 5.2$ to calculate the dimension of $V_{m}$. We find that
$$
\operatorname{dim} V_{m} \geq \frac{(m+1)(m+2)}{2}-\sum \frac{r_{P}\left(r_{P}-1\right)}{2},
$$
with equality if $m$ is large. (Note that $V_{m}$ is the vector space of forms, not the projective space of curves.)
Let $\varphi: V_{m} \rightarrow L\left(E_{m}\right)$ be defined by $\varphi(G)=G / Z^{m}$. Then $\varphi$ is a linear map, and $\varphi(G)=0$ if and only if $G$ is divisible by $F$.
We claim that $\varphi$ is onto. For if $f \in L\left(E_{m}\right)$, write $f=R / S$, with $R, S$ forms of the same degree. Then $\operatorname{div}\left(R Z^{m}\right) \geq \operatorname{div}(S)+E$. By Proposition 3 of $\S 7.5$, there is an equation $R Z^{m}=A S+B F$. So $R / S=A / Z^{m}$ in $k(F)$, and so $\varphi(A)=f$. (Note that $\operatorname{div}(A)=\operatorname{div}\left(R Z^{m}\right)-\operatorname{div}(S) \geq E$, so $\left.A \in V_{m} \cdot\right)$
It follows that the following sequence of vector spaces is exact:
$$
0 \longrightarrow W_{m-n} \stackrel{\psi}{\longrightarrow} V_{m} \stackrel{\varphi}{\longrightarrow} L\left(E_{m}\right) \longrightarrow 0
$$
where $W_{m-n}$ is the space of all forms of degree $m-n$, and $\psi(H)=F H$ for $H \in W_{m-n}$.
By Proposition 7 of $\$ 2.10$, we may calculate $\operatorname{dim} L\left(E_{m}\right)$, at least for $m$ large. It follows that
$$
l\left(E_{m}\right)=\operatorname{deg}\left(E_{m}\right)+1-\left(\frac{(n-1)(n-2)}{2}-\sum \frac{r_{P}\left(r_{P}-1\right)}{2}\right)
$$
for large $m$. But since $\operatorname{deg}\left(E_{m}\right)$ increases as $m$ increases, Corollary 3 of Riemann's Theorem applies to finish the proof.
Corollary 1. Let $C$ be a plane curve of degree $n, r_{P}=m_{P}(C), P \in C$. Then
$$
g \leq \frac{(n-1)(n-2)}{2}-\sum \frac{r_{P}\left(r_{P}-1\right)}{2} .
$$
Proof. The number on the right is what we called $g^{*}(C)$ in Chapter 7, Section 4 . We saw there that $g^{*}$ decreases under quadratic transformations, so Theorem 2 of $\S 7.4$ concludes the proof. Corollary 2. If $\sum \frac{r_{P}\left(r_{P}-1\right)}{2}=\frac{(n-1)(n-2)}{2}$, then $C$ is rational.
Corollary 3. (a) With $E_{m}$ as in the proof of the proposition, any $h \in L\left(E_{m}\right)$ may be written $h=H / Z^{m}$, where $H$ is an adjoint of degree $m$.
(b) $\operatorname{deg}\left(E_{n-3}\right)=2 g-2$, and $l\left(E_{n-3}\right) \geq g$.
Proof. This follows from the exact sequence constructed in proving the proposition. Note that if $m<n$, then $V_{m}=L\left(E_{m}\right)$.
Examples. Lines and conics are rational. Nonsingular cubics have genus one. Singular cubics are rational. Since a nonsingular curve of degree $n$ has genus $\frac{(n-1)(n-2)}{2}$, not every curve is birationally equivalent to a nonsingular plane curve. For example, $Y^{2} X Z=X^{4}+Y^{4}$ has one node, so is of genus 2 , and no nonsingular plane curve has genus 2 .
## Problems
8.14. Calculate the genus of each of the following curves:
(a) $X^{2} Y^{2}-Z^{2}\left(X^{2}+Y^{2}\right)$.
(b) $\left(X^{3}+Y^{3}\right) Z^{2}+X^{3} Y^{2}-X^{2} Y^{3}$.
(c) The two curves of Problem 7.12.
(d) $\left(X^{2}-Z^{2}\right)^{2}-2 Y^{3} Z-3 Y^{2} Z^{2}$.
8.15. Let $D=\sum n_{P} P$ be an effective divisor, $S=\left\{P \in C \mid n_{P}>0\right\}, U=X \backslash S$. Show that $L(r D) \subset \Gamma\left(U, \mathscr{O}_{X}\right)$ for all $r \geq 0$.
8.16. Let $U$ be any open set on $X, \varnothing \neq U \neq X$. Then $\Gamma\left(U, \mathscr{O}_{X}\right)$ is infinite dimensional over $k$.
8.17. Let $X, Y$ be nonsingular projective curves, $f:: X \rightarrow Y$ a dominating morphism. Prove that $f(X)=Y$. (Hint: If $P \in Y \backslash f(X)$, then $\tilde{f}(\Gamma(Y \backslash\{P\})) \subset \Gamma(X)=k$; apply Problem 8.16.)
8.18. Show that a morphism from a projective curve $X$ to a curve $Y$ is either constant or surjective; if it is surjective, $Y$ must be projective.
8.19. If $f: C \rightarrow V$ is a morphism from a projective curve to a variety $V$, then $f(C)$ is a closed subvariety of $V$. (Hint: Consider $C^{\prime}=$ closure of $f(C)$ in $V$.)
8.20. Let $C$ be the curve of Problem 8.14(b), and let $P$ be a simple point on $C$. Show that there is a $z \in \Gamma(C \backslash\{P\})$ with $\operatorname{ord}_{P}(z) \geq-12, z \notin k$.
8.21. Let $C_{0}(X)$ be the divisor class group of $X$. Show that $C_{0}(X)=0$ if and only if $X$ is rational.
### Derivations and Differentials
This section contains the algebraic background needed to study differentials on a curve. Let $R$ be a ring containing $k$, and let $M$ be an $R$-module. A derivation of $R$ into $M$ over $k$ is a $k$-linear map $D: R \rightarrow M$ such that $D(x y)=x D(y)+y D(x)$ for all $x, y \in R$. It follows that for any $F \in k\left[X_{1}, \ldots, X_{n}\right]$ and $x_{1}, \ldots, x_{n} \in R$,
$$
D\left(F\left(X_{1}, \ldots, X_{n}\right)\right)=\sum_{i=1}^{n} F_{X_{i}}\left(x_{1}, \ldots, x_{n}\right) D\left(x_{i}\right) .
$$
Since all rings will contain $k$, we will omit the phrase "over $k$ ".
Lemma 2. If $R$ is a domain with quotient field $K$, and $M$ is a vector space over $K$, then any derivation $D: R \rightarrow M$ extends uniquely to a derivation $\tilde{D}: K \rightarrow M$.
Proof. If $z \in K$, and $z=x / y$, with $x$ and $y$ in $R$, then, since $x=y z$, we must have $D x=y \tilde{D} z+z D y$. So $\tilde{D}(z)=y^{-1}(D x-z D y)$, which shows the uniqueness. If we define $\tilde{D}$ by this formula, it is not difficult to verify that $\tilde{D}$ is a well-defined derivation from $K$ to $M$.
We want to define differentials of $R$ to be elements of the form $\sum x_{i} d y_{i}, x_{i}, y_{i} \in$ $R$; they should behave like the differentials of calculus. This is most easily done as follows:
For each $x \in R$ let $[x]$ be a symbol. Let $F$ be the free $R$-module on the set $\{[x] \mid x \in$ $R$ \}. Let $N$ be the submodule of $F$ generated by the following sets of elements:
(i) $\{[x+y]-[x]-[y] \mid x, y \in R\}$
(ii) $\{[\lambda x]-\lambda[x] \mid x \in R, \lambda \in k\}$
(iii) $\{[x y]-x[y]-y[x] \mid x, y \in R\}$
Let $\Omega_{k}(R)=F / N$ be the quotient module. Let $d x$ be the image of $[x]$ in $F / N$, and let $d: R \rightarrow \Omega_{k}(R)$ be the mapping that takes $x$ to $d x . \Omega_{k}(R)$ is an $R$-module, called the module of differentials of $R$ over $k$, and $d: R \rightarrow \Omega_{k}(R)$ is a derivation.
Lemma 3. For any $R$-module $M$, and any derivation $D: R \rightarrow M$, there is a unique homomorphism of $R$-module $\varphi: \Omega_{k}(R) \rightarrow M$ such that $D(x)=\varphi(d x)$ for all $x \in R$.
Proof. If we define $\varphi^{\prime}: F \rightarrow M$ by $\varphi^{\prime}\left(\sum x_{i}\left[y_{i}\right]\right)=\sum x_{i} D\left(y_{i}\right)$, then $\varphi^{\prime}(N)=0$, so $\varphi^{\prime}$ induces $\varphi: \Omega_{k}(R) \rightarrow M$.
If $x_{1}, \ldots, x_{n} \in R$, and $G \in k\left[X_{1}, \ldots, X_{n}\right]$, then
$$
d\left(G\left(x_{1}, \ldots, x_{n}\right)\right)=\sum_{i=1}^{n} G_{X_{i}}\left(x_{1}, \ldots, x_{n}\right) d x_{i} .
$$
It follows that if $R=k\left[x_{1}, \ldots, x_{n}\right]$, then $\Omega_{k}(R)$ is generated (as an $R$-module) by the differentials $d x_{1}, \ldots, d x_{n}$.
Likewise, if $R$ is a domain with quotient field $K$, and $z=x / y \in K, x, y \in R$, then $d z=y^{-1} d x-y^{-1} z d y$. In particular, if $K=k\left(x_{1}, \ldots, x_{n}\right)$, then $\Omega_{k}(K)$ is a vector space of finite dimension over $K$, generated by $d x_{1}, \ldots, d x_{n}$.
Proposition 6. (1) Let $K$ be an algebraic function field in one variable over $k$. Then $\Omega_{k}(K)$ is a one-dimensional vector space over $K$.
(2) $(\operatorname{char}(k)=0)$ If $x \in K, x \notin k$, then $d x$ is a basis for $\Omega_{k}(K)$ over $K$. Proof. Let $F \in k[X, Y]$ be an affine plane curve with function field $K$ (Corollary of §6.6). Let $R=k[X, Y] /(F)=k[x, y] ; K=k(x, y)$. We may assume $F_{Y} \neq 0$ (since $F$ is irreducible), so $F$ doesn't divide $F_{Y}$, i.e., $F_{Y}(x, y) \neq 0$. The above discussion shows that $d x$ and $d y$ generate $\Omega_{k}(K)$ over $K$. But $0=d(F(x, y))=F_{X}(x, y) d x+$ $F_{Y}(x, y) d y$, so $d y=u d x$, where $u=-F_{X}(x, y) / F_{Y}(x, y)$. Therefore $d x$ generates $\Omega_{k}(K)$, so $\operatorname{dim}_{K}\left(\Omega_{k}(K)\right) \leq 1$.
So we must show that $\Omega_{k}(K) \neq 0$. By Lemmas 2 and 3, it suffices to find a nonzero derivation $D: R \rightarrow M$ for some vector space $M$ over $K$. Let $M=K$, and, for $G \in$ $k[X, Y], \bar{G}$ its image in $R$, let $D(\bar{G})=G_{X}(x, y)-u G_{Y}(x, y)$, with $u$ as in the preceding paragraph. It is left to the reader to verify that $D$ is a well-defined derivation, and that $D(x)=1$, so $D \neq 0$.
It follows ( $\operatorname{char}(k)=0)$ that for any $f, t \in K, t \notin k$, there is a unique element $v \in K$ such that $d f=v d t$. It is natural to write $v=\frac{d f}{d t}$, and call $v$ the derivative of $f$ with respect to $t$.
Proposition 7. With $K$ as in Proposition 6 , let $\mathscr{O}$ be a discrete valuation ring of $K$, and let $t$ be a uniformizing parameter in $\mathscr{O}$. If $f \in \mathscr{O}$, then $\frac{d f}{d t} \in \mathscr{O}$.
Proof. Using the notation of the proof of Proposition 6, we may assume $\mathscr{O}=\mathscr{O}_{P}(F)$, $P=(0,0)$ a simple point on $F$. For $z \in K$, write $z^{\prime}$ instead of $\frac{d z}{d t}, t$ being fixed throughout.
Choose $N$ large enough that $\operatorname{ord}_{P}\left(x^{\prime}\right) \geq-N, \operatorname{ord}_{P}\left(y^{\prime}\right) \geq-N$. Then if $f \in R=$ $k[x, y], \operatorname{ord}_{P}\left(f^{\prime}\right) \geq-N$, since $f^{\prime}=f_{X}(x, y) x^{\prime}+f_{Y}(x, y) y^{\prime}$. $-N$.
If $f \in \mathscr{O}$, write $f=g / h, g, h \in R, h(P) \neq 0$. Then $f^{\prime}=h^{-2}\left(h g^{\prime}-g h^{\prime}\right)$, so $\operatorname{ord}_{P}\left(f^{\prime}\right) \geq$
We can now complete the proof. Let $f \in \mathscr{O}$. Write $f=\sum_{i<N} \lambda_{i} t^{i}+t^{N} g, \lambda_{i} \in k$, $g \in \mathscr{O}$ (Problem 2.30). Then $f^{\prime}=\sum i \lambda_{i} t^{i-1}+g N t^{N-1}+t^{N} g^{\prime}$. Since $\operatorname{ord}_{P}\left(g^{\prime}\right) \geq-N$, each term belongs to $\mathscr{O}$, so $f^{\prime} \in \mathscr{O}$, as required.
## Problems
8.22. Generalize Proposition 6 to function fields in $n$ variables.
8.23. With $\mathscr{O}, t$ as in Proposition 7, let $\varphi: \mathscr{O} \rightarrow k[[T]]$ be the corresponding homomorphism (Problem 2.32). Show that, for $f \in \mathscr{O}, \varphi$ takes the derivative of $f$ to the "formal derivative" of $\varphi(f)$. Use this to give another proof of Proposition 7, and of the fact that $\Omega_{k}(K) \neq 0$ in Proposition 6 .
### Canonical Divisors
Let $C$ be a projective curve, $X$ its nonsingular model, $K$ their function field as before. We let $\Omega=\Omega_{k}(K)$ be the space of differentials of $K$ over $k$; elements $\omega \in \Omega$ may also be called differentials on $X$, or on $C$.
Let $\omega \in \Omega, \omega \neq 0$, and let $P \in X$ be a place. We define the $\operatorname{order}$ of $\omega$ at $P$, $\operatorname{ord}_{P}(\omega)$, as follows: Choose a uniformizing parameter $t$ in $\mathscr{O}_{P}(X)$, write $\omega=f d t, f \in K$, and set $\operatorname{ord}_{P}(\omega)=\operatorname{ord}_{P}(f)$. To see that this is well-defined, suppose $u$ were another uniformizing parameter, and $f d t=g d u$; then $f / g=\frac{d u}{d t} \in \mathscr{O}_{P}(X)$ by Proposition 7, and likewise $g / f \in \mathscr{O}_{P}(X)$, so $\operatorname{ord}_{P}(f)=\operatorname{ord}_{P}(g)$.
If $0 \neq \omega \in \Omega$, the divisor of $\omega$, $\operatorname{div}(\omega)$, is defined to be $\sum_{P \in X} \operatorname{ord}_{P}(\omega) P$. In Proposition 8 we shall show that only finitely many $\operatorname{ord}_{P}(\omega) \neq 0$ for a given $\omega$, so that $\operatorname{div}(\omega)$ is a well-defined divisor.
Let $W=\operatorname{div}(\omega) . W$ is called a canonical divisor. If $\omega^{\prime}$ is another nonzero differential in $\Omega$, then $\omega^{\prime}=f \omega, f \in K$, so $\operatorname{div}\left(\omega^{\prime}\right)=\operatorname{div}(f)+\operatorname{div}(\omega)$, and $\operatorname{div}\left(\omega^{\prime}\right) \equiv \operatorname{div}(\omega)$. Conversely if $W^{\prime} \equiv W$, say $W^{\prime}=\operatorname{div}(f)+W$, then $W^{\prime}=\operatorname{div}(f \omega)$. So the canonical divisors form an equivalence class under linear equivalence. In particular, all canonical divisors have the same degree.
Proposition 8. Assume $C$ is a plane curve of degree $n \geq 3$ with only ordinary multiple points. Let $E=\sum_{Q \in X}\left(r_{Q}-1\right) Q$, as in Section 1. Let $G$ be any plane curve of degree $n-3$. Then $\operatorname{div}(G)-E$ is a canonical divisor. (If $n=3, \operatorname{div}(G)=0$.)
Proof. Choose coordinates $X, Y, Z$ for $\mathbb{P}^{2}$ in such a way that: $Z \cdot C=\sum_{i=1}^{n} P_{i}, P_{i}$ distinct; $[1: 0: 0] \notin C$; and no tangent to $C$ at a multiple point passes through $[1: 0: 0]$. Let $x=X / Z, y=Y / Z \in K$. Let $F$ be the form defining $C$, and let $f_{x}=F_{X}(x, y, 1)$, $f_{y}=F_{Y}(x, y, 1)$. Let $E_{m}=m \sum_{i=1}^{n} P_{i}-E$.
Let $\omega=d x$. Since divisors of the form $\operatorname{div}(G)-E, \operatorname{deg}(G)=n-3$, are linearly equivalent, it suffices to show that $\operatorname{div}(\omega)=E_{n-3}+\operatorname{div}\left(f_{y}\right)$. Since $f_{y}=F_{Y} / Z^{n-1}$, this is the same as showing
$$
\operatorname{div}(d x)-\operatorname{div}\left(F_{Y}\right)=-2 \sum_{i=1}^{n} P_{i}-E .
$$
$\operatorname{ord}_{Q}\left(F_{X}\right)$ for all $Q \in X$.
Suppose $Q$ is a place centered at $P_{i} \in Z \cap C$. Then $y^{-1}=Z / Y$ is a uniformizing parameter in $\mathscr{O}_{P_{i}}(X)$, and $d y=-y^{2} d\left(y^{-1}\right)$, so $\operatorname{ord}_{Q}(d y)=-2$. Since $F_{X}\left(P_{i}\right) \neq 0$ (Problem 5.16), both sides of $(*)$ have order -2 at $Q$.
Suppose $Q$ is a place centered at $P=[a: b: 1] \in C$. We may assume $P=[0: 0: 1]$, since $d x=d(x-a)$, and derivatives aren't changed by translation.
Consider the case when $Y$ is tangent to $C$ at $P$. Then $P$ is not a multiple point (by hypothesis), so $x$ is a uniformizing parameter, and $F_{Y}(P) \neq 0$. Therefore $\operatorname{ord}_{Q}(d x)=$ $\operatorname{ord}_{Q}\left(F_{Y}\right)=0$, as desired.
If $Y$ is not tangent, then $y$ is a uniformizing parameter at $Q$ (Step (2) in $\$ 7.2$ ), so $\operatorname{ord}_{Q}(d y)=0$, and $\operatorname{ord}_{Q}\left(f_{x}\right)=r_{Q}^{-1}$ (Problem 7.4), as desired.
Corollary. Let $W$ be a canonical divisor. Then $\operatorname{deg}(W)=2 g-2$ and $l(W) \geq g$.
Proof. We may assume $W=E_{n-3}$. Then this is Corollary 3 (b) of $\S 8.2$.
## Problems
8.24. Show that if $g>0$, then $n \geq 3$ (notation as in Proposition 8). 8.25. Let $X=\mathbb{P}^{1}, K=k(t)$ as in Problem 8.1. Calculate $\operatorname{div}(d t)$, and show directly that the above corollary holds when $g=0$.
8.26. Show that for any $X$ there is a curve $C$ birationally equivalent to $X$ satisfying conditions of Proposition 8 (see Problem 7.21).
8.27. Let $X=C, x, y$ as in Problem 8.2. Let $\omega=y^{-1} d x$. Show that $\operatorname{div}(\omega)=0$.
8.28. Show that if $g>0$, there are effective canonical divisors.
### Riemann-Roch Theorem
This celebrated theorem finds the missing term in Riemann's Theorem. Our proof follows the classical proof of Brill and Noether.
RIEMANN-ROCH THEOREM. Let $W$ be a canonical divisor on $X$. Then for any divisor $D$,
$$
l(D)=\operatorname{deg}(D)+1-g+l(W-D) .
$$
Before proving the theorem, notice that we know the theorem for divisors of large enough degree. We can prove the general case if we can compare both sides of the equation for $D$ and $D+P, P \in X$; note that $\operatorname{deg}(D+P)=\operatorname{deg}(D)+1$, while the other two nonconstant terms change either by 0 or 1 . The heart of the proof is therefore
NOETHER'S REDUCTION LEMMA. If $l(D)>0$, and $l(W-D-P) \neq l(W-D)$, then $l(D+P)=l(D)$.
Proof. Choose $C$ as before with ordinary multiple points, and such that $P$ is a simple point on $C$ (Problem 7.21(a)), and so $Z \cdot C=\sum_{i=1}^{n} P_{i}$, with $P_{i}$ distinct, and $P \notin Z$. Let $E_{m}=m \sum P_{i}-E$. The terms in the statement of the lemma depend only on the linear equivalence classes of the divisors involved, so we may assume $W=E_{n-3}$, and $D \geq 0$ (Proposition 8 and Problem 8.11). So $L(W-D) \subset L\left(E_{n-3}\right)$.
Let $h \in L(W-D), h \notin L(W-D-P)$. Write $h=G / Z^{n-3}, G$ an adjoint of degree $n-3$ (Corollary 3 of $\S 8.3$ ). Then $\operatorname{div}(G)=D+E+A, A \geq 0$, but $A \nsupseteq P$.
Take a line $L$ such that $L . C=P+B$, where $B$ consists of $n-1$ simple points of $C$, all distinct from $P \cdot \operatorname{div}(L G)=(D+P)+E+(A+B)$.
Now suppose $f \in L(D+P)$; let $\operatorname{div}(f)+D=D^{\prime}$. We must show that $f \in L(D)$, i.e., $D^{\prime} \geq 0$.
Since $D+P \equiv D^{\prime}+P$, and both these divisors are effective, the Residue Theorem applies: There is a curve $H$ of degree $n-2$ with $\operatorname{div}(H)=\left(D^{\prime}+P\right)+E+(A+B)$.
But $B$ contains $n-1$ distinct collinear points, and $H$ is a curve of degree $n-2$. By Bézout's Theorem, $H$ must contain $L$ as a component. In particular, $H(P)=0$. Since $P$ does not appear in $E+A+B$, it follows that $D^{\prime}+P \geq P$, or $D^{\prime} \geq 0$, as desired.
We turn to the proof of the theorem. For each divisor $D$, consider the equation
$$
l(D)=\operatorname{deg}(D)+1-g+l(W-D) .
$$
Case 1: $l(W-D)=0$. We use induction on $l(D)$. If $l(D)=0$, applying Riemann's' Theorem to $D$ and $W-D$ gives $(*)_{D}$. If $l(D)=1$, we may assume $D \geq 0$. Then $g \leq$ $l(W)$ (Corollary of $\$ 8.5), l(W) \leq l(W-D)+\operatorname{deg}(D)$ (Problem 8.8), and $\operatorname{deg}(D) \leq g$ (Riemann's Theorem), proving $(*)_{D}$.
If $l(D)>1$, choose $P$ so that $l(D-P)=l(D)-1$ (Problem 8.13); then the Reduction Lemma implies that $l(W-(D-P))=0$, and $(*)_{D-P}$, which is true by induction, implies $(*)_{D}$.
Case 2: $l(W-D)>0$. This case can only happen if $\operatorname{deg}(D) \leq \operatorname{deg}(W)=2 g-2$ (Proposition 3 (2) of $\S 8.2$ ). So we can pick a maximal $D$ for which $(*)_{D}$ is false, i.e., $(*)_{D+P}$ is true for all $P \in X$. Choose $P$ so that $l(W-D-P)=l(W-D)-1$. If $l(D)=0$, applying Case 1 to the divisor $W-D$ proves it, so we may assume $l(D)>0$. Then the Reduction Lemma gives $l(D+P)=l(D)$. Since $(*)_{D+P}$ is true, $l(D)=l(D+P)=$ $\operatorname{deg}(D+P)+1-g+l(W-D-P)=\operatorname{deg}(D)+1-g+l(W-D)$, as desired.
Corollary 1. $l(W)=g$ if $W$ is a canonical divisor.
Corollary 2. If $\operatorname{deg}(D) \geq 2 g-1$, then $l(D)=\operatorname{deg}(D)+1-g$.
Corollary 3. If $\operatorname{deg}(D) \geq 2 g$, then $l(D-P)=l(D)-1$ for all $P \in X$.
Corollary 4 (Clifford's Theorem). If $l(D)>0$, and $l(W-D)>0$, then
$$
l(D) \leq \frac{1}{2} \operatorname{deg}(D)+1
$$
Proofs. The first three are straight-forward applications of the theorem, using Proposition 3 of $\$ 8.2$. For Corollary 4, we may assume $D \geq 0, D^{\prime} \geq 0, D+D^{\prime}=W$. And we may assume $l(D-P) \neq l(D)$ for all $P$, since otherwise we work with $D-P$ and get a better inequality.
Choose $g \in L(D)$ such that $g \notin L(D-P)$ for each $P \leq D^{\prime}$. Then it is easy to see that the linear map $\varphi: L\left(D^{\prime}\right) / L(0) \rightarrow L(W) / L(D)$ defined by $\varphi(\bar{f})=\overline{f g}$ (the bars denoting residues) is one-to-one. Therefore $l\left(D^{\prime}\right)-1 \leq g-l(D)$. Applying Riemann-Roch to $D^{\prime}$ concludes the proof.
The term $l(W-D)$ may also be interpreted in terms of differentials. Let $D$ be a divisor. Define $\Omega(D)$ to be $\{\omega \in \Omega \mid \operatorname{div}(\omega) \geq D\}$. It is a subspace of $\Omega$ (over $k$ ). Let $\delta(D)=\operatorname{dim}_{k} \Omega(D)$, the index of $D$. Differentials in $\Omega(0)$ are called differentials of the first kind (or holomorphic differentials, if $k=\mathbb{C}$ ).
Proposition 9. (1) $\delta(D)=l(W-D)$.
(2) There are $g$ linearly independent differentials of the first kind on $X$.
(3) $l(D)=\operatorname{deg}(D)+1-g+\delta(D)$.
Proof. Let $W=\operatorname{div}(\omega)$. Define a linear map $\varphi: L(W-D) \rightarrow \Omega(D)$ by $\varphi(f)=f \omega$. Then $\varphi$ is an isomorphism, which proves (1), and (2) and (3) follow immediately.
## Problems
8.29. Let $D$ be any divisor, $P \in X$. Then $l(W-D-P) \neq l(W-D)$ if and only if $l(D+P)=l(D)$. 8.30. (Reciprocity Theorem of Brill-Noether) Suppose $D$ and $D^{\prime}$ are divisors, and $D+D^{\prime}=W$ is a canonical divisor. Then $l(D)-l\left(D^{\prime}\right)=\frac{1}{2}\left(\operatorname{deg}(D)-\operatorname{deg}\left(D^{\prime}\right)\right)$.
8.31. Let $D$ be a divisor with $\operatorname{deg}(D)=2 g-2$ and $l(D)=g$. Show that $D$ is a canonical divisor. So these properties characterize canonical divisors.
8.32. Let $P_{1}, \ldots, P_{m} \in \mathbb{P}^{2}, r_{1}, \ldots, r_{m}$ nonnegative integers. Let $V\left(d ; r_{1} P_{1}, \ldots, r_{m} P_{m}\right)$ be the projective space of curves $F$ of degree $d$ with $m_{P_{i}}(F) \geq r_{i}$. Suppose there is a curve $C$ of degree $n$ with ordinary multiple points $P_{1}, \ldots, P_{m}$, and $m_{P_{i}}(C)=r_{i}+1$; and suppose $d \geq n-3$. Show that
$$
\operatorname{dim} V\left(d ; r_{1} P_{1}, \ldots, r_{m} P_{m}\right)=\frac{d(d+3)}{2}-\sum \frac{\left(r_{i}+1\right) r_{i}}{2} .
$$
Compare with Theorem 1 of $\$ 5.2$.
8.33. (Linear Series) Let $D$ be a divisor, and let $V$ be a subspace of $L(D)$ (as a vector space). The set of effective divisors $\{\operatorname{div}(f)+D \mid f \in V, f \neq 0\}$ is called a linear series. If $f_{1}, \ldots, f_{r+1}$ is a basis for $V$, then the correspondence $\operatorname{div}\left(\sum \lambda_{i} f_{i}\right)+D \mapsto\left(\lambda_{1}, \ldots, \lambda_{r+1}\right)$ sets up a one-to-one correspondence between the linear series and $\mathbb{P}^{r}$. If $\operatorname{deg}(D)=$ $n$, the series is often called a $g_{n}^{r}$. The series is called complete if $V=L(D)$, i.e., every effective divisor linearly equivalent to $D$ appears.
(a) Show that, with $C, E$ as in Section 1, the series $\{\operatorname{div}(G)-E \mid G$ is an adjoint of degree $n$ not containing $C$ \} is complete. (b) Assume that there is no $P$ in $X$ such that $\operatorname{div}(f)+D \geq P$ for all nonzero $f$ in $V$. (This can always be achieved by replacing $D$ by a divisor $D^{\prime} \leq D$.) For each $P \in X$, let $H_{P}=\{f \in V \mid \operatorname{div}(f)+D \geq P$ or $f=$ 0 , a hyperplane in $V$. Show that the mapping $P \mapsto H_{P}$ is a morphism $\varphi_{V}$ from $X$ to the projective space $\mathbb{P}^{*}(V)$ of hyperplanes in $V$. (c) A hyperplane $M$ in $\mathbb{P}^{*}(V)$ corresponds to a line $m$ in $V$. Show that $\varphi_{V}^{-1}(M)$ is the $\operatorname{divisor} \operatorname{div}(f)+D$, where $f$ spans the line $m$. Show that $\varphi_{V}(X)$ is not contained in any hyperplane of $\mathbb{P}^{*}(V)$. (d) Conversely, if $\varphi: X \rightarrow \mathbb{P}^{r}$ is any morphism whose image is not contained in any hyperplane, show that the divisors $\varphi^{-1}(M)$ form a linear system on $X$. (Hint: If $D=$ $\varphi^{-1}\left(M_{0}\right)$, then $\varphi^{-1}(M)=\operatorname{div}\left(M / M_{0}\right)+D$.) (e) If $V=L(D)$ and $\operatorname{deg}(D) \geq 2 g+1$, show that $\varphi_{V}$ is one-to-one. (Hint: See Corollary 3.)
Linear systems are used to map curves to and embed curves in projective spaces. 8.34. Show that there are curves of every positive genus. (Hint: Consider affine plane curves $y^{2} a(x)+b(x)=0$, where $\operatorname{deg}(a)=g, \operatorname{deg}(b)=g+2$.)
8.35. (a) Use linear systems to reprove that every curve of genus 1 is birationally equivalent to a plane cubic. (b) Show that every curve of genus 2 is birationally equivalent to a plane curve of degree 4 with one double point. (Hint: Use a $g_{4}^{3}$.)
8.36. Let $f: X \rightarrow Y$ be a nonconstant (therefore surjective) morphism of projective nonsingular curves, corresponding to a homomorphism $\tilde{f}$ of $k(Y)$ into $k(X)$. The integer $n=[k(X): k(Y)]$ is called the degree of $f$. If $P \in X, f(P)=Q$, let $t \in \mathscr{O}_{Q}(Y)$ be a uniformizing parameter. The integer $e(P)=\operatorname{ord}_{P}(t)$ is called the ramification index of $f$ at $P$.
(a) For each $Q \in Y$, show that $\sum_{f(P)=Q} e(P) P$ is an effective divisor of degree $n$ (see Proposition 4 of $\$ 8.2)$. (b) $(\operatorname{char}(k)=0)$ With $t$ as above, show that $\operatorname{ord}_{P}(d t)=$ $e(P)-1$. (c) $\left(\operatorname{char}(k)=0\right.$ ) If $g_{X}$ (resp. $g_{Y}$ ) is the genus of $X$ (resp. $Y$ ), prove the
## Hurwitz Formula
$$
2 g_{X}-2=\left(2 g_{Y}-2\right) n+\sum_{P \in X}(e(P)-1)
$$
(d) For all but a finite number of $P \in X, e(P)=1$. The points $P \in X$ (and $f(P) \in Y$ ) where $e(P)>1$ are called ramification points. If $Y=\mathbb{P}^{1}$ and $n>1$, show that there are always some ramification points.
If $k=\mathbb{C}$, a nonsingular projective curve has a natural structure of a one-dimensional compact complex analytic manifold, and hence a two-dimensional real analytic manifold. From the Hurwitz Formula (c) with $Y=\mathbb{P}^{1}$ it is easy to prove that the genus defined here is the same as the topological genus (the number of "handles") of this manifold. (See Lang's "Algebraic Functions" or my "Algebraic Topology", Part X.)
8.37. (Weierstrass Points; assume $\operatorname{char}(k)=0$ ) Let $P$ be a point on a nonsingular curve $X$ of genus $g$. Let $N_{r}=N_{r}(P)=l(r P)$. (a) Show that $1=N_{0} \leq N_{1} \leq \cdots \leq$ $N_{2 g-1}=g$. So there are exactly $g$ numbers $0<n_{1}<n_{2}<\cdots<n_{g}<2 g$ such that there is no $z \in k(X)$ with pole only at $P$, and $\operatorname{ord}_{P}(z)=-n_{i}$. These $n_{i}$ are called the Weierstrass gaps, and $\left(n_{1}, \ldots, n_{g}\right)$ the gap sequence, at $P$. The point $P$ is called a Weierstrass point if the gap sequence at $P$ is anything but $(1,2, \ldots, g)$ that is, if $\sum_{i=1}^{g}\left(n_{i}-i\right)>0$. (b) The following are equivalent: (i) $P$ is a Weierstrass point; (ii) $l(g P)>1$; (iii) $l(W-g P)>0$; (iv) There is a differential $\omega$ on $X$ with $\operatorname{div}(\omega) \geq g P$. (c) If $r$ and $s$ are not gaps at $P$, then $r+s$ is not a gap at $P$. (d) If 2 is not a gap at $P$, the gap sequence is $(1,3, \ldots, 2 g-1)$. Such a Weierstrass point (if $g>1$ ) is called hyperelliptic. The curve $X$ has a hyperelliptic Weierstrass point if and only if there is a morphism $f: X \rightarrow \mathbb{P}^{1}$ of degree 2. Such an $X$ is called a hyperelliptic curve. (e) An integer $n$ is a gap at $P$ if and only if there is a differential of the first kind $\omega$ with $\operatorname{ord}(\omega)=n-1$.
8.38. (char $k=0$ ) Fix $z \in K, z \notin k$. For $f \in K$, denote the derivative of $f$ with respect to $z$ by $f^{\prime}$; let $f^{(0)}=f, f^{(1)}=f^{\prime}, f^{(2)}=\left(f^{\prime}\right)^{\prime}$, etc. For $f_{1}, \ldots, f_{r} \in K$, let $W_{z}\left(f_{1}, \ldots, f_{r}\right)=$ $\operatorname{det}\left(f_{j}^{(i)}\right), i=0, \ldots, r-1, j=1, \ldots, r$ (the "Wronskian"). Let $\omega_{1}, \ldots, \omega_{g}$ be a basis of $\Omega(0)$. Write $\omega_{i}=f_{i} d z$, and let $h=W_{z}\left(f_{1}, \ldots, f_{g}\right)$. (a) $h$ is independent of choice of basis, up to multiplication by a constant. (b) If $t \in K$ and $\omega_{i}=e_{i} d t$, then $h=$ $W_{t}\left(e_{1}, \ldots, e_{g}\right)\left(t^{\prime}\right)^{1+\cdots+g}$. (c) There is a basis $\omega_{1}, \ldots, \omega_{g}$ for $\Omega(0)$ such that $\operatorname{ord}_{P}\left(\omega_{i}\right)=$ $n_{i}-1$, where $\left(n_{1}, \ldots, n_{g}\right)$ is the gap sequence at $P$. (d) Show that $\operatorname{ord}_{P}(h)=\sum\left(n_{i}-i\right)-$ $\frac{1}{2} g(g+1) \operatorname{ord}_{P}(d z)$ (Hint: Let $t$ be a uniformizing parameter at $P$ and look at lowest degree terms in the determinant.) (e) Prove the formula
$$
\sum_{P, i}\left(n_{i}(P)-i\right)=(g-1) g(g+1)
$$
so there are a finite number of Weierstrass points. Every curve of genus $>1$ has Weierstrass points.
More on canonical divisors, differentials, adjoints, and their relation to Max Noether's theorem and resolution of singularities, can be found in my "Adjoints and Max Noether's FundamentalSatz", which may be regarded as a ninth chapter to this book. It is available on the arXiv, math.AG/0209203.
## Appendix A
## Nonzero Characteristic
At several places, for simplicity, we have assumed that the characteristic of the field $k$ was zero. A reader with some knowledge of separable field extensions should have little difficulty extending the results to the case where $\operatorname{char}(k)=p \neq 0$. A few remarks might be helpful.
Proposition 9 (2) of Chapter 6, Section 5 is not true as stated in characteristic $p$. However, since $k$ is algebraically closed (hence perfect), it is possible to choose $x \in K$ so that $K$ is a finite separable extension of $k(x)$, and then $k=k(x, y)$ for some $y \in K$. The same comment applies to the study of $\Omega_{k}(K)$ in Chapter 8 . The differential $d x$ will be nonzero provided $K$ is separable over $k(x)$. From Problem 8.23 one can easily deduce that if $x$ is a uniformizing parameter at any point $P \in X$, then $d x \neq 0$.
A more serious difficulty is that encountered in Problem 5.26: Let $F$ be an irreducible projective plane curve of degree $n, P \in \mathbb{P}^{2}$, and $r=m_{P}(F) \geq 0$. Let us call the point $P$ terrible for $F$ if there are an infinite number of lines $L$ through $P$ that intersect $F$ in fewer than $n-r$ distinct points other than $P$. Note that $P$ can only be terrible if $p$ divides $n-r$ (see Problem 5.26). The point $[0: 1: 0]$ is terrible for $F=X^{p+1}-Y^{p} Z$ (see Problem 5.28). The point $[1: 0: 0]$ is terrible for $F=X^{p}-Y^{p-1} Z$.
The set of lines that pass through $P$ forms a hyperplane (i.e., a line) in the space $\mathbb{P}^{2}$ of all lines. If $P$ is terrible for $F$, the dual curve to $F$ contains an infinite number of points on a line (see Problem 6.47). Since the dual curve is irreducible, it must be a line. It follows that there can be at most one terrible point for $F$. In particular, one can always find lines that intersect $F$ in $n$ distinct points.
Lemma 1 of Chapter 7 , Section 4 is false in characteristic $p$. It may be impossible to perform a quadratic transformation centered at $P$ if $P$ is terrible. Theorem 2 of that section is still true, however. For if $P$ is terrible for $F, p$ must divide $n-r$. Take a quadratic transformation centered at some point $Q$ of multiplicity $m$, where $m=0$ or 1 , and so that $P$ is not on a fundamental line. Let $F^{\prime}$ be the quadratic transform of $F$. Then $n^{\prime}=\operatorname{deg}\left(F^{\prime}\right)=2 n-m$. Since $n^{\prime}-r \equiv n-m(\bmod p)$, one of the choices $m=0,1$ will insure that $p$ doesn't divide $n-r$. Then the point $P^{\prime}$ on $F^{\prime}$ corresponding to $P$ on $F$ will not be terrible for $F^{\prime}$, and we can proceed as before.
## Appendix B
## Suggestions for Further Reading
For algebraic background:
O. ZARISKI and P. SAMUEL. Commutative Algebra, Van Nostrand, Princeton, N. J., 1958.
For more on the classical theory of plane curves:
R. WALKER. Algebraic Curves, Dover, New York, 1962.
A. SEIDENBERG. Elements of the Theory of Algebraic Curves, Addison-Wesley, 1968.
For the analytic study of curves over the field of complex numbers:
S. LANG. Algebraic Functions, W. A. Benjamin, New York, 1965.
R. GUNNING. Lectures on Riemann Surfaces, Princeton Mathematical Notes, 1966.
For a purely algebraic treatment of curves over any field:
C. CHEVALLEY. Introduction to the Theory of Algebraic Functions in One Variable, Amer. Math. Soc., New York, 1952
For a classical treatment of general algebraic geometry:
S. LANG. Introduction to Algebraic Geometry, Interscience, New York, 1958.
A. WEIL. Foundations of Algebraic Geometry, Amer. Math. Soc., New York, 1946 and 1962.
B. L. VAN DER WAERDEN. Einführung in der algebraische Geometrie, Springer, Berlin, 1939.
For an introduction to abstract algebraic geometry:
D. MUMFORD. Introduction to Algebraic Geometry, Harvard lecture notes, 1967.
J.-P. SERRE. Faisceaux Algébriques Cohérents, Annals of Math., vol. 61, 1955, pp. 197-278. For other proofs of the Riemann-Roch Theorem:
J. TATE. Residues of Differentials on Curves, Annales Sci. de l'Ecole Normals Sup. 4th Ser. vol. 1, 1958, pp. 149-159.
J.-P. SERRE. Groupes Algébriques et Corps de Classes, Hermann, Paris, 1959.
For a modern treatment of multiplicity and intersection theory:
J.-P. SERRE. Algèbre Locale · Multiplicités, Springer-Verlag Berlin, Heidelberg, 1965.
## Appendix C
## Notation
References are to page numbers.
$\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}$
$\mathrm{UFD}$
$\mathrm{PID}$
$R / I$
$\frac{\partial F}{\partial X}, F_{X}, F_{X_{i}}$
$\mathbb{A}^{n}(k), \mathbb{A}^{n}$
$V(F), V(S)$
$I(X)$
$\operatorname{Rad}(I)$
$\left.\mathscr{F}^{(} V, k\right)$
$\Gamma(V)$
$\tilde{\varphi}$
$F^{T}, I^{T}, V^{T}$
$k(V)$
$\mathscr{O}_{P}(V)$
$f(P)$
$\mathfrak{m}_{P}(V)$
$D V R$, ord
$k[[X]]$
$F_{*}, f^{*}$
$I^{n}$
$m_{P}(F)$
$\operatorname{ord}_{P}$, ord $_{P}^{F}$
$I\left(P, F \cap G^{2}\right)$
$\mathbb{P}^{n}(k), \mathbb{P}^{n}$
$U_{i}, \varphi_{i}: \mathbb{A}^{n} \rightarrow U_{i} \subset \mathbb{P}^{n}$
$H_{\infty}$
| 1 | $V_{p}, I_{p}, V_{a}, I_{a}$ | 45 |
| :---: | :---: | :---: |
| 2 | $\Gamma_{h}(V)$ | 46 |
| 2 | $I^{*}, V^{*}, I_{*}, V_{*}$ | 48 |
| 2 | $\mathscr{O}_{P}(F)$ | 53 |
| 3 | $V\left(d ; r_{1} P_{1}, \ldots, r_{n} P_{n}\right)$ | 56 |
| 4 | $\sum n_{P} P$ | 61 |
| 4,45 | $\sum n_{P} P \geq \sum m_{P} P$ | 61,97 |
| 5,45 | $F \cdot G$ | 61 |
| 6 | $\Gamma(U), \Gamma\left(U, \mathscr{O}_{X}\right)$ | 69 |
| 17 | $(f, g), f \times g$ | 73 |
| 17,69 | $\Delta_{X}, G(f)$ | 74 |
| 18,70 | tr. deg | 75 |
| 19 | $\operatorname{dim}(X)$ | 75 |
| $\begin{array}{r}20,47 \\ 21 \quad 47,60\end{array}$ | $g^{*}(C)$ | 90 |
| $21,47,69$ | $\operatorname{deg}(D)$ | 97 |
| $\begin{array}{r}21,47,09 \\ 2147\end{array}$ | $\operatorname{div}(G), \operatorname{div}(z)$ | 97 |
| $\begin{array}{r}21,41 \\ 22\end{array}$ | $(z)_{0},(z)_{\infty}$ | 97 |
| $\begin{array}{l}22 \\ 23\end{array}$ | $D \equiv D^{\prime}$ | 98 |
| 24.53 | $E$ | 98 |
| 25 | $L(D), l(D)$ | 99 |
| 32 | $g$ | 101 |
| $35,53,93$ | $\Omega_{k}(R)$ | 105 |
| 36 | $d, d x$ | 105 |
| 43 | $\frac{d f}{d t}$ | 106 |
| 43 | $\operatorname{ord}_{P}(w), \operatorname{div}(w), W$ | 107 |
| 44 | $\delta(D)$ | 109 |
| Textbooks |
Cosmic formation and distribution
Biological role
Biochemical function
Food sources
Deficient intake
Detection by taste buds
Commercial production
Chemical extraction
Cation identification
Commercial uses
Medical use
Chemical element with atomic number 19
Chemical element, symbol K and atomic number 19
Potassium, 19K
Potassium pearls (in paraffin oil, ~5 mm each)
/pəˈtæsiəm/ (pə-TASS-ee-əm)
silvery gray
Standard atomic weight Ar, std(K)
39.0983(1)[1]
Potassium in the periodic table
Hydrogen Helium
Lithium Beryllium Boron Carbon Nitrogen Oxygen Fluorine Neon
Sodium Magnesium Aluminium Silicon Phosphorus Sulfur Chlorine Argon
Potassium Calcium Scandium Titanium Vanadium Chromium Manganese Iron Cobalt Nickel Copper Zinc Gallium Germanium Arsenic Selenium Bromine Krypton
Rubidium Strontium Yttrium Zirconium Niobium Molybdenum Technetium Ruthenium Rhodium Palladium Silver Cadmium Indium Tin Antimony Tellurium Iodine Xenon
Caesium Barium Lanthanum Cerium Praseodymium Neodymium Promethium Samarium Europium Gadolinium Terbium Dysprosium Holmium Erbium Thulium Ytterbium Lutetium Hafnium Tantalum Tungsten Rhenium Osmium Iridium Platinum Gold Mercury (element) Thallium Lead Bismuth Polonium Astatine Radon
Francium Radium Actinium Thorium Protactinium Uranium Neptunium Plutonium Americium Curium Berkelium Californium Einsteinium Fermium Mendelevium Nobelium Lawrencium Rutherfordium Dubnium Seaborgium Bohrium Hassium Meitnerium Darmstadtium Roentgenium Copernicium Nihonium Flerovium Moscovium Livermorium Tennessine Oganesson
argon ← potassium → calcium
Atomic number (Z)
group 1: hydrogen and alkali metals
s-block
Electron configuration
[Ar] 4s1
Electrons per shell
Phase at STP
336.7 K (63.5 °C, 146.3 °F)
1032 K (759 °C, 1398 °F)
Density (near r.t.)
when liquid (at m.p.)
0.828 g/cm3
Critical point
2223 K, 16 MPa[2]
Heat of fusion
76.9 kJ/mol
Molar heat capacity
29.6 J/(mol·K)
Atomic properties
Oxidation states
−1, +1 (a strongly basic oxide)
Electronegativity
1st: 418.8 kJ/mol
2nd: 3052 kJ/mol
3rd: 4420 kJ/mol
Atomic radius
Covalent radius
203±12 pm
Van der Waals radius
Spectral lines of potassium
Natural occurrence
body-centered cubic (bcc)
2000 m/s (at 20 °C)
Thermal expansion
83.3 µm/(m⋅K) (at 25 °C)
Thermal conductivity
102.5 W/(m⋅K)
72 nΩ⋅m (at 20 °C)
paramagnetic[3]
Molar magnetic susceptibility
+20.8×10−6 cm3/mol (298 K)[4]
Young's modulus
3.53 GPa
Shear modulus
Brinell hardness
0.363 MPa
CAS Number
Discovery and first isolation
Humphry Davy (1807)
"K": from New Latin kalium
Main isotopes of potassium
Isotope
Abundance
Half-life (t1/2)
Decay mode
Product
93.258% stable
0.012% 1.248×109 y β− 40Ca
ε 40Ar
β+ 40Ar
6.730% stable
Category: Potassium
Potassium is a chemical element with the symbol K (from Neo-Latin kalium) and atomic number 19. Potassium is a silvery-white metal that is soft enough to be cut with a knife with little force.[5] Potassium metal reacts rapidly with atmospheric oxygen to form flaky white potassium peroxide in only seconds of exposure. It was first isolated from potash, the ashes of plants, from which its name derives. In the periodic table, potassium is one of the alkali metals, all of which have a single valence electron in the outer electron shell, that is easily removed to create an ion with a positive charge – a cation, that combines with anions to form salts. Potassium in nature occurs only in ionic salts. Elemental potassium reacts vigorously with water, generating sufficient heat to ignite hydrogen emitted in the reaction, and burning with a lilac-colored flame. It is found dissolved in sea water (which is 0.04% potassium by weight[6][7]), and occurs in many minerals such as orthoclase, a common constituent of granites and other igneous rocks.[8]
Potassium is chemically very similar to sodium, the previous element in group 1 of the periodic table. They have a similar first ionization energy, which allows for each atom to give up its sole outer electron. It was suspected in 1702 that they were distinct elements that combine with the same anions to make similar salts,[9] and was proven in 1807 using electrolysis. Naturally occurring potassium is composed of three isotopes, of which 40
K is radioactive. Traces of 40
K are found in all potassium, and it is the most common radioisotope in the human body.
Potassium ions are vital for the functioning of all living cells. The transfer of potassium ions across nerve cell membranes is necessary for normal nerve transmission; potassium deficiency and excess can each result in numerous signs and symptoms, including an abnormal heart rhythm and various electrocardiographic abnormalities. Fresh fruits and vegetables are good dietary sources of potassium. The body responds to the influx of dietary potassium, which raises serum potassium levels, with a shift of potassium from outside to inside cells and an increase in potassium excretion by the kidneys.
Most industrial applications of potassium exploit the high solubility in water of potassium compounds, such as potassium soaps. Heavy crop production rapidly depletes the soil of potassium, and this can be remedied with agricultural fertilizers containing potassium, accounting for 95% of global potassium chemical production.[10]
The English name for the element potassium comes from the word potash,[11] which refers to an early method of extracting various potassium salts: placing in a pot the ash of burnt wood or tree leaves, adding water, heating, and evaporating the solution. When Humphry Davy first isolated the pure element using electrolysis in 1807, he named it potassium, which he derived from the word potash.
The symbol K stems from kali, itself from the root word alkali, which in turn comes from Arabic: القَلْيَه al-qalyah 'plant ashes'. In 1797, the German chemist Martin Klaproth discovered "potash" in the minerals leucite and lepidolite, and realized that "potash" was not a product of plant growth but actually contained a new element, which he proposed calling kali.[12] In 1807, Humphry Davy produced the element via electrolysis: in 1809, Ludwig Wilhelm Gilbert proposed the name Kalium for Davy's "potassium".[13] In 1814, the Swedish chemist Berzelius advocated the name kalium for potassium, with the chemical symbol K.[14]
The English and French-speaking countries adopted Davy and Gay-Lussac/Thénard's name Potassium, whereas the Germanic countries adopted Gilbert/Klaproth's name Kalium.[15] The "Gold Book" of the International Union of Pure and Applied Chemistry has designated the official chemical symbol as K.[16]
The flame test of potassium.
Potassium is the second least dense metal after lithium. It is a soft solid with a low melting point, and can be easily cut with a knife. Freshly cut potassium is silvery in appearance, but it begins to tarnish toward gray immediately on exposure to air.[17] In a flame test, potassium and its compounds emit a lilac color with a peak emission wavelength of 766.5 nanometers.[18]
Neutral potassium atoms have 19 electrons, one more than the configuration of the noble gas argon. Because of its low first ionization energy of 418.8 kJ/mol, the potassium atom is much more likely to lose the last electron and acquire a positive charge, although negatively charged alkalide K−
ions are not impossible.[19] In contrast, the second ionization energy is very high (3052 kJ/mol).
Potassium reacts with oxygen, water, and carbon dioxide components in air. With oxygen it forms potassium peroxide. With water potassium forms potassium hydroxide. The reaction of potassium with water can be violently exothermic, especially since the coproduced hydrogen gas can ignite. Because of this, potassium and the liquid sodium-potassium (NaK) alloy are potent desiccants, although they are no longer used as such.[20]
Structure of solid potassium superoxide (KO
2).
Three oxides of potassium are well studied: potassium oxide (K2O), potassium peroxide (K2O2), and potassium superoxide (KO2).[21] The binary potassium-oxygen binary compounds react with water forming potassium hydroxide.
Potassium hydroxide (KOH) is a strong base. Illustrating its hydrophilic character, as much as 1.21 kg of KOH can dissolve in a single liter of water.[22][23] Anhydrous KOH is rarely encountered. KOH reacts readily with carbon dioxide to produce potassium carbonate and in principle could be used to remove traces of the gas from air. Like the closely related sodium hydroxide, potassium hydroxide reacts with fats to produce soaps.
In general, potassium compounds are ionic and, owing to the high hydration energy of the K+
ion, have excellent water solubility. The main species in water solution are the aquated complexes [K(H
2O)
n]+
where n = 6 and 7.[24] Potassium heptafluorotantalate is an intermediate in the purification of tantalum from the otherwise persistent contaminant of niobium.[25]
Organopotassium compounds illustrate nonionic compounds of potassium. They feature highly polar covalent K---C bonds. Examples include benzyl potassium. Potassium intercalates into graphite to give a variety of compounds, including KC8.
There are 25 known isotopes of potassium, three of which occur naturally: 39
K (93.3%), 40
K (0.0117%), and 41
K (6.7%). Naturally occurring 40
K has a half-life of 1.250×109 years. It decays to stable 40
Ar by electron capture or positron emission (11.2%) or to stable 40
Ca by beta decay (88.8%).[26] The decay of 40
Ar is the basis of a common method for dating rocks. The conventional K-Ar dating method depends on the assumption that the rocks contained no argon at the time of formation and that all the subsequent radiogenic argon (40
Ar) was quantitatively retained. Minerals are dated by measurement of the concentration of potassium and the amount of radiogenic 40
Ar that has accumulated. The minerals best suited for dating include biotite, muscovite, metamorphic hornblende, and volcanic feldspar; whole rock samples from volcanic flows and shallow instrusives can also be dated if they are unaltered.[26][27] Apart from dating, potassium isotopes have been used as tracers in studies of weathering and for nutrient cycling studies because potassium is a macronutrient required for life.[28]
K occurs in natural potassium (and thus in some commercial salt substitutes) in sufficient quantity that large bags of those substitutes can be used as a radioactive source for classroom demonstrations. 40
K is the radioisotope with the largest abundance in the body. In healthy animals and people, 40
K represents the largest source of radioactivity, greater even than 14
C. In a human body of 70 kg mass, about 4,400 nuclei of 40
K decay per second.[29] The activity of natural potassium is 31 Bq/g.[30]
Potassium in feldspar
Potassium is formed in supernovae by nucleosynthesis from lighter atoms. Potassium is principally created in Type II supernovae via an explosive oxygen-burning process.[31] 40
K is also formed in s-process nucleosynthesis and the neon burning process.[32]
Potassium is the 20th most abundant element in the solar system and the 17th most abundant element by weight in the Earth. It makes up about 2.6% of the weight of the earth's crust and is the seventh most abundant element in the crust.[33] The potassium concentration in seawater is 0.39 g/L[6] (0.039 wt/v%), about one twenty-seventh the concentration of sodium.[34][35]
Potash is primarily a mixture of potassium salts because plants have little or no sodium content, and the rest of a plant's major mineral content consists of calcium salts of relatively low solubility in water. While potash has been used since ancient times, its composition was not understood. Georg Ernst Stahl obtained experimental evidence that led him to suggest the fundamental difference of sodium and potassium salts in 1702,[9] and Henri Louis Duhamel du Monceau was able to prove this difference in 1736.[36] The exact chemical composition of potassium and sodium compounds, and the status as chemical element of potassium and sodium, was not known then, and thus Antoine Lavoisier did not include the alkali in his list of chemical elements in 1789.[37][38] For a long time the only significant applications for potash were the production of glass, bleach, soap and gunpowder as potassium nitrate.[39] Potassium soaps from animal fats and vegetable oils were especially prized because they tend to be more water-soluble and of softer texture, and are therefore known as soft soaps.[10] The discovery by Justus Liebig in 1840 that potassium is a necessary element for plants and that most types of soil lack potassium[40] caused a steep rise in demand for potassium salts. Wood-ash from fir trees was initially used as a potassium salt source for fertilizer, but, with the discovery in 1868 of mineral deposits containing potassium chloride near Staßfurt, Germany, the production of potassium-containing fertilizers began at an industrial scale.[41][42][43] Other potash deposits were discovered, and by the 1960s Canada became the dominant producer.[44][45]
Sir Humphry Davy
Pieces of potassium metal
Potassium metal was first isolated in 1807 by Humphry Davy, who derived it by electrolysis of molten KOH with the newly discovered voltaic pile. Potassium was the first metal that was isolated by electrolysis.[46] Later in the same year, Davy reported extraction of the metal sodium from a mineral derivative (caustic soda, NaOH, or lye) rather than a plant salt, by a similar technique, demonstrating that the elements, and thus the salts, are different.[37][38][47][48] Although the production of potassium and sodium metal should have shown that both are elements, it took some time before this view was universally accepted.[38]
Because of the sensitivity of potassium to water and air, air-free techniques are normally employed for handling the element. It is unreactive toward nitrogen and saturated hydrocarbons such as mineral oil or kerosene.[49] It readily dissolves in liquid ammonia, up to 480 g per 1000 g of ammonia at 0 °C. Depending on the concentration, the ammonia solutions are blue to yellow, and their electrical conductivity is similar to that of liquid metals. Potassium slowly reacts with ammonia to form KNH
2, but this reaction is accelerated by minute amounts of transition metal salts.[50] Because it can reduce the salts to the metal, potassium is often used as the reductant in the preparation of finely divided metals from their salts by the Rieke method.[51] Illustrative is the preparation of magnesium:
MgCl
2 + 2 K → Mg + 2 KCl
Elemental potassium does not occur in nature because of its high reactivity. It reacts violently with water (see section Precautions below)[49] and also reacts with oxygen. Orthoclase (potassium feldspar) is a common rock-forming mineral. Granite for example contains 5% potassium, which is well above the average in the Earth's crust. Sylvite (KCl), carnallite (KCl·MgCl
2·6(H
2O)), kainite (MgSO
4·KCl·3H
2O) and langbeinite (MgSO
4·K
4) are the minerals found in large evaporite deposits worldwide. The deposits often show layers starting with the least soluble at the bottom and the most soluble on top.[35] Deposits of niter (potassium nitrate) are formed by decomposition of organic material in contact with atmosphere, mostly in caves; because of the good water solubility of niter the formation of larger deposits requires special environmental conditions.[52]
Potassium is the eighth or ninth most common element by mass (0.2%) in the human body, so that a 60 kg adult contains a total of about 120 g of potassium.[53] The body has about as much potassium as sulfur and chlorine, and only calcium and phosphorus are more abundant (with the exception of the ubiquitous CHON elements).[54] Potassium ions are present in a wide variety of proteins and enzymes.[55]
Potassium levels influence multiple physiological processes, including[56][57][58]
resting cellular-membrane potential and the propagation of action potentials in neuronal, muscular, and cardiac tissue. Due to the electrostatic and chemical properties, K+
ions are larger than Na+
ions, and ion channels and pumps in cell membranes can differentiate between the two ions, actively pumping or passively passing one of the two ions while blocking the other.[59]
hormone secretion and action
vascular tone
systemic blood pressure control
acid–base homeostasis
glucose and insulin metabolism
mineralocorticoid action
renal concentrating ability
Potassium homeostasis denotes the maintenance of the total body potassium content, plasma potassium level, and the ratio of the intracellular to extracellular potassium concentrations within narrow limits, in the face of pulsatile intake (meals), obligatory renal excretion, and shifts between intracellular and extracellular compartments.
Plasma levels
Plasma potassium is normally kept at 3.5 to 5.0 millimoles (mmol) [or milliequivalents (mEq)] per liter by multiple mechanisms. Levels outside this range are associated with an increasing rate of death from multiple causes,[60] and some cardiac, kidney,[61] and lung diseases progress more rapidly if serum potassium levels are not maintained within the normal range.
An average meal of 40–50 mmol presents the body with more potassium than is present in all plasma (20–25 mmol). However, this surge causes the plasma potassium to rise only 10% at most as a result of prompt and efficient clearance by both renal and extra-renal mechanisms.[62]
Hypokalemia, a deficiency of potassium in the plasma, can be fatal if severe. Common causes are increased gastrointestinal loss (vomiting, diarrhea), and increased renal loss (diuresis).[63] Deficiency symptoms include muscle weakness, paralytic ileus, ECG abnormalities, decreased reflex response; and in severe cases, respiratory paralysis, alkalosis, and cardiac arrhythmia.[64]
Control mechanisms
Potassium content in the plasma is tightly controlled by four basic mechanisms, which have various names and classifications. The four are 1) a reactive negative-feedback system, 2) a reactive feed-forward system, 3) a predictive or circadian system, and 4) an internal or cell membrane transport system. Collectively, the first three are sometimes termed the "external potassium homeostasis system";[65] and the first two, the "reactive potassium homeostasis system".
The reactive negative-feedback system refers to the system that induces renal secretion of potassium in response to a rise in the plasma potassium (potassium ingestion, shift out of cells, or intravenous infusion.)
The reactive feed-forward system refers to an incompletely understood system that induces renal potassium secretion in response to potassium ingestion prior to any rise in the plasma potassium. This is probably initiated by gut cell potassium receptors that detect ingested potassium and trigger vagal afferent signals to the pituitary gland.
The predictive or circadian system increases renal secretion of potassium during mealtime hours (e.g. daytime for humans, nighttime for rodents) independent of the presence, amount, or absence of potassium ingestion. It is mediated by a circadian oscillator in the suprachiasmatic nucleus of the brain (central clock), which causes the kidney (peripheral clock) to secrete potassium in this rhythmic circadian fashion.
The action of the sodium-potassium pump is an example of primary active transport. The two carrier proteins embedded in the cell membrane on the left are using ATP to move sodium out of the cell against the concentration gradient; The two proteins on the right are using secondary active transport to move potassium into the cell. This process results in reconstitution of ATP.
The ion transport system moves potassium across the cell membrane using two mechanisms. One is active and pumps sodium out of, and potassium into, the cell. The other is passive and allows potassium to leak out of the cell. Potassium and sodium cations influence fluid distribution between intracellular and extracellular compartments by osmotic forces. The movement of potassium and sodium through the cell membrane is mediated by the Na+/K+-ATPase pump.[66] This ion pump uses ATP to pump three sodium ions out of the cell and two potassium ions into the cell, creating an electrochemical gradient and electromotive force across the cell membrane. The highly selective potassium ion channels (which are tetramers) are crucial for hyperpolarization inside neurons after an action potential is triggered, to cite one example. The most recently discovered potassium ion channel is KirBac3.1, which makes a total of five potassium ion channels (KcsA, KirBac1.1, KirBac3.1, KvAP, and MthK) with a determined structure. All five are from prokaryotic species.[67]
Renal filtration, reabsorption, and excretion
Renal handling of potassium is closely connected to sodium handling. Potassium is the major cation (positive ion) inside animal cells [150 mmol/L, (4.8 g)], while sodium is the major cation of extracellular fluid [150 mmol/L, (3.345 g)]. In the kidneys, about 180 liters of plasma is filtered through the glomeruli and into the renal tubules per day.[68] This filtering involves about 600 g of sodium and 33 g of potassium. Since only 1–10 g of sodium and 1–4 g of potassium are likely to be replaced by diet, renal filtering must efficiently reabsorb the remainder from the plasma.
Sodium is reabsorbed to maintain extracellular volume, osmotic pressure, and serum sodium concentration within narrow limits. Potassium is reabsorbed to maintain serum potassium concentration within narrow limits.[69] Sodium pumps in the renal tubules operate to reabsorb sodium. Potassium must be conserved, but because the amount of potassium in the blood plasma is very small and the pool of potassium in the cells is about 30 times as large, the situation is not so critical for potassium. Since potassium is moved passively[70][71] in counter flow to sodium in response to an apparent (but not actual) Donnan equilibrium,[72] the urine can never sink below the concentration of potassium in serum except sometimes by actively excreting water at the end of the processing. Potassium is excreted twice and reabsorbed three times before the urine reaches the collecting tubules.[73] At that point, urine usually has about the same potassium concentration as plasma. At the end of the processing, potassium is secreted one more time if the serum levels are too high.[citation needed]
With no potassium intake, it is excreted at about 200 mg per day until, in about a week, potassium in the serum declines to a mildly deficient level of 3.0–3.5 mmol/L.[74] If potassium is still withheld, the concentration continues to fall until a severe deficiency causes eventual death.[75]
The potassium moves passively through pores in the cell membrane. When ions move through Ion transporters (pumps) there is a gate in the pumps on both sides of the cell membrane and only one gate can be open at once. As a result, approximately 100 ions are forced through per second. Ion channel have only one gate, and there only one kind of ion can stream through, at 10 million to 100 million ions per second.[76] Calcium is required to open the pores,[77] although calcium may work in reverse by blocking at least one of the pores.[78] Carbonyl groups inside the pore on the amino acids mimic the water hydration that takes place in water solution[79] by the nature of the electrostatic charges on four carbonyl groups inside the pore.[80]
The U.S. National Academy of Medicine (NAM), on behalf of both the U.S. and Canada, sets Estimated Average Requirements (EARs) and Recommended Dietary Allowances (RDAs), or Adequate Intakes (AIs) for when there is not sufficient information to set EARs and RDAs. Collectively the EARs, RDAs, AIs and ULs are referred to as Dietary Reference Intakes.
For both males and females under 9 years of age, the AIs for potassium are: 400 mg of potassium for 0-6-month-old infants, 860 mg of potassium for 7-12-month-old infants, 2,000 mg of potassium for 1-3-year-old children, and 2,300 mg of potassium for 4-8-year-old children.
For males 9 years of age and older, the AIs for potassium are: 2,500 mg of potassium for 9-13-year-old males, 3,000 mg of potassium for 14-18-year-old males, and 3,400 mg for males that are 19 years of age and older.
For females 9 years of age and older, the AIs for potassium are: 2,300 mg of potassium for 9-18-year-old females, and 2,600 mg of potassium for females that are 19 years of age and older.
For pregnant and lactating females, the AIs for potassium are: 2,600 mg of potassium for 14-18-year-old pregnant females, 2,900 mg for pregnant females that are 19 years of age and older; furthermore, 2,500 mg of potassium for 14-18-year-old lactating females, and 2,800 mg for lactating females that are 19 years of age and older. As for safety, the NAM also sets tolerable upper intake levels (ULs) for vitamins and minerals, but for potassium the evidence was insufficient, so no UL was established.[81][82]
As of 2004, most Americans adults consume less than 3,000 mg.[83]
Likewise, in the European Union, in particular in Germany and Italy, insufficient potassium intake is somewhat common.[84] The British National Health Service recommends a similar intake, saying that adults need 3,500 mg per day and that excess amounts may cause health problems such as stomach pain and diarrhoea.[85]
Previously the Adequate Intake for adults was set at 4,700 mg per day. In 2019, the National Academies of Sciences, Engineering, and Medicine revised the AI for potassium to 2,600 mg/day for females 19 years and older and 3,400 mg/day for males 19 years and older.[86]
Potassium is present in all fruits, vegetables, meat and fish. Foods with high potassium concentrations include yam, parsley, dried apricots, milk, chocolate, all nuts (especially almonds and pistachios), potatoes, bamboo shoots, bananas, avocados, coconut water, soybeans, and bran.[87]
The USDA lists tomato paste, orange juice, beet greens, white beans, potatoes, plantains, bananas, apricots, and many other dietary sources of potassium, ranked in descending order according to potassium content. A day's worth of potassium is in 5 plantains or 11 bananas.[88]
Diets low in potassium can lead to hypertension[89] and hypokalemia.
Supplements of potassium are most widely used in conjunction with diuretics that block reabsorption of sodium and water upstream from the distal tubule (thiazides and loop diuretics), because this promotes increased distal tubular potassium secretion, with resultant increased potassium excretion. A variety of prescription and over-the counter supplements are available. Potassium chloride may be dissolved in water, but the salty/bitter taste makes liquid supplements unpalatable.[90] Typical doses range from 10 mmol (400 mg), to 20 mmol (800 mg). Potassium is also available in tablets or capsules, which are formulated to allow potassium to leach slowly out of a matrix, since very high concentrations of potassium ion that occur adjacent to a solid tablet can injure the gastric or intestinal mucosa. For this reason, non-prescription potassium pills are limited by law in the US to a maximum of 99 mg of potassium.[citation needed]
Since the kidneys are the site of potassium excretion, individuals with impaired kidney function are at risk for hyperkalemia if dietary potassium and supplements are not restricted. The more severe the impairment, the more severe is the restriction necessary to avoid hyperkalemia.
A meta-analysis concluded that a 1640 mg increase in the daily intake of potassium was associated with a 21% lower risk of stroke.[91] Potassium chloride and potassium bicarbonate may be useful to control mild hypertension.[92] In 2017, potassium was the 37th most commonly prescribed medication in the United States, with more than 19 million prescriptions.[93][94]
Potassium can be detected by taste because it triggers three of the five types of taste sensations, according to concentration. Dilute solutions of potassium ions taste sweet, allowing moderate concentrations in milk and juices, while higher concentrations become increasingly bitter/alkaline, and finally also salty to the taste. The combined bitterness and saltiness of high-potassium solutions makes high-dose potassium supplementation by liquid drinks a palatability challenge.[90][95]
Sylvite from New Mexico
Monte Kali, a potash mining and beneficiation waste heap in Hesse, Germany, consisting mostly of sodium chloride.
Potassium salts such as carnallite, langbeinite, polyhalite, and sylvite form extensive evaporite deposits in ancient lake bottoms and seabeds,[34] making extraction of potassium salts in these environments commercially viable. The principal source of potassium – potash – is mined in Canada, Russia, Belarus, Kazakhstan, Germany, Israel, United States, Jordan, and other places around the world.[96][97][98] The first mined deposits were located near Staßfurt, Germany, but the deposits span from Great Britain over Germany into Poland. They are located in the Zechstein and were deposited in the Middle to Late Permian. The largest deposits ever found lie 1,000 meters (3,300 feet) below the surface of the Canadian province of Saskatchewan. The deposits are located in the Elk Point Group produced in the Middle Devonian. Saskatchewan, where several large mines have operated since the 1960s pioneered the technique of freezing of wet sands (the Blairmore formation) to drive mine shafts through them. The main potash mining company in Saskatchewan until its merge was the Potash Corporation of Saskatchewan, now Nutrien.[99] The water of the Dead Sea is used by Israel and Jordan as a source of potash, while the concentration in normal oceans is too low for commercial production at current prices.[97][98]
Several methods are used to separate potassium salts from sodium and magnesium compounds. The most-used method is fractional precipitation using the solubility differences of the salts. Electrostatic separation of the ground salt mixture is also used in some mines. The resulting sodium and magnesium waste is either stored underground or piled up in slag heaps. Most of the mined potassium mineral ends up as potassium chloride after processing. The mineral industry refers to potassium chloride either as potash, muriate of potash, or simply MOP.[35]
Pure potassium metal can be isolated by electrolysis of its hydroxide in a process that has changed little since it was first used by Humphry Davy in 1807. Although the electrolysis process was developed and used in industrial scale in the 1920s, the thermal method by reacting sodium with potassium chloride in a chemical equilibrium reaction became the dominant method in the 1950s.
The production of sodium potassium alloys is accomplished by changing the reaction time and the amount of sodium used in the reaction. The Griesheimer process employing the reaction of potassium fluoride with calcium carbide was also used to produce potassium.[35][100]
Na + KCl → NaCl + K (Thermal method)
2 KF + CaC
2 → 2 K + CaF
2 + 2 C (Griesheimer process)
Reagent-grade potassium metal costs about $10.00/pound ($22/kg) in 2010 when purchased by the tonne. Lower purity metal is considerably cheaper. The market is volatile because long-term storage of the metal is difficult. It must be stored in a dry inert gas atmosphere or anhydrous mineral oil to prevent the formation of a surface layer of potassium superoxide, a pressure-sensitive explosive that detonates when scratched. The resulting explosion often starts a fire difficult to extinguish.[101][102]
Potassium is now quantified by ionization techniques, but at one time it was quantitated by gravimetric analysis.
Reagents used to precipitate potassium salts include sodium tetraphenylborate, hexachloroplatinic acid, and sodium cobaltinitrite into respectively potassium tetraphenylborate, potassium hexachloroplatinate, and potassium cobaltinitrite.[49] The reaction with sodium cobaltinitrite is illustrative:
3K+ + Na3[Co(NO2)6] → K3[Co(NO2)6] + 3Na+
The potassium cobaltinitrite is obtained as a yellow solid.
Potassium sulfate/magnesium sulfate fertilizer
Potassium ions are an essential component of plant nutrition and are found in most soil types.[10] They are used as a fertilizer in agriculture, horticulture, and hydroponic culture in the form of chloride (KCl), sulfate (K
4), or nitrate (KNO
3), representing the 'K' in 'NPK'. Agricultural fertilizers consume 95% of global potassium chemical production, and about 90% of this potassium is supplied as KCl.[10] The potassium content of most plants ranges from 0.5% to 2% of the harvested weight of crops, conventionally expressed as amount of K
2O. Modern high-yield agriculture depends upon fertilizers to replace the potassium lost at harvest. Most agricultural fertilizers contain potassium chloride, while potassium sulfate is used for chloride-sensitive crops or crops needing higher sulfur content. The sulfate is produced mostly by decomposition of the complex minerals kainite (MgSO
4). Only a very few fertilizers contain potassium nitrate.[103] In 2005, about 93% of world potassium production was consumed by the fertilizer industry.[98] Furthermore, potassium can play a key role in nutrient cycling by controlling litter composition.[104]
Potassium, in the form of potassium chloride is used as a medication to treat and prevent low blood potassium.[105] Low blood potassium may occur due to vomiting, diarrhea, or certain medications.[106] It is given by slow injection into a vein or by mouth.[107]
Potassium sodium tartrate (KNaC
4O
6, Rochelle salt) is a main constituent of some varieties of baking powder; it is also used in the silvering of mirrors. Potassium bromate (KBrO
3) is a strong oxidizer (E924), used to improve dough strength and rise height. Potassium bisulfite (KHSO
3) is used as a food preservative, for example in wine and beer-making (but not in meats). It is also used to bleach textiles and straw, and in the tanning of leathers.[108][109]
Major potassium chemicals are potassium hydroxide, potassium carbonate, potassium sulfate, and potassium chloride. Megatons of these compounds are produced annually.[110]
Potassium hydroxide KOH is a strong base, which is used in industry to neutralize strong and weak acids, to control pH and to manufacture potassium salts. It is also used to saponify fats and oils, in industrial cleaners, and in hydrolysis reactions, for example of esters.[111][112]
Potassium nitrate (KNO
3) or saltpeter is obtained from natural sources such as guano and evaporites or manufactured via the Haber process; it is the oxidant in gunpowder (black powder) and an important agricultural fertilizer. Potassium cyanide (KCN) is used industrially to dissolve copper and precious metals, in particular silver and gold, by forming complexes. Its applications include gold mining, electroplating, and electroforming of these metals; it is also used in organic synthesis to make nitriles. Potassium carbonate (K
3 or potash) is used in the manufacture of glass, soap, color TV tubes, fluorescent lamps, textile dyes and pigments.[113] Potassium permanganate (KMnO
4) is an oxidizing, bleaching and purification substance and is used for production of saccharin. Potassium chlorate (KClO
3) is added to matches and explosives. Potassium bromide (KBr) was formerly used as a sedative and in photography.[10]
While potassium chromate (K
2CrO
4) is used in the manufacture of a host of different commercial products such as inks, dyes, wood stains (by reacting with the tannic acid in wood), explosives, fireworks, fly paper, and safety matches,[114]as well as in the tanning of leather, all of these uses are due to the chemistry of the chromate ion rather than to that of the potassium ion.[115]
Niche uses
There are thousands of uses of various potassium compounds. One example is potassium superoxide, KO
2, an orange solid that acts as a portable source of oxygen and a carbon dioxide absorber. It is widely used in respiration systems in mines, submarines and spacecraft as it takes less volume than the gaseous oxygen.[116][117]
4 KO
2 + 2 CO2 → 2 K
3 + 3 O
Another example is potassium cobaltinitrite, K
3[Co(NO
6], which is used as artist's pigment under the name of Aureolin or Cobalt Yellow.[118]
The stable isotopes of potassium can be laser cooled and used to probe fundamental and technological problems in quantum physics. The two bosonic isotopes possess convenient Feshbach resonances to enable studies requiring tunable interactions, while 40K is one of only two stable fermions amongst the alkali metals.[119]
Laboratory uses
An alloy of sodium and potassium, NaK is a liquid used as a heat-transfer medium and a desiccant for producing dry and air-free solvents. It can also be used in reactive distillation.[120] The ternary alloy of 12% Na, 47% K and 41% Cs has the lowest melting point of −78 °C of any metallic compound.[17]
Metallic potassium is used in several types of magnetometers.[121]
GHS labelling:
H260, H314
P223, P231+P232, P280, P305+P351+P338, P370+P378, P422[122]
NFPA 704 (fire diamond)
Potassium metal can react violently with water producing potassium hydroxide (KOH) and hydrogen gas.
2 K (s) + 2 H2O (l) → 2 KOH (aq) + H
2↑ (g)
A reaction of potassium metal with water. Hydrogen is produced, and with potassium vapor, burns with a pink or lilac flame. Strongly alkaline potassium hydroxide is formed in solution.
This reaction is exothermic and releases sufficient heat to ignite the resulting hydrogen in the presence of oxygen. Finely powdered potassium ignites in air at room temperature. The bulk metal ignites in air if heated. Because its density is 0.89 g/cm3, burning potassium floats in water that exposes it to atmospheric oxygen. Many common fire extinguishing agents, including water, either are ineffective or make a potassium fire worse. Nitrogen, argon, sodium chloride (table salt), sodium carbonate (soda ash), and silicon dioxide (sand) are effective if they are dry. Some Class D dry powder extinguishers designed for metal fires are also effective. These agents deprive the fire of oxygen and cool the potassium metal.[123]
During storage, potassium forms peroxides and superoxides. These peroxides may react violently with organic compounds such as oils. Both peroxides and superoxides may react explosively with metallic potassium.[124]
Because potassium reacts with water vapor in the air, it is usually stored under anhydrous mineral oil or kerosene. Unlike lithium and sodium, however, potassium should not be stored under oil for longer than six months, unless in an inert (oxygen free) atmosphere, or under vacuum. After prolonged storage in air dangerous shock-sensitive peroxides can form on the metal and under the lid of the container, and can detonate upon opening.[125]
Ingestion of large amounts of potassium compounds can lead to hyperkalemia, strongly influencing the cardiovascular system.[126][127] Potassium chloride is used in the United States for lethal injection executions.[126]
^ "Standard Atomic Weights: Potassium". CIAAW. 1979.
^ Haynes, William M., ed. (2011). CRC Handbook of Chemistry and Physics (92nd ed.). Boca Raton, FL: CRC Press. p. 4.122. ISBN 1-4398-5511-0.
^ Magnetic susceptibility of the elements and inorganic compounds, in Lide, D. R., ed. (2005). CRC Handbook of Chemistry and Physics (86th ed.). Boca Raton (FL): CRC Press. ISBN 0-8493-0486-5.
^ Weast, Robert (1984). CRC, Handbook of Chemistry and Physics. Boca Raton, Florida: Chemical Rubber Company Publishing. pp. E110. ISBN 0-8493-0464-4.
^ Augustyn, Adam. "Potassium/ Chemical element". Encyclopedia Britannica. Retrieved 2019-04-17. Potassium Physical properties
^ a b Webb, D. A. (April 1939). "The Sodium and Potassium Content of Sea Water" (PDF). The Journal of Experimental Biology (2): 183.
^ Anthoni, J. (2006). "Detailed composition of seawater at 3.5% salinity". seafriends.org.nz. Retrieved 2011-09-23.
^ Halperin, Mitchell L.; Kamel, Kamel S. (1998-07-11). "Potassium". The Lancet. 352 (9122): 135–140. doi:10.1016/S0140-6736(98)85044-7. ISSN 0140-6736. PMID 9672294. S2CID 208790031.
^ a b Marggraf, Andreas Siegmund (1761). Chymische Schriften. p. 167.
^ a b c d e Greenwood, p. 73
^ Davy, Humphry (1808). "On some new phenomena of chemical changes produced by electricity, in particular the decomposition of the fixed alkalies, and the exhibition of the new substances that constitute their bases; and on the general nature of alkaline bodies". Philosophical Transactions of the Royal Society. 98: 32. doi:10.1098/rstl.1808.0001.
^ Klaproth, M. (1797) "Nouvelles données relatives à l'histoire naturelle de l'alcali végétal" (New data regarding the natural history of the vegetable alkali), Mémoires de l'Académie royale des sciences et belles-lettres (Berlin), pp. 9–13 ; see p. 13. From p. 13: "Cet alcali ne pouvant donc plus être envisagé comme un produit de la végétation dans les plantes, occupe une place propre dans la série des substances primitivement simples du règne minéral, &I il devient nécessaire de lui assigner un nom, qui convienne mieux à sa nature.
La dénomination de Potasche (potasse) que la nouvelle nomenclature françoise a consacrée comme nom de tout le genre, ne sauroit faire fortune auprès des chimistes allemands, qui sentent à quel point la dérivation étymologique en est vicieuse. Elle est prise en effet de ce qu'anciennement on se servoit pour la calcination des lessives concentrées des cendres, de pots de fer (pott en dialecte de la Basse-Saxe) auxquels on a substitué depuis des fours à calciner.
Je propose donc ici, de substituer aux mots usités jusqu'ici d'alcali des plantes, alcali végétal, potasse, &c. celui de kali, & de revenir à l'ancienne dénomination de natron, au lieu de dire alcali minéral, soude &c."
(This alkali [i.e., potash] — [which] therefore can no longer be viewed as a product of growth in plants — occupies a proper place in the originally simple series of the mineral realm, and it becomes necessary to assign it a name that is better suited to its nature.
The name of "potash" (potasse), which the new French nomenclature has bestowed as the name of the entire species [i.e., substance], would not find acceptance among German chemists, who feel to some extent [that] the etymological derivation of it is faulty. Indeed, it is taken from [the vessels] that one formerly used for the roasting of washing powder concentrated from cinders: iron pots (pott in the dialect of Lower Saxony), for which roasting ovens have been substituted since then.
Thus I now propose to substitute for the until now common words of "plant alkali", "vegetable alkali", "potash", etc., that of kali ; and to return to the old name of natron instead of saying "mineral alkali", "soda", etc.)
^ Davy, Humphry (1809). "Ueber einige neue Erscheinungen chemischer Veränderungen, welche durch die Electricität bewirkt werden; insbesondere über die Zersetzung der feuerbeständigen Alkalien, die Darstellung der neuen Körper, welche ihre Basen ausmachen, und die Natur der Alkalien überhaupt" [On some new phenomena of chemical changes that are achieved by electricity; particularly the decomposition of flame-resistant alkalis [i.e., alkalies that cannot be reduced to their base metals by flames], the preparation of new substances that constitute their [metallic] bases, and the nature of alkalies generally]. Annalen der Physik. 31 (2): 113–175. Bibcode:1809AnP....31..113D. doi:10.1002/andp.18090310202. p. 157: In unserer deutschen Nomenclatur würde ich die Namen Kalium und Natronium vorschlagen, wenn man nicht lieber bei den von Herrn Erman gebrauchten und von mehreren angenommenen Benennungen Kali-Metalloid and Natron-Metalloid, bis zur völligen Aufklärung der chemischen Natur dieser räthzelhaften Körper bleiben will. Oder vielleicht findet man es noch zweckmässiger fürs Erste zwei Klassen zu machen, Metalle und Metalloide, und in die letztere Kalium und Natronium zu setzen. — Gilbert. (In our German nomenclature, I would suggest the names Kalium and Natronium, if one would not rather continue with the appellations Kali-metalloid and Natron-metalloid which are used by Mr. Erman [i.e., German physics professor Paul Erman (1764–1851)] and accepted by several [people], until the complete clarification of the chemical nature of these puzzling substances. Or perhaps one finds it yet more advisable for the present to create two classes, metals and metalloids, and to place Kalium and Natronium in the latter — Gilbert.)
^ Berzelius, J. Jacob (1814) Försök, att, genom användandet af den electrokemiska theorien och de kemiska proportionerna, grundlägga ett rent vettenskapligt system för mineralogien [Attempt, by the use of electrochemical theory and chemical proportions, to found a pure scientific system for mineralogy]. Stockholm, Sweden: A. Gadelius., p. 87.
^ 19. Kalium (Potassium) – Elementymology & Elements Multidict. vanderkrogt.net
^ McNaught, A. D. and Wilkinson,A. eds. (1997). Compendium of Chemical Terminology, 2nd ed. (the "Gold Book"). IUPAC. Blackwell Scientific Publications, Oxford.
^ a b Greenwood, p. 76
^ Greenwood, p. 75
^ Dye, J. L. (1979). "Compounds of Alkali Metal Anions". Angewandte Chemie International Edition. 18 (8): 587–598. doi:10.1002/anie.197905871.
^ Williams, D. Bradley G.; Lawton, Michelle (2010). "Drying of Organic Solvents: Quantitative Evaluation of the Efficiency of Several Desiccants". The Journal of Organic Chemistry. 75 (24): 8351–8354. doi:10.1021/jo101589h. PMID 20945830. S2CID 17801540.
^ Lide, David R. (1998). Handbook of Chemistry and Physics (87th ed.). Boca Raton, Florida, United States: CRC Press. pp. 477, 520. ISBN 978-0-8493-0594-8.
^ Lide, D. R., ed. (2005). CRC Handbook of Chemistry and Physics (86th ed.). Boca Raton (FL): CRC Press. p. 4–80. ISBN 0-8493-0486-5.
^ Schultz, p. 94
^ Lincoln, S. F.; Richens, D. T. and Sykes, A. G. "Metal Aqua Ions" in J. A. McCleverty and T. J. Meyer (eds.) Comprehensive Coordination Chemistry II, Vol. 1, pp. 515–555, ISBN 978-0-08-043748-4.
^ Anthony Agulyanski (2004). "Fluorine chemistry in the processing of tantalum and niobium". In Anatoly Agulyanski (ed.). Chemistry of Tantalum and Niobium Fluoride Compounds (1st ed.). Burlington: Elsevier. ISBN 9780080529028.
^ a b Audi, Georges; Bersillon, Olivier; Blachot, Jean; Wapstra, Aaldert Hendrik (2003), "The NUBASE evaluation of nuclear and decay properties", Nuclear Physics A, 729: 3–128, Bibcode:2003NuPhA.729....3A, doi:10.1016/j.nuclphysa.2003.11.001
^ Bowen, Robert; Attendorn, H. G. (1988). "Theory and Assumptions in Potassium–Argon Dating". Isotopes in the Earth Sciences. Springer. pp. 203–8. ISBN 978-0-412-53710-3.
^ Anaç, D. & Martin-Prével, P. (1999). Improved crop quality by nutrient management. Springer. pp. 290–. ISBN 978-0-7923-5850-3.
^ "Radiation and Radioactive Decay. Radioactive Human Body". Harvard Natural Sciences Lecture Demonstrations. Retrieved July 2, 2016.
^ Winteringham, F. P. W; Effects, F.A.O. Standing Committee on Radiation, Land And Water Development Division, Food and Agriculture Organization of the United Nations (1989). Radioactive fallout in soils, crops and food: a background review. Food & Agriculture Org. p. 32. ISBN 978-92-5-102877-3. CS1 maint: multiple names: authors list (link)
^ Shimansky, V.; Bikmaev, I. F.; Galeev, A. I.; Shimanskaya, N. N.; et al. (September 2003). "Observational constraints on potassium synthesis during the formation of stars of the Galactic disk". Astronomy Reports. 47 (9): 750–762. Bibcode:2003ARep...47..750S. doi:10.1134/1.1611216. S2CID 120396773.
^ The, L.-S.; Eid, M. F. El; Meyer, B. S. (2000). "A New Study of s-Process Nucleosynthesis in Massive Stars". The Astrophysical Journal. 533 (2): 998. arXiv:astro-ph/9812238. Bibcode:2000ApJ...533..998T. doi:10.1086/308677. ISSN 0004-637X. S2CID 7698683.
^ a b Micale, Giorgio; Cipollina, Andrea; Rizzuti, Lucio (2009). Seawater Desalination: Conventional and Renewable Energy Processes. Springer. p. 3. ISBN 978-3-642-01149-8.
^ a b c d Prud'homme, Michel; Krukowski, Stanley T. (2006). "Potash". Industrial minerals & rocks: commodities, markets, and uses. Society for Mining, Metallurgy, and Exploration. pp. 723–740. ISBN 978-0-87335-233-8.
^ du Monceau, H. L. D. (1702–1797). "Sur la Base de Sel Marin". Mémoires de l'Académie Royale des Sciences (in French): 65–68.
^ a b Weeks, Mary Elvira (1932). "The discovery of the elements. IX. Three alkali metals: Potassium, sodium, and lithium". Journal of Chemical Education. 9 (6): 1035. Bibcode:1932JChEd...9.1035W. doi:10.1021/ed009p1035.
^ a b c Siegfried, R. (1963). "The Discovery of Potassium and Sodium, and the Problem of the Chemical Elements". Isis. 54 (2): 247–258. doi:10.1086/349704. JSTOR 228541. PMID 14147904. S2CID 38152048.
^ Browne, C. A. (1926). "Historical notes upon the domestic potash industry in early colonial and later times". Journal of Chemical Education. 3 (7): 749–756. Bibcode:1926JChEd...3..749B. doi:10.1021/ed003p749.
^ Liebig, Justus von (1840). Die organische Chemie in ihrer Anwendung auf Agricultur und Physiologie (in German). F. Vieweg und Sohn.
^ Cordel, Oskar (1868). Die Stassfurter Kalisalze in der Landwirthschalt: Eine Besprechung ... (in German). L. Schnock.
^ Birnbaum, Karl (1869). Die Kalidüngung in ihren Vortheilen und Gefahren (in German).
^ United Nations Industrial Development Organization and Int'l Fertilizer Development Center (1998). Fertilizer Manual. pp. 46, 417. ISBN 978-0-7923-5032-3.
^ Miller, H. (1980). "Potash from Wood Ashes: Frontier Technology in Canada and the United States". Technology and Culture. 21 (2): 187–208. doi:10.2307/3103338. JSTOR 3103338.
^ Rittenhouse, P. A. (1979). "Potash and politics". Economic Geology. 74 (2): 353–7. doi:10.2113/gsecongeo.74.2.353.
^ Enghag, P. (2004). "11. Sodium and Potassium". Encyclopedia of the elements. Wiley-VCH Weinheim. ISBN 978-3-527-30666-4.
^ Davy, Humphry (1808). "On some new phenomena of chemical changes produced by electricity, in particular the decomposition of the fixed alkalies, and the exhibition of the new substances that constitute their bases; and on the general nature of alkaline bodies". Philosophical Transactions of the Royal Society. 98: 1–44. doi:10.1098/rstl.1808.0001.
^ Shaposhnik, V. A. (2007). "History of the discovery of potassium and sodium (on the 200th anniversary of the discovery of potassium and sodium)". Journal of Analytical Chemistry. 62 (11): 1100–2. doi:10.1134/S1061934807110160. S2CID 96141217.
^ a b c Holleman, Arnold F.; Wiberg, Egon; Wiberg, Nils (1985). "Potassium". Lehrbuch der Anorganischen Chemie (in German) (91–100 ed.). Walter de Gruyter. ISBN 978-3-11-007511-3.
^ Burkhardt, p. 32
^ Rieke, R. D. (1989). "Preparation of Organometallic Compounds from Highly Reactive Metal Powders". Science. 246 (4935): 1260–4. Bibcode:1989Sci...246.1260R. doi:10.1126/science.246.4935.1260. PMID 17832221. S2CID 92794.
^ Ross, William H. (1914). "The Origin of Nitrate Deposits". Popular Science. Bonnier Corporation. pp. 134–145.
^ Abdel-Wahab, M.; Youssef, S.; Aly, A.; el-Fiki, S.; et al. (1992). "A simple calibration of a whole-body counter for the measurement of total body potassium in humans". International Journal of Radiation Applications and Instrumentation A. 43 (10): 1285–9. doi:10.1016/0883-2889(92)90208-V. PMID 1330980.
^ Chang, Raymond (2007). Chemistry. McGraw-Hill Higher Education. p. 52. ISBN 978-0-07-110595-8.
^ Vašák, Milan; Schnabl, Joachim (2016). "Chapter 8. Sodium and Potassium Ions in Proteins and Enzyme Catalysis". In Astrid, Sigel; Helmut, Sigel; Roland K.O., Sigel (eds.). The Alkali Metal Ions: Their Role in Life. Metal Ions in Life Sciences. 16. Springer. pp. 259–290. doi:10.1007/978-3-319-21756-7_8. PMID 26860304.
^ Weiner ID, Linus S, Wingo CS (2014). "Disorders of potassium metabolism". In Freehally J, Johnson RJ, Floege J (eds.). Comprehensive clinical nephrology (5th ed.). St. Louis: Saunders. p. 118. ISBN 9780323242875.
^ Malnic G, Giebisch G, Muto S, Wang W, Bailey MA, Satlin LM (2013). "Regulation of K+ excretion". In Alpern RJ, Caplan MJ, Moe OW (eds.). Seldin and Giebisch's the kidney: physiology and pathophysiology (5th ed.). London: Academic Press. pp. 1659–1716. ISBN 9780123814630.
^ Mount DB, Zandi-Nejad K (2011). "Disorders of potassium balance". In Taal MW, Chertow GM, Marsden PA, Skorecki KL, Yu AS, Brenner BM (eds.). The kidney (9th ed.). Philadelphia: Elsevier. pp. 640–688. ISBN 9781455723041.
^ Lockless, S. W.; Zhou, M.; MacKinnon, R. (2007). "Structural and thermodynamic properties of selective ion binding in a K+ channel". PLOS Biol. 5 (5): e121. doi:10.1371/journal.pbio.0050121. PMC 1858713. PMID 17472437.
^ Goyal, Abhinav; Spertus, John A.; Gosch, Kensey; Venkitachalam, Lakshmi; Jones, Philip G.; Van den Berghe, Greet; Kosiborod, Mikhail (2012). "Serum Potassium Levels and Mortality in Acute Myocardial Infarction". JAMA. 307 (2): 157–164. doi:10.1001/jama.2011.1967. PMID 22235086.
^ Smyth, A.; Dunkler, D.; Gao, P.; et al. (2014). "The relationship between estimated sodium and potassium excretion and subsequent renal outcomes". Kidney Int. 86 (6): 1205–1212. doi:10.1038/ki.2014.214. PMID 24918156.
^ Moore-Ede, M. C. (1986). "Physiology of the circadian timing system: predictive versus reactive homeostasis". Am J Physiol. 250 (5 Pt 2): R737–R752. doi:10.1152/ajpregu.1986.250.5.R737. PMID 3706563.
^ Slonim, Anthony D.; Pollack, Murray M. (2006). "Potassium". Pediatric critical care medicine. Lippincott Williams & Wilkins. p. 812. ISBN 978-0-7817-9469-5.
^ Visveswaran, Kasi (2009). "hypokalemia". Essentials of Nephrology (2nd ed.). BI Publications. p. 257. ISBN 978-81-7225-323-3.
^ Gumz, Michelle L.; Rabinowitz, Lawrence; Wingo, Charles S. (2015-07-02). "An Integrated View of Potassium Homeostasis". The New England Journal of Medicine. 373 (1): 60–72. doi:10.1056/NEJMra1313341. ISSN 0028-4793. PMC 5675534. PMID 26132942.
^ Campbell, Neil (1987). Biology. Menlo Park, California: Benjamin/Cummings Pub. Co. p. 795. ISBN 978-0-8053-1840-1.
^ Hellgren, Mikko; Sandberg, Lars; Edholm, Olle (2006). "A comparison between two prokaryotic potassium channels (KirBac1.1 and KcsA) in a molecular dynamics (MD) simulation study". Biophysical Chemistry. 120 (1): 1–9. doi:10.1016/j.bpc.2005.10.002. PMID 16253415.
^ Potts, W. T. W.; Parry, G. (1964). Osmotic and ionic regulation in animals. Pergamon Press.
^ Lans, H. S.; Stein, I. F.; Meyer, K. A. (1952). "The relation of serum potassium to erythrocyte potassium in normal subjects and patients with potassium deficiency". American Journal of the Medical Sciences. 223 (1): 65–74. doi:10.1097/00000441-195201000-00011. PMID 14902792.
^ Bennett, C. M.; Brenner, B. M.; Berliner, R. W. (1968). "Micropuncture study of nephron function in the rhesus monkey". Journal of Clinical Investigation. 47 (1): 203–216. doi:10.1172/JCI105710. PMC 297160. PMID 16695942.
^ Solomon, A. K. (1962). "Pumps in the living cell". Scientific American. 207 (2): 100–8. Bibcode:1962SciAm.207b.100S. doi:10.1038/scientificamerican0862-100. PMID 13914986.
^ Kernan, Roderick P. (1980). Cell potassium (Transport in the life sciences). New York: Wiley. pp. 40, 48. ISBN 978-0-471-04806-0.
^ Wright, F. S. (1977). "Sites and mechanisms of potassium transport along the renal tubule". Kidney International. 11 (6): 415–432. doi:10.1038/ki.1977.60. PMID 875263.
^ Squires, R. D.; Huth, E. J. (1959). "Experimental potassium depletion in normal human subjects. I. Relation of ionic intakes to the renal conservation of potassium". Journal of Clinical Investigation. 38 (7): 1134–48. doi:10.1172/JCI103890. PMC 293261. PMID 13664789.
^ Fiebach, Nicholas H.; Barker, Lee Randol; Burton, John Russell & Zieve, Philip D. (2007). Principles of ambulatory medicine. Lippincott Williams & Wilkins. pp. 748–750. ISBN 978-0-7817-6227-4.
^ Gadsby, D. C. (2004). "Ion transport: spot the difference". Nature. 427 (6977): 795–7. Bibcode:2004Natur.427..795G. doi:10.1038/427795a. PMID 14985745. S2CID 5923529. ; for a diagram of the potassium pores are viewed, see Miller, C (2001). "See potassium run". Nature. 414 (6859): 23–24. Bibcode:2001Natur.414...23M. doi:10.1038/35102126. PMID 11689922. S2CID 4423041.
^ Jiang, Y.; Lee, A.; Chen, J.; Cadene, M.; et al. (2002). "Crystal structure and mechanism of a calcium-gated potassium channel" (PDF). Nature. 417 (6888): 515–22. Bibcode:2002Natur.417..515J. doi:10.1038/417515a. PMID 12037559. S2CID 205029269. Archived from the original on 2009-04-24. CS1 maint: bot: original URL status unknown (link)
^ Shi, N.; Ye, S.; Alam, A.; Chen, L.; et al. (2006). "Atomic structure of a Na+- and K+-conducting channel". Nature. 440 (7083): 570–4. Bibcode:2006Natur.440..570S. doi:10.1038/nature04508. PMID 16467789. S2CID 4355500. ; includes a detailed picture of atoms in the pump.
^ Zhou, Y.; Morais-Cabral, J. H.; Kaufman, A.; MacKinnon, R. (2001). "Chemistry of ion coordination and hydration revealed by a K+ channel-Fab complex at 2.0 A resolution". Nature. 414 (6859): 43–48. Bibcode:2001Natur.414...43Z. doi:10.1038/35102009. PMID 11689936. S2CID 205022645.
^ Noskov, S. Y.; Bernèche, S.; Roux, B. (2004). "Control of ion selectivity in potassium channels by electrostatic and dynamic properties of carbonyl ligands". Nature. 431 (7010): 830–4. Bibcode:2004Natur.431..830N. doi:10.1038/nature02943. PMID 15483608. S2CID 4414885.
^ National Academies of Sciences, Engineering and Medicine (2019). "Potassium: Dietary Reference Intakes for Adequacy". In Stallings, Virginia A; Harrison, Meghan; Oria, Maria (eds.). Dietary Reference Intakes for Sodium and Potassium. Washington, DC: The National Academies Press. doi:10.17226/25353. ISBN 978-0-309-48834-1. PMID 30844154.
^ Stallings, Virginia A; Harrison, Meghan; Oria, Maria, eds. (March 5, 2019). Dietary Reference Intakes for Sodium and Potassium – Publication. Health and Medicine Division. National Academies of Sciences, Engineering and Medicine. doi:10.17226/25353. ISBN 978-0-309-48834-1. PMID 30844154. S2CID 104464967. Retrieved May 13, 2019.
^ Panel on Dietary Reference Intakes for Electrolytes and Water, Standing Committee on the Scientific Evaluation of Dietary Reference Intakes, Food and Nutrition (2004). DRI, dietary reference intakes for water, potassium, sodium, chloride, and sulfate. Washington, D.C.: National Academies Press. ISBN 978-0-309-53049-1. Archived from the original on 2011-10-06. CS1 maint: multiple names: authors list (link)
^ Karger, S. (2004). "Energy and nutrient intake in the European Union". Annals of Nutrition and Metabolism. 48 (2 (suppl)): 1–16. doi:10.1159/000083041.
^ "Potassium" in NHS Choices – Other vitamins and minerals
^ "Potassium Fact Sheet". foodinsight.org. International Food Information Council Foundation. 7 April 2020. Retrieved May 16, 2020.
^ "Potassium Food Charts". Asia Pacific Journal of Clinical Nutrition. Archived from the original on 2021-04-29. Retrieved 2011-05-18.
^ "Potassium Content of Selected Foods per Common Measure, sorted by nutrient content" (PDF). USDA National Nutrient Database for Standard Reference, Release 20. Archived from the original (PDF) on December 17, 2008.
^ Whelton PK, He J, Cutler JA, Brancati FL, Appel LJ, Follmann D, Klag MJ (1997). "Effects of oral potassium on blood pressure. Meta-analysis of randomized controlled clinical trials". JAMA. 277 (20): 1624–32. doi:10.1001/jama.1997.03540440058033. PMID 9168293. S2CID 25937399.
^ a b Institute of Medicine (U.S.). Committee on Optimization of Nutrient Composition of Military Rations for Short-Term, High-Stress Situations; Institute of Medicine (U.S.). Committee on Military Nutrition Research (2006). Nutrient composition of rations for short-term, high-intensity combat operations. National Academies Press. pp. 287–. ISBN 978-0-309-09641-6.
^ D'Elia, L.; Barba, G.; Cappuccio, F.; Strazzullo (2011). "Potassium Intake, Stroke, and Cardiovascular Disease: A Meta-Analysis of Prospective Studies". J Am Coll Cardiol. 57 (10): 1210–9. doi:10.1016/j.jacc.2010.09.070. PMID 21371638.
^ He FJ, Marciniak M, Carney C, Markandu ND, Anand V, Fraser WD, Dalton RN, Kaski JC, MacGregor GA (2010). "Effects of potassium chloride and potassium bicarbonate on endothelial function, cardiovascular risk factors, and bone turnover in mild hypertensives". Hypertension. 55 (3): 681–8. doi:10.1161/HYPERTENSIONAHA.109.147488. PMID 20083724.
^ "The Top 300 of 2020". ClinCalc. Retrieved 11 April 2020.
^ "Potassium Drug Usage Statistics". ClinCalc. 23 December 2019. Retrieved 11 April 2020.
^ Shallenberger, R. S. (1993). Taste chemistry. Springer. pp. 120–. ISBN 978-0-7514-0150-9.
^ Garrett, Donald E. (1995-12-31). Potash: deposits, processing, properties and uses. Springer. ISBN 978-0-412-99071-7.
^ a b Ober, Joyce A. "Mineral Commodity Summaries 2008:Potash" (PDF). United States Geological Survey. Retrieved 2008-11-20.
^ a b c Ober, Joyce A. "Mineral Yearbook 2006:Potash" (PDF). United States Geological Survey. Retrieved 2008-11-20.
^ Wishart, David J. (2004). Encyclopedia of the Great Plains. U of Nebraska Press. p. 433. ISBN 978-0-8032-4787-1.
^ Chiu, Kuen-Wai (2000). "Potassium". Kirk-Othmer Encyclopedia of Chemical Technology. John Wiley & Sons, Inc. doi:10.1002/0471238961.1615200103080921.a01.pub2. ISBN 9780471238966.
^ Delahunt, J.; Lindeman, T. (2007). "Review of the safety of potassium and potassium oxides, including deactivation by introduction into water". Journal of Chemical Health and Safety. 14 (2): 21–32. doi:10.1016/j.jchas.2006.09.010.
^ Roy, Amit H. (2007). Kent and Riegel's handbook of industrial chemistry and biotechnology. Springer. pp. 1135–57. Bibcode:2007karh.book....... ISBN 978-0-387-27843-8.
^ Ochoa-Hueso, R; Delgado-Baquerizo, M; King, PTA; Benham, M; Arca, V; Power, SA (2019). "Ecosystem type and resource quality are more important than global change drivers in regulating early stages of litter decomposition". Soil Biology and Biochemistry. 129: 144–152. doi:10.1016/j.soilbio.2018.11.009. S2CID 92606851.
^ World Health Organization (2009). Stuart MC, Kouimtzi M, Hill SR (eds.). WHO Model Formulary 2008. World Health Organization. p. 491. hdl:10665/44053. ISBN 9789241547659.
^ "Potassium chloride medical facts from Drugs.com". www.drugs.com. Archived from the original on 18 January 2017. Retrieved 14 January 2017.
^ British national formulary : BNF 69 (69 ed.). British Medical Association. 2015. pp. 680, 684. ISBN 9780857111562.
^ Figoni, Paula I (2010). "Bleaching and Maturing Agents". How Baking Works: Exploring the Fundamentals of Baking Science. John Wiley and Sons. p. 86. ISBN 978-0-470-39267-6.
^ Chichester, C. O. (July 1986). "Uses and Exposure to Sulfites in Food". Advances in food research. Academic Press. pp. 4–6. ISBN 978-0-12-016430-1.
^ Schultz
^ Toedt, John; Koza, Darrell & Cleef-Toedt, Kathleen Van (2005). "Personal Cleansing Products: Bar Soap". Chemical composition of everyday products. Greenwood Publishing Group. ISBN 978-0-313-32579-3.
^ Siegel, Richard S. (1940). "Ignition of the safety match". Journal of Chemical Education. 17 (11): 515. Bibcode:1940JChEd..17..515S. doi:10.1021/ed017p515.
^ Anger, Gerd; Halstenberg, Jost; Hochgeschwender, Klaus; Scherhag, Christoph; Korallus, Ulrich; Knopf, Herbert; Schmidt, Peter; Ohlinger, Manfred. "Chromium Compounds". Ullmann's Encyclopedia of Industrial Chemistry. 9. Weinheim: Wiley-VCH. p. 178. doi:10.1002/14356007.a07_067.
^ Marx, Robert F. (1990). The history of underwater exploration. Courier Dover Publications. p. 93. ISBN 978-0-486-26487-5.
^ Gettens, Rutherford John & Stout, George Leslie (1966). Painting materials: A short encyclopaedia. Courier Dover Publications. pp. 109–110. ISBN 978-0-486-21597-6.
^ Modugno, G.; Benkő, C.; Hannaford, P.; Roati, G.; Inguscio, M. (1999-11-01). "Sub-Doppler laser cooling of fermionic ${}^{40}\mathrm{K}$ atoms". Physical Review A. 60 (5): R3373–R3376. arXiv:cond-mat/9908102. Bibcode:1999PhRvA..60.3373M. doi:10.1103/PhysRevA.60.R3373. S2CID 119001675.
^ Jackson, C. B.; Werner, R. C. (1957). "Ch. 18: The Manufacture of Potassium and NaK". Handling and uses of the alkali metals. Advances in Chemistry. 19. pp. 169–173. doi:10.1021/ba-1957-0019.ch018. ISBN 978-0-8412-0020-3.
^ Kearey, Philip; Brooks, M & Hill, Ian (2002). "Optical Pumped Magnetometer". An introduction to geophysical exploration. Wiley-Blackwell. p. 164. ISBN 978-0-632-04929-5.
^ "Potassium 244856". Sigma Aldrich.
^ Solomon, Robert E. (2002). Fire and Life Safety Inspection Manual. Jones & Bartlett Learning. p. 459. ISBN 978-0-87765-472-8.
^ "DOE Handbook-Alkali Metals Sodium, Potassium, NaK, and Lithium". Hss.doe.gov. Archived from the original on 2010-09-28. Retrieved 2010-10-16.
^ Wray, Thomas K. "Danger: peroxidazable chemicals" (PDF). Environmental Health & Public Safety, North Carolina State University. Archived from the original on 2016-07-29. CS1 maint: bot: original URL status unknown (link)
^ a b Schonwald, Seth (2004). "Potassium Chloride and Potassium Permanganate". Medical toxicology. Lippincott Williams & Wilkins. pp. 903–5. ISBN 978-0-7817-2845-4.
^ Markovchick, Vincent J. & Pons, Peter T. (2003). Emergency medicine secrets. Elsevier Health Sciences. p. 223. ISBN 978-1-56053-503-4.
Burkhardt, Elizabeth R. (2006). "Potassium and Potassium Alloys". Ullmann's Encyclopedia of Industrial Chemistry. A22. pp. 31–38. doi:10.1002/14356007.a22_031.pub2. ISBN 978-3-527-30673-2.
Greenwood, Norman N.; Earnshaw, Alan (1997). Chemistry of the Elements (2nd ed.). Butterworth-Heinemann. ISBN 978-0-08-037941-8.
Holleman, Arnold F.; Wiberg, Egon; Wiberg, Nils (2007). "Potassium". Lehrbuch der Anorganischen Chemie (in German) (91–100 ed.). Walter de Gruyter. ISBN 978-3110177701.
Schultz, H.; et al. (2006). "Potassium compounds". Ullmann's Encyclopedia of Industrial Chemistry. A22. pp. 39–103. doi:10.1002/14356007.a22_031.pub2. ISBN 978-3-527-30673-2.
National Nutrient Database at USDA Website
Potassiumat Wikipedia's sister projects
"Potassium". Drug Information Portal. U.S. National Library of Medicine.
KAsO2
K3AsO4
KBrO3
KCN
KCNO
KClO
KClO3
KHCO2
KHF2
KHSO3
KH2PO3
KIO3
KMnO4
KNH2
KN3
KNO2
KOCN
KPF6
KSCN
KCH3COO
K2Al2O4
K2Al2B2O7
K2CO3
KCrO3Cl
K3CrO4
K2Cr2O7
K2FeO4
K2HPO4
K2MnO4
K2O
K2O2
K2PtCl4
K2Pt(CN)4
K2SeO4
K2SO3
K2S2O3
K2Po
K2SiO3
K2SiF6
K3Fe(CN)6
K3Fe(C2O4)3
K3PO4
K4Mo2Cl8 | CommonCrawl |
\begin{document}
\begin{abstract}
We study stationary solutions to the Keller--Segel equation on curved planes.
We prove the necessity of the mass being $8 \pi$ and a sharp decay bound. Notably, our results do not require the solutions to have a finite second moment, and thus are novel already in the flat case.
Furthermore, we provide a correspondence between stationary solutions to the static Keller--Segel equation on curved planes and positively curved Riemannian metrics on the sphere. We use this duality to show the nonexistence of solutions in certain situations. In particular, we show the existence of metrics, arbitrarily close to the flat one on the plane, that do not support stationary solutions to the static Keller--Segel equation (with any mass).
Finally, as a complementary result, we prove a curved version of the logarithmic Hardy--Littlewood--Sobolev inequality and use it to show that the Keller--Segel free energy is bounded from below exactly when the mass is $8 \pi$, even in the curved case. \end{abstract}
\title{Stationary solutions to the Keller--Segel equation on curved planes}
\section{Introduction}
The Keller--Segel type equations describe \emph{chemotaxis}, that is the movement of organisms (typically bacteria) in the presence of a (chemical) substance. The simplest Keller--Segel system is a pair of equations on the density of the organisms, $\varrho$, and the concentration of the substance, $c$, both of which are functions on $[0, T) \times \mathbb{R}^n$. Furthermore, $\varrho$ is assumed to be nonnegative and integrable. Together they satisfy the (parabolic-elliptic) Keller--Segel equations: \begin{subequations} \begin{align}
\mathopen{}\mathclose\bgroup\originalleft( \partial_t + \Delta \aftergroup\egroup\originalright) \varrho &= \operatorname{d\!}{}^* \mathopen{}\mathclose\bgroup\originalleft( \varrho \operatorname{d\!}{} c \aftergroup\egroup\originalright), \label{eq:KS1} \\
\Delta c &= \varrho, \label{eq:KS2} \end{align} \end{subequations} where $\operatorname{d\!}{}$ is the gradient, $\operatorname{d\!}{}^*$ is its $L^2$-dual (the divergence), and $\Delta = \operatorname{d\!}{}^* \operatorname{d\!}{}$. The mass of $\varrho$ is \begin{equation}
m \mathrel{\vcenter{\baselineskip0.5ex\lineskiplimit0pt\hbox{.}\hbox{.}}}= \int\limits_{\mathbb{R}^d} \varrho (x) \operatorname{d\!}{}^n x \in \mathbb{R}_+, \end{equation} is a conserved quantity.
Stationary solutions to \cref{eq:KS1,eq:KS2} satisfy \begin{subequations} \begin{align}
\Delta \varrho &= \operatorname{d\!}{}^* \mathopen{}\mathclose\bgroup\originalleft( \varrho \operatorname{d\!}{} c \aftergroup\egroup\originalright), \label{eq:static_KS1} \\
\Delta c &= \varrho. \label{eq:static_KS2} \end{align} \end{subequations} There is some ambiguity in the choice of $c$ in \cref{eq:static_KS1,eq:static_KS2}, and the standard choice is to use the Green's function of the Laplacian to eliminate $c$ and \cref{eq:static_KS2} via \begin{equation}
c_\varrho (x) \mathrel{\vcenter{\baselineskip0.5ex\lineskiplimit0pt\hbox{.}\hbox{.}}}= - \frac{1}{2 \pi} \int\limits_{\mathbb{R}^2} \ln \mathopen{}\mathclose\bgroup\originalleft( |x - y| \aftergroup\egroup\originalright) \varrho (y) \operatorname{d\!}{}^2 y, \end{equation} and use the single equation \begin{equation}
\Delta \varrho = \operatorname{d\!}{}^* \mathopen{}\mathclose\bgroup\originalleft( \varrho \operatorname{d\!}{} c_\varrho \aftergroup\egroup\originalright). \label{eq:static_KS} \end{equation} There is a well-known family of solutions to \cref{eq:static_KS}: Let $\lambda \in \mathbb{R}_+$ and $x_\star \in \mathbb{R}^2$ be arbitrary, and define \begin{equation}
\varrho_{\lambda, x_\star} \mathrel{\vcenter{\baselineskip0.5ex\lineskiplimit0pt\hbox{.}\hbox{.}}}= \frac{8 \lambda^2}{\mathopen{}\mathclose\bgroup\originalleft( \lambda^2 + |x - x_\star|^2 \aftergroup\egroup\originalright)^2}. \label{eq:flat_solution} \end{equation} Then $\varrho_{\lambda, x_\star}$ is a solution to \cref{eq:static_KS} with $m = 8 \pi$.
When the metric is the standard, euclidean metric on $\mathbb{R}^2$, the literature of \cref{eq:KS1,eq:KS2} and \cref{eq:static_KS} is vast; the Reader may find good introductions in \cites{DP04,BDP06,CD12}. Very little is known about the curved case, that is, when the underlying space is not the (flat) plane. We remark here the work of \cite{MP20}, where the authors considered \cref{eq:KS1,eq:KS2} on the hyperbolic plane.
In this paper, we study the case when the metric is conformally equivalent to the flat metric and the conformal factor has the form $e^{2 \varphi}$, where $\varphi$ is smooth and compactly supported. Let us note that some of our results are novel already in the flat $(\varphi = 0)$ case. In particular, we prove that (under very mild hypotheses), solutions to \cref{eq:static_KS} have mass $8 \pi$.
\subsection*{Outline of the paper}
In \Cref{sec:static_KS}, we introduce the static Keller--Segel equation on the curved plane $\mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, e^{2 \varphi} g_0 \aftergroup\egroup\originalright)$. In \Cref{sec:reduction}, we prove in \Cref{theorem:reduced_KS_static} that, under mild hypothesis of the growth of $\varrho$, the static Keller--Segel equation can be reduced to a simpler equation (see in \cref{eq:reduced_KS_static}). Furthermore, in \Cref{cor:rho_bounds}, we give sharp bounds on the decay rate of $\varrho$ and in \Cref{theorem:critical_mass} we show that a (nonzero) solution must have $m = 8 \pi$. In \Cref{sec:KW}, we explore a connection between solutions to the (reduced) static Keller--Segel equation and Kazdan--Warner equation on the round sphere. As an application, we prove the nonexistence of solutions for certain conformal factors in \Cref{theorem:nonexistence}. Finally, in \Cref{sec:curved_log-HLS}, we prove the logarithmic Hardy--Littlewood--Sobolev for $\mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, e^{2 \varphi} g_0 \aftergroup\egroup\originalright)$ and in \Cref{sec:KS_fe}, as an application, we show that, as in the flat case, the Keller--Segel free energy on $\mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, e^{2 \varphi} g_0 \aftergroup\egroup\originalright)$ is bounded from below only when $m = 8 \pi$.
\begin{acknowledgment}
I thank Michael Sigal for introducing me to the topic and for his initial guidance. I also thank the referee for their helpful recommendations. \end{acknowledgment}
\section{The curved, static Keller--Segel equation} \label{sec:static_KS}
Let $g_0$ be the standard metric on $\mathbb{R}^2$, let $\varphi \in C_\mathrm{cpt}^\infty \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2 \aftergroup\egroup\originalright)$, let $g_\varphi \mathrel{\vcenter{\baselineskip0.5ex\lineskiplimit0pt\hbox{.}\hbox{.}}}= e^{2 \varphi} g_0$. Let $L_k^p \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, g_\varphi \aftergroup\egroup\originalright)$ be Banach space of functions on $\mathbb{R}^2$ that are $L_k^p$ with respect to $g_\varphi$. Note that the properties of being bounded in $L_{1, \mathrm{loc}}^2$ are independent of the chosen metric. Finally, let $L_+^1 \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, g_\varphi \aftergroup\egroup\originalright) \subseteq L^1 \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, g_\varphi \aftergroup\egroup\originalright)$ be the space of almost everywhere positive functions.
The area form and the Laplacian behave under a conformal change via \begin{equation}
\: \mathrm{dA}_\varphi = e^{2 \varphi} \: \mathrm{dA}_0 \quad \& \quad \Delta_\varphi = e^{- 2 \varphi} \Delta_0. \end{equation} Thus the Green's function is conformally invariant: \begin{equation}
G (x, y) = - \frac{1}{2 \pi} \ln \mathopen{}\mathclose\bgroup\originalleft( |x - y| \aftergroup\egroup\originalright). \end{equation} For any $\varrho \in L_+^1 \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, g_\varphi \aftergroup\egroup\originalright)$, let \begin{equation}
c_{\varphi, \varrho} \mathrel{\vcenter{\baselineskip0.5ex\lineskiplimit0pt\hbox{.}\hbox{.}}}= \int\limits_{\mathbb{R}^2} G (\cdot, y) \varrho (y) \: \mathrm{dA}_\varphi (y), \label{eq:c_def} \end{equation} when the integral exists. Assume that the function $\varrho \in L_+^1 \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, g_\varphi \aftergroup\egroup\originalright) \cap L_{1, \mathrm{loc}}^2$ is such that $c_{\varphi, \varrho}$ is defined on $\mathbb{R}^2$. Then $\varrho$ is a solution to the \emph{static Keller--Segel equation} on $\mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, g_\varphi \aftergroup\egroup\originalright)$ if it solves (the weak version of) \begin{equation}
\Delta_\varphi \varrho - \operatorname{d\!}{}^* \mathopen{}\mathclose\bgroup\originalleft( \varrho \operatorname{d\!}{} c_{\varphi, \varrho} \aftergroup\egroup\originalright) = 0. \label{eq:KS_static} \end{equation} In the next section we prove that, under mild hypotheses, \cref{eq:KS_static} is equivalent to the simpler \begin{equation}
\operatorname{d\!}{} \mathopen{}\mathclose\bgroup\originalleft( \ln \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) - c_{\varphi, \varrho} \aftergroup\egroup\originalright) = 0. \label{eq:reduced_KS_static} \end{equation} We call \cref{eq:reduced_KS_static} the \emph{reduced, static Keller--Segel equation}.
In applications it is always assumed that $\varrho$ has finite mass. Furthermore, the minimal regularity needed for the weak version of \cref{eq:KS_static} is $L_{1, \mathrm{loc}}^2$ and the fact that $c_{\varphi, \varrho}$ is defined. Finally, we impose the finiteness of the entropy: $\varrho \ln \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) \in L^1 \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, g_\varphi \aftergroup\egroup\originalright)$. This is implied by, for example, the finiteness of the Keller--Segel free energy; cf \Cref{sec:KS_fe}. With that in mind, we define the \emph{(curved) Keller--Segel configuration space} as: \begin{equation}
\mathcal{C}_\mathrm{KS} (m, \varphi) \mathrel{\vcenter{\baselineskip0.5ex\lineskiplimit0pt\hbox{.}\hbox{.}}}= \mathopen{}\mathclose\bgroup\originalleft\{ \ \varrho \in L_+^1 \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, g_\varphi \aftergroup\egroup\originalright) \cap L_{1, \mathrm{loc}}^2 \ \middle| \ \begin{array}{l} \varrho \ln \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) \in L^1 \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, g_\varphi \aftergroup\egroup\originalright), \\ \| \varrho \|_{L^1 \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, g_\varphi \aftergroup\egroup\originalright)} = m, \\ c_{\varphi, \varrho} \mbox{ is defined everywhere.} \end{array} \aftergroup\egroup\originalright\}. \label{eq:KS_config_space} \end{equation}
Let $r (x) \mathrel{\vcenter{\baselineskip0.5ex\lineskiplimit0pt\hbox{.}\hbox{.}}}= |x|$ be the euclidean radial function. First we prove a bound on $c_{\varphi, \varrho}$.
\begin{lemma}
\label{lemma:c_bound}
Let $\varrho \in \mathcal{C}_\mathrm{KS} (m, \varphi)$ be a solution of the static Keller--Segel \cref{eq:KS_static}. Then the function $c_{\varphi, \varrho} + \tfrac{m}{4 \pi} \ln \mathopen{}\mathclose\bgroup\originalleft( 1 + r^2 \aftergroup\egroup\originalright)$ is bounded. \end{lemma}
\begin{proof}
As $\Delta_\varphi c_{\varphi, \varrho} \in L^1 \mathopen{}\mathclose\bgroup\originalleft( B_1 (0), g_\varphi \aftergroup\egroup\originalright)$, it is enough to prove, without any loss of generality, the boundedness of $c_{\varphi, \varrho} + \tfrac{m}{2 \pi} \ln \mathopen{}\mathclose\bgroup\originalleft( r \aftergroup\egroup\originalright)$, when $r \geqslant 1$.
Since $c_{\varphi, \varrho} (0) = - \tfrac{1}{2 \pi} \int_{\mathbb{R}^2} \varrho \ln (r) \: \mathrm{dA}_\varphi$ is finite, we have that
\begin{align}
c_{\varphi, \varrho} (x) &\leqslant o (1) - \frac{1}{2 \pi} \int\limits_{B_{|x|/2} (x)} \ln \mathopen{}\mathclose\bgroup\originalleft( |x - y| \aftergroup\egroup\originalright) \varrho (y) \: \mathrm{dA}_\varphi (y) \\
&\leqslant o (1) - \frac{1}{2 \pi} \ln \mathopen{}\mathclose\bgroup\originalleft( |x| \aftergroup\egroup\originalright) \int\limits_{B_{|x|/2} (x)} \varrho \: \mathrm{dA}_\varphi \\
&\leqslant O (1) - \frac{m}{2 \pi} \ln \mathopen{}\mathclose\bgroup\originalleft( |x| \aftergroup\egroup\originalright) + \frac{1}{2 \pi} \int\limits_{\mathbb{R}^2 - B_{|x|/2} (x)} \ln (r) \varrho \: \mathrm{dA}_\varphi \\
&\leqslant O (1) - \frac{m}{2 \pi} \ln \mathopen{}\mathclose\bgroup\originalleft( |x| \aftergroup\egroup\originalright).
\end{align}
This proves the upper bound.
In order to get the lower bound, let us use Jensen's inequality to get
\begin{equation}
c_{\varphi, \varrho} (x) - c_{\varphi, \varrho} (0) = - \frac{m}{2 \pi} \int\limits_{\mathbb{R}^2} \ln \mathopen{}\mathclose\bgroup\originalleft( \frac{|x - y|}{|y|} \aftergroup\egroup\originalright) \frac{\varrho (y) \: \mathrm{dA}_\varphi (y)}{m} \geqslant - \frac{m}{2 \pi} \ln \mathopen{}\mathclose\bgroup\originalleft( \ \int\limits_{\mathbb{R}^2} \frac{|x - y|}{|y|} \varrho (y) \: \mathrm{dA}_\varphi (y) \aftergroup\egroup\originalright) + \frac{m}{2 \pi} \ln (m).
\end{equation}
Since $\varrho \in L_{1, \mathrm{loc}}^2$, we get that there exists $\delta > 0$, such that for all $p > 1$, $\varrho \in L^p \mathopen{}\mathclose\bgroup\originalleft( B_\delta (0) \aftergroup\egroup\originalright)$. We can assume that $\delta \leqslant 1$. Since for all $q \in [1, 2)$, $r^{- 1} \in L^q \mathopen{}\mathclose\bgroup\originalleft( B_\delta (0) \aftergroup\egroup\originalright)$ and $\tfrac{|x - y|}{|y|} \leqslant \tfrac{\sqrt{|x|^2 + \delta^2}}{\delta}$ on $\mathbb{R}^2 - B_\delta (0)$, we get that, for any $p > 1$, that
\begin{align}
\int\limits_{\mathbb{R}^2} \frac{|x - y|}{|y|} \varrho (y) \: \mathrm{dA}_\varphi (y) &= \mathopen{}\mathclose\bgroup\originalleft( \ \int\limits_{B_\delta (0)} + \int\limits_{\mathbb{R}^2 - B_\delta (0)} \aftergroup\egroup\originalright) \frac{|x - y|}{|y|} \varrho (y) \: \mathrm{dA}_\varphi (y) \\
&\leqslant \mathopen{}\mathclose\bgroup\originalleft( e^{2 \| \varphi \|_{L^\infty \mathopen{}\mathclose\bgroup\originalleft( B_\delta (0) \aftergroup\egroup\originalright)}} \| \varrho \|_{L^p \mathopen{}\mathclose\bgroup\originalleft( B_\delta (0) \aftergroup\egroup\originalright)} \| r^{- 1} \|_{L^{\frac{p}{p - 1}} \mathopen{}\mathclose\bgroup\originalleft( B_\delta (0) \aftergroup\egroup\originalright)} + m \aftergroup\egroup\originalright) \frac{\sqrt{|x|^2 + \delta^2}}{\delta}.
\end{align}
Thus, when $r \geqslant 1$, we get that
\begin{equation}
c_{\varphi, \varrho} + \frac{m}{2 \pi} \ln \mathopen{}\mathclose\bgroup\originalleft( r \aftergroup\egroup\originalright) \geqslant C (\varphi, \varrho),
\end{equation}
which completes the proof. \end{proof}
\section{Reduction of order and the necessity of $m = 8 \pi$} \label{sec:reduction}
\begin{theorem}
\label{theorem:reduced_KS_static}
Let $\varrho \in \mathcal{C}_\mathrm{KS} (m, \varphi)$ be a solution of the static Keller--Segel \cref{eq:KS_static}. Furthermore assume the following bound: there exists a positive number $C$, such that on $\mathbb{R}^2 - B_C (0)$, we have
\begin{equation}
\varrho \leqslant C r^{Cr^2}. \label[cond]{cond:hypothesis}
\end{equation}
Then the reduced, static Keller--Segel \cref{eq:reduced_KS_static} holds, that is $\operatorname{d\!}{} \mathopen{}\mathclose\bgroup\originalleft( \ln \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) - c_{\varphi, \varrho} \aftergroup\egroup\originalright) = 0$. \end{theorem}
\begin{remark}
If $\varrho \in L^\infty \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2 \aftergroup\egroup\originalright)$, then \cref{cond:hypothesis} is trivially satisfied with $C = \max \mathopen{}\mathclose\bgroup\originalleft( 1, \| \varrho \|_{L^\infty \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2 \aftergroup\egroup\originalright)} \aftergroup\egroup\originalright)$. We conjecture that \cref{cond:hypothesis} is not necessary in general for the conclusion \Cref{theorem:reduced_KS_static} to hold. \end{remark}
\begin{remark}
A corollary of the reduced, static Keller--Segel \cref{eq:reduced_KS_static} is that the (nonreduced) static Keller--Segel \cref{eq:KS_static} is no longer nonlocal, as $c_{\varrho, \varphi}$ can be eliminated using $\operatorname{d\!}{} c_{\varphi, \varrho} = \operatorname{d\!}{} \mathopen{}\mathclose\bgroup\originalleft( \ln \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) \aftergroup\egroup\originalright) = \tfrac{\operatorname{d\!}{} \varrho}{\varrho}$, and get
\begin{equation}
\operatorname{d\!}{}^* \mathopen{}\mathclose\bgroup\originalleft( \varrho \operatorname{d\!}{} c_{\varphi, \varrho} \aftergroup\egroup\originalright) = - g_\varphi \mathopen{}\mathclose\bgroup\originalleft( \operatorname{d\!}{} \varrho, \operatorname{d\!}{} c_{\varphi, \varrho} \aftergroup\egroup\originalright) + \varrho \Delta_\varphi c_{\varphi, \varrho} = - g_\varphi \mathopen{}\mathclose\bgroup\originalleft( \operatorname{d\!}{} \varrho, \tfrac{\operatorname{d\!}{} \varrho}{\varrho} \aftergroup\egroup\originalright) + \varrho^2 = - \frac{|\operatorname{d\!}{} \varrho|_\varphi^2}{\varrho} + \varrho^2.
\end{equation}
Thus the static Keller--Segel \cref{eq:KS_static} becomes
\begin{equation}
\Delta_\varphi \varrho + \frac{|\operatorname{d\!}{} \varrho|_\varphi^2}{\varrho} - \varrho^2 = 0.
\end{equation} \end{remark}
\begin{proof}[Proof of \Cref{theorem:reduced_KS_static}]
Let $f \mathrel{\vcenter{\baselineskip0.5ex\lineskiplimit0pt\hbox{.}\hbox{.}}}= \ln \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) - c_{\varphi, \varrho}$. The static Keller--Segel \cref{eq:KS_static} implies that
\begin{equation}
\forall R \in \mathbb{R}_+ : \forall \phi \in L_{1, 0}^2 \mathopen{}\mathclose\bgroup\originalleft( B_R (0), g_\varphi \aftergroup\egroup\originalright) : \quad \int\limits_{\mathbb{R}^2} \varrho g_0 \mathopen{}\mathclose\bgroup\originalleft( \operatorname{d\!}{} \phi, \operatorname{d\!}{} f \aftergroup\egroup\originalright) \: \mathrm{dA}_0 = 0. \label{eq:new_KS_static}
\end{equation}
We now apply an Agmon-trick type argument: Let $\chi$ be a smooth and compactly supported function. Then, using $\phi = f \chi^2$ in the second row, we get
\begin{align}
\int\limits_{\mathbb{R}^2} \varrho \mathopen{}\mathclose\bgroup\originalleft| \operatorname{d\!}{} \mathopen{}\mathclose\bgroup\originalleft( \chi f \aftergroup\egroup\originalright) \aftergroup\egroup\originalright|^2 \: \mathrm{dA}_0 &= \int\limits_{\mathbb{R}^2} \varrho \mathopen{}\mathclose\bgroup\originalleft| \operatorname{d\!}{} \chi \aftergroup\egroup\originalright|^2 f^2 \: \mathrm{dA}_0 + 2 \int\limits_{\mathbb{R}^2} \varrho f \chi g_0 \mathopen{}\mathclose\bgroup\originalleft( \operatorname{d\!}{} \chi, \operatorname{d\!}{} f \aftergroup\egroup\originalright) \: \mathrm{dA}_0 + \int\limits_{\mathbb{R}^2} \varrho \chi^2 \mathopen{}\mathclose\bgroup\originalleft| \operatorname{d\!}{} f \aftergroup\egroup\originalright|^2 \: \mathrm{dA}_0 \\
&= \int\limits_{\mathbb{R}^2} \varrho \mathopen{}\mathclose\bgroup\originalleft| \operatorname{d\!}{} \chi \aftergroup\egroup\originalright|^2 f^2 \: \mathrm{dA}_0 + \int\limits_{\mathbb{R}^2} \varrho g_0 \mathopen{}\mathclose\bgroup\originalleft( \operatorname{d\!}{} \mathopen{}\mathclose\bgroup\originalleft( f \chi^2 \aftergroup\egroup\originalright), \varrho \operatorname{d\!}{} f \aftergroup\egroup\originalright) \: \mathrm{dA}_0 - \int\limits_{\mathbb{R}^2} \varrho \chi^2 \mathopen{}\mathclose\bgroup\originalleft| \operatorname{d\!}{} f \aftergroup\egroup\originalright|^2 \: \mathrm{dA}_0 + \int\limits_{\mathbb{R}^2} \varrho \chi^2 \mathopen{}\mathclose\bgroup\originalleft| \operatorname{d\!}{} f \aftergroup\egroup\originalright|^2 \: \mathrm{dA}_0 \\
&= \int\limits_{\mathbb{R}^2} \varrho \mathopen{}\mathclose\bgroup\originalleft| \operatorname{d\!}{} \chi \aftergroup\egroup\originalright|^2 f^2 \: \mathrm{dA}_0. \label[ineq]{ineq:Agmon}
\end{align}
Now for each $R \gg 1$, let $\chi = \chi_R$ be a smooth cut-off function that is 1 on $B_R (0)$, vanishes on $\mathbb{R}^2 - B_{2 R} (0)$, and (for some $K \in \mathbb{R}_+$) $|\operatorname{d\!}{} \chi_R| = \tfrac{K}{R}$. Let $A_R = B_{2 R} (0) - B_R (0)$. Then we get that
\begin{equation}
\int\limits_{\mathbb{R}^2} \varrho |\operatorname{d\!}{} f|^2 \: \mathrm{dA}_0 \leqslant \liminf\limits_{R \rightarrow \infty} \int\limits_{\mathbb{R}^2} \varrho \mathopen{}\mathclose\bgroup\originalleft| \operatorname{d\!}{} \mathopen{}\mathclose\bgroup\originalleft( \chi_R f \aftergroup\egroup\originalright) \aftergroup\egroup\originalright|^2 \: \mathrm{dA}_0 = \liminf\limits_{R \rightarrow \infty} \int\limits_{\mathbb{R}^2} \varrho \mathopen{}\mathclose\bgroup\originalleft| \operatorname{d\!}{} \chi_R \aftergroup\egroup\originalright|^2 f^2 \: \mathrm{dA}_0 \leqslant \liminf\limits_{R \rightarrow \infty} \frac{K^2}{R^2} \int\limits_{A_R} \varrho f^2 \: \mathrm{dA}_0.
\end{equation}
To complete the proof, we show now that the last limit inferior is zero. Since
\begin{equation}
\int\limits_{A_R} \varrho f^2 \: \mathrm{dA}_0 \leqslant \mathopen{}\mathclose\bgroup\originalleft( \sqrt{\int\limits_{A_R} \varrho \ln \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright)^2 \: \mathrm{dA}_0} + \sqrt{\int\limits_{A_R} \varrho c_{\varphi, \varrho}^2 \: \mathrm{dA}_0} \aftergroup\egroup\originalright)^2,
\end{equation}
it is enough to show that both terms under the square roots are $o \mathopen{}\mathclose\bgroup\originalleft( R^2 \aftergroup\egroup\originalright)$, at least for some divergent sequence of radii. This is immediate for the second term by \Cref{lemma:c_bound}. To bound the first term, let $C$ be the constant from \cref{cond:hypothesis} and break up $A_R$ into 2 pieces:
\begin{align}
A_{R, I} &\mathrel{\vcenter{\baselineskip0.5ex\lineskiplimit0pt\hbox{.}\hbox{.}}}= \mathopen{}\mathclose\bgroup\originalleft\{ \ x \in A_R \ \middle| \ \varrho (x) \leqslant r (x)^{- C r(x)^2} \ \aftergroup\egroup\originalright\}, \\
A_{R, II} &\mathrel{\vcenter{\baselineskip0.5ex\lineskiplimit0pt\hbox{.}\hbox{.}}}= \mathopen{}\mathclose\bgroup\originalleft\{ \ x \in A_R \ \middle| \ r (x)^{- C r(x)^2} \leqslant \varrho (x) \leqslant r (x)^{C r(x)^2} \ \aftergroup\egroup\originalright\}.
\end{align}
By \cref{cond:hypothesis}, $A_R = A_{R, I} \cup A_{R, II}$. Let is first inspect
\begin{equation}
0 \leqslant \int\limits_{A_{R, I}} \varrho \ln \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright)^2 \: \mathrm{dA}_0 \leqslant C (2 R)^{- C (2 R)^2} \ln \mathopen{}\mathclose\bgroup\originalleft( C (2 R)^{- C (2 R)^2} \aftergroup\egroup\originalright)^2 \mathrm{Area} \mathopen{}\mathclose\bgroup\originalleft( A_{R, I}, g_0 \aftergroup\egroup\originalright) = o \mathopen{}\mathclose\bgroup\originalleft( R^2 \aftergroup\egroup\originalright).
\end{equation}
Finally, note that on $A_{R, II}$, we have $\mathopen{}\mathclose\bgroup\originalleft| \ln \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) \aftergroup\egroup\originalright| = O \mathopen{}\mathclose\bgroup\originalleft( R^2 \ln (R) \aftergroup\egroup\originalright)$. Thus, for $R \gg 1$, we have
\begin{equation}
0 \leqslant \int\limits_{A_{R, II}} \varrho \ln \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright)^2 \: \mathrm{dA}_0 \leqslant \| \ln \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) \|_{L^\infty \mathopen{}\mathclose\bgroup\originalleft( A_{R, II} \aftergroup\egroup\originalright)} \int\limits_{A_{R, II}} \varrho \mathopen{}\mathclose\bgroup\originalleft| \ln \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) \aftergroup\egroup\originalright| \: \mathrm{dA}_0 \leqslant 8 C R^2 \ln \mathopen{}\mathclose\bgroup\originalleft( R \aftergroup\egroup\originalright) \int\limits_{A_{R, II}} \varrho \mathopen{}\mathclose\bgroup\originalleft| \ln \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) \aftergroup\egroup\originalright| \: \mathrm{dA}_0.
\end{equation}
Now let $R_k \mathrel{\vcenter{\baselineskip0.5ex\lineskiplimit0pt\hbox{.}\hbox{.}}}= 2^k$, and then
\begin{equation}
0 \leqslant \frac{1}{R_k^2} \int\limits_{A_{R_k, II}} \varrho \ln \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright)^2 \: \mathrm{dA}_0 \leqslant 8 C \ln (2) k \int\limits_{A_{R_k, II}} \varrho \mathopen{}\mathclose\bgroup\originalleft| \ln \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) \aftergroup\egroup\originalright| \: \mathrm{dA}_0.
\end{equation}
Since $\varrho \ln \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) \in L^1 \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, g_0 \aftergroup\egroup\originalright)$ we have that
\begin{equation}
\liminf\limits_{k \rightarrow \infty} \: \mathopen{}\mathclose\bgroup\originalleft( k \int\limits_{A_{R_k, II}} \varrho \mathopen{}\mathclose\bgroup\originalleft| \ln \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) \aftergroup\egroup\originalright| \: \mathrm{dA}_0 \aftergroup\egroup\originalright) = 0,
\end{equation}
and thus
\begin{equation}
0 \leqslant \int\limits_{\mathbb{R}^2} \varrho \mathopen{}\mathclose\bgroup\originalleft| \operatorname{d\!}{} f \aftergroup\egroup\originalright|^2 \: \mathrm{dA}_0 \leqslant \liminf\limits_{k \rightarrow \infty} \frac{K^2}{R_k^2} \int\limits_{A_{R_k}} \varrho f^2 \: \mathrm{dA}_0 = 0,
\end{equation}
and hence
\begin{equation}
\int\limits_{\mathbb{R}^2} \varrho \mathopen{}\mathclose\bgroup\originalleft| \operatorname{d\!}{} f \aftergroup\egroup\originalright|^2 \: \mathrm{dA}_0 = 0,
\end{equation}
which implies \cref{eq:reduced_KS_static}, and thus completes the proof. \end{proof}
\begin{corollary} \label{cor:rho_bounds}
If $\varrho \in \mathcal{C}_\mathrm{KS}$ is a solution of the static Keller--Segel \cref{eq:KS_static} and satisfies \cref{cond:hypothesis}, then there is a number $K = K (\varphi, \varrho) \geqslant 1$, such that
\begin{equation}
K \geqslant \varrho \mathopen{}\mathclose\bgroup\originalleft( 1 + r^2 \aftergroup\egroup\originalright)^{\frac{m}{4 \pi}} \geqslant K^{- 1}. \label[ineq]{ineq:rho_bounds}
\end{equation}
In particular, $\varrho \sim r^{- \frac{m}{2 \pi}}$ and $m > 4 \pi$. \end{corollary}
\begin{proof}
We have
\begin{equation}
\ln \mathopen{}\mathclose\bgroup\originalleft( \varrho \mathopen{}\mathclose\bgroup\originalleft( 1 + r^2 \aftergroup\egroup\originalright)^{\frac{m}{4 \pi}} \aftergroup\egroup\originalright) = \ln \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) + \tfrac{m}{4 \pi} \ln \mathopen{}\mathclose\bgroup\originalleft( 1 + r^2 \aftergroup\egroup\originalright) = \underbrace{\ln \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) - c_{\varphi, \varrho}}_{\mbox{constant by \Cref{theorem:reduced_KS_static}}} + \underbrace{c_{\varphi, \varrho} + \tfrac{m}{4 \pi} \ln \mathopen{}\mathclose\bgroup\originalleft( 1 + r^2 \aftergroup\egroup\originalright)}_{\mbox{bounded by \Cref{lemma:c_bound}}},
\end{equation}
which concludes the proof. \end{proof}
\begin{remark}
\Cref{theorem:reduced_KS_static} remains true (with the same proof) even when $g_\varphi$ is replaced by any compactly supported, smooth perturbation of $g_0$. However proving \Cref{lemma:c_bound} becomes more complicated in that case, although conjecturally, that claim should still hold, and thus so should \Cref{cor:rho_bounds}. \end{remark}
\begin{remark}
Before stating our next theorem, let us recall a few facts, commonly used in literature of the Keller--Segel equations.
First of all, and to the best of our knowledge, the only known solutions in the flat case are the ones given in \cref{eq:flat_solution}. Note that they all have mass $8 \pi$.
A complementary fact, supporting the conjecture that static solutions must have mass $8 \pi$, is the the following "Virial Theorem" that applies to the time-dependent equation as well: Assume that $\varrho$ is a solution to the (time-dependent) Keller--Segel \cref{eq:KS1,eq:KS2}, such that for all $t$ in the domain of $\varrho$ the following quantity is finite
\begin{equation}
W (t) \mathrel{\vcenter{\baselineskip0.5ex\lineskiplimit0pt\hbox{.}\hbox{.}}}= \int\limits_{\mathbb{R}^2} |x|^2 \varrho (t, x) \: \mathrm{dA}_0 (x).
\end{equation}
Then $W$ satisfies the following equation (cf. \cite{BDP06}*{Lemma~22} for the proof):
\begin{equation}
\dot{W} (t) = 4 m - \frac{m}{2 \pi}.
\end{equation}
In particular, if $\varrho$ is a (positive) solution to the static Keller--Segel \cref{eq:KS_static} with finite $W$, then $m = 8 \pi$. Note that for each $\varrho_{\lambda, x_\star}$ in \cref{eq:flat_solution}, we get $W = \infty$, so the above two results are indeed complementary. \end{remark}
In the next theorem we prove that, under \cref{cond:hypothesis}, all (positive) solutions to the static Keller--Segel \cref{eq:KS_static} must have mass $8 \pi$.
\begin{theorem} \label{theorem:critical_mass}
If $\varrho \in \mathcal{C}_\mathrm{KS}$ is a solution of the static Keller--Segel \cref{eq:KS_static} and satisfies \cref{cond:hypothesis}, then its mass is necessarily $8 \pi$. \end{theorem}
\begin{proof}
By \Cref{cor:rho_bounds}, we have that $m > 4 \pi$ and thus, for some $\epsilon > 0$, we have $\varrho = O \mathopen{}\mathclose\bgroup\originalleft( r^{- 2 - \epsilon} \aftergroup\egroup\originalright)$.
Let now $v = (v_1, v_2)$ be a smooth, compactly supported vector field. Let us pair both sides of \cref{eq:reduced_KS_static} with $- \varrho v$, integrate over $\mathbb{R}^2$ with respect to $\: \mathrm{dA}_0$ and then integrate by parts in the first term to get
\begin{equation}
\sum\limits_{i = 1}^2 \mathopen{}\mathclose\bgroup\originalleft( \ \int\limits_{\mathbb{R}^2} \varrho \mathopen{}\mathclose\bgroup\originalleft( \partial_i v_i + v_i \partial_i c_{\varphi, \varrho} \aftergroup\egroup\originalright) \: \mathrm{dA}_0 \ \aftergroup\egroup\originalright) = 0. \label{eq:virial1}
\end{equation}
For any smooth, real function $f$, let
\begin{equation}
v^f (x) \mathrel{\vcenter{\baselineskip0.5ex\lineskiplimit0pt\hbox{.}\hbox{.}}}= \mathopen{}\mathclose\bgroup\originalleft( 2 x_1 e^{2 \varphi (x)} + \partial_1 f, 2 x_2 e^{2 \varphi (x)} + \partial_2 f \aftergroup\egroup\originalright),
\end{equation}
and let $\chi_R$ as in the proof of \Cref{theorem:reduced_KS_static}. Let us assume that $\mathopen{}\mathclose\bgroup\originalleft| \operatorname{d\!}{} f \aftergroup\egroup\originalright| \in L^2 \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, g_\varphi \aftergroup\egroup\originalright)$. Then for $v = \chi_R v^f$ \cref{eq:virial1} becomes
\begin{equation}
\begin{aligned}
0 &= \sum\limits_{i = 1}^2 \mathopen{}\mathclose\bgroup\originalleft( \ \int\limits_{\mathbb{R}^2} \varrho \mathopen{}\mathclose\bgroup\originalleft( \chi_R \partial_i v_i^f + \chi_R \varrho v_i^f \partial_i c_{\varphi, \varrho} + \partial_i \chi_R v_i^f \aftergroup\egroup\originalright) \: \mathrm{dA}_0 \ \aftergroup\egroup\originalright) \\
&= \sum\limits_{i = 1}^2 \mathopen{}\mathclose\bgroup\originalleft( \ \int\limits_{\mathbb{R}^2} \chi_R (x) \varrho (x) \mathopen{}\mathclose\bgroup\originalleft( 2 e^{2 \varphi (x)} + 4 x_i \partial_i \varphi (x) e^{2 \varphi (x)} + \partial_i^2 f (x) + \mathopen{}\mathclose\bgroup\originalleft( 2 x_i e^{2 \varphi (x)} + \partial_i f (x) \aftergroup\egroup\originalright) \partial_i c_{\varphi, \varrho} (x) \aftergroup\egroup\originalright) \: \mathrm{dA}_0 (x) \ \aftergroup\egroup\originalright) \\
& \quad + O \mathopen{}\mathclose\bgroup\originalleft( \ \int\limits_{B_{2R} (0) - B_R (0)} |\operatorname{d\!}{} \chi_R| \mathopen{}\mathclose\bgroup\originalleft| v^f \aftergroup\egroup\originalright| \varrho \: \mathrm{dA}_\varphi \aftergroup\egroup\originalright) \\
&= 4 \underbrace{\int\limits_{\mathbb{R}^2} \chi_R \varrho \: \mathrm{dA}_\varphi}_{\mathcal{I}_1 (R)} + 2 \underbrace{\sum\limits_{i = 1}^2 \ \int\limits_{\mathbb{R}^2} \chi_R (x) \varrho(x) x_i \partial_i c_{\varphi, \varrho} (x) \: \mathrm{dA}_\varphi (x)}_{\mathcal{I}_2 (R)} + \underbrace{\sum\limits_{i = 1}^2 \mathopen{}\mathclose\bgroup\originalleft( \ \int\limits_{\mathbb{R}^2} \chi_R \varrho \mathopen{}\mathclose\bgroup\originalleft( 4 r \partial_r \varphi - \Delta_\varphi f - g_\varphi \mathopen{}\mathclose\bgroup\originalleft( \operatorname{d\!}{} f, \operatorname{d\!}{} c_{\varphi, \varrho} \aftergroup\egroup\originalright) \aftergroup\egroup\originalright) \: \mathrm{dA}_0 \ \aftergroup\egroup\originalright)}_{\mathcal{I}_3 (R)} \\
& \quad + O \mathopen{}\mathclose\bgroup\originalleft( R^{- 1} \mathopen{}\mathclose\bgroup\originalleft( R + \| \operatorname{d\!}{} f \|_{L^2 \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, g_\varphi \aftergroup\egroup\originalright)} \aftergroup\egroup\originalright) R^{- 2 - \epsilon} R^2 \aftergroup\egroup\originalright). \label{eq:virial2}
\end{aligned}
\end{equation}
As $R \rightarrow \infty$ the last term goes to zero, by definition, $\mathcal{I}_1 (R) \rightarrow m$. Using \cref{eq:c_def}, we get
\begin{align}
\mathcal{I}_2 (R) &= \sum\limits_{i = 1}^2 \ \int\limits_{\mathbb{R}^2} \chi_R (x) x_i \partial_i c_{\varphi, \varrho} (x) \: \mathrm{dA}_\varphi (x) \\
&= - \frac{1}{2 \pi} \sum\limits_{i = 1}^2 \ \iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} \chi_R (x) \varrho (x) x_i \partial_i \ln \mathopen{}\mathclose\bgroup\originalleft( |x - y| \aftergroup\egroup\originalright) \varrho (y) \: \mathrm{dA}_\varphi (y) \: \mathrm{dA}_\varphi (x) \\
&= - \frac{1}{2 \pi} \sum\limits_{i = 1}^2 \ \iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} \chi_R (x) \varrho (x) x_i \frac{x_i - y_i}{|x - y|^2} \varrho (y) \: \mathrm{dA}_\varphi (y) \: \mathrm{dA}_\varphi (x),
\end{align}
thus
\begin{align}
\lim\limits_{R \rightarrow \infty} \mathcal{I}_2 (R) &= - \frac{1}{2 \pi} \sum\limits_{i = 1}^2 \ \iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} \varrho (x) x_i \frac{x_i - y_i}{|x - y|^2} \varrho (y) \: \mathrm{dA}_\varphi (y) \: \mathrm{dA}_\varphi (x) \\
&= - \frac{1}{2 \pi} \sum\limits_{i = 1}^2 \ \iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} \varrho (x) \mathopen{}\mathclose\bgroup\originalleft( \frac{x_i - y_i}{2} + \frac{x_i + y_i}{2} \aftergroup\egroup\originalright) \frac{x_i - y_i}{|x - y|^2} \varrho (y) \: \mathrm{dA}_\varphi (y) \: \mathrm{dA}_\varphi (x) \\
&= - \frac{1}{4 \pi} \iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} \varrho (x) \varrho (y) \: \mathrm{dA}_\varphi (y) \: \mathrm{dA}_\varphi (x) + \underbrace{0}_{\mbox{due to antisymmetry}} \\
&= - \frac{m^2}{4 \pi}.
\end{align}
Finally, if we can choose a smooth $f$ so that
\begin{equation}
\Delta_\varphi f + g_\varphi \mathopen{}\mathclose\bgroup\originalleft( \operatorname{d\!}{} f, \operatorname{d\!}{} c_{\varphi, \varrho} \aftergroup\egroup\originalright) = 4 r \partial_r \varphi, \label{eq:aux_PDE}
\end{equation}
and $|\operatorname{d\!}{} f| \in L^2 \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, g_\varphi \aftergroup\egroup\originalright)$, then $\mathcal{I}_3 (R) = 0$, for all $R$. For any smooth, compactly supported function $\phi$, let
\begin{equation}
\| \phi \|_{\varphi, \varrho} \mathrel{\vcenter{\baselineskip0.5ex\lineskiplimit0pt\hbox{.}\hbox{.}}}= \sqrt{ \| \operatorname{d\!}{} \phi \|_{L^2 \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, g_\varphi \aftergroup\egroup\originalright)}^2 + \tfrac{1}{2} \| \sqrt{\varrho} \phi \|_{L^2 \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, g_\varphi \aftergroup\egroup\originalright)}^2},
\end{equation}
and let $\mathopen{}\mathclose\bgroup\originalleft( \mathcal{H}_{\varphi, \varrho}, \langle - | - \rangle_{\varphi, \varrho} \aftergroup\egroup\originalright)$ the corresponding Hilbert space. Clearly $\mathcal{H}_{\varphi, \varrho} \subseteq L_{1, \mathrm{loc}}^2$. The weak formulation of \cref{eq:aux_PDE} on $\mathcal{H}_{\varphi, \varrho}$ is
\begin{equation}
\forall \phi \in C_{\mathrm{cpt}}^\infty \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2 \aftergroup\egroup\originalright) : \quad \underbrace{\langle \operatorname{d\!}{} \phi | \operatorname{d\!}{} f \rangle_{L^2 \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, g_\varphi \aftergroup\egroup\originalright)} + \int\limits_{\mathbb{R}^2} \phi g_\varphi \mathopen{}\mathclose\bgroup\originalleft( \operatorname{d\!}{} f, \operatorname{d\!}{} c_{\varphi, \varrho} \aftergroup\egroup\originalright) \: \mathrm{dA}_\varphi}_{B \mathopen{}\mathclose\bgroup\originalleft( f, \phi \aftergroup\egroup\originalright)} = \underbrace{\int\limits_{\mathbb{R}^2} \phi r \partial_r \varphi \: \mathrm{dA}_\varphi}_{\Phi_\varphi \mathopen{}\mathclose\bgroup\originalleft( \phi \aftergroup\egroup\originalright)}.
\end{equation}
Now if $f = \phi \in C_{\mathrm{cpt}}^\infty \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2 \aftergroup\egroup\originalright)$, then
\begin{align}
B \mathopen{}\mathclose\bgroup\originalleft( \phi, \phi \aftergroup\egroup\originalright) &= \langle \operatorname{d\!}{} \phi | \operatorname{d\!}{} \phi \rangle_{L^2 \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, g_\varphi \aftergroup\egroup\originalright)} + \int\limits_{\mathbb{R}^2} \phi g_\varphi \mathopen{}\mathclose\bgroup\originalleft( \operatorname{d\!}{} \phi, \operatorname{d\!}{} c_{\varphi, \varrho} \aftergroup\egroup\originalright) \: \mathrm{dA}_\varphi \\
&= \| \operatorname{d\!}{} \phi \|_{L^2 \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, g_\varphi \aftergroup\egroup\originalright)}^2 + \frac{1}{2} \int\limits_{\mathbb{R}^2} g_\varphi \mathopen{}\mathclose\bgroup\originalleft( \operatorname{d\!}{} \phi^2, \operatorname{d\!}{} c_{\varphi, \varrho} \aftergroup\egroup\originalright) \: \mathrm{dA}_\varphi \\
&= \| \operatorname{d\!}{} \phi \|_{L^2 \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, g_\varphi \aftergroup\egroup\originalright)}^2 + \frac{1}{2} \int\limits_{\mathbb{R}^2} \phi^2 \Delta_\varphi c_{\varphi, \varrho} \: \mathrm{dA}_\varphi \\
&= \| \operatorname{d\!}{} \phi \|_{L^2 \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, g_\varphi \aftergroup\egroup\originalright)}^2 + \frac{1}{2} \int\limits_{\mathbb{R}^2} \phi^2 \varrho \: \mathrm{dA}_\varphi \\
&= \| \phi \|_{\varphi, \varrho}^2,
\end{align}
and, using that $\varphi$ has compact support and \cref{ineq:rho_bounds}, we have
\begin{align}
\mathopen{}\mathclose\bgroup\originalleft| \Phi_\varphi \mathopen{}\mathclose\bgroup\originalleft( \phi \aftergroup\egroup\originalright) \aftergroup\egroup\originalright| &= \int\limits_{\mathbb{R}^2} \phi r \partial_r \varphi \: \mathrm{dA}_\varphi \\
&= \int\limits_{\mathbb{R}^2} \mathopen{}\mathclose\bgroup\originalleft( \phi \sqrt{\varrho} \aftergroup\egroup\originalright) \mathopen{}\mathclose\bgroup\originalleft( \frac{r \partial_r \varphi}{\sqrt{\varrho}} \aftergroup\egroup\originalright) \: \mathrm{dA}_\varphi \\
&\leqslant \| \phi \sqrt{\varrho} \|_{L^2 \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, g_\varphi \aftergroup\egroup\originalright)} \sqrt{\int\limits_{\mathbb{R}^2} \tfrac{r^2 \mathopen{}\mathclose\bgroup\originalleft( \partial_r \varphi \aftergroup\egroup\originalright)^2}{\varrho} \: \mathrm{dA}_\varphi} \\
&\leqslant K (\varphi, \varrho) \| \phi \|_{\varphi, \varrho}.
\end{align}
Thus the conditions of the Lax--Milgram theorem are satisfied and hence there is a unique $f \in \mathcal{H}_{\varphi, \varrho}$ that solves \cref{eq:aux_PDE}. By elliptic regularity, $f$ is in fact smooth and by the definition $\mathcal{H}_{\varphi, \varrho}$, $|\operatorname{d\!}{} f| \in L^2 \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, g_\varphi \aftergroup\egroup\originalright)$. Hence \cref{eq:virial2} becomes $0 = 4 m - \tfrac{m^2}{2 \pi}$, which concludes the proof. \end{proof}
\section{Connection to the critical Kazdan--Warner equation on the round sphere} \label{sec:KW}
Let us assume that $\varrho \in \mathcal{C}_\mathrm{KS}$ is a solution of the static Keller--Segel \cref{eq:KS_static} and satisfies \cref{cond:hypothesis}, and thus $m = 8 \pi$. Fix $\lambda \in \mathbb{R}_+$ and $x_\star \in \mathbb{R}^2$, and let $\varrho_{\lambda, x_\star}$ as in \cref{eq:flat_solution}. Pick the unique stereographic projection $p_{\lambda, x_\star} : \mathbb{S}^2 - \{ \mbox{ North pole } \} \rightarrow \mathbb{R}^2$, so that $g_{\mathbb{S}^2} \mathrel{\vcenter{\baselineskip0.5ex\lineskiplimit0pt\hbox{.}\hbox{.}}}= \mathopen{}\mathclose\bgroup\originalleft( p_{\lambda, x_\star} \aftergroup\egroup\originalright)^* \mathopen{}\mathclose\bgroup\originalleft( \tfrac{1}{2} \varrho_{\lambda, x_\star} g_0 \aftergroup\egroup\originalright)$ is the round metric of unit radius. By \Cref{cor:rho_bounds}, the function $\widetilde{u} \mathrel{\vcenter{\baselineskip0.5ex\lineskiplimit0pt\hbox{.}\hbox{.}}}= \tfrac{1}{2} \ln \mathopen{}\mathclose\bgroup\originalleft( \tfrac{\varrho}{\varrho_{\lambda, x_\star}} \aftergroup\egroup\originalright)$ is bounded on $\mathbb{R}^2$. Let $u \mathrel{\vcenter{\baselineskip0.5ex\lineskiplimit0pt\hbox{.}\hbox{.}}}= \widetilde{u} \circ p_{\lambda, x_\star} \in L^\infty \mathopen{}\mathclose\bgroup\originalleft( \mathbb{S}^2 \aftergroup\egroup\originalright)$. Then (omitting obvious pullbacks and computations) we have \begin{align}
\Delta_{\mathbb{S}^2} u &= \frac{1}{\tfrac{1}{2} \varrho_{\lambda, x_\star}} \Delta_0 \mathopen{}\mathclose\bgroup\originalleft( \frac{1}{2} \ln \mathopen{}\mathclose\bgroup\originalleft( \frac{\varrho}{\varrho_{\lambda, x_\star}} \aftergroup\egroup\originalright) \aftergroup\egroup\originalright) \\
&= \frac{1}{\varrho_{\lambda, x_\star}} \Delta_0 \mathopen{}\mathclose\bgroup\originalleft( \mathopen{}\mathclose\bgroup\originalleft( \ln \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) - c_{\varphi, \varrho} \aftergroup\egroup\originalright) + c_{\varphi, \varrho} + \ln \mathopen{}\mathclose\bgroup\originalleft( \varrho_{\lambda, x_\star} \aftergroup\egroup\originalright) \aftergroup\egroup\originalright) \\
&= \frac{1}{\varrho_{\lambda, x_\star}} \mathopen{}\mathclose\bgroup\originalleft( 0 + e^{2 \varphi} \varrho - \varrho_{\lambda, x_\star} \aftergroup\egroup\originalright) \\
&= e^{2 \varphi} e^{2 u} - 1. \end{align} Since $\varphi$ is compactly supported, the pullback of $e^{2 \varphi}$ to $\mathbb{S}^2$ via $p_{\lambda, x_\star}$ extends smoothly over the North pole. Let us denote this extension by $h$. Then the equation on $u$ becomes \begin{equation}
\Delta_{\mathbb{S}^2} u = h e^{2 u} - 1. \label{eq:KW} \end{equation} This is the equation of Kazdan and Warner, \cite{kazdan_curvature_1974}*{Equation~(1.3)}, with $k = 1$ (note that they use the opposite sign convention for the Laplacian). When $\varphi$ vanishes identically, then $u = 0$ is a solution, which corresponds to the well-known $\varrho = \varrho_{\lambda, x_\star}$ solution on the flat plane. More generally, given any $\lambda \in \mathbb{R}_+$ and $x_\star \in \mathbb{R}^2$ and any positive scalar curvature metric $g$ on $\mathbb{S}^2$, one can construct a solutions to curved, static Keller-Segel \cref{eq:KS_static} as follows: by the uniformization theorem, $g$ and $g_{\mathbb{S}^2}$ are always conformally equivalent. Thus we have a function, $u$, that solves \cref{eq:KW} with $h$ being the scalar curvature of $g$ (pulled back under a diffeomorphism). Let now $\widetilde{u}$ and $\widetilde{h}$ be the pushforwards of $u$ and $h$, respectively, to $\mathbb{R}^2$ via $p_{\lambda, x_\star}$, and let $\varrho \mathrel{\vcenter{\baselineskip0.5ex\lineskiplimit0pt\hbox{.}\hbox{.}}}= \varrho_{\lambda, x_\star} e^{2 \widetilde{u}}$. Then $\varrho$ solves the curved, static Keller-Segel \cref{eq:KS_static} with $\varphi = \tfrac{1}{2} \ln \mathopen{}\mathclose\bgroup\originalleft( \widetilde{h} \aftergroup\egroup\originalright)$.
\begin{remark}
Using the reduced, static Keller--Segel \cref{eq:reduced_KS_static} also, equations similar to the Kazdan--Warner \cref{eq:KW} were studied in \cites{BCN17,WWY19}. These equations however are still on the plane so the geometric interpretation above is lost. \end{remark}
Unfortunately, \cref{eq:KW} is the critical version of the Kazdan--Warner equation in \cite{kazdan_curvature_1974}. Thus we cannot, in general, assume solvability for an arbitrary $h$. In fact, Kazdan and Warner found a necessary conditions for the existence of solutions: For each spherical harmonic of degree one, $u_1$, by \cite{kazdan_curvature_1974}*{Equation~(8.10)}, we have \begin{equation}
\int\limits_{\mathbb{S}^2} g_{\mathbb{S}^2} \mathopen{}\mathclose\bgroup\originalleft( \operatorname{d\!}{} u_1, \operatorname{d\!}{} h \aftergroup\egroup\originalright) e^{2 u} \omega_{\mathbb{S}^2} = 0, \label{eq:KS_obstruction} \end{equation} where $\omega_{\mathbb{S}^2}$ is the symplectic/area form of $g_{\mathbb{S}^2}$. We use \cref{eq:KS_obstruction} to prove the following:
\begin{theorem}
\label{theorem:nonexistence}
There exists $\varphi \in C_\mathrm{cpt}^\infty \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2 \aftergroup\egroup\originalright)$, arbitrarily close to the identically zero function, such that the static Keller--Segel \cref{eq:KS_static} has no solutions satisfying \cref{cond:hypothesis}. \end{theorem}
\begin{proof}
Let us assume that $\varphi$ is radial (with respect to $x_\star$). Then $h$ is only a function of the polar angle $\theta \in \mathopen{}\mathclose\bgroup\originalleft( - \tfrac{\pi}{2}, \tfrac{\pi}{2} \aftergroup\egroup\originalright)$, on $\mathbb{S}^2$. When $u_1 = \sin \mathopen{}\mathclose\bgroup\originalleft( \theta \aftergroup\egroup\originalright)$, then \cref{eq:KS_obstruction} becomes
\begin{equation}
\int\limits_{\mathbb{S}^2} \cos \mathopen{}\mathclose\bgroup\originalleft( \theta \aftergroup\egroup\originalright) \mathopen{}\mathclose\bgroup\originalleft( \partial_\theta h \aftergroup\egroup\originalright) e^{2 u} \omega_{\mathbb{S}^2} = 0. \label{eq:KW_obstraction_radial}
\end{equation}
Since $\partial_\theta h \sim e^{2 \varphi} \partial_r \varphi$, we get that if $\varphi$ is nonconstant and $\partial_r \varphi$ is either nonnegative or nonpositive, then \cref{eq:KW_obstraction_radial} cannot hold. This concludes the proof. \end{proof}
\section{The variation aspects of the Keller--Segel theory on curved planes}
We end this paper with a complementary result to \Cref{theorem:critical_mass}, showing that the energy functional (formally) corresponding to the Keller--Segel flow in \cref{eq:KS1,eq:KS2} is bounded from below only when $m = 8 \pi$. In order to do that, we first prove a curved version of the logarithmic Hardy--Littlewood--Sobolev inequality.
\subsection{Curved logarithmic Hardy--Littlewood--Sobolev inequality and the Keller--Segel free energy} \label{sec:curved_log-HLS}
Let $\lambda \in \mathbb{R}_+$ and $x_\star \in \mathbb{R}^2$, and define \begin{equation}
\mu_{\lambda, x_\star} (x) \mathrel{\vcenter{\baselineskip0.5ex\lineskiplimit0pt\hbox{.}\hbox{.}}}= \frac{\lambda^2}{\pi \mathopen{}\mathclose\bgroup\originalleft( \lambda^2 + |x - x_\star|^2 \aftergroup\egroup\originalright)^2}. \label{eq:mu_def} \end{equation} Then $\mu_{\lambda, x_\star}$ is everywhere positive, $\int_{\mathbb{R}^2} \mu_{\lambda, x_\star} \: \mathrm{dA}_0 = 1$, and for any $f \in C_{\mathrm{cpt}}^\infty \mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2 \aftergroup\egroup\originalright)$ \begin{equation}
\lim\limits_{\lambda \rightarrow 0} \ \int\limits_{\mathbb{R}^2} \mu_{\lambda, x_\star} f \: \mathrm{dA}_0 = f (x_\star). \label{eq:dirac} \end{equation} The following identities about $\mu_{\lambda, x_\star}$ are easy to verify: \begin{subequations} \begin{align}
\int\limits_{\mathbb{R}^2} m \mu_{\lambda, x_\star} \ln \mathopen{}\mathclose\bgroup\originalleft( m \mu_{\lambda, x_\star} \aftergroup\egroup\originalright) \: \mathrm{dA}_0 &= m \ln \mathopen{}\mathclose\bgroup\originalleft( \frac{m}{\pi e} \aftergroup\egroup\originalright) - 2 m \ln (\lambda), \label{eq:mu_entropy} \\
\int\limits_{\mathbb{R}^2} G (\cdot, y) \mu_{\lambda, x_\star} (y) \: \mathrm{dA}_0 (y) &= \frac{1}{8 \pi} \mathopen{}\mathclose\bgroup\originalleft( \ln \mathopen{}\mathclose\bgroup\originalleft( \mu_{\lambda, x_\star} \aftergroup\egroup\originalright) - 2 \ln (\lambda) + \ln (\pi) \aftergroup\egroup\originalright), \label{eq:G_mu} \\
\iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} \mu_{\lambda, x_\star} (x) G (x, y) \mu_{\lambda, x_\star} (y) \: \mathrm{dA}_0 (x) \: \mathrm{dA}_0 (y) &= - \frac{1}{2 \pi} \ln (\lambda) - \frac{1}{4 \pi}. \label{eq:mu_Coulomb_energy} \end{align} \end{subequations} Now we can state the \emph{logarithmic Hardy--Littlewood--Sobolev inequality} on $\mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, g_0 \aftergroup\egroup\originalright)$, which is a special case of \cite{beckner_sharp_1993}*{Theorem~2}.
\begin{theorem}
\label{theorem:classical_log-HLS}
Let $\varrho$ be an almost everywhere positive function on $\mathbb{R}^2$ and assume that
\begin{equation}
\int\limits_{\mathbb{R}^2} \varrho \: \mathrm{dA}_0 = m \in \mathbb{R}_+.
\end{equation}
Then for all $\lambda \in \mathbb{R}_+$, $x_\star \in \mathbb{R}^2$, we have
\begin{equation}
\int\limits_{\mathbb{R}^2} \varrho (x) \ln \mathopen{}\mathclose\bgroup\originalleft( \frac{\varrho (x)}{m \mu_{\lambda, x_\star} (x)} \aftergroup\egroup\originalright) \: \mathrm{dA}_0 \geqslant \frac{4 \pi}{m} \iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} (\varrho (x) - m \mu_{\lambda, x_\star} (x)) G (x, y) (\varrho (y) - m \mu_{\lambda, x_\star} (y)) \: \mathrm{dA}_0 (x) \: \mathrm{dA}_0 (y). \label[ineq]{ineq:classical_log-HLS}
\end{equation}
Moreover, equality holds exactly when $\varrho = m \mu_{\lambda, x_\star}$. \end{theorem}
\begin{proof}[Idea of the proof:]
Note that \cref{eq:mu_entropy,eq:mu_Coulomb_energy,eq:mu_entropy} imply that \cref{ineq:classical_log-HLS} is equivalent to
\begin{equation}
\int\limits_{\mathbb{R}^2} \varrho (x) \ln \mathopen{}\mathclose\bgroup\originalleft( \varrho (x) \aftergroup\egroup\originalright) \: \mathrm{dA}_0 + \frac{2}{m} \iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} \varrho (x) \ln \mathopen{}\mathclose\bgroup\originalleft( |x - y| \aftergroup\egroup\originalright) \varrho (y) \: \mathrm{dA}_0 (x) \: \mathrm{dA}_0 (y) + m \mathopen{}\mathclose\bgroup\originalleft( 1 + \ln (\pi) - \ln (m) \aftergroup\egroup\originalright) \geqslant 0. \label[ineq]{ineq:classical_log-HLS_alternative}
\end{equation}
Now \cref{ineq:classical_log-HLS_alternative} is the $n = 2$ and $f = g$ case of \cite{beckner_sharp_1993}*{inequality (27)}. \end{proof}
Let now $g$ be \emph{any} smooth Riemannian metric on $\mathbb{R}^2$, not necessarily conformally equivalent to $g_0$. There still exists a smooth function, $\varphi$, such that if the area form of $g$ is $\: \mathrm{dA}_g$, then \begin{equation}
\: \mathrm{dA}_g = e^{2 \varphi} \: \mathrm{dA}_0. \label{eq:area_form_change} \end{equation} For the remainder of this section (but this section only), let $\varphi$ be defined via \cref{eq:area_form_change}, and write, as before $\: \mathrm{dA}_\varphi \mathrel{\vcenter{\baselineskip0.5ex\lineskiplimit0pt\hbox{.}\hbox{.}}}= \: \mathrm{dA}_g$. When $g$ is not conformally equivalent to $g_0$, then $G$ is no longer the Green's function for $g$. Now let $\mu_{\lambda, x_\star}^\varphi \mathrel{\vcenter{\baselineskip0.5ex\lineskiplimit0pt\hbox{.}\hbox{.}}}= \mu_{\lambda, x_\star} e^{- 2 \varphi}$. Note that $\int_{\mathbb{R}^2} \mu_{\lambda, x_\star}^\varphi \: \mathrm{dA}_\varphi = 1$.
The next lemma is a generalization of \Cref{theorem:classical_log-HLS}.
\begin{lemma}
\label{lem:curved_log-HLS}
Let $\varrho$ be an almost everywhere positive function on $\mathbb{R}^2$ and assume that
\begin{equation}
\int\limits_{\mathbb{R}^2} \varrho \: \mathrm{dA}_\varphi = m \in \mathbb{R}_+.
\end{equation}
Then for all $\lambda \in \mathbb{R}_+$ and $x_\star \in \mathbb{R}^2$, we have
\begin{equation}
\int\limits_{\mathbb{R}^2} \varrho \ln \mathopen{}\mathclose\bgroup\originalleft( \frac{\varrho}{m \mu_{\lambda, x_\star}^\varphi} \aftergroup\egroup\originalright) \: \mathrm{dA}_\varphi \geqslant \frac{4 \pi}{m} \iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} \mathopen{}\mathclose\bgroup\originalleft( \varrho (x) - m \mu_{\lambda, x_\star}^\varphi (x) \aftergroup\egroup\originalright) G (x, y) \mathopen{}\mathclose\bgroup\originalleft( \varrho (y) - m \mu_{\lambda, x_\star}^\varphi (y) \aftergroup\egroup\originalright) \: \mathrm{dA}_\varphi (x) \: \mathrm{dA}_\varphi (y), \label[ineq]{ineq:curved_log-HLS}
\end{equation}
and equality holds exactly when $\varrho = m \mu_{\lambda, x_\star}^\varphi$. \end{lemma}
\begin{proof}
Let us first rewrite the left-hand side of \cref{ineq:curved_log-HLS}:
\begin{equation}
\int\limits_{\mathbb{R}^2} \varrho \ln \mathopen{}\mathclose\bgroup\originalleft( \frac{\varrho}{m \mu_{\lambda, x_\star}^\varphi} \aftergroup\egroup\originalright) \: \mathrm{dA}_\varphi = \int\limits_{\mathbb{R}^2} \varrho \ln \mathopen{}\mathclose\bgroup\originalleft( \frac{\varrho}{m \mu e^{- 2 \varphi}} \aftergroup\egroup\originalright) e^{2 \varphi} \: \mathrm{dA}_0 = \int\limits_{\mathbb{R}^2} \mathopen{}\mathclose\bgroup\originalleft( \varrho e^{2 \varphi} \aftergroup\egroup\originalright) \ln \mathopen{}\mathclose\bgroup\originalleft( \frac{\mathopen{}\mathclose\bgroup\originalleft( \varrho e^{2 \varphi} \aftergroup\egroup\originalright)}{m \mu} \aftergroup\egroup\originalright) \: \mathrm{dA}_0. \label[ineq]{ineq:first_ineq}
\end{equation}
Since $\varrho e^{2 \varphi}$ is almost everywhere positive and
\begin{equation}
\int\limits_{\mathbb{R}^2} \mathopen{}\mathclose\bgroup\originalleft( \varrho e^{2 \varphi} \aftergroup\egroup\originalright) \: \mathrm{dA}_0 = \int\limits_{\mathbb{R}^2} \varrho \: \mathrm{dA}_\varphi = m,
\end{equation}
we can use \cref{ineq:classical_log-HLS}, with $\varrho$ replaced by $\varrho e^{2 \varphi}$, and get
\begin{equation}
\int\limits_{\mathbb{R}^2} \mathopen{}\mathclose\bgroup\originalleft( \varrho e^{2 \varphi} \aftergroup\egroup\originalright) \ln \mathopen{}\mathclose\bgroup\originalleft( \frac{\mathopen{}\mathclose\bgroup\originalleft( \varrho e^{2 \varphi} \aftergroup\egroup\originalright)}{m \mu} \aftergroup\egroup\originalright) \: \mathrm{dA}_0 \geqslant \frac{4 \pi}{m} \iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} \mathopen{}\mathclose\bgroup\originalleft( \varrho (x) e^{2 \varphi (x)} - m \mu (x) \aftergroup\egroup\originalright) G (x, y) \mathopen{}\mathclose\bgroup\originalleft( \varrho (y) e^{2 \varphi (y)} - m \mu (y) \aftergroup\egroup\originalright) \: \mathrm{dA}_0 (x) \: \mathrm{dA}_0 (y). \label[ineq]{ineq:intermediate_ineq}
\end{equation}
Furthermore
\begin{multline}
\iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} \mathopen{}\mathclose\bgroup\originalleft( \varrho (x) e^{2 \varphi (x)} - m \mu (x) \aftergroup\egroup\originalright) G (x, y) \mathopen{}\mathclose\bgroup\originalleft( \varrho (y) e^{2 \varphi (y)} - m \mu (y) \aftergroup\egroup\originalright) \: \mathrm{dA}_0 (x) \: \mathrm{dA}_0 (y) \\
= \iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} \mathopen{}\mathclose\bgroup\originalleft( \varrho (x) - m \mu (x) e^{- 2 \varphi (x)} \aftergroup\egroup\originalright) G (x, y) \mathopen{}\mathclose\bgroup\originalleft( \varrho (y) - m \mu (y) e^{- 2 \varphi (y)} \aftergroup\egroup\originalright) \mathopen{}\mathclose\bgroup\originalleft( e^{2 \varphi (x)} \: \mathrm{dA}_0 (x) \aftergroup\egroup\originalright) \mathopen{}\mathclose\bgroup\originalleft( e^{2 \varphi (y)} \: \mathrm{dA}_0 (y) \aftergroup\egroup\originalright) \\
= \iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} \mathopen{}\mathclose\bgroup\originalleft( \varrho (x) - m \mu_{\lambda, x_\star}^\varphi (x) \aftergroup\egroup\originalright) G (x, y) \mathopen{}\mathclose\bgroup\originalleft( \varrho (y) - m \mu_{\lambda, x_\star}^\varphi (y) \aftergroup\egroup\originalright) \: \mathrm{dA}_\varphi (x) \: \mathrm{dA}_\varphi (y). \label{eq:last_eq}
\end{multline}
Combining \cref{ineq:first_ineq,ineq:intermediate_ineq,eq:last_eq} proves \cref{ineq:curved_log-HLS}. Finally, equality in \cref{ineq:intermediate_ineq} holds exactly when $\varrho e^{2 \varphi} = m \mu$, or equivalently, when $\varrho = m \mu_{\lambda, x_\star}^\varphi$, which conclude the proof. \end{proof}
\begin{remark}
As opposed to the flat case, when $\varphi$ is not identically zero, the $m = 8 \pi$ minimizer for the curved logarithmic Hardy--Littlewood--Sobolev \cref{ineq:curved_log-HLS}, $8 \pi \mu_{\lambda, x_\star}^\varphi$, is \emph{not} a solution to the static Keller--Segel \cref{eq:KS_static}, nor the reduced, static Keller--Segel \cref{eq:reduced_KS_static}. Instead, we get
\begin{equation}
\operatorname{d\!}{} \mathopen{}\mathclose\bgroup\originalleft( \ln \mathopen{}\mathclose\bgroup\originalleft( 8 \pi \mu_{\lambda, x_\star}^\varphi \aftergroup\egroup\originalright) - c_{\varphi, 8 \pi \mu_{\lambda, x_\star}^\varphi} \aftergroup\egroup\originalright) = \operatorname{d\!}{} \mathopen{}\mathclose\bgroup\originalleft( \ln \mathopen{}\mathclose\bgroup\originalleft( 8 \pi \mu_{\lambda, x_\star} \aftergroup\egroup\originalright) - 2 \varphi - c_{0, 8 \pi \mu_{\lambda, x_\star}} \aftergroup\egroup\originalright) = - 2 \operatorname{d\!}{} \varphi \not \equiv 0.
\end{equation} \end{remark}
\subsection{The Keller--Segel free energy} \label{sec:KS_fe}
The (flat) \emph{Keller--Segel free energy} of $\varrho \in \mathcal{C}_\mathrm{KS} (m, 0)$ is \begin{equation}
\mathcal{F}_0 \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) = \int\limits_{\mathbb{R}^2} \varrho \ln \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) \: \mathrm{dA}_0 - \frac{1}{2} \iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} \varrho (x) G (x, y) \varrho (y) \: \mathrm{dA}_0 (x) \: \mathrm{dA}_0 (y). \label{eq:KS_free_energy} \end{equation}
\begin{remark}
Formally, \cref{eq:KS1} is the negative gradient flow of the Keller--Segel free energy under the \emph{Wasserstein metric}. Formally this metric can be introduced as follows: If $\varrho \in \mathcal{C}_\mathrm{KS} (m, \varphi)$, then the operator $f \mapsto L_\varrho (f) \mathrel{\vcenter{\baselineskip0.5ex\lineskiplimit0pt\hbox{.}\hbox{.}}}= \operatorname{d\!}{}^* \mathopen{}\mathclose\bgroup\originalleft( \varrho \operatorname{d\!}{} f \aftergroup\egroup\originalright)$ is expected to be nondegenerate. Then if $\dot{\varrho}$ is a tangent vector to $\mathcal{C}_\mathrm{KS} (m, \varphi)$, then its Wasserstein norm is given by
\begin{equation}
\| \dot{\varrho} \|_W^2 \mathrel{\vcenter{\baselineskip0.5ex\lineskiplimit0pt\hbox{.}\hbox{.}}}= \int\limits_{\mathbb{R}^2} \dot{\varrho} L_\varrho^{- 1} \mathopen{}\mathclose\bgroup\originalleft( \dot{\varrho} \aftergroup\egroup\originalright) \: \mathrm{dA}_0.
\end{equation}
Then the Wasserstein norm is a Hilbert norm, thus can be used to define gradient flows. \end{remark}
\begin{remark}
The functional in \eqref{eq:KS_free_energy} is also the energy of self-gravitating Brownian dust; cf. \cite{CRRS04}. \end{remark}
Let us generalize $\mathcal{F}_0$ to $\mathopen{}\mathclose\bgroup\originalleft( \mathbb{R}^2, g_\varphi \aftergroup\egroup\originalright)$: For any $\varrho \in \mathcal{C}_\mathrm{KS} (m, \varphi)$, let the \emph{curved Keller--Segel free energy} be \begin{equation}
\mathcal{F}_\varphi \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) \mathrel{\vcenter{\baselineskip0.5ex\lineskiplimit0pt\hbox{.}\hbox{.}}}= \int\limits_{\mathbb{R}^2} \varrho \ln \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) \: \mathrm{dA}_\varphi - \frac{1}{2} \iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} \varrho (x) G (x, y) \varrho (y) \: \mathrm{dA}_\varphi (x) \: \mathrm{dA}_\varphi (y). \label{eq:curved_KS_free_energy} \end{equation}
Now we are ready to prove our last main result.
\begin{theorem}
\label{theorem:KS_bound}
The curved Keller--Segel free energy \eqref{eq:curved_KS_free_energy} is bounded from below on $\mathcal{C}_\mathrm{KS} (m, \varphi)$, exactly when $m = 8 \pi$. \end{theorem}
\begin{proof}
Let $m, \lambda \in \mathbb{R}_+$, and $\mu_{\lambda, 0}$ as in \cref{eq:mu_def} (with $x_\star = 0$). Now \cref{eq:mu_Coulomb_energy,eq:mu_entropy} imply that
\begin{align}
\mathcal{F}_\varphi \mathopen{}\mathclose\bgroup\originalleft( m \mu_{\lambda, x_\star} e^{- 2 \varphi} \aftergroup\egroup\originalright) &= \int\limits_{\mathbb{R}^2} m \mu_{\lambda, x_\star} e^{- 2 \varphi} \ln \mathopen{}\mathclose\bgroup\originalleft( m \mu_{\lambda, x_\star} e^{- 2 \varphi} \aftergroup\egroup\originalright) \: \mathrm{dA}_\varphi \\
& \quad - \frac{1}{2} \iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} m \mu_{\lambda, x_\star} (x) e^{- 2 \varphi (x)} G (x, y) m \mu_{\lambda, x_\star} (y) e^{- 2 \varphi (y)} \: \mathrm{dA}_\varphi (x) \: \mathrm{dA}_\varphi (y) \\
&= \int\limits_{\mathbb{R}^2} m \mu_{\lambda, x_\star} \ln \mathopen{}\mathclose\bgroup\originalleft( m \mu_{\lambda, x_\star} \aftergroup\egroup\originalright) \: \mathrm{dA}_0 - 2 m \int\limits_{\mathbb{R}^2} \mu_{\lambda, x_\star} \varphi \: \mathrm{dA}_0 \\
& \quad - \frac{m^2}{2} \iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} \mu_{\lambda, x_\star} (x) G (x, y) \mu_{\lambda, x_\star} (y) \: \mathrm{dA}_0 (x) \: \mathrm{dA}_0 (y) \\
&= \frac{m}{4 \pi} \mathopen{}\mathclose\bgroup\originalleft( m - 8 \pi \aftergroup\egroup\originalright) \ln (\lambda) + m \ln \mathopen{}\mathclose\bgroup\originalleft( \frac{m}{\pi e} \aftergroup\egroup\originalright) - 2 m \int\limits_{\mathbb{R}^2} \mu_{\lambda, x_\star} \varphi \: \mathrm{dA}_0.
\end{align}
As $\lambda \rightarrow 0^+$, the last term goes to $\varphi (x_\star)$. Thus, when $m > 8 \pi$, then
\begin{equation}
\lim\limits_{\lambda \rightarrow 0^+} \mathcal{F}_\varphi \mathopen{}\mathclose\bgroup\originalleft( m \mu_{\lambda, x_\star} e^{- 2 \varphi} \aftergroup\egroup\originalright) = - \infty.
\end{equation}
Similarly, as $\lambda \rightarrow \infty$, the last term goes to zero. Thus, when $m < 8 \pi$, then
\begin{equation}
\lim\limits_{\lambda \rightarrow \infty} \mathcal{F}_\varphi \mathopen{}\mathclose\bgroup\originalleft( m \mu_{\lambda, x_\star} e^{- 2 \varphi} \aftergroup\egroup\originalright) = - \infty.
\end{equation}
This proves the claim for $m \neq 8 \pi$.
When $m = 8 \pi$, then for any $\varrho \in \mathcal{C}_\mathrm{KS} (m, \varphi)$, we have
\begin{align}
\mathcal{F}_\varphi \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) &= \int\limits_{\mathbb{R}^2} \varrho \ln \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) \: \mathrm{dA}_\varphi - \frac{1}{2} \iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} \varrho (x) G (x, y) \varrho (y) \: \mathrm{dA}_\varphi (x) \: \mathrm{dA}_\varphi (y) \\
&= \int\limits_{\mathbb{R}^2} \varrho \ln \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) \: \mathrm{dA}_\varphi - \frac{4 \pi}{m} \iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} \varrho (x) G (x, y) \varrho (y) \: \mathrm{dA}_\varphi (x) \: \mathrm{dA}_\varphi (y) \\
&= \int\limits_{\mathbb{R}^2} \varrho \ln \mathopen{}\mathclose\bgroup\originalleft( \frac{\varrho}{m \mu_{\lambda, x_\star}^\varphi} \aftergroup\egroup\originalright) \: \mathrm{dA}_\varphi - \frac{4 \pi}{m} \iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} \mathopen{}\mathclose\bgroup\originalleft( \varrho (x) - m \mu_{\lambda, x_\star}^\varphi (x) \aftergroup\egroup\originalright) G (x, y) \mathopen{}\mathclose\bgroup\originalleft( \varrho (y) - m \mu_{\lambda, x_\star}^\varphi (y) \aftergroup\egroup\originalright) \: \mathrm{dA}_\varphi (x) \: \mathrm{dA}_\varphi (y) \\
& \quad + m \ln (m) - 2 \int\limits_{\mathbb{R}^2} \varrho \varphi \: \mathrm{dA}_\varphi + \int\limits_{\mathbb{R}^2} \varrho \ln \mathopen{}\mathclose\bgroup\originalleft( \mu_{\lambda, x_\star} \aftergroup\egroup\originalright) \: \mathrm{dA}_\varphi - 8 \pi \iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} \varrho (x) G (x, y) \mu_{\lambda, x_\star} (y) \: \mathrm{dA}_\varphi (x) \: \mathrm{dA}_0 (y) \\
& \quad + 4 \pi m^2 \iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} \mu_{\lambda, x_\star} (x) G (x, y) \mu_{\lambda, x_\star} (y) \: \mathrm{dA}_0 (x) \: \mathrm{dA}_0 (y) \\
&= \int\limits_{\mathbb{R}^2} \varrho \ln \mathopen{}\mathclose\bgroup\originalleft( \frac\varrho{m \mu_{\lambda, x_\star}^\varphi} \aftergroup\egroup\originalright) \: \mathrm{dA}_\varphi - \frac{4 \pi}{m} \iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} \mathopen{}\mathclose\bgroup\originalleft( \varrho (x) - m \mu_{\lambda, x_\star}^\varphi (x) \aftergroup\egroup\originalright) G (x, y) \mathopen{}\mathclose\bgroup\originalleft( \varrho (y) - m \mu_{\lambda, x_\star}^\varphi (y) \aftergroup\egroup\originalright) \: \mathrm{dA}_\varphi (x) \: \mathrm{dA}_\varphi (y) \\
& \quad + m \ln (m) - 2 \int\limits_{\mathbb{R}^2} \varrho \varphi \: \mathrm{dA}_\varphi + \int\limits_{\mathbb{R}^2} \varrho (x) \mathopen{}\mathclose\bgroup\originalleft( \ln \mathopen{}\mathclose\bgroup\originalleft( \mu_{\lambda, x_\star} (x) \aftergroup\egroup\originalright) - 8 \pi \int\limits_{\mathbb{R}^2} G (x, y) \mu_{\lambda, x_\star} (y) \: \mathrm{dA}_0 (y) \aftergroup\egroup\originalright) \: \mathrm{dA}_\varphi (x) \\
& \quad + 4 \pi m \iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} \mu_{\lambda, x_\star} (x) G (x, y) \mu_{\lambda, x_\star} (y) \: \mathrm{dA}_0 (x) \: \mathrm{dA}_0 (y).
\end{align}
Now, using \cref{ineq:curved_log-HLS,eq:G_mu,eq:mu_Coulomb_energy,eq:dirac}, and plugging back $m = 8 \pi$, we get
\begin{equation}
\inf \: \mathopen{}\mathclose\bgroup\originalleft( \mathopen{}\mathclose\bgroup\originalleft\{ \ \mathcal{F}_\varphi \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) \ \middle| \ \varrho \in \mathcal{C}_\mathrm{KS} (m, \varphi) \ \aftergroup\egroup\originalright\} \aftergroup\egroup\originalright) = 8 \pi \ln \mathopen{}\mathclose\bgroup\originalleft( \tfrac{8}{e} \aftergroup\egroup\originalright) - 16 \pi \sup \mathopen{}\mathclose\bgroup\originalleft( \mathopen{}\mathclose\bgroup\originalleft\{ \varphi (x) \middle| x \in \mathbb{R}^2 \aftergroup\egroup\originalright\} \aftergroup\egroup\originalright),
\end{equation}
which completes the proof. \end{proof}
\begin{remark}
It is not entirely obvious if the relevant generalization of Keller--Segel free energy \eqref{eq:KS_free_energy} is the functional, $\mathcal{F}_\varphi$, in \cref{eq:curved_KS_free_energy}. There is an generalization that is minimally coupled to the metric: Let $\kappa_\varphi \mathrel{\vcenter{\baselineskip0.5ex\lineskiplimit0pt\hbox{.}\hbox{.}}}= \Delta_\varphi \varphi$ be the Gauss curvature of $g_\varphi$ and $q \in \mathbb{R}$ be a coupling constant. Then let us define
\begin{align}
\mathcal{F}_{\varphi, q} \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) &\mathrel{\vcenter{\baselineskip0.5ex\lineskiplimit0pt\hbox{.}\hbox{.}}}= \int\limits_{\mathbb{R}^2} \varrho \ln \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) \: \mathrm{dA}_\varphi - \frac{1}{2} \iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} \varrho (x) G (x, y) \varrho (y) \: \mathrm{dA}_\varphi (x) \: \mathrm{dA}_\varphi (y) \\
& \quad + q \iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} \kappa_\varphi (x) G (x, y) \varrho (y) \: \mathrm{dA}_\varphi (x) \: \mathrm{dA}_\varphi (y). \label{eq:coupled_KS_free_energy}
\end{align}
When $m \neq 8 \pi$, the proof of \Cref{theorem:KS_bound} can still be used the prove the unboundedness of $\mathcal{F}_{\varphi, q}$, and when $m = 8 \pi$, we get
\begin{align}
\mathcal{F}_{\varphi, q} \mathopen{}\mathclose\bgroup\originalleft( \varrho \aftergroup\egroup\originalright) &\geqslant \int\limits_{\mathbb{R}^2} \varrho \ln \mathopen{}\mathclose\bgroup\originalleft( \frac\varrho{m \mu_{\lambda, x_\star}^\varphi} \aftergroup\egroup\originalright) \: \mathrm{dA}_\varphi - \frac{4 \pi}{m} \iint\limits_{\mathbb{R}^2 \times \mathbb{R}^2} \mathopen{}\mathclose\bgroup\originalleft( \varrho (x) - m \mu_{\lambda, x_\star}^\varphi (x) \aftergroup\egroup\originalright) G (x, y) \mathopen{}\mathclose\bgroup\originalleft( \varrho (y) - m \mu_{\lambda, x_\star}^\varphi (y) \aftergroup\egroup\originalright) \: \mathrm{dA}_\varphi (x) \: \mathrm{dA}_\varphi (y) \\
& \quad + (q - 2) \int\limits_{\mathbb{R}^2} \varrho \varphi \: \mathrm{dA}_\varphi \: \mathrm{dA}_\varphi + 8 \pi \ln \mathopen{}\mathclose\bgroup\originalleft( \tfrac{8}{e} \aftergroup\egroup\originalright).
\end{align}
In particular, when $q = 2$, then $\varrho = 8 \pi \mu_{\lambda, x_\star}^\varphi$ is an absolute minimizer of $\mathcal{F}_{\varphi, q}$. \end{remark}
\end{document} | arXiv |
Likelihood-ratio test
In statistics, the likelihood-ratio test assesses the goodness of fit of two competing statistical models, specifically one found by maximization over the entire parameter space and another found after imposing some constraint, based on the ratio of their likelihoods. If the constraint (i.e., the null hypothesis) is supported by the observed data, the two likelihoods should not differ by more than sampling error.[1] Thus the likelihood-ratio test tests whether this ratio is significantly different from one, or equivalently whether its natural logarithm is significantly different from zero.
The likelihood-ratio test, also known as Wilks test,[2] is the oldest of the three classical approaches to hypothesis testing, together with the Lagrange multiplier test and the Wald test.[3] In fact, the latter two can be conceptualized as approximations to the likelihood-ratio test, and are asymptotically equivalent.[4][5][6] In the case of comparing two models each of which has no unknown parameters, use of the likelihood-ratio test can be justified by the Neyman–Pearson lemma. The lemma demonstrates that the test has the highest power among all competitors.[7]
Definition
General
Suppose that we have a statistical model with parameter space $\Theta $. A null hypothesis is often stated by saying that the parameter $\theta $ is in a specified subset $\Theta _{0}$ of $\Theta $. The alternative hypothesis is thus that $\theta $ is in the complement of $\Theta _{0}$, i.e. in $\Theta ~\backslash ~\Theta _{0}$, which is denoted by $\Theta _{0}^{\text{c}}$. The likelihood ratio test statistic for the null hypothesis $H_{0}\,:\,\theta \in \Theta _{0}$ is given by:[8]
$\lambda _{\text{LR}}=-2\ln \left[{\frac {~\sup _{\theta \in \Theta _{0}}{\mathcal {L}}(\theta )~}{~\sup _{\theta \in \Theta }{\mathcal {L}}(\theta )~}}\right]$
where the quantity inside the brackets is called the likelihood ratio. Here, the $\sup $ notation refers to the supremum. As all likelihoods are positive, and as the constrained maximum cannot exceed the unconstrained maximum, the likelihood ratio is bounded between zero and one.
Often the likelihood-ratio test statistic is expressed as a difference between the log-likelihoods
$\lambda _{\text{LR}}=-2\left[~\ell (\theta _{0})-\ell ({\hat {\theta }})~\right]$
where
$\ell ({\hat {\theta }})\equiv \ln \left[~\sup _{\theta \in \Theta }{\mathcal {L}}(\theta )~\right]~$
is the logarithm of the maximized likelihood function ${\mathcal {L}}$, and $\ell (\theta _{0})$ is the maximal value in the special case that the null hypothesis is true (but not necessarily a value that maximizes ${\mathcal {L}}$ for the sampled data) and
$\theta _{0}\in \Theta _{0}\qquad {\text{ and }}\qquad {\hat {\theta }}\in \Theta ~$
denote the respective arguments of the maxima and the allowed ranges they're embedded in. Multiplying by −2 ensures mathematically that (by Wilks' theorem) $\lambda _{\text{LR}}$ converges asymptotically to being χ²-distributed if the null hypothesis happens to be true.[9] The finite sample distributions of likelihood-ratio tests are generally unknown.[10]
The likelihood-ratio test requires that the models be nested – i.e. the more complex model can be transformed into the simpler model by imposing constraints on the former's parameters. Many common test statistics are tests for nested models and can be phrased as log-likelihood ratios or approximations thereof: e.g. the Z-test, the F-test, the G-test, and Pearson's chi-squared test; for an illustration with the one-sample t-test, see below.
If the models are not nested, then instead of the likelihood-ratio test, there is a generalization of the test that can usually be used: for details, see relative likelihood.
Case of simple hypotheses
A simple-vs.-simple hypothesis test has completely specified models under both the null hypothesis and the alternative hypothesis, which for convenience are written in terms of fixed values of a notional parameter $\theta $:
${\begin{aligned}H_{0}&:&\theta =\theta _{0},\\H_{1}&:&\theta =\theta _{1}.\end{aligned}}$
In this case, under either hypothesis, the distribution of the data is fully specified: there are no unknown parameters to estimate. For this case, a variant of the likelihood-ratio test is available:[11][12]
$\Lambda (x)={\frac {~{\mathcal {L}}(\theta _{0}\mid x)~}{~{\mathcal {L}}(\theta _{1}\mid x)~}}$
Some older references may use the reciprocal of the function above as the definition.[13] Thus, the likelihood ratio is small if the alternative model is better than the null model.
The likelihood-ratio test provides the decision rule as follows:
If $~\Lambda >c~$, do not reject $H_{0}$;
If $~\Lambda <c~$, reject $H_{0}$;
If $~\Lambda =c~$, reject $H_{0}$ with probability $~q~$.
The values $c$ and $q$ are usually chosen to obtain a specified significance level $\alpha $, via the relation
$~q~$ $\operatorname {P} (\Lambda =c\mid H_{0})~+~\operatorname {P} (\Lambda <c\mid H_{0})~=~\alpha ~.$
The Neyman–Pearson lemma states that this likelihood-ratio test is the most powerful among all level $\alpha $ tests for this case.[7][12]
Interpretation
The likelihood ratio is a function of the data $x$; therefore, it is a statistic, although unusual in that the statistic's value depends on a parameter, $\theta $. The likelihood-ratio test rejects the null hypothesis if the value of this statistic is too small. How small is too small depends on the significance level of the test, i.e. on what probability of Type I error is considered tolerable (Type I errors consist of the rejection of a null hypothesis that is true).
The numerator corresponds to the likelihood of an observed outcome under the null hypothesis. The denominator corresponds to the maximum likelihood of an observed outcome, varying parameters over the whole parameter space. The numerator of this ratio is less than the denominator; so, the likelihood ratio is between 0 and 1. Low values of the likelihood ratio mean that the observed result was much less likely to occur under the null hypothesis as compared to the alternative. High values of the statistic mean that the observed outcome was nearly as likely to occur under the null hypothesis as the alternative, and so the null hypothesis cannot be rejected.
An example
The following example is adapted and abridged from Stuart, Ord & Arnold (1999, §22.2).
Suppose that we have a random sample, of size n, from a population that is normally-distributed. Both the mean, μ, and the standard deviation, σ, of the population are unknown. We want to test whether the mean is equal to a given value, μ0 .
Thus, our null hypothesis is H0: μ = μ0 and our alternative hypothesis is H1: μ ≠ μ0 . The likelihood function is
${\mathcal {L}}(\mu ,\sigma \mid x)=\left(2\pi \sigma ^{2}\right)^{-n/2}\exp \left(-\sum _{i=1}^{n}{\frac {(x_{i}-\mu )^{2}}{2\sigma ^{2}}}\right)\,.$
With some calculation (omitted here), it can then be shown that
$\lambda =\left(1+{\frac {t^{2}}{n-1}}\right)^{-n/2}$
where t is the t-statistic with n − 1 degrees of freedom. Hence we may use the known exact distribution of tn−1 to draw inferences.
Asymptotic distribution: Wilks’ theorem
Main article: Wilks' theorem
If the distribution of the likelihood ratio corresponding to a particular null and alternative hypothesis can be explicitly determined then it can directly be used to form decision regions (to sustain or reject the null hypothesis). In most cases, however, the exact distribution of the likelihood ratio corresponding to specific hypotheses is very difficult to determine.
Assuming H0 is true, there is a fundamental result by Samuel S. Wilks: As the sample size $n$ approaches $\infty $, the test statistic $\lambda _{\text{LR}}$ defined above will be asymptotically chi-squared distributed ($\chi ^{2}$) with degrees of freedom equal to the difference in dimensionality of $\Theta $ and $\Theta _{0}$.[14] This implies that for a great variety of hypotheses, we can calculate the likelihood ratio $\lambda $ for the data and then compare the observed $\lambda _{\text{LR}}$ to the $\chi ^{2}$ value corresponding to a desired statistical significance as an approximate statistical test. Other extensions exist.
See also
• Akaike information criterion
• Bayes factor
• Johansen test
• Model selection
• Vuong's closeness test
• Sup-LR test
• Error exponents in hypothesis testing
References
1. King, Gary (1989). Unifying Political Methodology : The Likelihood Theory of Statistical Inference. New York: Cambridge University Press. p. 84. ISBN 0-521-36697-6.
2. Li, Bing; Babu, G. Jogesh (2019). A Graduate Course on Statistical Inference. Springer. p. 331. ISBN 978-1-4939-9759-6.
3. Maddala, G. S.; Lahiri, Kajal (2010). Introduction to Econometrics (Fourth ed.). New York: Wiley. p. 200.
4. Buse, A. (1982). "The Likelihood Ratio, Wald, and Lagrange Multiplier Tests: An Expository Note". The American Statistician. 36 (3a): 153–157. doi:10.1080/00031305.1982.10482817.
5. Pickles, Andrew (1985). An Introduction to Likelihood Analysis. Norwich: W. H. Hutchins & Sons. pp. 24–27. ISBN 0-86094-190-6.
6. Severini, Thomas A. (2000). Likelihood Methods in Statistics. New York: Oxford University Press. pp. 120–121. ISBN 0-19-850650-3.
7. Neyman, J.; Pearson, E. S. (1933), "On the problem of the most efficient tests of statistical hypotheses" (PDF), Philosophical Transactions of the Royal Society of London A, 231 (694–706): 289–337, Bibcode:1933RSPTA.231..289N, doi:10.1098/rsta.1933.0009, JSTOR 91247
8. Koch, Karl-Rudolf (1988). Parameter Estimation and Hypothesis Testing in Linear Models. New York: Springer. p. 306. ISBN 0-387-18840-1.
9. Silvey, S.D. (1970). Statistical Inference. London: Chapman & Hall. pp. 112–114. ISBN 0-412-13820-4.
10. Mittelhammer, Ron C.; Judge, George G.; Miller, Douglas J. (2000). Econometric Foundations. New York: Cambridge University Press. p. 66. ISBN 0-521-62394-4.
11. Mood, A.M.; Graybill, F.A.; Boes, D.C. (1974). Introduction to the Theory of Statistics (3rd ed.). McGraw-Hill. §9.2.
12. Stuart, A.; Ord, K.; Arnold, S. (1999), Kendall's Advanced Theory of Statistics, vol. 2A, Arnold, §§20.10–20.13
13. Cox, D. R.; Hinkley, D. V. (1974), Theoretical Statistics, Chapman & Hall, p. 92, ISBN 0-412-12420-3
14. Wilks, S.S. (1938). "The large-sample distribution of the likelihood ratio for testing composite hypotheses". Annals of Mathematical Statistics. 9 (1): 60–62. doi:10.1214/aoms/1177732360.
Further reading
• Glover, Scott; Dixon, Peter (2004), "Likelihood ratios: A simple and flexible statistic for empirical psychologists", Psychonomic Bulletin & Review, 11 (5): 791–806, doi:10.3758/BF03196706, PMID 15732688
• Held, Leonhard; Sabanés Bové, Daniel (2014), Applied Statistical Inference—Likelihood and Bayes, Springer
• Kalbfleisch, J. G. (1985), Probability and Statistical Inference, vol. 2, Springer-Verlag
• Perlman, Michael D.; Wu, Lang (1999), "The emperor's new tests", Statistical Science, 14 (4): 355–381, doi:10.1214/ss/1009212517
• Perneger, Thomas V. (2001), "Sifting the evidence: Likelihood ratios are alternatives to P values", The BMJ, 322 (7295): 1184–5, doi:10.1136/bmj.322.7295.1184, PMC 1120301, PMID 11379590
• Pinheiro, José C.; Bates, Douglas M. (2000), Mixed-Effects Models in S and S-PLUS, Springer-Verlag, pp. 82–93
• Solomon, Daniel L. (1975), "A note on the non-equivalence of the Neyman-Pearson and generalized likelihood ratio tests for testing a simple null versus a simple alternative hypothesis" (PDF), The American Statistician, 29 (2): 101–102, doi:10.1080/00031305.1975.10477383, hdl:1813/32605
External links
• Practical application of likelihood ratio test described
• R Package: Wald's Sequential Probability Ratio Test
• Richard Lowry's Predictive Values and Likelihood Ratios Online Clinical Calculator
Statistics
• Outline
• Index
Descriptive statistics
Continuous data
Center
• Mean
• Arithmetic
• Arithmetic-Geometric
• Cubic
• Generalized/power
• Geometric
• Harmonic
• Heronian
• Heinz
• Lehmer
• Median
• Mode
Dispersion
• Average absolute deviation
• Coefficient of variation
• Interquartile range
• Percentile
• Range
• Standard deviation
• Variance
Shape
• Central limit theorem
• Moments
• Kurtosis
• L-moments
• Skewness
Count data
• Index of dispersion
Summary tables
• Contingency table
• Frequency distribution
• Grouped data
Dependence
• Partial correlation
• Pearson product-moment correlation
• Rank correlation
• Kendall's τ
• Spearman's ρ
• Scatter plot
Graphics
• Bar chart
• Biplot
• Box plot
• Control chart
• Correlogram
• Fan chart
• Forest plot
• Histogram
• Pie chart
• Q–Q plot
• Radar chart
• Run chart
• Scatter plot
• Stem-and-leaf display
• Violin plot
Data collection
Study design
• Effect size
• Missing data
• Optimal design
• Population
• Replication
• Sample size determination
• Statistic
• Statistical power
Survey methodology
• Sampling
• Cluster
• Stratified
• Opinion poll
• Questionnaire
• Standard error
Controlled experiments
• Blocking
• Factorial experiment
• Interaction
• Random assignment
• Randomized controlled trial
• Randomized experiment
• Scientific control
Adaptive designs
• Adaptive clinical trial
• Stochastic approximation
• Up-and-down designs
Observational studies
• Cohort study
• Cross-sectional study
• Natural experiment
• Quasi-experiment
Statistical inference
Statistical theory
• Population
• Statistic
• Probability distribution
• Sampling distribution
• Order statistic
• Empirical distribution
• Density estimation
• Statistical model
• Model specification
• Lp space
• Parameter
• location
• scale
• shape
• Parametric family
• Likelihood (monotone)
• Location–scale family
• Exponential family
• Completeness
• Sufficiency
• Statistical functional
• Bootstrap
• U
• V
• Optimal decision
• loss function
• Efficiency
• Statistical distance
• divergence
• Asymptotics
• Robustness
Frequentist inference
Point estimation
• Estimating equations
• Maximum likelihood
• Method of moments
• M-estimator
• Minimum distance
• Unbiased estimators
• Mean-unbiased minimum-variance
• Rao–Blackwellization
• Lehmann–Scheffé theorem
• Median unbiased
• Plug-in
Interval estimation
• Confidence interval
• Pivot
• Likelihood interval
• Prediction interval
• Tolerance interval
• Resampling
• Bootstrap
• Jackknife
Testing hypotheses
• 1- & 2-tails
• Power
• Uniformly most powerful test
• Permutation test
• Randomization test
• Multiple comparisons
Parametric tests
• Likelihood-ratio
• Score/Lagrange multiplier
• Wald
Specific tests
• Z-test (normal)
• Student's t-test
• F-test
Goodness of fit
• Chi-squared
• G-test
• Kolmogorov–Smirnov
• Anderson–Darling
• Lilliefors
• Jarque–Bera
• Normality (Shapiro–Wilk)
• Likelihood-ratio test
• Model selection
• Cross validation
• AIC
• BIC
Rank statistics
• Sign
• Sample median
• Signed rank (Wilcoxon)
• Hodges–Lehmann estimator
• Rank sum (Mann–Whitney)
• Nonparametric anova
• 1-way (Kruskal–Wallis)
• 2-way (Friedman)
• Ordered alternative (Jonckheere–Terpstra)
• Van der Waerden test
Bayesian inference
• Bayesian probability
• prior
• posterior
• Credible interval
• Bayes factor
• Bayesian estimator
• Maximum posterior estimator
• Correlation
• Regression analysis
Correlation
• Pearson product-moment
• Partial correlation
• Confounding variable
• Coefficient of determination
Regression analysis
• Errors and residuals
• Regression validation
• Mixed effects models
• Simultaneous equations models
• Multivariate adaptive regression splines (MARS)
Linear regression
• Simple linear regression
• Ordinary least squares
• General linear model
• Bayesian regression
Non-standard predictors
• Nonlinear regression
• Nonparametric
• Semiparametric
• Isotonic
• Robust
• Heteroscedasticity
• Homoscedasticity
Generalized linear model
• Exponential families
• Logistic (Bernoulli) / Binomial / Poisson regressions
Partition of variance
• Analysis of variance (ANOVA, anova)
• Analysis of covariance
• Multivariate ANOVA
• Degrees of freedom
Categorical / Multivariate / Time-series / Survival analysis
Categorical
• Cohen's kappa
• Contingency table
• Graphical model
• Log-linear model
• McNemar's test
• Cochran–Mantel–Haenszel statistics
Multivariate
• Regression
• Manova
• Principal components
• Canonical correlation
• Discriminant analysis
• Cluster analysis
• Classification
• Structural equation model
• Factor analysis
• Multivariate distributions
• Elliptical distributions
• Normal
Time-series
General
• Decomposition
• Trend
• Stationarity
• Seasonal adjustment
• Exponential smoothing
• Cointegration
• Structural break
• Granger causality
Specific tests
• Dickey–Fuller
• Johansen
• Q-statistic (Ljung–Box)
• Durbin–Watson
• Breusch–Godfrey
Time domain
• Autocorrelation (ACF)
• partial (PACF)
• Cross-correlation (XCF)
• ARMA model
• ARIMA model (Box–Jenkins)
• Autoregressive conditional heteroskedasticity (ARCH)
• Vector autoregression (VAR)
Frequency domain
• Spectral density estimation
• Fourier analysis
• Least-squares spectral analysis
• Wavelet
• Whittle likelihood
Survival
Survival function
• Kaplan–Meier estimator (product limit)
• Proportional hazards models
• Accelerated failure time (AFT) model
• First hitting time
Hazard function
• Nelson–Aalen estimator
Test
• Log-rank test
Applications
Biostatistics
• Bioinformatics
• Clinical trials / studies
• Epidemiology
• Medical statistics
Engineering statistics
• Chemometrics
• Methods engineering
• Probabilistic design
• Process / quality control
• Reliability
• System identification
Social statistics
• Actuarial science
• Census
• Crime statistics
• Demography
• Econometrics
• Jurimetrics
• National accounts
• Official statistics
• Population statistics
• Psychometrics
Spatial statistics
• Cartography
• Environmental statistics
• Geographic information system
• Geostatistics
• Kriging
• Category
• Mathematics portal
• Commons
• WikiProject
| Wikipedia |
User association for energy harvesting relay stations in cellular networks
Zhe Wang1,
Xiaodong Wang1,
Motasem Aldiab2 &
Tareq Jaber2
We consider a cellular wireless network enhanced by relay stations that are powered by renewable energy sources. Such a network consists of the macro base stations (BS), relay stations (RSs), and many mobile stations (MSs). In addition to the traditional data/voice transmission between the BS and the MSs, a higher service tier may be provided by using the energy harvesting RSs for some MSs. We propose a network scenario utilizing the energy harvesting relay stations to improve the service quality without taking the additional licensed frequency band and transmission power, and design a user association algorithm for the energy harvesting RSs in such a network. The goal is to assign each MS an RS for relaying its signal to minimize the probability of the relay service outage, i.e, the probability that an MS's relay service request is rejected. First, we propose a network scenario and develop a mathematical model to estimate the rejection probability for a given user association. We then propose a low-complexity local search algorithm, which balances the computational complexity and the performance, to obtain a locally optimal user association. Simulation results are provided to demonstrate the superior performance of the proposed techniques over the traditional methods.
Presently, broadband data services have become an increasingly significant source for mobile operators' businesses [1]. Such broadband services require much higher data throughput performance than the traditional voice services to meet the end users' quality-of-service (QoS) requirements, which bring challenges to the existing cellular wireless infrastructure [2]. One effective and low-cost solution is to employ relay stations in the existing cellular network. Moreover, with the rapid development of energy harvesting technologies, a new paradigm of wireless communications that employs energy harvesting transmitters has become a reality [3, 4]. In particular, energy harvesting relay stations are of great interest since they can improve service quality of the cellular network without taking the additional resources of the macro base station.
Typically, a relay station is a low-power and flexibly deployable base station that is able to provide complementary wireless access to the macrocell network within a short range [5]. In both the WiMAX and the long-term evolution (LTE) systems, the deployment of relay station is a cost-effective solution to improving downlink data rate as well as to extending cell coverages [2, 6]. In particular, when the relay station communicates with the nearby mobile stations using a low transmission power and only uses the non-conflict unlicensed band, it can work as a network enhancement device to provide enhanced service without taking additional resources of the macro base station, e.g., the frequency band and the transmission power. On the other hand, green communication is a new evolution trend in the telecommunication industry, which calls for the use of renewable energy, and improved energy efficiency [7]. The rapid development of the renewable energy technology makes the deployment of the energy harvesting (EH) relay station possible [8]. Utilizing the renewable energy, e.g., the energy harvested from the sun or wind, the EH relay station will not only have the flexible deployment capability but also consumes less amount of the traditional energy, leading to lower carbon emission [5, 9]. To optimally utilize the renewable energy, scheduling for the energy harvesting transmitters is also discussed [10–12].
Due to the significant difference to the traditional femto station powered by the grid and connected to the backbone network, the new EH relay station and the corresponding network operations need to be carefully designed by incorporating the energy budget due to the limited energy harvesting capability and the channel budget due to the limited capacity of the wireless macro base station-relay station link. In [13], the resource management was discussed based on a relay model from the fairness perspective; and in [14], distributed resource allocation algorithms were proposed for coexisting femto- and macrocell networks under an interference model. With the consideration of energy saving, the handover policy was studied for LTE networks with the use of femto stations [15]. However, the resource management problem under the renewable energy source becomes even more challenging under the QoS constraint, due to the dynamic nature of the energy harvesting process. In [16], the optimal energy usage was discussed for the base stations (BSs) powered by both on-grid energy and renewable energy. A cooperative system with the energy harvesting relays was analyzed in [17]. The authors of [18] studied the resource management problem for a mesh topology network with renewable energy source based on a queueing model, taking into account the relay path selection and the admission control. In [19], the energy management for a relay system was considered from the efficiency perspective under the battery constraints. In this paper, we consider a relay-assisted cellular network where the relay stations are powered solely by renewable energy sources and address the user association problem under the constraints of the energy and channel resources.
We assume that the relay station is small and light so that it can be deployed/redeployed flexibly, indoor or outdoor, according to the users' demands. Specifically, the voice service and the uplink data transmission are provided by the macro base station via the direct links between the BS and the mobile stations (MSs). For the downlink data transmission service, it can be provided either by the BS via the direct link or by the EH relay station (RS) via the relay link (i.e., BS-RS-MS). In particular, due to the channel condition, the best feasible service tier (or the best feasible transmission rate) provided by the relay link may be higher than that provided by the directly link for some MSs. In this case, when the demand tier of the MS cannot be fulfilled via the direct BS-MS link, the relay service is requested and then will be admitted whenever the RS's energy and channel resources are enough. We use the probability that a relay service request is rejected by the RS, i.e., rejection probability, to evaluate the performance of the relay service. In addition, any rejected relay service request would be redirected to the BS for the best available tier service. Moreover, we assume that the BS is powered by the grid and has enough channel and energy resources but the energy and channel resources of the EH RS are limited due to the energy harvesting and wireless BS-RS link, respectively. Also, we assume that the RS uses the non-conflict unlicensed frequency band for the RS-MS transmission and may serve multiple MSs concurrently.
In case of RS coverage overlap, the user association can be introduced to optimize the quality of the RS service, i.e., to let each MS register at certain RS to maximize the overall network service quality. Such a user association problem can be formulated as a stochastic combinational optimization problem. Solving such a problem is extremely difficult in case that the profiles of the RSs, e.g., their resource limitations and the channel conditions, are not identical. We first develop a model to describe the relationship between the user association and the rejection probability, based on a queueing model and a Markov chain model. Using this model, the original stochastic optimization problem is converted to a deterministic combinational optimization problem. To solve the problem, a local search method is proposed to balance the performance and the complexity. Specifically, the user association specifies the RS assigned to each MS for relaying service in the subsequent time slots, which is static over certain period and updated by the proposed algorithm when a new MS enters/leaves the BS's coverage area or the profile of some MS changes. Simulation results are provided to demonstrate the effectiveness of the proposed method.
The remainder of the paper is organized as follows. In Section 2, we describe the system of the hierarchical relay-assisted cellular network and propose the network scenario. In Section 3, we provide a mathematical model that describes the relationship between the user association and the service quality. In Section 4, we provide a local search method to find the near-optimal user association. Simulation results are provided in Section 5. Finally, Section 6 concludes the paper.
Network scenario
We consider a conventional cellular network consisting of BSs and MSs, where the BS is connected to the backbone network via cable or fiber and powered by the grid, and the MS may be a cellphone, a tablet, or a laptop, etc., and is powered by battery. The BS and MS can communicate with each other in duplex mode on the licensed frequency band with the proper channel-access control. We assume that the BS has enough energy and bandwidth resources.
We now introduce energy harvesting RSs as the network enhancement devices, which are solely powered by some renewable energy source (along with a buffer battery) and equipped with the high gain antennas, relaying part of the downlink data transmission. We assume that the transmissions from BS to RS, and from RS to MS, are simplex, where the BS-RS transmission uses the licensed frequency band, and the RS-BS transmission uses the non-conflict unlicensed frequency band, e.g., the available frequency band detected by cognitive radio techniques or the frequency band assigned to another cell. Since the high-gain antennas are equipped by the RS, a higher data transmission speed can be achieved for the transmission from BS to RS with the limited bandwidth and transmission power, as compared to the BS-MS link. Also, when the MS is close to the BS, the high speed transmission from the MS to the BS can also be achieved without taking any additional licensed frequency band by using a low transmission power. As a result, as compared with the BS-MS link, assisted by the EH RS, multiple MSs may be served simultaneously with the bandwidth and transmission power used for a single BS-MS link, achieving higher downlink transmission speed, i.e., improve the service quality without taking the additional resources of the BS.
We assume that each RS is associated with a fixed BS, i.e., relays the downlink transmission from the associated BS, and each MS is associated with a designated RS, i.e., receives the data from the associated RS. When a MS enters the BS's coverage area, it registers at the BS and searches the nearby RSs which is associated to the BS. For each discovered RS, if the channel condition of the RS-MS link is better than that of the BS-MS link by certain degree, we consider it as a feasible RS, i.e., by using the RS, a high service tier becomes available without using the additional licensed frequency band. In particular, for the MS, multiple RSs may be available. In order to give a proper RS-MS association, the MS needs to submit its profiles, including the channel conditions between it and different feasible RSs and the expected arrival rate of the (downlink) relay service requests, to the BS.
Note that since the RS is solely powered by the renewable energy source, its incoming energy is limited, and we assume a simple structure and operation mode for the RS. Also, due to the use of the non-conflict unlicensed frequency band, the RS can only serve the MSs in a limited range and the channel conditions may be significantly better than those of the BS-MS links. Then, we can assume that the RS provides the relay service for all associated MSs at the same and fixed service tier (transmission rate), e.g., the top tier. Moreover, due to the bottleneck of the BS-RS link and the availability of the non-conflict unlicensed frequency band, the RS may only serve a limited number of the MSs concurrently, i.e., the RS has limited channel resource for RS-MS links. Then, we assume that the interference between different RS-MS links can be neglected.
The structure of the RS-assisted cellular network is given in Fig. 1.
The structure of the RS-assisted cellular network
User association
When a new MS enters/leaves the BS's coverage area or the profile of some MS changes, the BS generates a new user association, indicating the RS assigned to each MS for relaying service in the subsequent time slots. We denote j∈{1,2,…,J} as the index of each MS registered at the BS and i∈{1,2,…,I} as the index of each RS connected to the BS. We define s=[s 1,s 2,…,s J ] as the user association where s j is the index of the assigned RS for MS j.
We assume that the channel conditions of different RS-MS pairs (i,j) may be different and denote P i,j as the discrete energy consumption level per time slot to achieve the required service tier. For each RS-MS pair, let P i denote the individual maximum energy consumption per time slots by RS i. Then, if P i,j >P i , the required service tier cannot be achieved by the RS; therefore, we consider that RS i is not feasible for MS j and cannot be assigned to RS i, i.e., s j ≠i. The set of feasible user association can then be defined as
$$ \mathcal{S}(\mathcal{P})\triangleq\left\{\boldsymbol{s}\;:\;P_{s_{j},j}<P_{s_{j}},\ (P_{s_{j},j},P_{s_{j}})\in\mathcal{P},\ j=1,2,\ldots, J\right\}\, $$
where \(\mathcal {P}\triangleq \{P_{i,j},P_{i}\;|\;i=1,2,\ldots, I,\, j=1,2,\ldots, J\}\).
Note that the user association is static over a certain number of time slots and is updated when a new MS enters/leaves the BS's coverage area or some MS's profile changes. Once the user association is updated, the BS immediately informs the MSs and RSs to follow the new user association. Although the MS may be covered by multiple RSs, each MS can only be assigned to one RS. Moreover, since the BS collects all of the information about the associated RSs and the MSs in its coverage area, including the parameters of the RS's channel and energy resources, the channel condition of each RS-MS pair, and the MS's transmission profile, we can perform the algorithms in the BS to obtain the user association and then send them to the related RSs and MSs, as shown in Fig. 2.
An illustration of the user association
MS admission control
After an MS is assigned an RS, the relay service request of the MS is then handled by the assigned RS. In particular, if there are some large data packages queueing for transmitting to the MS from the BS and the direct link cannot provide the required service quality, the relay service is requested by the MS for these packages, and then the RS decides whether the request will be admitted or rejected, based on the channel and energy resource availabilities. If the request is admitted, these packages are transmitted from the BS to the MS via the assigned RS at high service tier. Otherwise, the packages have to be transmitted through the regular BS-MS link at the best available service tier, which is lower than that of the relay service.
To start an relay service, the MS j needs to initiate an relay service request ("request" for short in the remainder of the paper) first, indicating the requested transmission duration, denoted by \({D_{j}^{k}}\), where k is the index of the time slot when the request is made. In addition, we use \({D_{j}^{k}}=0\) to indicate that no request is made in the kth time slot. The relay service request is processed by the assigned RS and an admission control decision is made instantaneously. Specifically, whenever both the uncommitted channel resource and energy resource of the RS are sufficient to fulfill the MS's relay service request, this request is admitted. Otherwise, the request is rejected.
The RS's channel resource, denoted by C i , restricts the number of concurrently served MSs. Denote the channel resource state vector for RS i by \(\boldsymbol {c}_{i} = \left [c_{i}^{1}, {c_{i}^{2}},\ldots,{c_{i}^{K}}\right ]\) where \({c_{i}^{k}}\) is the number of MSs that RS i is committed to serve at time slot k. Then, in order for RS i to admit the request \({D_{j}^{k}}>0\), we must have
$$ c_{i}^{\ell} < C_{i},\ \ell = k+1, k+2, \ldots, k+ \left\lceil {D_{j}^{k}}/T \right\rceil \, $$
where T is the duration of a time slot. If the request \({D_{j}^{k}}\) is admitted, c i needs to be updated immediately as follows:
$$ c_{i}^{\ell} \leftarrow c_{i}^{\ell} + 1,\ \ell =k+1, k+2, \ldots, k+ \left\lceil {D_{j}^{k}}/T \right\rceil. $$
The RS should also have sufficient energy to admit an MS relay service request. Assuming that the stochastic energy harvesting process is stationary within the assignment interval, e.g., K time slots, we denote \({E_{i}^{k}}\) as the amount of the energy harvested by RS i in time slot k. Note that, since the energy harvesting is a stochastic process, to guarantee the service reliability for the admitted request, we evaluate the energy availability in terms of the uncommitted energy in each slot. Specifically, upon admitting a relay service request by the RS, certain amount of energy is committed, and we denote the uncommitted energy level stored in the battery of RS i at the beginning of the kth time slot by \({B_{i}^{k}}\), where \(0\leq {B_{i}^{k}}\leq B_{i}^{\text{max}}\), and \(B_{i}^{\text{max}}\) is the maximum allowed uncommitted energy, which is mainly determined by the battery capacity. Moreover, we denote \({B_{i}^{0}}\) as the initial energy in the battery.
Another necessary condition for admitting the request \({D_{j}^{k}} >0\) by RS i is then
$$ {B_{i}^{k}} \geq P_{i,j}\left\lceil {D_{j}^{k}}/T\right\rceil. $$
If the request \({D_{j}^{k}}\) is admitted, then we update the uncommitted energy as follows:
$$ B_{i}^{k+1} = \min\left\{{B_{i}^{k}} - P_{i,j}\left\lceil {D_{j}^{k}}/T\right\rceil + {E_{i}^{k}},B_{i}^{\text{max}}\right\}. $$
Moreover, at the end of each time slot when no request is admitted, i.e., when \({D_{j}^{k}} > 0\) is rejected or \({D_{j}^{k}}=0\), \({B_{i}^{k}}\) is updated according to
$$ B_{i}^{k+1} = \min\left\{{B_{i}^{k}} + {E_{i}^{k}},B_{i}^{\text{max}}\right\}. $$
In summary, (2) and (4) constitute the necessary and sufficient conditions for RS i to admit the request \({D_{j}^{k}} > 0\) by MS j at slot k, where s j =i.
We assume that the energy harvesting \({E_{i}^{k}}\) is a discrete independent and identically distributed (i.i.d.) random variable, whose probability density function (PDF) is denoted as \(\text {Prob}\,({E_{i}^{k}}=E)\), where \(E\in \left \{0,1,\ldots,E_{i}^{\text{max}}\right \}\) is the harvested energy and E max is the maximum energy harvesting capability of RS i. We further assume that the number of relay service requests made by MS j in a time slot follows a Poisson distribution with rate λ j (requests per time slot), i.e.,
$$ \text{Prob}(m~ \textrm{requests by MS}~j~\mathrm{in a slot}) = \frac{{\lambda_{j}^{m}}}{m!}e^{-\lambda_{j}}. $$
Also, since the RSs provide the same service tier for all MSs, e.g., the top tier, we assume that the requested transmission duration \({D_{j}^{k}}\) follows an exponential distribution with the average transmission duration μ, i.e., the PDF of \({D_{j}^{k}}\) is given by
$$ f_{D}(x) = \mu e^{-\mu x}. $$
Each relay service request \({D_{j}^{k}} > 0\) by MS j is processed by the pre-assigned RS s j and is either admitted or rejected based on the channel and energy resource availabilities. Our objective is to design the user association s to minimize the average probability R(s) of a relay service request by an MS in the cellular network being rejected, where \(R(\boldsymbol {s}) \triangleq \text {Prob} ({D_{j}^{k}} >0\textrm { is rejected, }\forall j, k)\). In what follows, we will develop models that lead to an approximate expression for R(s).
Model decomposition
Define \(\mathcal {R}\) as the event that a relay service request by an MS is rejected. Further define \(\mathcal {R}_{c}\) and \(\mathcal {R}_{e}\) as the events that a relay service request is rejected due to conditions (2) and (4) that are violated, respectively. Then, we have \(\mathcal {R} = \mathcal {R}_{c} \cup \mathcal {R}_{e}\). By supposing that the channel resource or energy resource is unlimited first, we can get two independent models, the model with unlimited energy resource (UE model), where \(\mathcal {R}= \mathcal {R}_{c}\), and the model with unlimited channel resource (UC model), where \(\mathcal {R} =\mathcal {R}_{e}\).
First, we suppose that the energy resource is unlimited and we have the UE model. For a specific RS i, its probability of rejecting a relay service request can be obtained by resorting to an M/M/S/S queueing model [20, 21], where the customer arrival process follows the Poisson distribution, the service time follows the exponential distribution, there are S servers, and there is no waiting room.
Since the request arrivals of MS j follow the Poisson distribution with the arrival rate λ j , the request arrivals at each RS also follow a Poisson distribution with the arrival rate [20]
$$ \tilde \lambda_{i}(\boldsymbol{s},\alpha_{c}) = \alpha_{c} \sum_{j\in\mathcal{J}_{i}(\boldsymbol{s})}\lambda_{j}\, $$
where α c is a discount parameter (α c =1 when the energy resource is unlimited) and \(\mathcal {J}_{i}(\boldsymbol {s}) \triangleq \{j\;|\;s_{j}=i\}\) is the set of assigned MSs to RS i. Moreover, since the transmission duration follows the exponential distribution, so does the service time in this queueing model. Also, the channel resource capacity C i characterizes the number of servers.
Then, according to this queueing model, the rejection probability caused by the violation of (2) in RS i corresponds to the probability of a new service request arrival when all servers are busy and is given by [20, 21],
$$\begin{array}{*{20}l} {R_{c}^{i}}(\boldsymbol{s},\alpha_{c})= \left[\frac{\tilde\lambda_{i}(\boldsymbol{s},\alpha_{c})}{\mu}\right]^{C_{i}}/\left[C_{i}!\sum_{c=0}^{C_{i}}\frac{\tilde \lambda_{i}(\boldsymbol{s},\alpha_{c})^{c}}{\mu^{c}c!}\right]. \end{array} $$
((10))
Next, suppose that the channel resource is unlimited and we have the UC model. We use a Markov chain to model the evolution of the uncommitted energy at RS i, as shown in Fig. 3. Denote \(\mathcal {B}_{i} \triangleq \left \{0,1,\ldots,B_{i}^{\text{max}}\right \}\) as the set of the states, where each state \(m\in \mathcal {B}_{i}\) represents the uncommitted energy level in the battery. According to (5), when a request is admitted, the energy required to fulfill this request is committed. Then, after each time slot, the uncommitted energy level and therefore the state of the chain may change depending on whether a relay service request is admitted or not, based only on the state, i.e., the uncommitted energy, in the last time slot. In particular, if the energy consumption of the requested transmission is larger than the uncommitted energy in the battery, the request would be rejected. We next compute the probability of this event.
A Markov chain for modeling the dynamics of the uncommitted energy at each RS
Since the duration of the time slot T and the request arrival rate λ i are small enough, the probability that more than one relay service requests are made in a time slot is small, e.g., when T=100 ms, 1/λ=100 s/request, Prob(request=1)/Prob(request>1)>5×104, and Prob(request>1)<5×10−7. We note that, even if at most one relay service request is made in a time slot, the RS can serve multiple MSs currently since a service request corresponds to a transmission that may last several time slots. Then, in the UC model, we make the approximate assumption that at most one relay service request is made in a time slot. The probability that MS j makes a relay service request to RS s j in a time slot is then
$$ p_{j}(\boldsymbol{s},\alpha_{e})\! = \!1 - \text{Prob}(0~\textrm{request by MS}~j~\textrm{in a slot}) \,=\, 1 - e^{-\alpha_{e}\lambda_{j}} $$
where α e is a discount factor (α e =1 when the channel resource is unlimited).
Since \({D_{j}^{k}}\) follows the exponential distribution and the probability that the request duration \({D_{j}^{k}}\) is d time slots is
$$\begin{array}{*{20}l} \text{Prob}\left(\left\lceil {D_{j}^{k}}/T\right\rceil =d\right) &= \int_{(d-1)T}^{dT}\mu e^{-\mu x} dx\\ &= e^{-\mu dT}\left(e^{\mu T}-1\right). \end{array} $$
Moreover, given specific average transmission duration μ, we can find d max such that \(\text {Prob}\left ({D_{n}^{k}} > d^{\text{max}}T\right)\) is small enough. Then, we assume d max as the maximum requested transmission duration in the UC model. Specifically, the MS can only submit the request with \({D_{i}^{k}}=d^{\text{max}}T\) even if more time slots are required to complete the package transmission. Mapping the MS's request to the energy commitment, the probability that w units of energy are committed for MS j's request in a time slot is given by
$$ {\fontsize{8.9}{6} \begin{aligned} \beta_{j}(w) = \left\{ \begin{array}{ll} e^{-\mu dT}(e^{\mu}-1),& \textrm{if }w=P_{s_{j},j}d,\ d=1,2,\ldots,d^{\text{max}}-1,\\ e^{-\mu dT},& \textrm{if }w=P_{s_{j},j}d,\ d = d^{\text{max}},\\ 0,& \text{otherwise}.\\ \end{array}\right. \end{aligned}} $$
Then, we can further write the probability that w units of energy are committed by RS i in a time slot as
$$ \gamma_{i}(\boldsymbol{s},w,\alpha_{e}) = \sum_{j\in{\mathcal J}_{i}(\boldsymbol{s})} p_{j}(\boldsymbol{s},\alpha_{e}) \beta_{j}(w). $$
Define the state transition probability matrix as \(\boldsymbol {Q}_{i}(\boldsymbol {s},\alpha _{e})\triangleq \left [q^{i}_{m,n}(\boldsymbol {s},\alpha _{e})\right ]\in \left [0,1\right ]^{\left (B_{i}^{\text{max}}+1\right)\times \left (B_{i}^{\text{max}}+1\right)}\), where \(q^{i}_{m,n}(\boldsymbol {s},\alpha _{e})\) is the state transition probability for RS i. Since the state of the Markov chain represents the uncommitted energy level in the battery, its state transition follows the energy commitment process in (4)–(5), with the transition probabilities given by (14). In particular, at the end of each time slot, if the energy commitment w is such that the uncommitted energy exceeds the battery capacity, i.e., \({B_{i}^{k}} + {E_{i}^{k}} - w > B_{i}^{\text{max}}\), the chain transits to state \(B_{i}^{\text{max}}\), corresponding to energy overflow. On the other hand, if w is such that \({B_{i}^{k}} + {E_{i}^{k}} - w < 0\), the chain transits to state \(\min \left \{B_{i}^{\text{max}},{B_{i}^{k}} + {E_{i}^{k}}\right \}\), corresponding to request rejection.
Note that, for each RS, we can form a Markov chain, whose transition probability is dependent on two i.i.d. random variables, i.e., the committed energy w and the harvested energy E. Specifically, for given any pair of states m and n, \(q^{i}_{m,n}(\boldsymbol {s},\alpha _{e})\) is the sum probabilities of all possible energy combination of w and E that leads the chain to transit from state m to state n. Then, the state transition probability matrix Q i (s,α e ) can be calculated as follows.
In the above procedures, we initially set the state transition matrix as 0. For each starting state \(m=0,1,2,\ldots,B_{i}^{\text{max}} \), we calculate the ending state and the corresponding transition probability based on the MS admission control rules with all possible combinations of the realizations of the random variables. Specifically, for a specific starting state m and the realizations (w,E). We first check if the current battery level can afford the committed energy w of the current relay service request. If so, the request is admitted and then the state is transit to n following (5); otherwise, the request is rejected and then the state is transit to n following (6). Also, the state transition is associated with a probability, which is the product of γ i (s,w,α e ) and \(\text {Prob}({E_{i}^{k}} = E)\). Finally, we accumulate the probabilities for all possible (w,E) that leads the chain to transit from state m to state n.
Moreover, for a special case that the energy harvesting process is static, i.e., E i =E, the only random variable in the chain is w. Then, using the above method, we can easily give the elements of the state transition matrix in an explicit form:
$$ {\fontsize{7.7}{6} \begin{aligned} q^{i}_{m,n}(\boldsymbol{s},\alpha_{e}) = \left\{ \begin{array}{ll} \gamma_{i}(\boldsymbol{s},m-n+E,\alpha_{e}), & m - n\geq -E,\ n \neq B_{i}^{\text{max}},\ n \neq 0,\\ \sum_{w=m+E}^{W_{i}}\gamma_{i}(\boldsymbol{s},w,\alpha_{e}),& m - n\geq -E,\ n = 0,\\ \sum_{w=0}^{m+E-B_{i}^{\text{max}}}\gamma_{i}(\boldsymbol{s},w,\alpha_{e}), & m - n\geq -E,\ n =B_{i}^{\text{max}},\\ 0, & \text{otherwise}, \end{array} \right. \end{aligned}} $$
where \(W_{i} \triangleq P_{i} d^{\text{max}}\) is the maximum possible committed energy of RS i for each relay service request.
Since this finite-state Markov chain is irreducible and aperiodic, there exists a stationary distribution [22], denoted by \(\boldsymbol {\pi }_{i}(\boldsymbol {s},\alpha _{e})= \left [\pi _{0}^{i}(\boldsymbol {s},\alpha _{e}),{\pi _{1}^{i}}(\boldsymbol {s},\alpha _{e}), \ldots,\pi _{B_{i}^{\text{max}}}^{i}(\boldsymbol {s},\alpha _{e})\right ]\) that can be obtained by solving the equation π i =π i Q i . Then, given a user association s, the probability that an MS relay service request is declined by RS i due to the shortage of uncommitted energy, is given by
$$ {R_{e}^{i}}(\boldsymbol{s},\alpha_{e})=\frac{1}{\gamma_{i}(\boldsymbol{s},0,\alpha_{e})} \sum_{m=0}^{B_{i}^{\text{max}}}{\pi_{m}^{i}}(\boldsymbol{s},\alpha_{e})\sum_{w=m+1}^{w\leq W_{i} } \gamma_{i}(\boldsymbol{s},w,\alpha_{e}) \, $$
where γ i (s,0,α e ) is the probability that no relay service request is made in a time slot.
Rejection probability estimation
We find that the events \({\mathcal R}_{e}\) and \({\mathcal R}_{c}\) are highly correlated only in case that multiple energy-consuming requests are made within a short period, leading to the exhaustion of both channel and energy resources. However, if the RS has adequate resources, e.g., the channel capacity, the battery capacity and energy harvesting capability are high, we may assume that \({\mathcal R}_{e}\) and \({\mathcal R}_{c}\) are approximately independent. Indeed, in order to maintain an acceptable quality of service and system reliability, the RSs need to be equipped with sufficient resources. Then, we have
$$\begin{array}{*{20}l} R^{i}(\boldsymbol{s}) &= {R_{c}^{i}}(\boldsymbol{s},\alpha_{c}) + {R_{e}^{i}}(\boldsymbol{s},\alpha_{e}) - {R_{c}^{i}}(\boldsymbol{s},\alpha_{c}){R_{e}^{i}}(\boldsymbol{s},\alpha_{e})\, \end{array} $$
with the proper discount factors α c and α e .
When \(\mathcal {R}_{e}\) (or \(\mathcal {R}_{c}\)) occurs, a relay service request is rejected, which can be viewed as equivalent to removing a request from the relay service request arrival process, without committing the energy or channel resource. To approximate the model with \(\mathcal {R}_{e}\) and \(\mathcal {R}_{c}\), we consider such removal, which is caused by \(\mathcal {R}_{e}\) (or \(\mathcal {R}_{c}\)), as a process that randomly sampling the relay service request arrival, resulting in another Poisson process with a discounted arrival rate [20], where the discount factor is denoted by α c (or α e ). Specifically, since we assume that \(\mathcal {R}_{e}\) and \(\mathcal {R}_{c}\) are approximately independent, when the energy resource and the channel resource are both limited, i.e., a request may be rejected by \(\mathcal {R}_{e}\), \(\mathcal {R}_{c}\), or both, we may use the probability that a request is not solely rejected by \(\mathcal {R}_{e}\) (or \(\mathcal {R}_{c}\)) to characterize the discount factor α c (or α e ), i.e.,
$$ \alpha_{c} = 1 - {R_{e}^{i}}(\boldsymbol{s},\alpha_{e}) + {R_{c}^{i}}(\boldsymbol{s},\alpha_{c}) {R_{e}^{i}}(\boldsymbol{s},\alpha_{e})\, $$
$$ \alpha_{e} =1- {R_{c}^{i}}(\boldsymbol{s},\alpha_{c})+ {R_{e}^{i}}(\boldsymbol{s},\alpha_{e}){R_{c}^{i}}(\boldsymbol{s},\alpha_{c}). $$
By solving the fixed-point Eqs. (18)–(19), we can obtain the values of \({R^{i}_{c}}(\boldsymbol {s},\alpha _{c})\) and \({R^{i}_{e}}(\boldsymbol {s},\alpha _{e})\). Finally, an estimate of the probability that a relay service request is rejected is given by
$$ R(\boldsymbol{s}) = \sum_{i=1}^{I} \frac{\sum_{j\in {\mathcal J}_{i}(\boldsymbol{s})}\lambda_{j}}{\sum_{j=1}^{J}\lambda_{j}}{R}^{i}(\boldsymbol{s}). $$
Finally, we summarize the procedure for estimating the rejection probability as follows.
Computing the user association
We note that the probability that a relay service request is rejected in (20) is a function of the user association s. We would like to find the optimal user association that minimizes the rejection probability, i.e.,
$$ \boldsymbol{s}^{*} = \text{arg}\min_{\boldsymbol{s}\in{\mathcal S}(\mathcal{P})} R(\boldsymbol{s}). $$
Specifically, the user association is static over a certain number of time slots, and we solve the problem in (21) when the user association requires to be updated, e.g., a new MS enters/leaves the BS's coverage area or some MS's profile changes.
Since the problem in (21) is a deterministic combinational optimization problem, it can be solved using the exhaustive search over the set of feasible assignments \(\mathcal {S}(\mathcal {P})\). To perform the exhaustive search, we can first generate the entier set of the feasible user associations \(\mathcal {S}(\mathcal {P})\) in (1) for the given channel conditions \(\mathcal {P}\). Then, we estimate the rejection probability R(s) for each user association \(\boldsymbol {s}\in \mathcal {S}(\mathcal {P})\). Finally, the user association s ∗ with the minimum R(s) is the optimal one.
Local search algorithm
When the cardinality of the feasible set \({\mathcal S}(\mathcal{P})\) is large, the complexity of the exhaustive search becomes prohibitive. To strike a balance between the computational complexity and the performance, we propose a low-complexity local search algorithm to compute the sub-optimal user association. The basic idea is to start from a feasible assignment and, in each iteration, consider assignments that are within a distance d to the current best assignment and pick the one with the lowest rejection probability, until no improvement can be made.
We define the distance between two assignments, s 1 and s 2, denoted by D(s 1,s 2), as the Hamming distance between s 1 and s 2, i.e.,
$$ D(\boldsymbol{s}^{1},\boldsymbol{s}^{2}) \triangleq \sum_{j=1}^{J} \mathbb{I}\left({s_{j}^{1}}\neq {s_{j}^{2}}\right)\ $$
and define \(\mathcal{F}(\boldsymbol {s},d,\mathcal{P})\triangleq \{\tilde {\boldsymbol {s}}\;|\;D(\tilde {\boldsymbol {s}},\boldsymbol {s}) \leq d,\tilde {\boldsymbol {s}}\in S(\mathcal P)\}\) as the set of feasible user associations with distance to s no more than d. Starting from an initial feasible user association s (0), our proposed algorithm iteratively performs the following two steps: assignment composition that generates the local feasible set \(\mathcal{F}(\boldsymbol {s},d,\mathcal{P})\) based on the current feasible assignments and assignment selection that evaluates the rejection probabilities of the assignments in the local feasible set and selects the one with the lowest rejection probability, i.e.,
$$ \tilde{\boldsymbol{s}}^{*} = \arg\min_{\bar{\boldsymbol{s}}}\left\{R(\bar{\boldsymbol{s}})\;|\;\bar{\boldsymbol{s}}\in \mathcal{F}(\boldsymbol{s},d,\mathcal{P})\right\}\, $$
until \(R(\tilde {\boldsymbol {s}}^{*})\) no longer changes.
Specifically, in the assignment composition step, a local feasible set \(\mathcal{F}(\boldsymbol {s},d,\mathcal{P})\) needs to be generated. Denote
$$ \mathcal{U}(d)\triangleq \{U\;|\;U\subseteq\{1,2,\ldots,J\}, |U|\leq d\} $$
as the set of MS groups with no more than d MSs. For each \(U\in \mathcal {U}(d)\), we can try all possible user association changes for the MSs in U. For each change, if the resulting user association is still in \(\mathcal {S}(\mathcal {P})\), this assignment can be added to \( \mathcal {F}(\boldsymbol {s},d,\mathcal {P})\). In addition, since different local feasible sets may contain the common user associations, to avoid the repeated calculation, we can cache the result of each calculated user association in a lookup table, especially when d is large.
The local search algorithm is summarized as follows.
Note that the distance parameter d in the above algorithm controls the balance between the system performance and computational complexity. For example, when d=1, we search only among the assignments that differ from the current solution by one element. As d increases, the search space grows with improved performance. When d=J, the algorithm becomes the exhaustive search that finds the optimal assignment.
Denote \(\mathcal{P}\) as the cardinality of the set \(\mathcal{P}\), i.e., the number of elements in \(\mathcal{P}\). We define \(F_{j} \triangleq \left |\left \{P_{i,j}\;|\;P_{i,j} \leq P_{i},i=1,2,\ldots,I{\vphantom {P_{i,j}}}\right \}\right |\). We use the number of operations in calculating R(s) to measure the computational complexity. In the local search algorithm, in each iteration, \((|\mathcal{F}(\boldsymbol {s},d,\mathcal{P})|-1)\) user associations are evaluated and the number of iterations is upper bounded by \(M(|\mathcal{F}(\boldsymbol {s},d,\mathcal{P})|-1)\). On the other hand, since totally there are \(\prod _{j=1}^{J} F_{j}\) different feasible user associations, the computational complexity, i.e., the maximum number of R(s) calculations is bounded by
$$\begin{array}{*{20}l} T(d) &\leq \min\left(\prod_{j=1}^{J} F_{j},M (|\mathcal{F}(\boldsymbol{s},d,\mathcal{P})|-1)\right). \end{array} $$
Moreover, given the distance parameter d and the initial user association s, we can change the user association at most d different MSs for local search in each iteration. There are \(J \choose d\) possible MS subsets U and each can form at most \(\prod _{j\in U}F_{j}-1\) different user associations. Denoting \(F_{\text{max}}\triangleq \text{max}\left \{F_{j}\;|\;j\in {\mathcal J}\right \}\), we then have the following relaxations,
$$\begin{array}{*{20}l} \prod_{j\in U}F_{j}-1 \leq F_{\text{max}}^{d},\ \textrm{for any } U\, \end{array} $$
$$\begin{array}{*{20}l} |\mathcal{F}(\boldsymbol{s},d,\mathcal{P})| &\leq \sum_{U\in\mathcal{U}(d)}\left(\prod_{j\in u} F_{j} -1\right)\\ &\leq \sum_{U\in\mathcal{U}(d)} F_{\text{max}}^{d} \\ & \leq {J \choose d} F_{\text{max}}^{d}. \end{array} $$
Then, (25) can be further upper bounded by
$$\begin{array}{*{20}l} T(d) &\leq \min\left(\prod_{j=1}^{J} F_{j},M {J \choose d}F_{\text{max}}^{d}\right)\\ &\leq \min\left(\prod_{j=1}^{J} F_{j},MJ! F_{\text{max}}^{d}\right)\\ &\sim {\mathcal O}\left(F_{\text{max}}^{d}\right). \end{array} $$
Specifically, when we set d=1, we have \(T(1) \leq {\mathcal O}(F_{\text{max}})\), and we obtain a sub-optimal assignment. On the other hand, when we set d=J, we have \(T(J)\leq {\mathcal O}\left (F_{\text{max}}^{J}\right)\), which is the complexity of the exhaustive search. By adjusting the parameter d, we can balance the system performance and the computational complexity.
Simulation results
We consider a sub-network of the cellular system consisting of I=3 RSs and J=7 MSs with the request arrival rates λ 1=0.009,λ 2=0.008,λ 3=0.006,λ 4=0.005,λ 5=0.004,λ 6=0.003, and λ 7=0.002 requests per time slot, the average transmission duration parameter μ=0.05 requests per time slot, and the maximum requested duration is D max=250 time slots. For each RS, we assume that the battery capacity is \(B_{i}^{\text{max}}=200\) units, the initial energy level is \({B_{i}^{0}} = 100\) units, the energy harvesting capacity \({E_{i}^{k}}\) follows the truncated Gaussian distribution (between 0 and 20) with the mean of 10 and the variance of 2, and the power threshold is P i =4 units per time slot. For the channel set realizations \(\mathcal{P}=\{{P_{i}^{j}},P_{i}\;|\;i=1,2,\ldots,I,j=1,2,\ldots,J\}\), we generate them following the uniform distribution between 1 and 4 and pick up the ones with mean between [ 2.45,2.55] for simulations. In addition, we set d=1 in the proposed local search algorithm and the simulation horizon is set as 107 time slots.
For performance comparison purpose, we consider two simple user association strategies: the load-balanced strategy where the MSs are assigned to each RS (nearly) evenly regardless of the channel conditions, e.g., s=[ 1,1,1,2,2,3,3] when there are J=7 MSs; the best-channel strategy where each MS is assigned to the RS with the best channel condition. Moreover, to evaluate the accuracy of the modeling techniques developed in Section 3, we also consider the "optimal assignment" which is obtained by exhaustively search for the user association that has the lowest rejection probability, where the rejection probability is evaluated through simulations rather than based on the proposed rejection probability estimation procedure.
To evaluate the performance, we randomly generate 100 sets of channel realizations and sort the simulation results in ascending order by the optimal rejection probability. In Fig. 4, the rejection probabilities of the assignments by the four methods are plotted. It is seen that, in most cases, the rejection probability of the assignment obtained by the proposed algorithm (d=1) is the same or close to the optimal value while the rejection probabilities of the other two simple strategies are far away from the optimal one.
Performance comparisons among different user associations (J=5,C i =2)
Moreover, we compare the performances of the optimal assignment, the model-based exhaustive search, the local search algorithm (d=1), the load-balanced strategy, and the best-channel strategy, over different number of MSs, for C i =2 and C i =3, shown in Figs. 5 and 6, respectively. In this comparison, the performances are averaged over 200 sets of random channel realizations (we simulate 107 time slots for each channel realizations set). It is seen that the performance of the model-based exhaustive search almost coincides with that of the optimal assignment, while the tiny gap is caused by the approximation error of the rejection probability approximation model. We also see that the performance of the local search algorithm is only slightly worse than that of the exhaustive search. Compared to the load-balance assignment and the best channel assignment, a significant improvement is gained by using our proposed user association algorithm. Moreover, as compared in the two figures, the performances shown in Fig. 6 for the larger channel capacity C i =3 is better than that in Fig. 5 for the smaller channel capacity C i =2.
Performance comparisons among different user associations over different number of MSs (C i =2)
Recall that we use the number of the operations involved in the rejection probability estimation algorithm as the metric to measure the search complexity. Next, we compare the search complexity between our proposed algorithm and the exhaustive search. In this simulation, we consider that the first J=2,3,4,5,6,7 MS candidates are active, and we average the complexity, which is measured by the number of operations in calculating R(s), over 200 random channel realizations. It is seen from Fig. 7 that our proposed local search algorithm has a significantly lower complexity than the exhaustive search method, especially when the number of MSs is large.
Computational complexity comparison between the local search algorithm (d=1) and the exhaustive search (C i =2)
We have considered a EH relay station to improve the service quality of the cellular network without taking the additional licensed frequency band and transmission power of the macro base station, where the RSs powered by renewable energy sources relay the downlink data transmissions from the macro base station to the mobile stations in some conditions. We have formulated the user association problem, which is to assign each MS an RS to minimize the rejection probability of the requested relay service, as a combinatorial optimization problem based on a rejection probability estimation model. We proposed a local search algorithm to efficiently obtain a locally optimal assignment. Simulations have demonstrated that, even with the simplest local search (d=1), the algorithm can still provide substantial service quality and reliability improvement over conventional methods. Finally, we note that the proposed framework can also be applied to an heterogeneous network (HetNet) consisting of macrocells and picocells where the picocells are powered by energy harvesting devices.
J Weitzen, T Grosch, Comparing coverage quality for femtocell and macrocell broadband data services. IEEE Commun. Mag. 48(1), 40–44 (2010).
X Chu, Y Wu, D Lopez-Perez, X Tao, On providing downlink services in collocated spectrum-sharing macro and femto networks. IEEE Trans. Wireless Commun. 10(12), 4306–4315 (2011).
JA Paradiso, T Starner, Energy scavenging for mobile and wireless electronics. IEEE Trans. Pervasive Comput. 4:, 18–27 (2005).
S Sudevalayam, P Kulkarni, Energy harvesting sensor nodes: survey and implications. IEEE Commun. Surveys Tuts. 13(3), 443–461 (2011).
D Calin, H Claussen, H Uzunalioglu, On femto deployment architectures and macrocell offloading benefits in joint macro-femto deployments. IEEE Commun. Mag. 48(1), 26–32 (2010).
Y Tseng, C Huang, Analysis of femto base station network deployment. IEEE Trans. Veh. Technol. 61(2), 748–757 (2012).
S Yeh, in Proc. 2010 Global Mobile Congress. Green 4G communications: renewable-energy-based architectures and protocols (IEEEShanghai, P.R. China, 2010), pp. 1–5.
Z Hasan, H Boostanimehr, V Hhargava, Green cellular networks: a survey, some research issues and challenges. IEEE Commun. Surveys Tuts. 13(4), 524–540 (2011).
A Fehske, G Fettweis, J Malmodin, G Biczok, The global footprint of mobile communications: the ecological and economic perspective. IEEE Commun. Mag. 49(8), 55–62 (2011).
O Ozel, K Tutuncuoglu, J Yang, S Ulukus, A Yener, Transmission with Energy Harvesting Nodes in Fading Wireless Channels: Optimal Policies. IEEE J.Sel. Areas Commun. 29(8), 1732–1743 (2011).
J Yang, S Ulukus, Optimal packet scheduling in an energy harvesting communication system. IEEE Trans. Commun. 60(1), 220–230 (2011).
K Tutuncuoglu, A Yener, Optimum transmission policies for battery limited energy harvesting nodes. IEEE Trans. Wireless Commun. 21(4), 1117–1130 (2013).
E Liu, Q Zhang, K Leung, Relay-assisted transmission with fairness constraint for cellular networks. IEEE Trans. Mobile Comput. 11(2), 230–239 (2012).
Y Shi, A MacKenzie, in Proc. GLOBECOM 2011. Distributed algorithms for resource allocation in cellular networks with coexisting femto- and macrocells (IEEEHouston, TX, USA, 2011), pp. 1–6.
D Xenakis, N Passas, C Verikoukis, in Proc. ICC 2012. A novel handover decision policy for reducing power transmissions in the two-tier LTE network (IEEEOttawa, ON, Canada, 2012), pp. 1352–1356.
T Han, N Ansari, On optimizing green energy utilization for cellular networks with hybrid energy supplies. IEEE Trans. Wireless Commun. 12(8), 3872–3882 (2013).
B Medepally, N Mehta, Voluntary energy harvesting relays and selection in cooperative wireless networks. IEEE Trans. Wireless Commun. 9(11), 3543–3553 (2010).
L Cai, Y Liu, T Luan, X Shen, J Mark, H Poor, in Proc. GLOBECOM 2011. Adaptive resource management in sustainable energy powered wireless mesh networks (IEEEHouston, TX, USA, 2011), pp. 1–5.
W Zhang, D Duan, L Yang, Relay selection from a battery energy efficiency perspective. IEEE Trans. Commun. 59(6), 1525–1529 (2011).
S Ross, Introduction to Probability Models (Academic Press, San Diego, CA, 2000).
G Giambene, Queuing Theory and Telecommunications Networks and Applications (Springer, New York, 2005).
M Puterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming (John Wiley & Sons, New York, 1994).
Electrical Engineering Department, Columbia University, New York, 10027, NY, USA
Zhe Wang
& Xiaodong Wang
Faculty of Computing & Information Technology, University of Jeddah, Jeddah, 21589, Saudi Arabia
Motasem Aldiab
& Tareq Jaber
Search for Zhe Wang in:
Search for Xiaodong Wang in:
Search for Motasem Aldiab in:
Search for Tareq Jaber in:
Correspondence to Xiaodong Wang.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Wang, Z., Wang, X., Aldiab, M. et al. User association for energy harvesting relay stations in cellular networks. J Wireless Com Network 2015, 264 (2015) doi:10.1186/s13638-015-0489-9
Relay station
Markov chain
Queueing model | CommonCrawl |
\begin{definition}[Definition:Euclidean Metric/Riemannian Manifold]
Let $x \in \R^n$ be a point.
Let $\tuple {x_1, \ldots, x_n}$ be the standard coordinates.
Let $T_x \R^n$ be the tangent space of $\R^n$ at $x$.
Let $T_x \R^n$ be identified with $\R^n$:
:$T_x \R^n \cong \R^n$
{{Research|why this is needed?}}
Let $v, w \in T_x \R^n$ be vectors such that:
:$\ds v = \sum_{i \mathop = 1}^n v^i \valueat {\partial_i} x$
:$\ds w = \sum_{i \mathop = 1}^n w^i \valueat {\partial_i} x$
Let $g$ be a Riemannian metric such that:
:$\ds g_x = \innerprod v w_x = \sum_{i \mathop = 1}^n v^i w^i$
Then $g$ is called the '''Euclidean metric'''.
{{NamedforDef|Euclid|cat = Euclid}}
\end{definition} | ProofWiki |
•https://doi.org/10.1364/OE.390518
Performance of real-time adaptive optics compensation in a turbulent channel with high-dimensional spatial-mode encoding
Jiapeng Zhao, Yiyu Zhou, Boris Braverman, Cong Liu, Kai Pang, Nicholas K. Steinhoff, Glenn A. Tyler, Alan E. Willner, and Robert W. Boyd
Jiapeng Zhao,1,* Yiyu Zhou,1 Boris Braverman,2 Cong Liu,3 Kai Pang,3 Nicholas K. Steinhoff,4 Glenn A. Tyler,4 Alan E. Willner,3 and Robert W. Boyd1,2
1The Institute of Optics, University of Rochester, Rochester, New York 14627, USA
2Department of Physics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada
3Department of Electrical and Computer Engineering, University of Southern California, California 90007, USA
4The Optical Science Company, Anaheim, California 92806, USA
*Corresponding author: [email protected]
Jiapeng Zhao https://orcid.org/0000-0002-7851-4648
Cong Liu https://orcid.org/0000-0003-0089-0763
Robert W. Boyd https://orcid.org/0000-0002-1234-2265
Y Zhou
B Braverman
C Liu
K Pang
N Steinhoff
G Tyler
A Willner
R Boyd
Jiapeng Zhao, Yiyu Zhou, Boris Braverman, Cong Liu, Kai Pang, Nicholas K. Steinhoff, Glenn A. Tyler, Alan E. Willner, and Robert W. Boyd, "Performance of real-time adaptive optics compensation in a turbulent channel with high-dimensional spatial-mode encoding," Opt. Express 28, 15376-15391 (2020)
Mitigating the effect of atmospheric turbulence on orbital angular momentum-based quantum key...
Zhiwei Tao, et al.
Evaluation of channel capacities of OAM-based FSO link with real-time wavefront correction by...
Ming Li, et al.
Integrating deep learning to achieve phase compensation for free-space...
Xingyu Wang, et al.
Photon. Res. 9(2) B9-B17 (2021)
Optical Communications and Interconnects
Atmospheric turbulence
Optical vortices
Quantum cryptography
Turbulence effects
Wave front sensing
Original Manuscript: February 17, 2020
Revised Manuscript: April 17, 2020
Manuscript Accepted: April 18, 2020
Method and results
The orbital angular momentum (OAM) of photons is a promising degree of freedom for high-dimensional quantum key distribution (QKD). However, effectively mitigating the adverse effects of atmospheric turbulence is a persistent challenge in OAM QKD systems operating over free-space communication channels. In contrast to previous works focusing on correcting static simulated turbulence, we investigate the performance of OAM QKD in real atmospheric turbulence with real-time adaptive optics (AO) correction. We show that even though our AO system provides a limited correction, it is possible to mitigate the errors induced by weak turbulence and establish a secure channel. The crosstalk induced by turbulence and the performance of AO systems is investigated in two configurations: a lab-scale link with controllable turbulence, and a 340 m long cross-campus link with dynamic atmospheric turbulence. Our experimental results suggest that an advanced AO system with fine beam tracking, reliable beam stabilization, precise wavefront sensing, and accurate wavefront correction is necessary to adequately correct turbulence-induced error. We also propose and demonstrate different solutions to improve the performance of OAM QKD with turbulence, which could enable the possibility of OAM encoding in strong turbulence.
Quantum key distribution (QKD), which assures unconditionally secure communication between multiple parties, is one of the most promising and encouraging applications of quantum physics [1–3]. Instead of relying on mathematical complexity, the security of QKD is guaranteed by fundamental physical laws, which indicate that the encrypted keys will remain secure even against eavesdroppers with unlimited computation power [1–3].
Since its birth in 1984 [4], the concepts of QKD have been demonstrated in various platforms, including fiber-based networks [5,6], free-space communication links [7–9], underwater [10,11] and over-marine channels [12,13]. However, in most QKD systems, the information is encoded in the polarization degree of freedom, which is a two-dimensional Hilbert space limiting the information capacity to 1 bit per photon. Even through a single-photon source with a high brightness has been developed [14,15], the two-dimensional QKD systems are still photon-inefficient.
As a comparison, high-dimensional QKD systems are more photon-efficient and robust to eavesdropping [16–18]. In recent decades, many new protocols involving high-dimensional encoding have emerged. Encoding information with orbital angular momentum (OAM) states, which can span an infinite-dimensional Hilbert space, has been experimentally demonstrated to be advantageous in both high-dimensional quantum cryptography [19–23] and classical communication [24,25]. By definition, an OAM state $\vert {\ell \rangle}$ carrying $\ell \hbar$ units of OAM has $\ell$ intertwined helical wavefronts, where $\ell$ denotes the OAM quantum number and is an integer [26]. While efficient and high-fidelity fibers for high-order spatial modes are under investigation [27,28], OAM QKD in free-space links remains desirable due to the greater flexibility in applications and the lower loss. Since the information is carried by the phase profile, OAM states are vulnerable to atmospheric turbulence. Even though the behavior of OAM states in a turbulent channel has been studied both theoretically and experimentally [11,29–42], realizing high-dimensional OAM-based QKD still remains challenging.
To reduce the crosstalk induced by turbulence, most works either rely on post-selection of data or increasing the mode spacing (i.e. not using successive states for encoding) [11,21,35]. For a given free-space link, although these methods can reduce the quantum symbol error rates (QSER), they lead to a reduction of photon rate and size of encoding space. Therefore, the advantage of high-dimensional encoding on information capacity cannot be fully realized. Moreover, for an OAM encoding space with mode spacing equal to one, the OAM basis and angular (ANG) basis form the mutually unbiased bases (MUBs). The ANG basis can be described as:
(1)$$\vert{j}\rangle = \frac{1}{\sqrt{d}}\sum_{\ell = -L}^{L}\vert{\ell}\rangle\textrm{exp}(-i2\pi j\ell/(2L+1)),$$
where $L$ is the maximal OAM quantum number in use. A high-fidelity sorter for efficiently measuring these MUBs has been developed, and its effectiveness has been demonstrated in QKD systems as well [20,43,44]. However, its counterparts for mode spacing larger than one have not yet been demonstrated, and an inefficicent measurement device may introduce additional security loopholes [45]. Therefore, an efficient approach which can both take the full advantages of high-dimensional encoding and reduce the crosstalk from atmospheric turbulence is still under investigation.
Conceptually, the simplest technique for overcoming turbulence-induced errors is to use adaptive optics (AO) to correct the phase errors and recover the benefits of using the high-dimensional encoding QKD system. However, since OAM states are very sensitive to wavefront errors, any imperfect correction may actually lead to an increase rather than a decrease in QSER, and hence the failure of QKD system. Most previous experimental works focused on correcting static turbulence simulated by single or multiple random phase screens [31–34,36–38,42]. This is based on the fact that the time scale of the atmospheric turbulence, which is usually much longer than the travel time of laser pulses, can be considered as static. However, dynamically correcting realistic turbulence will lead to additional challenges which cannot be revealed in a static system. In addition, some theoretical simulations predict that a simple AO system may not be adequate for turbulence correction [46–48]. Therefore, under real dynamic turbulence, using AO systems to correct errors in OAM states remains very challenging, and the performance of AO correction in real atmospheric turbulence is still unknown and needs to be investigated.
To thoroughly study the effect of AO correction on OAM states, we investigate the performance of an OAM QKD system with real-time AO compensation in both a lab-scale and a cross-campus link. We first quantitatively investigate the performance of such a QKD system in the lab with a controllable source of turbulence. We find out that, even though AO only provides a limited correction, the quantum channel disturbed by weak turbulence can remain secure when the compensation is enabled. We then study the performance of OAM QKD in a 340 m long cross-campus link. Due to the relatively high turbulence level and modest performance of AO, we can reduce the QSER in the cross-campus link but it is still too high to guarantee the security of the channel. Based on our observation and previous simulation results, advanced AO system with fine beam tracking, reliable beam stabilization, precise wavefront sensing and accurate wavefront correction is necessary to correct the error induced by moderate or strong turbulence. In our summary, we propose three different solutions to improve the performance, and show that the performance of spatial mode QKD system can be improved if these methods are implemented.
2. Method and results
2.1 Lab-scale link under controllable turbulence
We first investigate the influence of atmospheric turbulence on OAM states and the conjugate ANG states under different levels of turbulence, and then perform real-time AO compensation on these states. The experimental setup is shown in Fig. 1. Alice prepares her states using a spatial light modulator (SLM) and a 633 nm He-Ne laser. The laser is first coupled into a single mode fiber (SMF) to generate a fundamental Gaussian state ($\ell = 0$), which is then collimated with an objective and illuminates SLM1 (SDE1024 from Cambridge Correlators Ltd). A pair of lenses ($f_1$ = 0.75 m and $f_2$ = 0.5 m) together with an iris are used to select the desired state of light, carried by the first-order diffraction from the SLM. A polarizer and a half-wave plate (HWP1) after these lenses are used to prepare four different polarization states: horizontal ($\vert {H}\rangle$), vertical ($\vert {V}\rangle$), diagonal ($\vert {D}\rangle$) and anti-diagonal ($\vert {A}\rangle$). The beacon beam, which comes from a 532 nm green laser, is collimated using an aspheric lens from a SMF. Since the atmospheric turbulence can be considered as nondispersive in visible range, the beacon beam should have similar distortion to the 633 nm beam. Therefore, using a beacon beam whose wavelength is 100 nm shorter than signal beam is acceptable [47,49–51]. The signal beam (the beam encoded by SLM1) is combined with the beacon beam through the use of a beam splitter (BS). Afterwards, both beams propagate collinearly through the turbulent channel consisting of a turbulence cell (TC) and three mirrors. The TC is a ring heater (RH) blown on by a fan. We adjust the level of turbulence by changing three parameters: the temperature of RH, the fan speed and the number of times that the beams go through the TC. The separation between BS and RH is 1.5 m while the separations between RH and the first two mirrors (M4 and M5) are both 0.15 m. For the strongest turbulence, the beams go through the TC four times and are then reflected to the deformable mirror (DM) by a fast steering mirror (FSM, OIM5002 from Optics In Motion LLC). For the weakest turbulence, RH is moved 0.3 m away from the beams so that the beams simply bypass the RH but still experience some turbulence from the edge of RH.
Fig. 1. The configuration of the lab-scale link with a controllable turbulence cell. Both signal and beacon beams go through the center of the RH. The size of the signal beam is selected to cover the central part of the DM (3$\times$3 actuators) to avoid the cutoff from the edges. The size of the beacon beam overfills the DM aperture to provide a precise estimation of turbulence across the DM. The polarization of the signal beam is controlled by a polarizer to encode information while the polarization of the beacon beam is fixed in $\vert {H}\rangle$ state.
The AO compensation system consists of two parts. The first part has a FSM and a quad-cell position detector (PD), and is used to correct the beam wander induced by the 2nd and 3rd Zernike polynomials (in Noll index, i.e. tip and tilt). To redirect part of the green beacon beam to PD, a 488 nm 50/50 non-polarizing BS (#48-217 from Edmund Optics) is used as a dichroic mirror, which leads to 7.40$\%$ loss in the signal beam. Since one set of FSM and PD is involved, only two degrees of freedom can be corrected (either x-y position or the propagation direction on the DM, i.e. the x-y momentum). In our configuration, the beam position on the DM is corrected but not the propagation direction. To minimize the tip-tilt error on DM induced by the FSM, the separation between FSM and PD is much larger than the separation between FSM and TC. That is to say, the FSM is in the near-field of turbulence while the PD is in the far field. The second part of the AO system consists of a Shack-Hartmann wavefront sensor (WFS, WFS20-7AR from Thorlabs) and a DM with 32 actuators (DM32-35-UM01 from Boston Micromachine). WFS is working in the high-speed mode with 23$\times$23 microlens in use, and the measured Zernike coefficients are limited to the first 15 terms to give the best performance. The selection of this optimal specification will be discussed later. The compensation control is performed by Thorlabs AO kit software (Version 4.40). To get the optimal wavefront measurement, the beams at DM plane are imaged onto WFS plane using two Thorlabs best-form spherical singlet lenses ($f_3$ = 0.20 m and $f_4$ = 0.15 m). Before the WFS, a 605 nm dichroic mirror (#34-740 from Edmund Optics) is used to reflect the green beacon beam but transmit the red signal beam. To reduce the noise from beacon beam [52], a laser line filter (#68-943 from Edmund Optics) at 633 nm is used to filter out the residual green light. The DM plane is then imaged onto SLM2 using another imaging system, consisting of two Thorlabs best form spherical singlet lenses ($f_5$ = 0.2 m and $f_6$ = 0.2 m), to perform projective measurements [53]. To measure the polarization degree of freedom, a polarizer and a half-wave plate (HWP2) are used after the laser line filter. HWP2 is used to rotate the polarization state ($\vert {H}\rangle$, $\vert {V}\rangle$, $\vert {D}\rangle$ or $\vert {A}\rangle$, which can be used as another degree of freedom in hybrid encoding and will be discussed later) to $\vert {H}\rangle$ since liquid crystal SLM only affects horizontally polarized light.
To quantify the level of turbulence, we first introduce the Fried parameter $r_0$ which is the spatial coherence length of atmospheric eddies (called turbules), and then use the quantity $D/r_0$ to describe the level of turbulence, where $D$ is the beam diameter because the beacon Gaussian beam underfills the collection aperture [54,55]. Therefore, the quantity $(D/r_0)^2$ denotes the number of turbules inside the beam cross section. A large $D/r_0$ indicates strong turbulence ($D/r_0>1$) while a small $D/r_0$ stands for weak turbulence ($D/r_0<1$) [56]. In our experimental setup, the OAM states with quantum number $\ell$ from $-2$ to 2 comprise our encoding space with dimension $d = 5$. The $r_0$ under different levels of turbulence are estimated from the beam wander at the receiver side by analyzing the centroid of the received Gaussian states [21]. This leads to an experimental $D/r_0$ ranging from 0.11 to 3.06, which corresponds to turbulence levels ranging from weak turbulence to strong turbulence.
We measure the crosstalk matrices between the prepared and received states under different turbulence conditions, and calculate the measured fidelity ($F$) as a function of turbulence strength $D/r_0$, which are shown in Fig. 2 [53]. With the turbulence turned off, we measure an average fidelity $F$ = 93.69$\%$ of the MUBs (the blue dot in Fig. 2(c)). The fidelity of the OAM basis and ANG basis can be found in Figs. 2(a) and (b) respectively. Each measurement takes 1.5 mins so that the total measurement time for the average fidelity of the MUBs under one specific level of turbulence is about 150 mins. The measured fidelity of the MUBs is above the fidelity threshold $F$ = 79.01$\%$ for this $d = 5$ system, which indicates that a secure quantum channel can be established [16,17]. As $D/r_0$ increases, $F$ drops quickly due to an increase in fluctuation levels, and the fidelity in the OAM basis matches well with the theoretical prediction $F = 1- [1+c(D/r_0)^2]^{-1/2}$, where coefficient $c$ is 3.404 for the no turbulence case [29]. The yellow and red curves are least square fitting results of the experimental data. Even under weak turbulence with $D/r_0$ = 0.11, the average fidelity in the ANG basis (78.04$\%$) is below the threshold. After we turn on the AO, the fidelity is improved to 80.07$\%$ in the ANG basis and the fidelity in the OAM basis is improved from 86.69$\%$ to 90.35$\%$. Therefore, a secure channel, which could not have been otherwise established, becomes possible after the AO correction is applied.
Fig. 2. Measured fidelity of lab-scale OAM QKD as functions of turbulence strength and time. (a) measured fidelity of the OAM basis. (b) measured fidelity of the ANG basis. (c) measured fidelity of the MUBs which is the average of (a) and (b). The data point with $D/r_0 = 0.01$ corresponds to the no turbulence case. All the yellow and red curves are least-square fitting results of the measured data using the same model with different coefficient $c$. All the error bars are the measured standard deviation of the fidelity distribution over the measurement time. Each error bar is the average standard deviation of all states in the corresponding basis. One example of the fidelity histogram distributions is shown in (d). The standard deviation without AO correction is 11.24$\%$. When AO is turned on, this is reduced to 8.68$\%$. As a comparison, the measurement uncertainty without turbulence is 1.46$\%$, which is mainly induced by the laser power fluctuation. (d) histogram of the fidelity of the OAM state $\ell$ = 1 with and without AO correction. $D/r_0$ is 0.884.
The modest improvement in fidelity mainly comes from the limited performance of the AO with an insufficient number of micromirrors on DM and the trade-off between speed and accuracy of the WFS. Our DM only has 32 actuators (6$\times$6 square grid without corners). To avoid being cutoff by the edge, the signal beams are aligned to fall within only the central actuators (3$\times$3 actuators for $\vert {\ell = 1\rangle}$) while the beacon beam fills the entire aperture. The inadequate number of actuators can result in a low complexity of the wavefront that can be corrected and a poor accuracy, which means that only the first few orders of Zernike terms can be corrected with limited precision.
The trade-off between correction speed and accuracy exists in most AO systems. To provide a fast wavefront measurement, our WFS is set to operate in the high-speed mode. This specific mode can provide a measurement speed up to kHz levels by sacrificing the number of lenslets used to estimate the Zernike coefficients, which indicates that a more accurate wavefront measurement leads to a slower speed of AO compensation. In our case, WFS is working at 350 Hz, which is also the speed of our AO system according to the manual of our AO kit (AOK5-UM01-Manual). To optimize the performance, one needs to first measure the bandwidth of turbulence to select an AO speed that is fast enough to sample the turbulence. This goal can be achieved by measuring the fidelity fluctuations as a function of time. For an AO system to be effective in compensating for turbulence, its speed must exceed the fluctuation rate of the atmosphere, a frequency around 60 Hz known as the Greenwood frequency [57–61]. Our AO system, with a sampling speed of 350 Hz, satisfies this condition. One also needs to consider the complexity of the Zernike terms that can be corrected by the DM. If the DM has a small number of actuators, one can limit the order of Zernikes measured by the WFS to further improve the speed. Otherwise, the number of Zernike terms measured by WFS should be large enough to avoid the waste of DM correction power. From the number of actuators on the DM, the speed of WFS, and the corresponding number of lenslets, one can determine the beam sizes on the WFS and then use a telescope to relay the field at DM to WFS. Through this procedure, the optimal combination of AO speed and the number of spots on WFS can be found, which will provide the best performance for each specific system. In our case, since the complexity of the DM (6$\times$6 actuators) limits the performance of our AO system, the number of pixels of the WFS is set to a lower level (23$\times$23 lenslets) to provide a faster sampling rate.
Another observation is that for a given AO system, the effect of AO compensation varies with the level of turbulence. For our system, the optimal performance occurs when the turbulence is moderate ($D/r_0$ = 0.884). As shown in Fig. 2(d), when the correction is enabled, the average fidelity of the OAM state $\vert {\ell = 1\rangle}$ is improved from 64.68$\%$ to 72.24$\%$, and the standard deviation is reduced from 11.24$\%$ to 8.68$\%$ as shown in Fig. 2(d). The probability of events which have an instantaneous fidelity larger than the threshold is increased by a factor of 3.48. In contrast, the improvement in either weak turbulence or strong turbulence is small. This is caused by different reasons in weak turbulence case and strong turbulence case.
When the turbulence is weak, the Zernike terms that introduce the majority of the error are tip and tilt, and the high-order terms are usually small and may be negligible compared to the WFS noise. Even though tip and tilt can in principle be easily corrected by two sets of FSM and PD, only one set is involved in our system. Therefore, the propagation direction of the beams leaving the FSM is not under control. After being imaged onto the DM plane, the error in the propagation direction should be corrected by the DM. Apart from tip and tilt, other aberrations in the weak turbulence are not so strong that thus not much error needs to be corrected. This combination of errors seems easy to be corrected. However, due to the inadequate number of actuators on the DM and the noise of the WFS, the AO may introduce some errors to the system that limits the potential fidelity improvement.
When strong or deep turbulence (extremely strong turbulence) is present, the high-order Zernike terms contribute more to the errors compared to low-order terms. Not only does the transverse phase profile get disturbed, but also the intensity profile is highly distorted. For example, when strong astigmatism (the 5th and 6th Zernike polynominals in Noll index) is present, the phase singularity at the center of OAM states $\vert {\ell }\rangle$ will get fractured into $\ell$ new singularities, and the OAM states will get stretched to elliptical shapes. Similar distortion also happens in our cross-campus link as shown in Fig. 4(a). This phenomenon is observed when $D/r_0$ equals to 1.90 and 3.06, which indicates that the turbulence is strong. In such a case, one simple AO system cannot sufficiently correct the errors in both phase and intensity profiles. Moreover, considering the inadequate number of actuators on the DM, the complexity of the wavefront that the DM can provide is not good enough to correct high-order terms. Therefore, an advanced AO system including multiple conjugate DMs and WFS with fast speed and high resolution is essential to correct strong and deep turbulence, while the exact specifications depends on the level of turbulence [48]. In contrast, for moderate turbulence, Zernike coefficients are usually large enough so that the WFS can provide a precise measurement, and the DM can also provide a relatively accurate correction. In the meantime, the wavefront complexity is not too great. Considering both effects, our AO correction has adequate performance in the moderate turbulence regime.
2.2 Free-space link across the UR campus
We next investigate the performance of AO compensation in a 340 m long free-space link across the University of Rochester (UR). The experimental setup is shown in Fig. 3. The state preparation stage is the same as the setup shown in Fig. 1. After BS, the combined beam is expanded using an achromatic 3$\times$ beam expander (GBE03-A from Thorlabs) and launched to the roof of Bausch & Lomb Hall through the use of one pair of mirrors (M4 and M5). As shown in the photograph, the hollow retroreflector (#49-672 from Edmund Optics) and the rotation stage are mounted on an optical breadboard, which is then mounted on the steel and aluminum frames on the roof. To protect the retroreflector, a double protection scheme is used. A high efficiency AR coated protection window is used to seal the front aperture of the retroreflector, which can prevent the formation of dew on the mirrors. The entire system, which is about 35 m above the ground, is also covered by an acrylic protection box (the front side of this box is replaced with a high efficiency window (#43-975 from Edmund Optics)) to protect the retroreflector and stage from weather. The reflected beams are collected by a pair of mirrors (3 inch clear aperture, M6 and M7), giving a Fresnel number product $N_f$ of the system equal to 4.89. An achromatic lens with 3 inch diameter (L3, $f_3 = 200$ mm) and a negative achromatic lens (L4, $f_4 = -40$ mm) are used to reduce the beam size. After this, the beams are sent to the AO system, which is almost the same as what is shown in Fig. 1 except that the DM has 140 actuators (12$\times$12 without corners, DM140A-35-UM01 from Boston Micromachine). The large DM has more actuators which can provide a better accuracy and complexity in the wavefront correction. To match the size of the clear aperture of DM and WFS, the beam size is reduced by several imaging systems which are not shown in the figure. All the lenses used in the imaging systems are best form spherical singlet lenses to reduce the spherical aberration.
Fig. 3. The configuration of a 340 m long cross-campus link. Both Alice and Bob are on the optical table. Since the turbulence in the cross-campus link is not controllable, the turbulence structure number varies between 5.4$\times 10^{-15}$ $\textrm {m}^{-2/3}$ and 3.2$\times 10^{-14}$ $\textrm {m}^{-2/3}$.
The turbulence structure number $C_n^2$ of our cross-campus link is estimated by calculating the beam wander of the returned $\vert {\ell }\rangle$ beam, which yields a $C_n^2$ from 5.4$\times 10^{-15}$ $\textrm {m}^{-2/3}$ to 3.2$\times 10^{-14}$ $\textrm {m}^{-2/3}$. The corresponding $D/r_0$ is from 1.03 to 2.98. This indicates a moderate to strong turbulence [56]. The intensity profiles of the prepared and received states under a turbulence strength $C_n^2 = 1.9\times 10^{-14}$ $\textrm {m}^{-2/3}$ are shown in Fig. 4(a), and the corresponding $D/r_0$ is 2.18. The intensity profiles are strongly distorted by turbulence so that the original donut shapes of the OAM states are not maintained. One can also see the effects induced by different Zernike terms from these images. For example, the first column of the received states have relatively good intensity profiles but are not at the center of receiver's aperture. These shifts are the result of tip and tilt. The received states in the 4th column show the effects induced by astigmatism. The received states are elongated into elliptical shapes, and the phase singularities are fractured into multiple vortices. In scenarios where multiple Zernike terms dominate the effect, the received states can be highly distorted leading to indistinguishable intensity profiles. For instance, as shown in the 5th column, the $\vert {\ell = 0}\rangle$ and $\vert {\ell = 1}\rangle$ states are split into 2 separate spots so that the two intensity profiles are similar to each other. To quantitatively show the statistic of such cases, we define the charge number of the received OAM states. This quantity denotes the number of phase singularities in the received states. When the state is so distorted as to break down, the charge number will become 0. After carefully analyzing all the received states, we find that only 0.11$\%$ of the received $\vert {\ell = 0}\rangle$ has a charge number more than 0. In principle, the received Gaussian state should have 0 charge number. Therefore, even though very rare under this turbulence level, Gaussian state can possibly break down into a non-Gaussian shape (as shown in the top right figure of Fig. 4(a)). For $\vert {\ell = 1}\rangle$, 22.21$\%$ of the received states has a charge number equal to 0. This number becomes 49.52$\%$ and 78.74$\%$ for $\vert {\ell = 2, 3}\rangle$ respectively. This surprising result comes from the fact that when the beam wander is strong, beams are easily cut off by the aperture due to the lack of precise beam tracking. When the cut off happens, the donut shape of OAM states cannot be maintained because of the aperture edge.
Fig. 4. (a) The prepared and received states in the cross-campus link with different OAM values. All the images acquired under same turbulence but at different timepoints during the night showing the distortions on OAM states. Note that the images of the prepared states have been scaled up by a factor of 3. To clearly show the details of the received states, the images of the received states only show the field in the collection aperture, and the intensities of the received $\vert {\ell = 2, 3}\rangle$ states are enhanced by a factor of 1.5. (b) crosstalk matrix of the OAM basis after AO correction. The average fidelity is found to be 22.57$\%$. (c) The theoretical and measured mode transmission efficiency. The error bars correspond to the standard deviation in transmission efficiencies. The effect of mode-dependent diffraction has been taken into consideration. All the data shown above are measured in turbulence with $D/r_0 = 2.18$.
The crosstalk matrix and the measured transmission efficiency are shown in Fig. 4(b) and (c). With AO compensation, the fidelity in the OAM basis is only improved from 19.74$\%$ to 22.57$\%$ which is far below the fidelity threshold (76.30$\%$ for $d = 7$ systems), and the improvement in fidelity provided by AO is modest compared to the lab-scale data with a similar turbulence level (for $D/r_0 = 1.90$, the lab-scale fidelity can be improved from 35.53$\%$ to 43.47$\%$). Meanwhile, the transmission efficiency of OAM states, especially the high order terms, also fluctuates significantly. As shown in Fig. 3(c), the theoretical efficiency of the $\vert {\ell = 3}\rangle$ state, including the effect of mode-dependent diffraction, should be 86.83$\%$. However, the measured efficiency in the cross-campus link under AO correction is only 73.95$\%$ with 7.32$\%$ standard deviation, which is 12.88$\%$ lower than the expected efficiency. As a comparison, the measured efficiency of a Gaussian state is 89.21$\%$ with little fluctuation, which is only 1.58$\%$ lower than the theoretical efficiency predicted by the simulation. Considering that we are not in the strong or deep turbulence regimes, the low link efficiency with considerable fluctuations and modest improvements with AO are mainly caused by the distorted beacon beam, finite collection aperture size and mode-dependent diffraction [53,62]. These effects are only observed in the cross-campus link since the lab-scale link has a sufficiently large Fresnel number product (in our case the $N_f > 270$ in the lab-scale link) and a more stable turbulence strength. The breakdown of the Gaussian state is usually observed in strong and deep turbulence regime but is also occasionally observed in our cross-campus link (the last photograph of the received $\vert {\ell = 0}\rangle$ state in Fig. 4(a)), which leads to the failure of precise beam tracking for low Fresnel number product channels. This will then lead to a low link efficiency and a problematic tip-tilt correction, which may introduce additional errors. This effect becomes more severe for a limited Fresnel number product of the link due to finite collection aperture and long propagation distance. In this case, a high-order OAM state arriving at the receiver's aperture will be have a much larger size than the beacon beam due to mode-dependent diffraction. In our case, due to the spherical aberration and defocus, the size of the $\vert {\ell = 3\rangle}$ states can vary from 6 cm to more than 7.62 cm, which exceeds the size of collection aperture (effective size is less than 7.62 cm). This indicates that, for a given Fried parameter $r_0$, higher order OAM states with larger cross sections are more distorted than the lower order states. Therefore, a slight mistracking caused by a distorted beacon beam will lead to the cutoff of high-order OAM state and hence a lower efficiency and a much larger error. Meanwhile, due to the mode-dependent diffraction, the size of beacon beam (usually in $\vert {\ell = 0\rangle}$ state) cannot match the size of high-order states across the whole link. This indicates that the beacon beam cannot capture all the aberrations that the high-order states experience in the link, which leads to an inefficient and inaccurate AO correction. To solve this, one may need to introduce new protocols to mitigate mode-dependent diffraction, for example as discussed in Ref. [53].
Hence, for free-space OAM QKD channels, a fine beam tracking with a reliable beam stabilization is necessary, which usually requires multiple FSMs and PDs. However, precise beam tracking is still not sufficient to correct the turbulence-induced errors since, in principle, it can only correct lowest Zernike terms (tip and tilt). As we show in the lab-scale link, high-order Zernike terms, which can only be corrected by multiple DMs and WFSs, become dominant under moderate or strong turbulence. That is to say, an advanced AO system should include two basic compositions: at least one set of beam stabilization and tracking system, and one set of advanced wavefront correction system consisting of multiple conjugate DMs and WFSs [46–48].
3. Discussion
To enhance the quality of OAM QKD, one has to mitigate the defects induced by the turbulence and improve the fidelity of the states. One widely ussed solution is increasing the mode spacing in the encoding space. As shown in Figs. 5(a)–(d), under turbulence with $D/r_0 = 1.90$ and no AO correction, we can improve the average fidelity from 35.53$\%$ to 45.72$\%$ by simply encoding information with $\vert {\ell = -4,-2,0,2,4}\rangle$ states, which corresponds to a mode spacing of 2. If AO correction is introduced, the average fidelity can be improved from 43.47 to 57.54$\%$. This improvement can be further enhanced if the mode spacing increases. Under the same turbulence, when the mode spacing becomes 4 (i.e. using $\vert {\ell = -4}\rangle$, $\vert {\ell = 0}\rangle$ and $\vert {\ell = 4}\rangle$ states), the fidelity can be improved from 71.49$\%$ to 84.33$\%$ in a $d = 3$ system, which is above the fidelity threshold ($F = 84.05\%$) and a secure channel is achievable (not shown in the figure). However, this solution has two limitations. For a fixed dimension $d$, a large mode spacing involves states with larger $\vert {\ell }\vert\rangle$, which will exacerbate the defects induced by mode-dependent diffraction [53] and suffer more turbulence due to the larger beam size. This will result in a lower data rate compared to an encoding system with the same dimensionality but consecutive states. The other limitation is the lack of an efficient sorter to measure the corresponding MUBs. Even though a generic quantum sorter for an arbitrary system has been proposed, implementing such an idea usually requires multiple phase screens, which results in a low overall efficiency and hence a lower key rate and more security loopholes [63,64]. Even though this approach can alleviate the crosstalk problem to some extent, it might be unsuited for OAM encoding unless these limitations are resolved.
Fig. 5. (a) and (b) measured crosstalk matrix of the OAM basis in the lab-scale link without and with AO correction respectively. The average fidelity in (a) is 35.53$\%$ and the fidelity in (b) is 43.47$\%$. The mode spacing is 1 in both cases. (c) and (d) measured crosstalk matrix of OAM basis in the lab-scale link without and with AO correction respectively. The average fidelity in (c) is 45.72$\%$ and the fidelity in (d) is 57.54$\%$. The mode spacing is 2 in both cases. The turbulence level in all figures is $D/r_0 = 1.90$.
Another approach is to introduce a new degree of freedom, which is robust to turbulence, as the ancillary basis. By doing so, the dimension $d$ can be increased significantly, leading to a more robust encoding system. The most straightforward idea is using polarization as the ancillary basis. The possibility of encoding information on polarization and spatial degrees of freedom has been demonstrated in Ref. [65,66], in which the authors cascade the spatial mode sorter (including radial and OAM) after HWPs and polarizing beam splitters (PBSs) to efficiently measure the received photons. Under turbulence with $D/r_0 = 1.90$, the average fidelity over polarization states is 98.23$\%$. After AO compensation, the average fidelity becomes 98.17$\%$. This indicates that the polarization degree of freedom is robust to turbulence, and will not introduce much additional error to the combined OAM-polarization MUBs. The new joint encoding system has a threshold of 73.78$\%$ for $d = 10$, which is 5.23$\%$ lower than the error threshold for a $d = 5$ system (79.01$\%$) [16]. Even though this improvement can be achieved by doubling the OAM encoding space, OAM states, especially the high order states, are not as robust as polarization states under turbulence [33,36,37]. Therefore, it is more convenient and robust to enlarge the encoding space by using polarization degree of freedom. Moreover, as shown in Fig. 2(c), the fidelity under a turbulence level $D/r_0 = 0.30$ can be improved from 70.14$\%$ to 75.63$\%$ with AO correction. Considering the new error threshold of 73.78$\%$, this improvement will allow two parties to establish a secure channel in our lab-scale link, which should have been impossible without enlarging the encoding space by introducing the polarization basis. The downside of this solution is that the improvement will become relatively small when the original $d$ is large, since the error threshold is a logarithmic function of $d$.
The last solution is to improve the performance of the entire AO correction system, which is conceptually most straightforward but also most complicated in engineering implementation. As we showed above, a simple AO system consisting of one DM, one WFS, one FSM and one PD will not be sufficient, and the performance of each device should also be improved. Note that the specifications of each device depend on the specifics of the free-space link, and the exact numbers can only be determined when the link parameters are fixed. The DM needs a large number of actuators to provide a high complexity of reconstructed wavefront, which should match the number of lenslets on the WFS and be at least complex enough to correct the dominant Zernike terms. The WFS should have an adequate number of lenslets and a high SNR for an accurate measurement. The AO system should operate at a high enough speed, which should be ten times faster than the Greenwood frequency (around 60 Hz), to comprehensively sample the time-varying wavefront and control the compensation loop [60]. The bandwidth of the FSM and PD is usually large enough and the more challenging requirement is the resolution and sensitivity. The FSM should be able to provide a sufficiently small step size but a considerable angular range so that it can accurately direct the beams toward the center of receiver's aperture even when the displacement is large. In a real free-space link, an advanced AO system with multiple devices is necessary to compensate the wavefront error introduced by atmospheric turbulence. The first section of the system should consist of two FSMs, located at the Alice's side, to point the beams at the center of collection aperture. At the Bob's side, the second section of correction system needs two FSMs and two PDs to accurately stabilize the received beams. These two sections jointly provide a precise beam tracking and control, which is essential to the advanced wavefront corrections afterwards [46–48]. An advanced wavefront correction system consisting of multiple conjugate DMs and WFSs with high resolution and large bandwidths is required to mitigate the error in intensity and phase profiles induced by moderate or stronger turbulence. Based on our experimental data and simulation results [46,47], we believe that an advanced AO correction system as described above is necessary, and in theory it should be able to correct the errors even under strong turbulence. However, the performance of such a system on OAM QKD still needs to be experimentally investigated.
Considering the complexity and cost of an advanced AO system as the one described above, it might be difficult to comprehensively build such a system. Therefore, we would suggest the following priority order under a limited budget. Based on our experimental observation, precise beam tracking is usually the first priority of the entire AO system. This conclusion comes from the fact that the OAM states are very sensitive to lateral displacement at the receiver's aperture, and such a tracking system is essential to the rest of the AO system. In addition, we believe the noise level and bandwidth are more important than the number of actuators on DM and the number of lenslets on WFS. This conclusion comes from the laboratory observation that in weak turbulence cases a pair of low-noise and high-bandwidth DM and WFS should provide better measurement and correction accuracy than our results. The bandwidth for a comprehensive correction should be ten times larger than the Greenwood frequency of the turbulence [60]. However, for strong atmospheric turbulence, both of the pairs will not work and one has to use high-resolution and bandwidth DM and WFS.
In summary, we study the performance and the possibility of OAM QKD in a turbulent channel under real-time AO correction. The effect of turbulence and the performance of the AO system are quantitatively studied in a lab-scale link under a controllable level of turbulence. We find that, under weak turbulence, real-time AO correction can mitigate the error induced by turbulence and recover the security of the channel. For moderate and strong turbulence, a simple AO system will not be adequate to mitigate the error and an advanced AO system as we described above is required. The performance of an AO system and OAM QKD in a 340 m long free-space link with finite collection aperture is also studied. In additional to the effects we observed in the laboratory measurements, additional errors are induced by the finite collection aperture and mode-dependent diffraction. These errors require one to improve the performance of AO system so that a precise beam tracking and control can be achieved. Finally, we propose three different solutions to improve the performance of OAM QKD over a free-space communication link. We experimentally validate the effectiveness of the first two of the proposed solutions in discussion session, and discuss the possible layout of an advanced AO system.
Office of Naval Research; Canada Excellence Research Chairs, Government of Canada; Natural Sciences and Engineering Research Council of Canada; Banting Postdoctoral Fellowship.
We acknowledge the helpful discussion with Mohammad Mirhosseini, Seyed Mohammad Hashemi Rafsanjani, Omar S. Maga$\tilde {\textrm {n}}$a-Loaiza and Boshen Gao. J.Z. thanks Myron W. Culver, Tony DiMino, Eric Hebert and Weichen Yao for mounting and aligning the retroreflector. B.B. acknowledges the support of the Banting Postdoctoral Fellowship.
1. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74(1), 145–195 (2002). [CrossRef]
2. V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, "The security of practical quantum key distribution," Rev. Mod. Phys. 81(3), 1301–1350 (2009). [CrossRef]
3. H.-K. Lo, M. Curty, and K. Tamaki, "Secure quantum key distribution," Nat. Photonics 8(8), 595–604 (2014). [CrossRef]
4. C. H. Bennett and G. Brassard, "Quantum cryptography: public key distribution and coin tossing, Proceedings of the IEEE International, Conference on Computers, Systems and Signal Processing pp. 175–179 (1984).
5. B. Korzh, C. C. W. Lim, R. Houlmann, N. Gisin, M. J. Li, D. Nolan, B. Sanguinetti, R. Thew, and H. Zbinden, "Provably secure and practical quantum key distribution over 307 km of optical fibre," Nat. Photonics 9(3), 163–168 (2015). [CrossRef]
6. H.-L. Yin, T.-Y. Chen, Z.-W. Yu, H. Liu, L.-X. You, Y.-H. Zhou, S.-J. Chen, Y. Mao, M.-Q. Huang, W.-J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X.-B. Wang, and J.-W. Pan, "Measurement-device-independent quantum key distribution over a 404 km optical fiber," Phys. Rev. Lett. 117(19), 190501 (2016). [CrossRef]
7. J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, and J.-W. Pan, "Direct and full-scale experimental verifications towards ground–satellite quantum key distribution," Nat. Photonics 7(5), 387–393 (2013). [CrossRef]
8. S.-K. Liao, W.-Q. Cai, W.-Y. Liu, L. Zhang, Y. Li, J.-G. Ren, J. Yin, Q. Shen, Y. Cao, Z.-P. Li, F.-Z. Li, X.-W. Chen, L.-H. Sun, J.-J. Jia, J.-C. Wu, X.-J. Jiang, J.-F. Wang, Y.-M. Huang, Q. Wang, Y.-L. Zhou, L. Deng, T. Xi, L. Ma, T. Hu, Q. Zhang, Y.-A. Chen, N.-L. Liu, X.-B. Wang, Z.-C. Zhu, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, "Satellite-to-ground quantum key distribution," Nature 549(7670), 43–47 (2017). [CrossRef]
9. J. Jin, J.-P. Bourgoin, R. Tannous, S. Agne, C. J. Pugh, K. B. Kuntz, B. L. Higgins, and T. Jennewein, "Genuine time-bin-encoded quantum key distribution over a turbulent depolarizing free-space channel," Opt. Express 27(26), 37214–37223 (2019). [CrossRef]
10. L. Ji, J. Gao, A.-L. Yang, Z. Feng, X.-F. Lin, Z.-G. Li, and X.-M. Jin, "Towards quantum communications in free-space seawater," Opt. Express 25(17), 19795–19806 (2017). [CrossRef]
11. F. Bouchard, A. Sit, F. Hufnagel, A. Abbas, Y. Zhang, K. Heshami, R. Fickler, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, "Quantum cryptography with twisted photons through an outdoor underwater channel," Opt. Express 26(17), 22563–22573 (2018). [CrossRef]
12. R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Amer, M. Furst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, "Entanglement-based quantum communication over 144 km," Nat. Phys. 3(7), 481–486 (2007). [CrossRef]
13. X. Sun, I. B. Djordjevic, and M. A. Neifeld, "Multiple spatial modes based qkd over marine free-space optical channels in the presence of atmospheric turbulence," Opt. Express 24(24), 27663–27673 (2016). [CrossRef]
14. A. Dousse, J. Suffczyński, A. Beveratos, O. Krebs, A. Lemaître, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, "Ultrabright source of entangled photon pairs," Nature 466(7303), 217–220 (2010). [CrossRef]
15. F. Steinlechner, P. Trojek, M. Jofre, H. Weier, D. Perez, T. Jennewein, R. Ursin, J. Rarity, M. W. Mitchell, J. P. Torres, H. Weinfurter, and V. Pruneri, "A high-brightness source of polarization-entangled photons optimized for applications in free space," Opt. Express 20(9), 9640–9649 (2012). [CrossRef]
16. N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, "Security of quantum key distribution using d-level systems," Phys. Rev. Lett. 88(12), 127902 (2002). [CrossRef]
17. L. Sheridan and V. Scarani, "Security proof for quantum key distribution using qudit systems," Phys. Rev. A 82(3), 030301 (2010). [CrossRef]
18. K. Brádler, M. Mirhosseini, R. Fickler, A. Broadbent, and R. W. Boyd, "Finite-key security analysis for multilevel quantum key distribution," New J. Phys. 18(7), 073030 (2016). [CrossRef]
19. G. Vallone, V. D'Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, "Free-space quantum key distribution by rotation-invariant twisted photons," Phys. Rev. Lett. 113(6), 060503 (2014). [CrossRef]
20. M. Mirhosseini, O. S. Magana-Loaiza, M. N. O'Sullivan, B. Rodenburg, M. Malik, M. P. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, "High-dimensional quantum cryptography with twisted light," New J. Phys. 17(3), 033033 (2015). [CrossRef]
21. A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Günthner, B. Heim, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, "High-dimensional intracity quantum cryptography with structured photons," Optica 4(9), 1006–1010 (2017). [CrossRef]
22. F. Bouchard, K. Heshami, D. England, R. Fickler, R. W. Boyd, B.-G. Englert, L. L. Sánchez-Soto, and E. Karimi, "Experimental investigation of high-dimensional quantum key distribution protocols with twisted photons," Quantum 2, 111111 (2018). [CrossRef]
23. F. Bouchard, A. Sit, K. Heshami, R. Fickler, and E. Karimi, "Round-robin differential-phase-shift quantum key distribution with twisted photons," Phys. Rev. A 98(1), 010301 (2018). [CrossRef]
24. J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, "Terabit free-space data transmission employing orbital angular momentum multiplexing," Nat. Photonics 6(7), 488–496 (2012). [CrossRef]
25. Z. Zhu, M. Janasik, A. Fyffe, D. Hay, Y. Zhou, B. Kantor, T. Winder, R. W. Boyd, G. Leuchs, and Z. Shi, "Compensation-free high-capacity free-space optical communication using turbulence-resilient vector beams," arXiv preprint arXiv:1910.05406 (2019).
26. L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]
27. D. E. B. Flaes, J. Stopka, S. Turtaev, J. F. de Boer, T. Tyc, and T. Čižmár, "Robustness of light-transport processes to bending deformations in graded-index multimode waveguides," Phys. Rev. Lett. 120(23), 233901 (2018). [CrossRef]
28. D. Cozzolino, D. Bacco, B. Da Lio, K. Ingerslev, Y. Ding, K. Dalgaard, P. Kristensen, M. Galili, K. Rottwitt, S. Ramachandran, and L. K. Oxenløwe, "Orbital angular momentum states enabling fiber-based high-dimensional quantum communication," Phys. Rev. Appl. 11(6), 064058 (2019). [CrossRef]
29. G. A. Tyler and R. W. Boyd, "Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum," Opt. Lett. 34(2), 142–144 (2009). [CrossRef]
30. G. A. Tyler, "Spatial bandwidth considerations for optical communication through a free space propagation link," Opt. Lett. 36(23), 4650–4652 (2011). [CrossRef]
31. Y. Ren, H. Huang, G. Xie, N. Ahmed, Y. Yan, B. I. Erkmen, N. Chandrasekaran, M. P. Lavery, N. K. Steinhoff, M. Tur, S. Dolinar, M. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, "Atmospheric turbulence effects on the performance of a free space optical link employing orbital angular momentum multiplexing," Opt. Lett. 38(20), 4062–4065 (2013). [CrossRef]
32. Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, M. P. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, "Adaptive-optics-based simultaneous pre-and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link," Optica 1(6), 376–382 (2014). [CrossRef]
33. B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magana-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, "Simulating thick atmospheric turbulence in the lab with application to orbital angular momentum communication," New J. Phys. 16(3), 033020 (2014). [CrossRef]
34. Y. Zhang, S. Prabhakar, A. H. Ibrahim, F. S. Roux, A. Forbes, and T. Konrad, "Experimentally observed decay of high-dimensional entanglement through turbulence," Phys. Rev. A 94(3), 032310 (2016). [CrossRef]
35. Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, I. B. Djordjevic, M. A. Neifeld, and A. E. Willner, "Experimental characterization of a 400 Gbit/s orbital angular momentum multiplexed free-space optical link over 120 m," Opt. Lett. 41(3), 622–625 (2016). [CrossRef]
36. N. Leonhard, G. Sorelli, V. N. Shatokhin, C. Reinlein, and A. Buchleitner, "Protecting the entanglement of twisted photons by adaptive optics," Phys. Rev. A 97(1), 012321 (2018). [CrossRef]
37. G. Sorelli, N. Leonhard, V. N. Shatokhin, C. Reinlein, and A. Buchleitner, "Entanglement protection of high-dimensional states by adaptive optics," New J. Phys. 21(2), 023003 (2019). [CrossRef]
38. C. Liu, K. Pang, Z. Zhao, P. Liao, R. Zhang, H. Song, Y. Cao, J. Du, L. Li, H. Song, Y. Ren, G. Xie, Y. Zhao, J. Zhao, S. M. H. Rafsanjani, A. N. Willner, J. H. Shapiro, R. W. Boyd, M. Tur, and A. E. Willner, "Single-end adaptive optics compensation for emulated turbulence in a bi-directional 10-Mbit/s per channel free-space quantum communication link using orbital-angular-momentum encoding," Research 2019, 1–10 (2019). [CrossRef]
39. S. Zhao, W. Li, Y. Shen, Y. Yu, X. Han, H. Zeng, M. Cai, T. Qian, S. Wang, Z. Wang, Y. Xiao, and Y. Gu, "Experimental investigation of quantum key distribution over a water channel," Appl. Opt. 58(14), 3902–3907 (2019). [CrossRef]
40. F. Hufnagel, A. Sit, F. Grenapin, F. Bouchard, K. Heshami, D. England, Y. Zhang, B. J. Sussman, R. W. Boyd, G. Leuchs, and E. Karimi, "Characterization of an underwater channel for quantum communications in the Ottawa river," Opt. Express 27(19), 26346–26354 (2019). [CrossRef]
41. L. Li, R. Zhang, P. Liao, Y. Cao, H. Song, Y. Zhao, J. Du, Z. Zhao, C. Liu, K. Pang, H. Song, A. Almaiman, D. Starodubov, B. Lynn, R. Bock, M. Tur, A. F. Molisch, and A. E. Willner, "Mitigation for turbulence effects in a 40-gbit/s orbital-angular-momentum-multiplexed free-space optical link between a ground station and a retro-reflecting uav using mimo equalization," Opt. Lett. 44(21), 5181–5184 (2019). [CrossRef]
42. C. M. Mabena and F. S. Roux, "Quantum channel correction with twisted light using compressive sensing," Phys. Rev. A 101(1), 013807 (2020). [CrossRef]
43. M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, "Efficient separation of the orbital angular momentum eigenstates of light," Nat. Commun. 4(1), 2781 (2013). [CrossRef]
44. D. Fu, Y. Zhou, R. Qi, S. Oliver, Y. Wang, S. M. H. Rafsanjani, J. Zhao, M. Mirhosseini, Z. Shi, P. Zhang, and R. W. Boyd, "Realization of a scalable Laguerre–Gaussian mode sorter based on a robust radial mode sorter," Opt. Express 26(25), 33057–33065 (2018). [CrossRef]
45. A. Acín, D. Cavalcanti, E. Passaro, S. Pironio, and P. Skrzypczyk, "Necessary detection efficiencies for secure quantum key distribution and bound randomness," Phys. Rev. A 93(1), 012319 (2016). [CrossRef]
46. G. A. Tyler, J. L. Vaughn, and N. K. Steinhoff, "High-capacity, free-space quantum key distribution based on spatial and polarization encoding: Atmospheric considerations interim review," The Optical Science Company (2017).
47. G. A. Tyler, P. Merritt, R. Q. Fugate, and T. J. Brennan, "Adaptive opitcs: Theory and applications," The Optical Science Company (2006).
48. M. P. Lavery, "Vortex instability in turbulent free-space propagation," New J. Phys. 20(4), 043023 (2018). [CrossRef]
49. N. Woolf, J. R. P. Angel, and J. Williams, "Options for a new arizona observatory: Configurations, costs and site," in International Astronomical Union Colloquium, vol. 79 (Cambridge University, 1984), pp. 831–843.
50. J. M. Beckers, "Adaptive optics for astronomy: principles, performance, and applications," Annu. Rev. Astron. Astrophys. 31(1), 13–62 (1993). [CrossRef]
51. F. Roddier, Adaptive optics in astronomy (Cambridge University, 1999).
52. Y. Ren, C. Liu, K. Pang, J. Zhao, Y. Cao, G. Xie, L. Li, P. Liao, Z. Zhao, M. Tur, R. W. Boyd, and A. E. Willner, "Spatially multiplexed orbital-angular-momentum-encoded single photon and classical channels in a free-space optical communication link," Opt. Lett. 42(23), 4881–4884 (2017). [CrossRef]
53. J. Zhao, M. Mirhosseini, B. Braverman, Y. Zhou, S. M. H. Rafsanjani, Y. Ren, N. K. Steinhoff, G. A. Tyler, A. E. Willner, and R. W. Boyd, "Performance analysis of d-dimensional quantum cryptography under state-dependent diffraction," Phys. Rev. A 100(3), 032319 (2019). [CrossRef]
54. D. L. Fried, "Statistics of a geometric representation of wavefront distortion," J. Opt. Soc. Am. 55(11), 1427–1435 (1965). [CrossRef]
55. D. L. Fried, "Optical resolution through a randomly inhomogeneous medium for very long and very short exposures," J. Opt. Soc. Am. 56(10), 1372–1379 (1966). [CrossRef]
56. L. C. Andrews and R. L. Phillips, Laser beam propagation through random media, vol. 152 (SPIE, 2005).
57. D. P. Greenwood, "Bandwidth specification for adaptive optics systems," J. Opt. Soc. Am. 67(3), 390–393 (1977). [CrossRef]
58. D. L. Fried, "Greenwood frequency measurements," J. Opt. Soc. Am. A 7(5), 946–947 (1990). [CrossRef]
59. G. A. Tyler, "Bandwidth considerations for tracking through turbulence," J. Opt. Soc. Am. A 11(1), 358–367 (1994). [CrossRef]
60. R. K. Tyson and B. W. Frazier, Field guide to adaptive optics (SPIE, 2012).
61. W. Liu, K. Yao, D. Huang, X. Lin, L. Wang, and Y. Lv, "Performance evaluation of coherent free space optical communications with a double-stage fast-steering-mirror adaptive optics system depending on the greenwood frequency," Opt. Express 24(12), 13288–13302 (2016). [CrossRef]
62. G. A. Tyler, "Adaptive optics compensation for propagation through deep turbulence: a study of some interesting approaches," Opt. Eng. 52(2), 021011 (2012). [CrossRef]
63. R. Ionicioiu, "Sorting quantum systems efficiently," Sci. Rep. 6(1), 25356 (2016). [CrossRef]
64. N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, "Laguerre-Gaussian mode sorter," Nat. Commun. 10(1), 1865 (2019). [CrossRef]
65. B. Ndagano, I. Nape, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, M. P. Lavery, and A. Forbes, "A deterministic detector for vector vortex states," Sci. Rep. 7(1), 13882–8 (2017). [CrossRef]
66. Y. Zhou, M. Mirhosseini, S. Oliver, J. Zhao, S. M. H. Rafsanjani, M. P. Lavery, A. E. Willner, and R. W. Boyd, "Using all transverse degrees of freedom in quantum communications based on a generic mode sorter," Opt. Express 27(7), 10383–10394 (2019). [CrossRef]
N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74(1), 145–195 (2002).
V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, "The security of practical quantum key distribution," Rev. Mod. Phys. 81(3), 1301–1350 (2009).
H.-K. Lo, M. Curty, and K. Tamaki, "Secure quantum key distribution," Nat. Photonics 8(8), 595–604 (2014).
C. H. Bennett and G. Brassard, "Quantum cryptography: public key distribution and coin tossing, Proceedings of the IEEE International, Conference on Computers, Systems and Signal Processing pp. 175–179 (1984).
B. Korzh, C. C. W. Lim, R. Houlmann, N. Gisin, M. J. Li, D. Nolan, B. Sanguinetti, R. Thew, and H. Zbinden, "Provably secure and practical quantum key distribution over 307 km of optical fibre," Nat. Photonics 9(3), 163–168 (2015).
H.-L. Yin, T.-Y. Chen, Z.-W. Yu, H. Liu, L.-X. You, Y.-H. Zhou, S.-J. Chen, Y. Mao, M.-Q. Huang, W.-J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X.-B. Wang, and J.-W. Pan, "Measurement-device-independent quantum key distribution over a 404 km optical fiber," Phys. Rev. Lett. 117(19), 190501 (2016).
J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, and J.-W. Pan, "Direct and full-scale experimental verifications towards ground–satellite quantum key distribution," Nat. Photonics 7(5), 387–393 (2013).
S.-K. Liao, W.-Q. Cai, W.-Y. Liu, L. Zhang, Y. Li, J.-G. Ren, J. Yin, Q. Shen, Y. Cao, Z.-P. Li, F.-Z. Li, X.-W. Chen, L.-H. Sun, J.-J. Jia, J.-C. Wu, X.-J. Jiang, J.-F. Wang, Y.-M. Huang, Q. Wang, Y.-L. Zhou, L. Deng, T. Xi, L. Ma, T. Hu, Q. Zhang, Y.-A. Chen, N.-L. Liu, X.-B. Wang, Z.-C. Zhu, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, "Satellite-to-ground quantum key distribution," Nature 549(7670), 43–47 (2017).
J. Jin, J.-P. Bourgoin, R. Tannous, S. Agne, C. J. Pugh, K. B. Kuntz, B. L. Higgins, and T. Jennewein, "Genuine time-bin-encoded quantum key distribution over a turbulent depolarizing free-space channel," Opt. Express 27(26), 37214–37223 (2019).
L. Ji, J. Gao, A.-L. Yang, Z. Feng, X.-F. Lin, Z.-G. Li, and X.-M. Jin, "Towards quantum communications in free-space seawater," Opt. Express 25(17), 19795–19806 (2017).
F. Bouchard, A. Sit, F. Hufnagel, A. Abbas, Y. Zhang, K. Heshami, R. Fickler, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, "Quantum cryptography with twisted photons through an outdoor underwater channel," Opt. Express 26(17), 22563–22573 (2018).
R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Amer, M. Furst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, "Entanglement-based quantum communication over 144 km," Nat. Phys. 3(7), 481–486 (2007).
X. Sun, I. B. Djordjevic, and M. A. Neifeld, "Multiple spatial modes based qkd over marine free-space optical channels in the presence of atmospheric turbulence," Opt. Express 24(24), 27663–27673 (2016).
A. Dousse, J. Suffczyński, A. Beveratos, O. Krebs, A. Lemaître, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, "Ultrabright source of entangled photon pairs," Nature 466(7303), 217–220 (2010).
F. Steinlechner, P. Trojek, M. Jofre, H. Weier, D. Perez, T. Jennewein, R. Ursin, J. Rarity, M. W. Mitchell, J. P. Torres, H. Weinfurter, and V. Pruneri, "A high-brightness source of polarization-entangled photons optimized for applications in free space," Opt. Express 20(9), 9640–9649 (2012).
N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, "Security of quantum key distribution using d-level systems," Phys. Rev. Lett. 88(12), 127902 (2002).
L. Sheridan and V. Scarani, "Security proof for quantum key distribution using qudit systems," Phys. Rev. A 82(3), 030301 (2010).
K. Brádler, M. Mirhosseini, R. Fickler, A. Broadbent, and R. W. Boyd, "Finite-key security analysis for multilevel quantum key distribution," New J. Phys. 18(7), 073030 (2016).
G. Vallone, V. D'Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, "Free-space quantum key distribution by rotation-invariant twisted photons," Phys. Rev. Lett. 113(6), 060503 (2014).
M. Mirhosseini, O. S. Magana-Loaiza, M. N. O'Sullivan, B. Rodenburg, M. Malik, M. P. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, "High-dimensional quantum cryptography with twisted light," New J. Phys. 17(3), 033033 (2015).
A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Günthner, B. Heim, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, "High-dimensional intracity quantum cryptography with structured photons," Optica 4(9), 1006–1010 (2017).
F. Bouchard, K. Heshami, D. England, R. Fickler, R. W. Boyd, B.-G. Englert, L. L. Sánchez-Soto, and E. Karimi, "Experimental investigation of high-dimensional quantum key distribution protocols with twisted photons," Quantum 2, 111111 (2018).
F. Bouchard, A. Sit, K. Heshami, R. Fickler, and E. Karimi, "Round-robin differential-phase-shift quantum key distribution with twisted photons," Phys. Rev. A 98(1), 010301 (2018).
J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, "Terabit free-space data transmission employing orbital angular momentum multiplexing," Nat. Photonics 6(7), 488–496 (2012).
Z. Zhu, M. Janasik, A. Fyffe, D. Hay, Y. Zhou, B. Kantor, T. Winder, R. W. Boyd, G. Leuchs, and Z. Shi, "Compensation-free high-capacity free-space optical communication using turbulence-resilient vector beams," arXiv preprint arXiv:1910.05406 (2019).
L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45(11), 8185–8189 (1992).
D. E. B. Flaes, J. Stopka, S. Turtaev, J. F. de Boer, T. Tyc, and T. Čižmár, "Robustness of light-transport processes to bending deformations in graded-index multimode waveguides," Phys. Rev. Lett. 120(23), 233901 (2018).
D. Cozzolino, D. Bacco, B. Da Lio, K. Ingerslev, Y. Ding, K. Dalgaard, P. Kristensen, M. Galili, K. Rottwitt, S. Ramachandran, and L. K. Oxenløwe, "Orbital angular momentum states enabling fiber-based high-dimensional quantum communication," Phys. Rev. Appl. 11(6), 064058 (2019).
G. A. Tyler and R. W. Boyd, "Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum," Opt. Lett. 34(2), 142–144 (2009).
G. A. Tyler, "Spatial bandwidth considerations for optical communication through a free space propagation link," Opt. Lett. 36(23), 4650–4652 (2011).
Y. Ren, H. Huang, G. Xie, N. Ahmed, Y. Yan, B. I. Erkmen, N. Chandrasekaran, M. P. Lavery, N. K. Steinhoff, M. Tur, S. Dolinar, M. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, "Atmospheric turbulence effects on the performance of a free space optical link employing orbital angular momentum multiplexing," Opt. Lett. 38(20), 4062–4065 (2013).
Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, M. P. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, "Adaptive-optics-based simultaneous pre-and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link," Optica 1(6), 376–382 (2014).
B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magana-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, "Simulating thick atmospheric turbulence in the lab with application to orbital angular momentum communication," New J. Phys. 16(3), 033020 (2014).
Y. Zhang, S. Prabhakar, A. H. Ibrahim, F. S. Roux, A. Forbes, and T. Konrad, "Experimentally observed decay of high-dimensional entanglement through turbulence," Phys. Rev. A 94(3), 032310 (2016).
Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, I. B. Djordjevic, M. A. Neifeld, and A. E. Willner, "Experimental characterization of a 400 Gbit/s orbital angular momentum multiplexed free-space optical link over 120 m," Opt. Lett. 41(3), 622–625 (2016).
N. Leonhard, G. Sorelli, V. N. Shatokhin, C. Reinlein, and A. Buchleitner, "Protecting the entanglement of twisted photons by adaptive optics," Phys. Rev. A 97(1), 012321 (2018).
G. Sorelli, N. Leonhard, V. N. Shatokhin, C. Reinlein, and A. Buchleitner, "Entanglement protection of high-dimensional states by adaptive optics," New J. Phys. 21(2), 023003 (2019).
C. Liu, K. Pang, Z. Zhao, P. Liao, R. Zhang, H. Song, Y. Cao, J. Du, L. Li, H. Song, Y. Ren, G. Xie, Y. Zhao, J. Zhao, S. M. H. Rafsanjani, A. N. Willner, J. H. Shapiro, R. W. Boyd, M. Tur, and A. E. Willner, "Single-end adaptive optics compensation for emulated turbulence in a bi-directional 10-Mbit/s per channel free-space quantum communication link using orbital-angular-momentum encoding," Research 2019, 1–10 (2019).
S. Zhao, W. Li, Y. Shen, Y. Yu, X. Han, H. Zeng, M. Cai, T. Qian, S. Wang, Z. Wang, Y. Xiao, and Y. Gu, "Experimental investigation of quantum key distribution over a water channel," Appl. Opt. 58(14), 3902–3907 (2019).
F. Hufnagel, A. Sit, F. Grenapin, F. Bouchard, K. Heshami, D. England, Y. Zhang, B. J. Sussman, R. W. Boyd, G. Leuchs, and E. Karimi, "Characterization of an underwater channel for quantum communications in the Ottawa river," Opt. Express 27(19), 26346–26354 (2019).
L. Li, R. Zhang, P. Liao, Y. Cao, H. Song, Y. Zhao, J. Du, Z. Zhao, C. Liu, K. Pang, H. Song, A. Almaiman, D. Starodubov, B. Lynn, R. Bock, M. Tur, A. F. Molisch, and A. E. Willner, "Mitigation for turbulence effects in a 40-gbit/s orbital-angular-momentum-multiplexed free-space optical link between a ground station and a retro-reflecting uav using mimo equalization," Opt. Lett. 44(21), 5181–5184 (2019).
C. M. Mabena and F. S. Roux, "Quantum channel correction with twisted light using compressive sensing," Phys. Rev. A 101(1), 013807 (2020).
M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, "Efficient separation of the orbital angular momentum eigenstates of light," Nat. Commun. 4(1), 2781 (2013).
D. Fu, Y. Zhou, R. Qi, S. Oliver, Y. Wang, S. M. H. Rafsanjani, J. Zhao, M. Mirhosseini, Z. Shi, P. Zhang, and R. W. Boyd, "Realization of a scalable Laguerre–Gaussian mode sorter based on a robust radial mode sorter," Opt. Express 26(25), 33057–33065 (2018).
A. Acín, D. Cavalcanti, E. Passaro, S. Pironio, and P. Skrzypczyk, "Necessary detection efficiencies for secure quantum key distribution and bound randomness," Phys. Rev. A 93(1), 012319 (2016).
G. A. Tyler, J. L. Vaughn, and N. K. Steinhoff, "High-capacity, free-space quantum key distribution based on spatial and polarization encoding: Atmospheric considerations interim review," The Optical Science Company (2017).
G. A. Tyler, P. Merritt, R. Q. Fugate, and T. J. Brennan, "Adaptive opitcs: Theory and applications," The Optical Science Company (2006).
M. P. Lavery, "Vortex instability in turbulent free-space propagation," New J. Phys. 20(4), 043023 (2018).
N. Woolf, J. R. P. Angel, and J. Williams, "Options for a new arizona observatory: Configurations, costs and site," in International Astronomical Union Colloquium, vol. 79 (Cambridge University, 1984), pp. 831–843.
J. M. Beckers, "Adaptive optics for astronomy: principles, performance, and applications," Annu. Rev. Astron. Astrophys. 31(1), 13–62 (1993).
F. Roddier, Adaptive optics in astronomy (Cambridge University, 1999).
Y. Ren, C. Liu, K. Pang, J. Zhao, Y. Cao, G. Xie, L. Li, P. Liao, Z. Zhao, M. Tur, R. W. Boyd, and A. E. Willner, "Spatially multiplexed orbital-angular-momentum-encoded single photon and classical channels in a free-space optical communication link," Opt. Lett. 42(23), 4881–4884 (2017).
J. Zhao, M. Mirhosseini, B. Braverman, Y. Zhou, S. M. H. Rafsanjani, Y. Ren, N. K. Steinhoff, G. A. Tyler, A. E. Willner, and R. W. Boyd, "Performance analysis of d-dimensional quantum cryptography under state-dependent diffraction," Phys. Rev. A 100(3), 032319 (2019).
D. L. Fried, "Statistics of a geometric representation of wavefront distortion," J. Opt. Soc. Am. 55(11), 1427–1435 (1965).
D. L. Fried, "Optical resolution through a randomly inhomogeneous medium for very long and very short exposures," J. Opt. Soc. Am. 56(10), 1372–1379 (1966).
L. C. Andrews and R. L. Phillips, Laser beam propagation through random media, vol. 152 (SPIE, 2005).
D. P. Greenwood, "Bandwidth specification for adaptive optics systems," J. Opt. Soc. Am. 67(3), 390–393 (1977).
D. L. Fried, "Greenwood frequency measurements," J. Opt. Soc. Am. A 7(5), 946–947 (1990).
G. A. Tyler, "Bandwidth considerations for tracking through turbulence," J. Opt. Soc. Am. A 11(1), 358–367 (1994).
R. K. Tyson and B. W. Frazier, Field guide to adaptive optics (SPIE, 2012).
W. Liu, K. Yao, D. Huang, X. Lin, L. Wang, and Y. Lv, "Performance evaluation of coherent free space optical communications with a double-stage fast-steering-mirror adaptive optics system depending on the greenwood frequency," Opt. Express 24(12), 13288–13302 (2016).
G. A. Tyler, "Adaptive optics compensation for propagation through deep turbulence: a study of some interesting approaches," Opt. Eng. 52(2), 021011 (2012).
R. Ionicioiu, "Sorting quantum systems efficiently," Sci. Rep. 6(1), 25356 (2016).
N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, "Laguerre-Gaussian mode sorter," Nat. Commun. 10(1), 1865 (2019).
B. Ndagano, I. Nape, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, M. P. Lavery, and A. Forbes, "A deterministic detector for vector vortex states," Sci. Rep. 7(1), 13882–8 (2017).
Y. Zhou, M. Mirhosseini, S. Oliver, J. Zhao, S. M. H. Rafsanjani, M. P. Lavery, A. E. Willner, and R. W. Boyd, "Using all transverse degrees of freedom in quantum communications based on a generic mode sorter," Opt. Express 27(7), 10383–10394 (2019).
Abbas, A.
Acín, A.
Agne, S.
Ahmed, N.
Allen, L.
Almaiman, A.
Amer, B.
Andrews, L. C.
Angel, J. R. P.
Ashrafi, N.
Ashrafi, S.
Bacco, D.
Bao, C.
Barbieri, C.
Bechmann-Pasquinucci, H.
Beckers, J. M.
Beijersbergen, M. W.
Bennett, C. H.
Beveratos, A.
Blauensteiner, B.
Bloch, J.
Bock, R.
Bouchard, F.
Bourennane, M.
Bourgoin, J.-P.
Boyd, R. W.
Brádler, K.
Brassard, G.
Braverman, B.
Brennan, T. J.
Broadbent, A.
Buchleitner, A.
Cai, M.
Cai, W.-Q.
Cao, Y.
Carpenter, J.
Cavalcanti, D.
Cerf, N. J.
Chandrasekaran, N.
Chen, K.
Chen, S.-J.
Chen, T.-Y.
Chen, X.-W.
Chen, Y.-A.
Cižmár, T.
Cozzolino, D.
Curty, M.
D'Ambrosio, V.
Da Lio, B.
Dalgaard, K.
de Boer, J. F.
Ding, Y.
Djordjevic, I. B.
Dolinar, S.
Dousse, A.
Du, J.
Dušek, M.
Elser, D.
England, D.
Englert, B.-G.
Erkmen, B. I.
Fazal, I. M.
Feng, Z.
Fickler, R.
Flaes, D. E. B.
Fontaine, N. K.
Frazier, B. W.
Fried, D. L.
Fu, D.
Fugate, R. Q.
Furst, M.
Fyffe, A.
Gagnon-Bischoff, J.
Galili, M.
Gauthier, D. J.
Gisin, N.
Greenwood, D. P.
Grenapin, F.
Gu, Y.
Günthner, K.
Han, X.
Hay, D.
Heim, B.
Hernandez-Aranda, R. I.
Heshami, K.
Higgins, B. L.
Houlmann, R.
Hu, T.
Hu, X.-F.
Hu, Y.-H.
Huang, D.
Huang, M.-Q.
Huang, Y.-M.
Hufnagel, F.
Ibrahim, A. H.
Ingerslev, K.
Ionicioiu, R.
Janasik, M.
Jennewein, T.
Ji, L.
Jia, J.-J.
Jiang, H.
Jiang, X.
Jiang, X.-J.
Jin, J.
Jin, X.-M.
Jofre, M.
Kantor, B.
Karimi, E.
Karlsson, A.
Kim, K.
Konrad, T.
Korzh, B.
Krebs, O.
Kristensen, P.
Kuntz, K. B.
Larocque, H.
Lavery, M. P.
Lavery, M. P. J.
Lemaître, A.
Leonhard, N.
Leuchs, G.
Li, F.-Z.
Li, M. J.
Li, Z.-G.
Li, Z.-P.
Liang, H.
Liao, P.
Liao, S.-K.
Lim, C. C. W.
Lin, X.
Lin, X.-F.
Lindenthal, M.
Liu, C.
Liu, N.-L.
Liu, W.
Liu, W.-Y.
Lo, H.-K.
Lu, C.-Y.
Lütkenhaus, N.
Lv, Y.
Lynn, B.
Ma, L.
Mabena, C. M.
Magana-Loaiza, O. S.
Maher, L.
Malik, M.
Marquardt, C.
Marrucci, L.
Merritt, P.
Meyenburg, M.
Mirhosseini, M.
Mitchell, M. W.
Molisch, A. F.
Nape, I.
Ndagano, B.
Neifeld, M.
Neifeld, M. A.
Neilson, D. T.
Nolan, D.
O'Sullivan, M. N.
Oliver, S.
Oxenløwe, L. K.
Padgett, M. J.
Pan, G.-S.
Pan, J.-W.
Pang, K.
Passaro, E.
Peev, M.
Peng, C.-Z.
Perdigues, J.
Perez, D.
Perez-Garcia, B.
Peuntinger, C.
Phillips, R. L.
Pironio, S.
Prabhakar, S.
Pruneri, V.
Pugh, C. J.
Qi, B.
Qi, R.
Qian, T.
Rafsanjani, S. M. H.
Ramachandran, S.
Rarity, J.
Reinlein, C.
Ren, J.-G.
Ren, Y.
Ribordy, G.
Roddier, F.
Rodenburg, B.
Rottwitt, K.
Roux, F. S.
Ryf, R.
Sagnes, I.
Sánchez-Soto, L. L.
Sanguinetti, B.
Scarani, V.
Scheidl, T.
Schmitt-Manderbach, T.
Scholes, S.
Sciarrino, F.
Senellart, P.
Shapiro, J. H.
Shatokhin, V. N.
Shen, Q.
Shen, Y.
Sheridan, L.
Shi, Z.
Shu, R.
Sit, A.
Skrzypczyk, P.
Slussarenko, S.
Sodnik, Z.
Song, H.
Sorelli, G.
Sponselli, A.
Spreeuw, R.
Starodubov, D.
Steinhoff, N. K.
Steinlechner, F.
Stopka, J.
Suffczynski, J.
Sun, L.-H.
Sun, X.
Sussman, B. J.
Tamaki, K.
Tang, Y.-L.
Tannous, R.
Thew, R.
Tiefenbacher, F.
Tittel, W.
Torres, J. P.
Trojek, P.
Tur, M.
Turtaev, S.
Tyc, T.
Tyler, G. A.
Tyson, R. K.
Ursin, R.
Vallone, G.
Vaughn, J. L.
Villoresi, P.
Voisin, P.
Wang, J.-F.
Wang, J.-Y.
Wang, X.-B.
Weier, H.
Weinfurter, H.
Williams, J.
Willner, A.
Willner, A. E.
Willner, A. N.
Winder, T.
Woerdman, J.
Woolf, N.
Wu, J.-C.
Xi, T.
Xiao, Y.
Xie, G.
Yan, Y.
Yanakas, M.
Yang, A.-L.
Yang, B.
Yang, J.-Y.
Yang, S.-J.
Yao, K.
Yin, H.-L.
Yin, J.
You, L.-X.
Yu, Y.
Yu, Z.-W.
Yue, Y.
Zbinden, H.
Zeilinger, A.
Zeng, H.
Zhang, L.
Zhang, Q.
Zhang, R.
Zhang, W.-J.
Zhao, Y.
Zhao, Z.
Zhong, B.
Zhou, F.
Zhou, Y.
Zhou, Y.-H.
Zhou, Y.-L.
Zhu, Z.
Zhu, Z.-C.
Annu. Rev. Astron. Astrophys. (1)
Nat. Commun. (2)
Nat. Photonics (4)
Nat. Phys. (1)
New J. Phys. (5)
Opt. Eng. (1)
Phys. Rev. Appl. (1)
Rev. Mod. Phys. (2)
Sci. Rep. (2)
(1) | j ⟩ = 1 d ∑ ℓ = − L L | ℓ ⟩ exp ( − i 2 π j ℓ / ( 2 L + 1 ) ) , | CommonCrawl |
\begin{document}
\title{\LARGE\textbf{Invariant Manifolds for Non-differentiable Operators} \author \textcolor{blue}{}\global\long\def\sbr#1{\left[#1\right] } \textcolor{blue}{}\global\long\def\cbr#1{\left\{ #1\right\} } \textcolor{blue}{}\global\long\def\rbr#1{\left(#1\right)} \textcolor{blue}{}\global\long\def\ev#1{\mathbb{E}{#1}} \textcolor{blue}{}\global\long\def\mathbb{R}{\mathbb{R}} \textcolor{blue}{}\global\long\def\mathbb{E}{\mathbb{E}} \textcolor{blue}{}\global\long\def\norm#1#2#3{\Vert#1\Vert_{#2}^{#3}} \textcolor{blue}{}\global\long\def\pr#1{\mathbb{P}\rbr{#1}} \textcolor{blue}{}\global\long\def\mathbb{Q}{\mathbb{Q}} \textcolor{blue}{}\global\long\def\mathbb{A}{\mathbb{A}} \textcolor{blue}{}\global\long\def\ind#1{1_{#1}} \textcolor{blue}{}\global\long\def\mathbb{P}{\mathbb{P}} \textcolor{blue}{}\global\long\def\lesssim{\lesssim} \textcolor{blue}{}\global\long\def\eqsim{\eqsim} \textcolor{blue}{}\global\long\def\Var#1{\text{Var}(#1)} \textcolor{blue}{}\global\long\def\TDD#1{{\color{red}To\, Do(#1)}} \textcolor{blue}{}\global\long\def\dd#1{\textnormal{d}#1} \textcolor{blue}{}\global\long\def:={:=} \textcolor{blue}{}\global\long\def\ddp#1#2{\left\langle #1,#2\right\rangle } \textcolor{blue}{}\global\long\def\mathcal{E}_{n}{\mathcal{E}_{n}} \textcolor{blue}{}\global\long\def\mathbb{Z}{\mathbb{Z}} \textcolor{blue}{{} }
\textcolor{blue}{}\global\long\def\nC#1{\newconstant{#1}} \textcolor{blue}{}\global\long\def\C#1{\useconstant{#1}} \textcolor{blue}{}\global\long\def\nC#1{\newconstant{#1}\text{nC}_{#1}} \textcolor{blue}{}\global\long\def\C#1{C_{#1}} \textcolor{blue}{}\global\long\def\mathcal{M}{\mathcal{M}} \textcolor{blue}{}\global\long\def\mathcal{C}{\mathcal{C}} \textcolor{blue}{}\global\long\def\mathcal{P}{\mathcal{P}}
\begin{abstract} A general invariant manifold theorem is needed to study the topological classes of smooth dynamical systems. These classes are often invariant under renormalization. The classical invariant manifold theorem cannot be applied, because the renormalization operator for smooth systems is not differentiable and sometimes does not have an attractor. Examples are the renormalization operator for general smooth dynamics, such as unimodal dynamics, circle dynamics, Cherry dynamics, Lorenz dynamics, H\'enon dynamics, etc. A general method to construct invariant manifolds of non-differentiable non-linear operators is presented. An application is that the $\mathcal C^{4+\epsilon}$ Fibonacci Cherry maps form a $\mathcal C^1$ codimension one manifold. \end{abstract}
\section{Introduction} The classical invariant manifold theorem assures the existence of a submanifold preserved by a given map. The map has to satisfy two crucial conditions, namely, it has to be differentiable and it has to have an hyperbolic invariant set (for exampe an hyperbolic fixed point). The invariant manifold theorem has many applications in smooth hyperbolic dynamics where it supplies the framework for a complete topological description of the dynamics in phase space.
The invariant manifold theorem also plays an important role in the description of parameter space of families of dynamical systems. Namely, in one and two dimensional dynamics, the topology of systems can sometimes be characterized in terms of renormalization schemes. This is the case for circle diffeomorphisms, critical circle maps, unimodal maps, quadratic-like maps and dissipative H\'enon maps at the boundary of chaos. A topological class is a collection of systems which share a topological property. For example, systems which are conjugate on their attractors form topological classes. A topological class is often an invariant manifold of renormalization. The renormalization operator acts on parts of the corresponding spaces of systems.
In the analytic setting of circle diffeomorphisms, unimodal maps, quadratic-like maps or strongly dissipative H\'enon-like maps at the boundary of chaos, the classical invariant manifold theorem can be applied and it assures that many topological classes are indeed finite codimension analytic manifolds. In these contexts the renormalization operators are indeed differentiable and have hyperbolic attractors.
One expects that many topological classes are also smooth submanifolds in the context of smooth systems, like the ones mentioned before, as well as Lorenz dynamics, H\'enon dynamics, Cherry dynamics, etc.
Unfortunately, in the smooth context, the renormalization operators are not differentiable and sometimes they don't have attractors at all. The only results in this direction so far has been realized for the particular case of smooth unimodal dynamics, see \cite{Davie, dFdMP}. In this case the renormalization operator is not differentiable, however it has an hyperbolic attractor which is part of the space of analytic maps. The convergence of renormalization towards the space of analytic systems allows to extend a topological class of analytic unimodal maps into the corresponding topological class of smooth unimodal maps.
Anticipating that renormalization will be a powerful tool to describe the topological classes of dynamical systems in many different contexts, one would like to have a general invariant manifold theorem which gives a method to prove that the topological classes are smooth manifolds also when the renormalization operator is not differentiable, like in smooth dynamics and when the renormalization does not have an attractor, like in Fibonacci unimodal dynamics, \cite{MilnorLyubic}, Lorenz dynamics, \cite{MW} and Cherry dynamics, \cite{5aut, P1, P2}. The main theorem presented here, namely Theorem \ref{InvMan}, provides such a method.
The method has two parts. The first one is very general and is the same in all different contexts. The second part has a quantitative aspect which depends on the specific setting. We illustrate it in the most difficult situation where the renormalization operator is not differentiable and does not have an attractor, namely in Fibonacci Cherry dynamics. A very similar situation occurs for Fibonacci unimodal dynamics and Lorenz dynamics. However, the method is applicable in much broader contexts, for example smooth one dimensional dynamics and H\'enon dynamics. One has to adapt the quantitative aspect of the method following the guidline given in the application to Cherry dynamics.
The two main ingredients of the classical invariant manifold theorem, such as the differentiability and the hyperbolicity of the operator are substituted by weaker versions: jump-out differentiability, see Definition \ref{def:jumpoutdiff} and topological hyperbolicity, see Theorem \ref{InvMan}. One of the reason why the renormalization operator is not differentiable in the smooth context is that composition is not differentiable. We replace the space of systems by decomposed systems in which renomalization is jump-out differentiable and the jump-out derivative is described, see Section \ref{deco}. The construction of the invariant manifold uses the classical graph transform which is studied with the non conventional method of curve dynamics. Curve dynamics does rely on Lipschitz regularity instead of differentiability.
The main theorem, Theorem \ref{InvMan}, is stated in the context of Cherry dynamics. Renormalization of Cherry dynamics is introduced in Section \ref{section:renorm}. This concerns the dynamics of circle maps with a flat spot and critical exponents greater than one. For those maps whose critical exponent is between one and two and the rotation number is the Fibonacci number, the renormalizations diverge and in particular the Fibonacci renormalization operator does not have an attractor. The specific degeneration of the renormalizations is quantified by three invariants given in Proposition \ref{superformula}. These invariants determine the specific form of the quantitative aspects in the Invariant Manifold Theorem \ref{InvMan}, which applied in this case gives the following. Refer to Theorem \ref{manifold}. \paragraph{Theorem A.}\label{manifoldintr}The $\mathcal C^{4+\epsilon}$ Fibonacci Cherry maps form a ${{\mathcal C}^1}$ codimension one manifold.
The ingredients developed to construct the smooth Fibonacci class of Theorem A. can also be used to study one of the fundamental questions in low-dimensional dynamics whether two systems with the same topological properties have also the same geometrical properties. More precisely, consider two dynamical systems defined by two functions $f$ and $g$ and suppose that there exists an homeomorphism $h$ which conjugates $f$ and $g$ on their attractors. Is $h$ a $\mathcal C^{1+\beta}$, $\beta>0$, diffeomorphism? Such a regularity of the conjugacy implies that the geometry of the two systems is rigid, it is not possible to modify it on asymptotical small scales. The fact that the conjugacy has any regularity is by itself very surprising. It tells that as soon the topology of a systems is known, the asymptotic small scale geometry is also determined. The rigidity question has been studied for circle diffeomorphisms \cite{H79} \cite{Yo}, critical circle homeomorphisms \cite{dFdM}, \cite{Y}, \cite{Y1}, \cite{GdM}, \cite{GMdM}, unimodal maps \cite{Lan}, \cite{S}, \cite{Mc}, \cite{Lyu1}, \cite{Lyu2}, \cite{dMP}, \cite{dFdMP}, circle maps with breakpoints \cite{KT1}, \cite{KK14} and for Kleinian groups \cite{Mo}.
In all these cases the topological conjugacy is indeed smooth, $\mathcal C^{1+\beta}$, $\beta>0$. In the context of Fibonacci Cherry dynamics one encounter a different behavior, also detected in unimodal Fibonacci dynamics, see \cite{MilnorLyubic}. Given a Fibonacci Cherry map $f$ with critical exponent bigger than one and smaller than two, the smoothness of the conjugacy is determined by the three geometrical invariants, $C(f)=\left(C_u(f), C_-(f),C_s(f)\right)$. Refer to Theorem \ref{LM}.
\paragraph{Theorem B.} Two $\mathcal C^{4+\epsilon}$ Fibonacci Cherry maps, $f$ and $g$ with the same critical exponent bigger than one and smaller than two, are $\mathcal C^{1+\beta}$, $\beta>0$, conjugate if and only if the three geometrical invariants are the same, $C(f)=C(g)$.
Different phenomena concerning rigidity of systems has been detected also for strongly dissipative H\'enon-like maps, see \cite{CLM05, ML} and Lorenz maps, see \cite{MW}. A new rigidity conjecture, taking into account the recent phenomena detected for one and two dimensional dynamical systems, has been formulated in \cite{MPW}.
\paragraph {Standing notation.} Let $\alpha_n$ and $\beta_n$ be two sequences of positive numbers. We say that $\alpha_n$ is of the order of $\beta_n$ if there exists an uniform constant $K>0$ such that, for $n$ big enough $\alpha_n<K \beta_n$. We will use the notation $\alpha_n=O(\beta_n).$ Moreover we denote by $[a, b] = [b, a]$ the shortest interval between $a$ and $b$ regardless of the order of these two points. The length of that interval in the natural metric will be denoted by $\left|[a , b]\right|$. \paragraph{Acknowledgements.} The authors would like to thank IMPAN for its hospitality. The work was initiated and mostly developed at IMPAN. The first author was partially supported by the NSF grant 1600554 and the second author was partially supported by the Leverhulme Trust through the Leverhulme Prize of C. Ulcigrai and by the Trygger foundation, Project CTS 17:50. \section{Smooth invariant manifolds} The classical invariant manifold theorem assures the existence of a submanifold preserved by a given map. The map has to satisfy two crucial conditions, namely, it has to be differentiable and it has to have an hyperbolic invariant set. Here a general invariant manifold theorem is given. The two crucial conditions needed for the classical invariant manifold theorem are substituted by weaker ones: jump-out differentiability and topological hyperbolocity, see Theorem \ref{InvMan}.
\subsection{Jump-out differentiability} \begin{defin}\label{def:jumpoutdiff}
Let $\left(B_0, \|\cdot\|_0\right)$ and $\left(B_1, \|\cdot\|_1\right)$ be two normed spaces such that $B_1\subset B_0$ and $\|b\|_0\leq\|b\|_1$ for all $b\in B_1$. Let $U\subset B_1$ be an open subset and let $$F:U\subset B_1\to B_1\subset B_0.$$ We say that $F$ is jump-out-differentiable if, for every $p\in U$, $$F:U\subset B_1\to B_0$$ is differentiable in $p$. Moreover the derivative $DF_p$ extends to a bounded operator $$DF_p:B_0\to B_0,$$ and $DF_p$ depends continuously on $p$. \end{defin} \begin{rem} The word "jump-out-differentiability" describes the following situation. The differentiable map $F:U\to B_0$ is not necessarily differentiable as map $F:U\to B_1$, although $F(U)\subset B_1$. The derivative jumps-out $B_1$, namely for some $p\in B_1$, $\text{Image} (DF_p(B_1))\not\subset B_1$. Observe that in finite dimensional vector spaces all norms are equivalent. As consequence the notion of jump-out-differentiability and differentiability are in fact equivalent when the dimension of $B_0$ is finite. The jump-out-differentiability is purely an infinite dimensional phenomenon.
The following example will illustrate jump-out differentiability. It is similar to what we will encounter when discussing renormalization in Section \ref{jumpoutdiff}. Let $\eta\in C^1(\mathbb{R},\mathbb{R})$ and $Z:\mathbb{R}\to C^1(\mathbb{R},\mathbb{R})$ defined by $$ Z(t)(x)=t\eta(tx). $$ Then $Z$ is not differentiable. However, $$ Z:\mathbb{R}\to C^0(\mathbb{R},\mathbb{R}) $$ is differentialble with derivative $DZ_t: \mathbb{R}\to C^0(\mathbb{R},\mathbb{R})$ given by $$ DZ_t(x)=txD\eta(tx)+\eta(tx). $$ Observe that $$ \text{Image}(DZ_t)\not\subset C^1(\mathbb{R},\mathbb{R}). $$ \end{rem} \begin{rem}\label{chainrule} The chain rule holds for jump-out differentiable maps. More in detail, the composition of two jump-out differentiable maps $F$ and $G$ is again jump-out differentiable and the extension of the derivative of the composition is the composition of the extended derivatives of $F$ and $G$. \end{rem} \subsection{Cone field} In the following we introduce a cone field which will be used to study the graph transform in the proof of the main theorem. The role of the cone field here is the same as in the proof of the classical invariant manifold theorem. Observe that the cone field in Definition \ref{degconefield} is degenerating when $y$ goes to infinity. The form of the cone is inspired by the specific degeneration of renormalization of Fibonacci Cherry dynamics. This specific choice of the cone field is one of the quantitative aspect of the method discussed in the introduction which has to be adapted in other applications.
We fix two normed spaces $\left(B_0, \|\cdot\|_0\right)$ and $\left(B_1, \|\cdot\|_1\right)$ with $B_1\subset B_0$ and $\|b\|_0\leq\|b\|_1$ for all $b\in B_1$. If $\left(B, \|\cdot\|\right)$ is a normed space then
we will use the norm $|p|=|y|+\|b\|$ with $p=(y,b)\in \mathbb{R}\times B$ on $\mathbb{R}\times B$. Moreover, we fix $\theta>0$, $0<\kappa<1$, we chose an open set $U\subset B_1$ and we define a cone field on $\mathbb{R}\times B_0$.
\begin{defin}\label{degconefield} Let $p=(y,b)\in \mathbb{R}\times B_0$. The cone at $p$ is defined as
$$C_p=\left\{(\Delta y,\Delta b)\in \mathbb{R}\times B_0 | \text{ }\theta |y|^{\kappa}\|\Delta b \|_0 < | \Delta y |\right\}.$$ \end{defin}
\subsection{Almost horizontal curves and almost vertical graps} \begin{defin} An almost horizontal curve $\gamma$ is the graph of a continuous function $\hat\gamma:[t_-,t_+]\subset [0,1]\to B_1$ having the following properties: \begin{itemize} \item $\hat\gamma:[t_-,t_+]\to B_0$ is continuously differentiable, \item for every $p\in\gamma$, the tangent space satisfies $T_p\gamma\subset C_p$. \end{itemize} The set of almost horizontal curves is denoted by $\Gamma_0$. \end{defin} \begin{lem}\label{perturbation} Let $\gamma\in\Gamma_0$ and $p_1,p_2\in\gamma$. Then there exist $V_1$ neighborhood of $p_1$ and $V_2$ neighborhood of $p_2$ such that, for all $p'_1\in V_1$ and $p'_2\in V_2$ there exists $\gamma'\in\Gamma_0$ such that $p'_1,p'_2\in\gamma'$. \end{lem} \begin{proof} Let $p'_1=(y_1, b_1)$ and $p'_2=(y_2, b_2)$ any two points. Let $\varphi$ be the affine function in $y$ with $\varphi(y_1)=b_1-\hat\gamma(y_1)$ and $\varphi(y_2)=b_2-\hat\gamma(y_2)$. Consider the function $$\hat\gamma'(y)=\hat\gamma(y)+\varphi(y).$$ Then $\gamma'=\text{graph}(\hat\gamma')$ is a smooth curve passing trough $p'_1$ and $p'_2$.
Let $y$ be in the domain of $\hat\gamma$ and $\left(\Delta y, \Delta b\right)$ be a tangent vector to the curve $\gamma'$ in the point $\left(y, \hat\gamma'(y)\right)$. Then $\Delta b=\left(D\hat\gamma+D\varphi\right)\Delta y$. By continuity, there exists $\epsilon>0$, independent on $y$, such that $\theta y^{\kappa}\|D\hat\gamma\|_0\leq 1-\epsilon$. As consequence $$\theta y^{\kappa}\|\Delta b\|_0\leq \left(1-\epsilon +\theta y^{\kappa} \|D\varphi\|_0\right)|\Delta y|.$$ Notice that if $p'_1=p_1$ and $p'_2=p_2$, then $\varphi\equiv 0$. Hence, for $p'_1$ close enough to $p_1$ and $p'_2$ close enough to $p_2$, $\theta y^{\kappa}\|\Delta b\|_0\leq |\Delta y|.$ As consequence $\gamma'\in\Gamma_0$. \end{proof}
\begin{defin} An almost vertical graph $\omega$ is the graph of a continuous function $\hat\omega: U\subset B_1\to\mathbb{R}$ such that, for all almost horizontal curve $\gamma\in\Gamma_0$, $\gamma\cap\omega$ is at most one point. The set of almost vertical graphs is denoted by $\Omega_0$. \end{defin} \subsection{Invariant Manifold Theorem} We are now ready to state our main theorem. \begin{theo}[\bf Invariant Manifold Theorem]\label{InvMan}
Let $\left(B_0, \|\cdot\|_0\right)$ and $\left(B_1, \|\cdot\|_1\right)$ be two normed spaces with $B_1\subset B_0$ and $\|b\|_0\leq\|b\|_1$ for all $b\in B_1$. Let $U\subset B_1$ be open, $\partial_\pm\in \Omega_0$ be the almost vertical graphs corresponding to $\hat\partial_\pm$ with $1>\hat\partial_+(b)>\hat\partial_-(b)>0$, for $b\in U$ and $$
D=\{(y,b)\in [0,1]\times U| \hat\partial_-(b)\le y \le \hat\partial_+(b)\}. $$ For all $p=(y,b)\in D$ with $F(p)=(\tilde y,\tilde b)$ assume the map $$F:D\to \mathbb{R}\times B_1\subset \mathbb{R}\times B_0$$ has the following properties. \begin{itemize} \item $F$ is jump-out-differentiable. \item $F$ has derivatives of the form: there exist $E\neq 0$ and $0\leq\kappa<1$ such that, if $DF_p\left(\Delta y,\Delta b\right)=\left(\Delta\tilde y,\Delta\tilde b\right)$, then \begin{equation}\label{cond2} \left\{\begin{matrix}
\Delta\tilde y&=&\frac{E_p}{y}\Delta y+O\left(\|\Delta b\|_0\right)\\ &&\\
\|\Delta\tilde b \|_0&=&O\left(\frac{1}{\tilde y^{\kappa}}|\Delta y|+\|\Delta b\|_0\right) \end{matrix}\right. \end{equation}
where ${1}/{E}\leq |E_p|\leq E$. \end{itemize} Let ${1}/{4}>\delta>0$ and let $U_{\delta}\subset U$ such that, for all $b\in U_{\delta}$, $\hat\partial_{+}(b)<\delta$. Let $D_{\delta}\subset D$ be the set of $(y,b)\in D$, with $b\in U_{\delta}$.
\begin{itemize} \item $F:D_{\delta}\to\mathbb{R}\times B_1\subset \mathbb{R}\times B_0$ is topologically hyperbolic for all $\delta>0$, i.e. \begin{eqnarray*} \text{If } F(\hat\partial_+(b),b)&=&(\tilde{y},\tilde{b}) \text{ then } \tilde{y}\le \hat\partial_-(\tilde{b}),\\ \text{If } F(\hat\partial_-(b),b)&=&(\tilde{y},\tilde{b})\text{ then }\tilde{y}\ge \hat\partial_+(\tilde{b}),\\ \text{If } F(y,b)&=&(\tilde{y},\tilde{b})\text{ then }\tilde{b}\in U_{\delta}. \end{eqnarray*} \item $F$ has vertical $\xi$-expansion, $\xi>0$, i.e. \begin{eqnarray} \label{cond3} \frac{y}{\tilde y^{\kappa}}&\geq &\xi. \end{eqnarray}
\item $F$ has $\eta$-dominating horizontal expansion, $\eta>0$, i.e. \begin{eqnarray} \label{cond4} \frac{y^2}{\tilde y^{\kappa}}&\leq &\eta. \end{eqnarray}
\end{itemize}
For $\theta>0$, $\delta>0$, $\eta>0$ small enough and $\xi>0$ large enough, the invariant set $$W=\left\{p\in D | \forall n\in{\mathbb N}\text{ }F^n(p)\in D\right\}$$ is an almost vertical graph of a ${{\mathcal C}^1}$ function $\hat\omega:U_{\delta}\to \mathbb{R}$. \end{theo} \begin{rem} Lemma \ref{lemma1} states that the expansion along vertical graphs is of the order ${y}/{\tilde y^{\kappa}}$. This motivates the name of "vertical $\xi$-expansion". Lemma \ref{lemma1} and Lemma \ref{yexpansion} state that the expansion along almost horizontal curves dominates the expansion along almost vertical graphs by a factor of order ${y^2}/{\tilde y^{\kappa}}$. This motivates the name of "$\eta$-dominating horizontal expansion". \end{rem} \begin{rem} Observe that $W$ is not necessarily the stable manifold of an hyperbolic fixed point, as in the most classical context. Theorem \ref{InvMan} will in fact be applied to the context of Cherry maps whose renormalization does not have a fixed point. $W$ will correspond to the class of maps with Fibonacci rotation number. The renormalizations diverge to infinity. \end{rem} From now on we assume the conditions of the Theorem \ref{InvMan}. The proof involves adjustments of parameters $\delta$, $\theta$, $\chi$, $\xi$ and $\eta$. When these adjustments are required in a proof of a lemma, then they are expressed in the statement of the lemma and assumed in the sequel. \begin{lem}\label{lemma1} For $\theta>0$ small enough and $\xi>0$ large enough, there exists $K>0$ such that if $p=(y,b)\in D$ with $\tilde p=F(p)=(\tilde y, \tilde b)\in D$ and $\Delta q=\left(\Delta y, \Delta b\right)$ with $\Delta\tilde q=DF_p\left(\Delta q\right)\notin C_{F(p)}$, then \begin{enumerate}
\item $|\Delta y|\leq K y\|\Delta b\|_{0}$,
\item $|\Delta\tilde q|\leq K\left(\frac{y}{\tilde y^{\kappa}}\right)|\Delta q|$. \end{enumerate} \end{lem} \begin{proof} Let $\Delta\tilde q=\left(\Delta \tilde y, \Delta\tilde b\right)$. By (\ref{cond2}) and the fact that $\Delta\tilde q\notin C_{F(p)}$ we get \begin{eqnarray*}
\frac{E_p}{ y}|\Delta y|+ O\left(\| \Delta b\|_0\right)\leq |\Delta\tilde y|< \theta\tilde y^{\kappa} \| \Delta\tilde b\|_0
= \theta\tilde y^{\kappa}O\left(\frac{1}{\tilde y^{\kappa}}|\Delta y|+\| \Delta b\|_0\right). \end{eqnarray*} As consequence, for $\theta>0$ small enough we get property $1$. For proving property $2$, it is enought to use (\ref{cond2}), (\ref{cond3}) and property $1$, namely
$$|\Delta\tilde q|\leq\frac{E_p}{y}|\Delta y|+O\left(\frac{1}{\tilde y^{\kappa}}|\Delta y|+\|\Delta_b\|_0\right)=O\left(\frac{y}{\tilde y^{\kappa}}\|\Delta b\|_0\right)=O\left(\frac{y}{\tilde y^{\kappa}}|\Delta q|\right).$$ \end{proof} The following lemma states that the cone field $C_p$ is expanding and invariant. \begin{lem}\label{yexpansion}
For ${1}/{4}>\delta>0$ small enough, the following holds. Let $p=(y,b)\in D$ and $\Delta q=(\Delta y, \Delta b)\in C_p$ with $\Delta \tilde q=DF_p(\Delta x)=(\Delta\tilde y, \Delta\tilde b)$, then $\Delta \tilde q\in C_{F(p)}$ and $$|\Delta\tilde y|\geq\frac{1}{2Ey}|\Delta y|\geq 2 |\Delta y|.$$ \end{lem} \begin{proof} By (\ref{cond2}) and by the fact that $\Delta x\in C_p$ we get
$$|\Delta\tilde y|\geq \frac{1}{Ey}|\Delta y|+O\left(|\Delta b\|_{0}\right)=\frac{1}{Ey}|\Delta y|+O\left(\frac{1}{\theta y^{\kappa}}|\Delta y|\right)=\frac{1}{Ey}|\Delta y|\left(1+O\left(\frac{y^{1-\kappa}}{\theta}\right)\right).$$ The expansion estimate follows by taking $\delta>0$ small enough. Left is to show the cone invariance. From $(\ref{cond2})$, the fact that $\Delta q\in C_p$ and the expansion estimate we get
$$\theta \tilde y^{\kappa}\|\Delta\tilde b\|_0=O\left(1+\frac{\tilde y^{\kappa}}{y^{\kappa}}\right)|\Delta y|=O\left(y+{\tilde y^{\kappa}}{y^{1-\kappa}}\right)|\Delta\tilde y|.$$ The cone invariance follows by taking $\delta$ small enough. \end{proof}
We denote by $\Gamma$ the subset of $\Gamma_0$ of all almost horizontal curves having the additional property that $\gamma\cap\partial_-$ and $\gamma\cap\partial_+$ consist each of exactly one point. We denote by $\Omega$ the subset of $\Omega_0$ of all almost vertical graphs having the additional property that $\omega\subset D_{\delta}$. The next lemma states that the set of almost vertical graphs is invariant. This crucial property reduces the method to curve dynamics. \begin{lem}\label{Fgamma} If $\gamma\in\Gamma$, then $F(\gamma\cap D_{\delta})\in\Gamma$. \end{lem} \begin{proof} Observe that, by the definition of $\Omega$, $\gamma\cap D_{\delta}$ is itself and almost horizontal curve, it only contains one component. Hence, we may assume that $\gamma\subset D_{\delta}$. We introduce the map $$F{\gamma}:[t_-,t_+]\ni t\mapsto F\left(\left(t,\hat\gamma(t)\right)\right)\subset B_1.$$ By Remark \ref{chainrule}, $F\gamma:[t_-,t_+]\to B_0$ is continuously differentiable and by the cone invariance, Lemma \ref{yexpansion}, for every $p\in\gamma$, $T_{F(p)}F\left(\gamma\right)=DF_p\left(T_p\gamma\right)\subset C_{F(p)}$. Suppose now that $F(\gamma)$ is not a graph. Then, there exists $\tilde p\in F(\gamma)$ such that the tangent vector of $F(\gamma)$ in $\tilde p$ is of the form $(0,\Delta b)$ which is not in the cone $C_{\tilde p}$. This contradict the cone invariance. We proved that $F(\gamma)\in\Gamma_0$. Because $F$ is topologically hyperbolic we have that $F(\gamma)\cap\partial_{\pm}\neq\emptyset$. Hence $F(\gamma)\in\Gamma$. \end{proof} \begin{cor}\label{Fgamma0} If $\gamma\in\Gamma_0$, then $F(\gamma\cap D_{\delta})\in\Gamma_0$. \end{cor} \begin{cor}\label{Finj}
If $\gamma\in\Gamma$, then $F_{|\gamma\cap D_{\delta}}$ is injective. \end{cor} The proof of Theorem \ref{InvMan} is divided into $2$ steps inspired by the graph transform method. In the following $2$ subsections, discussing these steps, we assume the hypotheses of Theorem \ref{InvMan}.
\subsection{The graph transform} In this section we introduce the graph transform. The invariant manifold of $F$ will be the fixed point of the graph transform.
We define a distance $d$ on $\Omega$. Let $\omega_1,\omega_2\in\Omega$ and $\gamma\in\Gamma$. Let $p_1=(y_1, b_1)=\gamma\cap\omega_1$ and let $p_2=(y_2, b_2)=\gamma\cap\omega_2$. We define the distance $$d\left(\omega_1,\omega_2\right)=\sup_{\gamma\in\Gamma}|y_1-y_2|\leq 1.$$
\begin{lem} $\left(\Omega, d\right)$ is complete. \end{lem} \begin{proof} Notice that the uniform distance on the functions $\hat\omega$ is bounded by $d$. As consequence, any Cauchy sequence $\omega_n$ in $\Omega$ has a limit $\omega$ which is the graph of a continuous function $\hat\omega$. Because $D_{\delta}$ is closed, $\hat\omega\in D_{\delta}$. Moreover, for any $\gamma\in\Gamma$, $\gamma\cap\omega\neq\emptyset$. It remains to prove that this intersection consists of only one point. By contradiction, suppose there is a $\gamma\in\Gamma$ such that $\gamma\cap\omega\supset \left\{p_1,p_2\right\}$. By Lemma \ref{perturbation}, there exists $\tilde\gamma\in\Gamma$ such that, for $n$ large enough, $\tilde\gamma\cap\omega_n$ contains at least two points. This is a contradiction and hence $\omega\in\Omega$. \end{proof}
We are ready to define the graph transform $T:\Omega\to \Omega$. Let $\omega\in\Omega$, $b\in U_{\delta}$ and $\gamma_b$ the (almost) horizontal curve at $b$, given by $\hat\gamma_b(t)=(t,b)$. By Lemma \ref{Fgamma}, we know that $F\gamma_b\in\Gamma$, hence $F\gamma_b\cap\omega=\left\{\tilde p\right\}$ and by Corollary \ref{Finj}, $F^{-1}(\omega)$ is the graph of a function. We call this graph ${T\omega}$.
\begin{lem}
$T{\omega}\in\Omega$. \end{lem} \begin{proof} Let $\gamma\in\Gamma$. We have to prove that $\gamma\cap T\omega$ is a point. By Lemma \ref{Fgamma}, $F\gamma\in\Gamma$, hence $F\gamma\cap\omega=\left\{\tilde p\right\}$. As a consequence, $$\gamma\cap T\omega=F^{-1}(\tilde p),$$ which is a unique point by Corollary \ref{Finj}. \end{proof} The graph transform $T:\Omega\to \Omega$ is defined by $\omega\mapsto T\omega$. \begin{lem} For $\delta>0$ small enough, $T$ is a contraction. \end{lem} \begin{proof}
Let $\omega_1,\omega_2\in\Omega$ and $\gamma\in\Gamma$. Denote by $p_1=(y_1,b_1)=\gamma\cap T\omega_1$, by $p_2=(y_2,b_2)=\gamma\cap T\omega_2$, by $\tilde p_1=(\tilde y_1,\tilde b_1)=F\gamma\cap \omega_1$ and by $\tilde p_2=(\tilde y_2,\tilde b_2)=F\gamma\cap \omega_2$.
By Lemma \ref{yexpansion}, for $\delta$ small enough, $$|y_1-y_2|\leq {2Ey} |\tilde y_1-\tilde y_2| \leq\frac{1}{2} |\tilde y_1-\tilde y_2|,$$ and as a consequence
$$d\left(T\omega_1,T\omega_2\right)\leq\frac{1}{2} d\left(\omega_1,\omega_2\right).$$
\end{proof}
The graph transform is then a contraction on a complete space. It follows the next proposition:
\begin{prop}
$T$ has a unique fixed point $\omega^*\in\Omega$.
\end{prop} \begin{lem}\label{omegaW}
Let $\omega^*$ be the fixed point of $T$. Then $$\omega^*=W=\left\{p\in D_{\delta} | \forall n\in{\mathbb N}\text{ }F^n(p)\in D_{\delta}\right\}.$$ \end{lem} \begin{proof}
Take $p\in\omega^*\subset D_{\delta}$. Because $\omega^*$ is a fixed point for $T$, $\omega^*=F^{-1}\omega^*$. As a consequence, $F(p)\in\omega^*\subset D_{\delta}.$ Hence $\omega^*\subset W$. Take now $p=(y,b)\in W$. Notice that, for all $n\in{\mathbb N}$, $p_n=F^n(p)=(y_{n},b_n)\in D_{\delta}$. Let $\hat p=(\hat y,b)\in \omega^*\subset D_{\delta}$ with $\hat y=\omega^*(b)$. Observe that, because $\omega^*$ is a fixed point for $T$, for all $n\in{\mathbb N}$, $\hat p_n=F^n(\hat p)=(\hat y_n,b_n)\in\omega^*\subset D_{\delta}$. In particular $\hat y_n\leq \frac{1}{4}$. We want to prove that $p=\hat p\in\omega^*$. Suppose not. Let $\gamma$ be the line segment between $p$ and $\hat p$ and let $\gamma_n=F^n\gamma$. Observe that $\gamma_n\subset D_{\delta}$. By Lemma \ref{yexpansion}, for all $n\in{\mathbb N}$, $$|y_n-\hat y_n|\geq 2^n |y-\omega^*(b)|.$$ We get to a contradiction, noticing that, for $n$ big enough, either $p_n$ or $\hat p_n$ is not in $D_{\delta}$.
\end{proof}
\subsection{Differentiability} In this section we prove that $W=\omega^*$ is a ${{\mathcal C}^1}$ codimension one manifold. This will conclude the proof of Theorem \ref{InvMan}.
A {\it plane} is a codimension one subspace of $\mathbb{R}\times B_0$ which is a the graph of a functional $b^*\in \text{Dual}(B_0)$. We identify the plane with the corresponding functional $b^*$. In particular, $\text{Dual}(B_0)$ is the space of planes and carries a corresponding complete distance $d^*_{B_0}$. Fix a constant $\chi>0$. \begin{defin} Let $p=(y,b)\in\omega^*$. A plane $V_p$ is admissible for $p$ if it has the following properties: \begin{enumerate}
\item if $\left(\Delta y, \Delta b\right)\in V_p$ then $|\Delta y|\leq
\chi y\|\Delta b\|_0$, \item $V_p$ depends continuously on $p$ with respect to $d^*_{B_0}$. \end{enumerate} The set of planes admissible for $p$ is denoted by $\text{Dual}_p(B_0)$. A plane field is a continuous assignment $V :p\mapsto V_p$ where $V_p$ is admissible for $p$. The set of plane fields is denoted by $\Omega_1$. \end{defin} \begin{rem}\label{intlinecone} Observe that, for $\delta>0$ small enough, if $p\in\omega^*$ and $V_p$ is an admissible plane for $p$, then for all straight lines $\gamma$ with direction in $C_p$, $\gamma\cap V_p$ is a unique point. \end{rem} We fix $p=(y,b)\in\omega^*$ and we define a distance on $\text{Dual}_p(B_0)$. Let $V_p,V'_p\in \text{Dual}_p(B_0)$ and let $\gamma$ be a straight line with direction in $C_p$ not necessarily passing trough the origin. Let $\delta>0$ be small enough such that Remark \ref{intlinecone} can be applied. Denote by $\Delta q=(\Delta y,\Delta b)=V_p\cap\gamma$, $\Delta q'=(\Delta y',\Delta b')=V'_p\cap\gamma$. The distance $d_{1,p}$ on $\text{Dual}_p(B_0)$ is defined as
$$d_{1,p}\left(V_p,V'_p\right)=\sup_{\gamma}\frac{|\Delta y-\Delta y'|}{\min\left\{|\Delta q|,|\Delta q'|\right\}}.$$ This is a complete metric. The distance $d_1$ on $\Omega_1$ is the corresponding uniform distance. Namely, if $V,V'\in \Omega_1$ then $$d_1\left(V,V'\right)=\sup_{p}d_{1,p}\left(V_p,V'_p\right).$$
\begin{lem} For $\delta>0$ small enough, $d_1$ is a distance. \end{lem} \begin{proof}
We only have to show that $d_1$ is bounded. We use the notations from the definition of $d_1$. Without lose of generality we assume $|\Delta q|\leq|\Delta q'|$ and we estimate ${|\Delta y|+|\Delta y'|}/{|\Delta q|}$. By the definition of $V_p$ and $V'_p$ we get \begin{equation}\label{admisplane}
|\Delta y|+|\Delta y'|\leq \chi y \left(\|\Delta b\|_0+\|\Delta b'\|_0\right). \end{equation} Define $\Delta=\Delta b'-\Delta b$. Because the direction of $\gamma$ is in the cone $C_p$ and by (\ref{admisplane}), we get \begin{equation*}
\|\Delta\|_0\leq\chi\frac{y^{1-\kappa}}{\theta }\left(\|\Delta b\|_0+\|\Delta b'\|_0\right), \end{equation*} and \begin{equation*}
\|\Delta b'\|_0\leq \|\Delta b\|_0 +\chi\frac{y^{1-\kappa}}{\theta }\left(\|\Delta b\|_0+\|\Delta b'\|_0\right). \end{equation*} As a consequence, for $\delta$ small enough, \begin{equation}\label{admisplane1}
\frac{1}{2}\leq\frac{\|\Delta b'\|_0}{\|\Delta b\|_0}\leq 2. \end{equation} Observe that, for $\delta$ small enough, \begin{equation}\label{admisplane2}
|\Delta q|\geq\frac{1}{2}\|\Delta b\|_0, \end{equation} and by (\ref{admisplane}), (\ref{admisplane1}) and (\ref{admisplane2}) we get
$$\frac{|\Delta y|+|\Delta y'|}{|\Delta q|}\leq 2.$$ \end{proof} \begin{lem} For $\delta>0$ small enough, $\left(\Omega_1, d_1\right)$ is complete. \end{lem} \begin{proof} Let $V(n)\in\Omega_1$ be a $d_1$-Cauchy sequence. Then, for all $p$, $V_p(n)$ is a $d_{1,p}$-Cauchy sequence. Hence $V_p(n)$ converges to $V_p$ in $\text{Dual}_p(B_0)$. Let $V:\omega^*\to \text{Dual}(B_0)$ be the point-wise limit. Observe now that, for $\delta$ small enough, $$d^*_{B_0}\left(V_p,V'_p\right)\leq 2 d_{1,p}\left(V_p,V'_p\right),$$ for all $p\in\omega^*$. Hence $V(n)$ is also a Cauchy sequence in the space of continuous functions $\omega^*\to \text{Dual}(B_0)$ carrying the uniform distance corresponding to $d^*_{B_0}$. This space is complete, hence $V$ is continuous. \end{proof} The next step is to define the plane field transform $T_1: \Omega_1\to \Omega_1$. Let $V\in \Omega_1$ and $p\in\omega^*$, then $$T_1V_p=DF^{-1}_p\left(V_{F(p)}\right).$$ \begin{lem} For $\delta>0$ small enough and for $\chi>0$ large enough, $T_1V\in\Omega_1$, for all $V\in\Omega_1$. \end{lem} \begin{proof}
We prove that, for all $p\in\omega^*$, $T_1V_p$ is an admissible plane for $p$. Consider the vectors $\left(\Delta y, \Delta b\right)\in T_1V_p$ and $DF_p\left(\Delta y, \Delta b\right)=(\Delta\tilde y, \Delta\tilde b )\in V_{F(p)}$. From (\ref{cond2}) and from the fact that $V_{F(p)}$ is admissible in ${F(p)}$ we get $$\frac{1}{Ey}|\Delta y|+ \chi\tilde y^{1-\kappa}O\left(|\Delta y|\right)\leq \chi\tilde y O\left(\|\Delta b\|_0\right)+O\left(\|\Delta b\|_0\right).$$ As consequence, for $\delta>0$ small enough and $\chi>0$ large enough, we get
$$|\Delta y|\leq \chi y \|\Delta_b\|_0.$$ In particular $DF_p^{-1}\left(V_{F(p)}\right)$ is a codimension one subspace and this implies that $DF_p$ is transversal to $V_{F(p)}$. This transversality gives that $T_1V_p$ depends continuously on $p$. \end{proof} \begin{lem} For $\eta>0$ small enough, $T_1$ is a contraction. \end{lem} \begin{proof} We use the notations from the definition of $d_1$ and we denote by $\Delta\tilde q=DF_p\left(\Delta q\right)=\left(\Delta\tilde y, \Delta\tilde b\right)$ and by $\Delta\tilde q'=DF_p\left(\Delta q'\right)=\left(\Delta\tilde y', \Delta\tilde b'\right)$. By Lemma \ref{lemma1} and Lemma \ref{yexpansion} we get
$$\frac{|\Delta y'-\Delta y|}{\min\left\{|\Delta q|, |\Delta q'|\right\}}= O\left(\frac{y^2}{\tilde y^{\kappa}}\right)\frac{|\Delta\tilde y'-\Delta\tilde y|}{\min\left\{|\Delta\tilde q|, |\Delta\tilde q'|\right\}}\leq \frac{1}{2}d_{1, F(p)}\left(V_{F(p)},V'_{F(p)}\right),$$ when $\eta$ is small enough. Hence $d_{1}\left(T_1V, T_1V'\right)\leq\frac{1}{2}d_{1}\left(V, V'\right).$ \end{proof} \begin{prop} For $\delta>0$ small enough and $\chi>0$ large enough, $T_1$ has a unique fixed point $V^*\in\Omega_1$. \end{prop}
Although $V^*_p$ is a subspace of $\mathbb{R}\times B_0$, in the sequel we abuse the notation by denoting the set $\left\{p+v |v\in V^*_p\right\}$ also by $V^*_p$. Let $p\in\omega^*$ and take $\gamma\in\Gamma$ close enough to $p$ such that $\gamma\cap\omega^*=\left\{p+\Delta q=p+\left(\Delta y,\Delta b\right)\right\}$ and $\gamma\cap V^*_p=\left\{p+\Delta q'=p+\left(\Delta y',\Delta b'\right)\right\}$. We define by $$A=\sup_{p}\limsup_{\gamma\to p}\frac{|\Delta y-\Delta y'|}{|\Delta q|}.$$ Observe that $V^*$ describes the tangent bundle of $\omega^*$ if and only if $A=0$. \begin{lem} For $\delta>0$ small enough, $A\leq 1$. \end{lem} \begin{proof} Let $p=(y,b)$ and denote by $\Delta=\Delta b'-\Delta b$. From the definition of admissible planes we get \begin{equation}\label{fin1}
|\Delta y'|\leq\chi y\left(\|\Delta b'\|_0\right), \end{equation} and by the definition of almost horizontal lines, \begin{equation}\label{fin2}
\frac{1}{2}\theta y^{\kappa}\|\Delta \|_0\leq |\Delta y|+ |\Delta y'|, \end{equation} when $\gamma$ is close enough to $p$. Consider the straight line $L$ going to $p$ and $p+\Delta q$. $L$ intersect $\omega^*$ in two points. Hence, $L$ is not an almost horizontal curve and $\Delta q\notin C_p$. As consequence, when $\gamma$ is close enough to $p$. \begin{equation}\label{fin3}
|\Delta y|\leq 2\theta y^{\kappa}\|\Delta b \|_0. \end{equation} By (\ref{fin1}), (\ref{fin2}) and (\ref{fin3}) we get
$$\|\Delta \|_0\leq \frac{2\theta+\chi y^{1-\kappa}}{\frac{1}{2}\theta-\chi y^{1-\kappa}}\|\Delta b\|_0,$$ and \begin{equation}\label{fin4}
\|\Delta b'\|_0\leq 6\|\Delta b\|_0, \end{equation} when $\delta>0$ is small enough. By (\ref{fin1}), (\ref{fin3}) and (\ref{fin4}) we get
$$\frac{|\Delta y-\Delta y'|}{|\Delta q|}\leq\frac{|\Delta y|+|\Delta y'|}{\|\Delta b \|_0+|\Delta y|}\leq 12\chi y+4\theta y^{\kappa}\leq 1,$$ for $\delta>0$ is small enough. \end{proof} Lemma \ref{omegaW} and the following proposition conclude the proof of Theorem \ref{InvMan}. Namely $W=\omega^*$ is a ${{\mathcal C}^1}$ manifold. \begin{figure}
\caption{Notation for the proof of Proposition \ref{tgV}}
\label{FigP}
\end{figure} \begin{figure}
\caption{Notation for the proof of Proposition \ref{tgV}}
\label{FigI}
\end{figure} \begin{prop}\label{tgV} For $\delta>0$ and $\eta>0$ small enough, each point $p\in \omega^*$ has a tangent plane $T_p\omega^*=V^*_p$. \end{prop} \begin{proof} One has to show that $A=0$. We use the notation from the definition of $A$ and we introduce the following. \begin{eqnarray*} F(p)&=&(\tilde y, \tilde b),\\ F\left(\gamma\right)\cap\omega^*&=&\tilde q=F(p)+\Delta\tilde q=F(p)+\left(\Delta\tilde y,\Delta\tilde b\right),\\ F\left(\gamma\right)\cap V^*_{F(p)}&=&z=F(p)+\Delta z,\\ F\left(q'\right)&=&\tilde q'=F(p)+\Delta\tilde q'=F(p)+\left(\Delta\tilde y',\Delta\tilde b'\right),\\ z-\tilde q&=&\left(\Delta h_1,\Delta v\right),\\ \tilde q'-z&=&\left(\Delta h,\Delta v_1\right),\\ \Delta\tilde q'&=&DF_p\left(\Delta q'\right)+\Delta\epsilon,\\ DF_p\left(\Delta q'\right)-\Delta z&=&\left(\Delta h_2,\Delta v_2\right). \end{eqnarray*} For curves $\gamma\in\Gamma$ close enough to $p$ and by Lemma \ref{lemma1}, the differentiability of $F$, Lemma \ref{yexpansion}, $V^*_{F(p)}$ is admissible, $\gamma\in\Gamma$ and $F(\gamma)\in\Gamma$, we get \begin{eqnarray}\label{L1}
|\Delta\tilde q|&=&O\left(\frac{y}{\tilde y^{\kappa}}\right)|\Delta q|,\\\label{L2}
|\Delta\epsilon|&=&o\left(|\Delta q'|\right),\\ \label{L3}
|\Delta h_1|+|\Delta h|&\geq&\frac{1}{3y}|\Delta y-\Delta y'|,\\ \label{L7}
|\Delta h_2|&\leq& 2\chi\tilde y\|\Delta v_2\|_0,\\ \label{L4}
\frac{1}{2}\theta y^{\kappa}\|\Delta b'-\Delta b\|_0 &\leq& |\Delta y-\Delta y'|,\\ \label{L5}
\frac{1}{2}\theta \tilde y^{\kappa}\|\Delta v_1\|_0 &\leq& |\Delta h|,\\ \label{L6}
\|\Delta v_2-\Delta v_1\|_0 &\leq& |\Delta\epsilon |. \end{eqnarray} By (\ref{L7}), (\ref{L6}) and (\ref{L5}) we have
$$|\Delta h|\leq |\Delta\epsilon|+2\chi \tilde y\left(\frac{2}{\theta\tilde y^{\kappa}}|\Delta h|+|\Delta\epsilon|\right).$$ Hence, for $\delta>0$ small enough, \begin{equation}\label{L8}
|\Delta h|\leq 2|\Delta\epsilon|. \end{equation} By (\ref{L2}) and (\ref{L4}), we have \begin{equation*}
|\Delta\epsilon|=o\left(|\Delta q|+4\left(1+\frac{1}{\theta y^{\kappa}}\right)|\Delta y-\Delta y'|\right)=o\left(|\Delta q|\left(1+4\left(1+\frac{1}{\theta y^{\kappa}}\right) A\right)\right). \end{equation*} Hence \begin{equation}\label{L9}
|\Delta\epsilon|=o\left(|\Delta q|\right). \end{equation} By (\ref{L3}), (\ref{L1}), (\ref{L8}) and (\ref{L9}) we get $$
\frac{|\Delta y-\Delta y'|}{|\Delta q|}\leq O\left(3\frac{y^2}{\tilde y^{\kappa}}\right)\frac{|\Delta h_1|}{|\Delta\tilde q|}+6y\frac{|\Delta\epsilon|}{|\Delta q|}=\leq O\left(3\frac{y^2}{\tilde y^{\kappa}}\right)\frac{|\Delta h_1|}{|\Delta\tilde q|}+o(1). $$ As consequence, $$
\limsup_{\gamma\to p}\frac{|\Delta y-\Delta y'|}{|\Delta q|}\leq O\left(3\frac{y^2}{\tilde y^{\kappa}}\right) A. $$ Finally, from (\ref{cond4}) we get $A=0$ for $\eta$ small enough. \end{proof} \begin{lem}\label{pullbacklemma} Let $W\subset\mathbb{R}\times B_1$ with the following properties: \begin{itemize} \item[-] $W$ is a ${{\mathcal C}^1}$ codimension one manifold, \item[-] for all $p\in W$, the tangent space $T_pW$ extends to a plane in $\mathbb{R}\times B_0$ also denoted by $T_pW$, \item[-] the dependence of $p$ to $T_pW$ is continuous. \end{itemize} Let $V$ be a normed vector space and $H:V\to\mathbb{R}\times B_1$ be such that \begin{itemize} \item[-] $H:V\to\mathbb{R}\times B_0$ is continuously differentiable, \item[-] if $H(v)\in W$, then $DH_v\pitchfork T_{H(v)}W$. \end{itemize} Then $H^{-1}(W)$ is a ${{\mathcal C}^1}$ codimension one manifold. \end{lem} \begin{proof} Let $v\in H^{-1}(W)$ and $p=H(v)$. Observe that $DH^{-1}_v(T_{p}W)$ is a codimension one subspace which depends continuously on $v$. We denote it by $T_v H^{-1}(W)$. By a similar argument as in the proof of Proposition \ref{tgV}, it follows that $T_v H^{-1}(W)$ is the tangent space at $v$ to $H^{-1}(W)$. As consequence $H^{-1}(W)$ is a ${{\mathcal C}^1}$ codimension one manifold. \end{proof}
\begin{rem} Observe that Lemma \ref{pullbacklemma} is not the usual pull back lemma. Namely, $H:V\to\mathbb{R}\times B_0$ is continuously differentiable but $W\subset R\times B_0$ has infinite codimension. It has codimension one in $\mathbb{R}\times B_1$. \end{rem}
\section{Renormalization of Cherry dynamics} \label{section:renorm} This section presents the dynamical systems of interest, namely the circle maps with a flat interval and the action of the renormalization operator. Because of their close connection with Cherry flows, we call circle maps with a flat interval, Cherry maps. \subsection{The class of functions} We fix $1<\ell<2$ and we denote by $\Sigma^{(X)}$ the simplex \begin{equation*}
\Sigma^{(X)}=\{(x_1,x_2,x_3,x_4,s)\in\mathbb R^5 | x_1<0<x_3<x_4<1, 0<x_2<1\text{ and } 0<s<1 \}, \end{equation*} by $\text{ Diff }^r([0,1])$ the space of ${{\mathcal C}^r}$, $r\ge 2$, orientation preserving diffeomorphisms of $[0,1]$. The space of ${{\mathcal C}^r}$ circle maps with a flat interval is denoted by $${\mathscr L}^{(X,r)}=\Sigma^{(X)}\times \text{Diff }^r([0,1])\times \text{Diff }^r([0,1])\times \text{Diff }^r([0,1]).$$ The space ${\mathscr L}^{(X,r)}$ is equipped by a distance which is the sum of the usual distances: the euclidian distance on $\Sigma^{(X)}$ and the sum of the ${{\mathcal C}^r}$ distance on $\text{Diff }^r([0,1])$. Usually we will suppress the index indicating the smoothness of the maps and simply use the notation ${\mathscr L}^{(X)}$. A point $f=(x_1,x_2,x_3,x_4, s,\varphi,\varphi^{l},\varphi^{r})\in {\mathscr L}^{(X)}$ represents the following interval map $f:[x_1,1]\to[x_1,1]$:
\begin{center} \begin{equation}\label{eqfun} f(x):=\left\{
\begin{aligned}
& (1-x_2)q_s\circ\varphi\left(1-\frac{x}{x_1}\right)+x_2 & \text{ if } x\in[x_1,0) \\
& x_1\left(\varphi^{l}\left(\frac{x_3-x}{x_3}\right)\right)^{\ell} & \text{ if } x\in[0,x_3] \\
& 0 & \text{ if } x\in(x_3,x_4)\\ &x_2\left(\varphi^{r}\left(\frac{x-x_4}{1-x_4}\right)\right)^{\ell} & \text{ if } x\in[x_4,1] \\
\end{aligned} \right. \end{equation} \end{center} and $q_s:[0,1]\to [0,1]$ is a diffeomorphic part of $x^{\ell}$ parametrized by $s\in (0,1)$, namely $$q_s(x)=\frac{\left[(1-s)x+s\right]^{\ell}-s^{\ell}}{1-s^{\ell}}.$$ The meaning of the parts of $f$ is illustrated in figures \ref{Fig1}, \ref{Fig1.1}, \ref{Fig1.2} and \ref{Fig1.3}. The role of $q_s$ will become clear in the study of the asymptotical behavior of the renormalization operator, see Section \ref{asymrenormalization}. \begin{figure}
\caption{A function in ${\mathscr L}^{(X)}$}
\label{Fig1}
\end{figure}
\begin{figure}
\caption{The left branch of a function in ${\mathscr L}^{(X)}$}
\label{Fig1.1}
\end{figure} \begin{figure}
\caption{The central branch of a function in ${\mathscr L}^{(X)}$}
\label{Fig1.2}
\end{figure} \begin{figure}
\caption{The right branch of a function in ${\mathscr L}^{(X)}$}
\label{Fig1.3}
\end{figure}
Depending on the situation, we use different coordinate systems. Given a system $f=(x_1,x_2,x_3,x_4, s, \varphi,\varphi^{l},\varphi^{r})\in {\mathscr L}^{(X)}$ we represent it in $S-$coordinates as follows: $f=(S_1,S_2,S_3,S_4, S_5, \varphi,\varphi^{l},\varphi^{r})$ where $$
\begin{aligned}
&S_1=\frac{x_3-x_2}{x_3}, & S_2=\frac{1-x_4}{1-x_2}, & &S_3=\frac{x_3}{1-x_4}, && S_4=\frac{x_2}{-x_1}, && S_5=s^{\ell-1}. \end{aligned} $$ As a consequence, we define \begin{equation*}
\Sigma^{(S)}=\{(S_1,S_2,S_3,S_4,S_5)\in\mathbb R^5 | 0<S_2, 0<S_3, 0<S_4\text{ and } 0<S_5<1 \}, \end{equation*} and $${\mathscr L}^{(S)}=\Sigma^{(S)}\times \text{Diff }^r([0,1])\times \text{Diff }^r([0,1])\times \text{Diff }^r([0,1]).$$ Similarly, given a system $f=(S_1,S_2,S_3,S_4, S_5, \varphi,\varphi^{l},\varphi^{r})\in {\mathscr L}^{(S)}$ we represent it in $Y-$coordinates as follows: $f=(y_1,y_2,y_3,y_4, y_5, \varphi,\varphi^{l},\varphi^{r})$ where $$
\begin{aligned} & y_1=S_1, & &y_2=\log S_2, && y_3=\log S_3, && y_4=\log S_4, & y_5=\log S_5. \end{aligned} $$ As a consequence, we define \begin{equation*}
\Sigma^{(Y)}=\{(y_1,y_2,y_3,y_4,y_5)\in\mathbb R^5 | y_5<0 \}, \end{equation*} and $${\mathscr L}^{(Y)}=\Sigma^{(Y)}\times \text{Diff }^r([0,1])\times \text{Diff }^r([0,1])\times \text{Diff }^r([0,1]).$$ Observe that these coordinates changes induce diffeomorphisms between ${\mathscr L}^{(X)}$, ${\mathscr L}^{(S)}$ and ${\mathscr L}^{(Y)}$. In particular by explicit calculations the following lemma holds. \begin{lem}\label{xtos} The inverse of $(x_1, x_2, x_3, x_4)\to (S_1, S_2, S_3, S_4)$ is given by \begin{enumerate}
\item $x_1=-\frac{S_3(1-S_1)S_2}{(1+S_3(1-S_1)S_2)S_4}$,
\item $x_2=\frac{S_3(1-S_1)S_2}{1+S_3(1-S_1)S_2}$,
\item $x_3=\frac{S_3S_2}{1+S_3(1-S_1)S_2}$,
\item $x_4=1-\frac{S_2}{1+S_3(1-S_1)S_2}$. \end{enumerate} \end{lem}
From the context and the notation it will be clear which parametrization of our space we are using. The space will then be simply denoted by ${\mathscr L}$ instead of ${\mathscr L}^{(X)}$, ${\mathscr L}^{(S)}$ or ${\mathscr L}^{(Y)}$. If we want to specify that our maps are $\mathcal C^r$ smooth, then we use the notation ${\mathscr L}^{r}$. When needed we use the notation $x_3(f)$ to denote the $x_3$ coordinate of $f$. Similarly for all others. \subsection{Renormalization} In this section we define the renormalization operator. The renormalization scheme that we use is adapted to study circle maps with Fibonacci rotation number. For basic concepts concerning circle maps, see \cite{H79}. \begin{defin} A map $f\in{\mathscr L}$ is renormalizable if $0<x_2<x_3$. The space of renormalizable maps is denoted by ${\mathscr L}_0$. \end{defin} Let $f\in{\mathscr L}_0$ and let $\text{pre}R(f)$ be the first return map of $f$ to the interval $[x_1,x_2]$. Let us consider the function $h:[x_1,x_2]\to[0,1]$ defined as $h(x)={x}/{x_1}$ for all $x\in [x_1,x_2]$. Then the function \begin{equation*} Rf:=h\circ \text{pre}R(f)\circ h^{-1} \end{equation*} is again a map in ${\mathscr L}$. Notice that $Rf$ is nothing else than the first return map of $f$ to the interval $[x_1,x_2]$ rescaled and flipped. This define the renormalization operator $$R:{\mathscr L}_0\to {\mathscr L}.$$ \begin{defin} A map $f\in{\mathscr L}$ is $\infty$-renormalizable if for every $n\geq 0$, $R^n f\in{\mathscr L}_0$. The set of $\infty$-renormalizable functions will be denoted by $\mathscr W\subset{\mathscr L}$. The maps in ${\mathscr W}$ are called Fibonacci maps. \end{defin} \begin{rem} Observe that, if $f\in{\mathscr W}$, by identifying $x_1$ with $1$ we obtain a map of the circle having Fibonacci rotation number. \end{rem} In Defintion \ref{zoomop} we introduce the concept of the zoom operator needed later to describe the action of the renormalization operator on the space of diffeomorphisms. \begin{defin}\label{zoomop} Let $I=[a,b]\subset[0,1]$. The \emph{zoom} operator $Z_{I}:\text{ Diff }^0([0,1])\to \text {Diff }^0([0,1])$ is defined as $$ Z_{I}\varphi(x)=\frac{\varphi((b-a)x+a)-\tilde a}{\tilde b-\tilde a} $$ where $\varphi\in \text{ Diff }^0([0,1])$, $x\in[0,1]$, $\tilde a=\varphi(a)$ and $\tilde b=\varphi(b)$. \end{defin}
The following two lemmas are a direct consequence of the definition of the renormalization operator. \begin{lem}\label{changexs} Let $f=(x_1,x_2,x_3,x_4, s, \varphi,\varphi^{l},\varphi^{r})\in {\mathscr L}_0$ and let\\ $Rf=(\tilde x_1,\tilde x_2,\tilde x_3, \tilde x_4, \tilde s, \tilde \varphi,\tilde\varphi^{l},\tilde\varphi^{r})$. Then $$ \begin{aligned}
1.&\text{ }\tilde x_{1}=\frac{x_2}{x_1},\\
2.&\text{ }\tilde x_{2}=\left(\varphi^{l}\left(\frac{x_3-x_2}{x_3}\right)\right)^{\ell},\\
3.&\text{ }\tilde x_3=1-\left(\varphi^{-1}\circ q_s^{-1}\right)\left(\frac{x_4-x_2}{1-x_2}\right),\\
4.&\text{ }\tilde x_4=1-\left(\varphi^{-1}\circ q_s^{-1}\right)\left(\frac{x_3-x_2}{1-x_2}\right),\\
5.&\text{ }\tilde s=\varphi^{l}\left(\frac{x_3-x_2}{x_3}\right),\\
6.&\text{ }\tilde \varphi =Z_{\left[\frac{x_3-x_2}{x_3},1\right]}\varphi^{l},\\
7.&\text{ } \tilde\varphi^{l}=\varphi^{r}\circ Z_{\left[1-\tilde x_3,1\right]}\left( q_s\circ \varphi\right),\\
8.&\text{ }\tilde \varphi^{r}=Z_{\left[0, \frac{x_3-x_{2}}{x_3}\right]}\varphi^{l}\circ Z_{\left[0,1-\tilde x_4\right]} \left(q_s\circ \varphi\right).
\end{aligned} $$ \end{lem} \begin{lem}\label{ss} Let $f=(S_1,S_2,S_3,S_4, S_5, \varphi,\varphi^{l},\varphi^{r})\in {\mathscr L}_0$ and \\ $Rf=(\tilde S_1, \tilde S_2, \tilde S_3, \tilde S_4, \tilde S_5, \tilde \varphi,\tilde\varphi^{l},\tilde\varphi^{r})$. Then \begin{eqnarray*} 1.&\tilde S_{1}&=1-\left(\frac{\ell S_{1}^{\ell}}{S_{2}}\right)\cdot\left[\frac{S_2}{1-\left(\varphi^{-1}\circ q_s^{-1}\right)\left(1- S_2\right)}\cdot\frac{\left(\varphi^{l}\left( S_1\right)\right)^{\ell}}{\ell S_{1}^{\ell}}\right],\\ 2.& \tilde S_{2}&=\frac{ S_{1}S_{2}S_{3}}{\ell S_{5}}\cdot\left[\frac{\ell S_5 \left(\varphi^{-1}\circ q_s^{-1}\right)\left(S_{1}S_{2}S_{3}\right)}{S_{1}S_{2}S_{3}}\cdot\frac{1}{1-\left(\varphi^{l}\left( S_1\right)\right)^{\ell}}\right],\\ 3.& \tilde S_{3}&=\frac{S_{5}}{ S_{1}S_{3}}\cdot\left[\frac{S_{1}S_{3} \left(1-\left(\varphi^{-1}\circ q_s^{-1}\right)\left(1-S_{2}\right)\right)}{S_5 \left(\varphi^{-1}\circ q_s^{-1}\right)\left(S_{1}S_{2}S_{3}\right)}\right],\\
4.& \tilde S_{4}&=\frac{S_{1}^{\ell}}{S_{4}}\cdot\left[\left(\frac{\varphi^{l}\left( S_1\right)}{S_{1}}\right)^{\ell}\right],\\ 5.& \tilde S_{5}&=S_{1}^{\ell-1}\cdot\left[\left(\frac{\varphi^{l}\left( S_1\right)}{S_{1}}\right)^{\ell -1}\right],\\ 6.&\tilde \varphi &=Z_{\left[S_1,1\right]}\varphi^{l},\\ 7.& \tilde\varphi^{l}&=\varphi^{r}\circ Z_{\left[q_s^{-1}\left(1-S_{2}\right),1\right]}\left( q_s\right)\circ Z_{\left[\varphi^{-1}\circ q_s^{-1}\left(1-S_{2}\right),1\right]}\left( \varphi\right),\\
8.& \tilde\varphi^{r}&=Z_{\left[0, S_1\right]}\left(\varphi^{l}\right)\circ Z_{\left[0,q_s^{-1}\left(S_1S_{2}S_3\right)\right]} \left(q_s\right)\circ Z_{\left[0,\varphi^{-1}\circ q_s^{-1}\left(S_1S_{2}S_3\right)\right]} \left(\varphi\right).
\end{eqnarray*} \end{lem} \subsection{Fibonacci rotation number}\label{Fib} In the sequel we fix the critical exponent $1<\ell<2$ and we use the following notation. If $f\in{\mathscr W}$ the coordinates of $R^n(f)$ are indicated with a lower index $n$, for example $$x_{3,n}=x_{3}(R^n f),$$ and $S_{2,n}=S_{2}(R^n f)$. Similarly for the other coordinates.
Moreover, let $U_f=[x_3, x_4]$ be the flat interval of $f$. Observe that $R^nf_{|[0,1]}$ is a rescaled version of $f^{q_n}$ where the sequence $\left(q_n\right)_{n\in{\mathbb N}}$ is the Fibonacci sequence satisfying that: $q_1=1$, $q_2=2$ and for all $n\geq 3$, $q_n=q_{n-1}+q_{n-2}$. Observe that if $R^nf=(x_{1,n},x_{2,n},x_{3,n},x_{4,n}, s_n,\varphi_{n},\varphi_{n}^l,\varphi_{n}^r)$ then the points $x_{1,n},x_{2,n},x_{3,n},x_{4,n}$ correspond to dynamical points of the original function $f$. Namely, \begin{itemize} \item[-] $\hat x_{1,n}=f^{q_n+1}(x_3)=f^{q_n+1}(x_4)=f^{q_n}(0)$, \item[-] $\hat x_{2,n}=f^{q_{n+1}+1}(x_3)=f^{q_{n+1}+1}(x_4)=f^{q_{n+1}}(0)$, \item[-] $\hat x_{3,n}=f^{-q_{n}+1}(x_3)$, \item[-] $\hat x_{4,n}=f^{-q_{n}+1}(x_4)$. \end{itemize}
\section{The asymptotics of renormalization}\label{asymrenormalization} This section explores the asymptotic behaviour of the renormalization operator. Let $f\in {\mathscr W}\subset{\mathscr L}^{(Y)}$. For all $n\in{\mathbb N}$ we define $$w_n(f)=\left(\begin{matrix} y_2(R^nf)\\y_3(R^nf)\\y_4(R^nf)\\y_5(R^nf) \end{matrix} \right),$$ $\varphi_{n}(f)=\varphi(R^nf)$, $\varphi^l_{n}(f)=\varphi^l(R^nf)$ and $\varphi^r_{n}(f)=\varphi^r(R^nf)$. \begin{defin} Let $\varphi: N\to N$ be a ${{\mathcal C}^1}$ map where $N$ is an interval. If $T\subset N$ is an interval such that $D\varphi(x)\neq 0$ for every $x\in T$, we define the \emph{distortion} of $\varphi$ in $T$ as:
$$\text{dist}(\varphi,T)=\sup_{x,y\in T}\log\frac{|D\varphi(x)|}{|D\varphi(y)|}.$$
Here $|D\varphi(x)|$ denoted the norm or absolute value of the derivative of $\varphi$ in $x$. \end{defin} \begin{prop}\label{superformula}
Let $1<\ell<2$. Then there exist $\lambda_u>1$, $|\lambda_s|<1$, $E_u,E_s, E_{-1},w_{fix}\in\mathbb{R}^4$ such that the following holds. Given $f\in {\mathscr W}\subset{\mathscr L}^{(Y)}$ with critical exponent $\ell$ then there exist $C_u(f)<0$, $C_s(f)$ and $C_{-}(f)$ such that for all $n\in{\mathbb N}$ $$w_n(f)=C_u(f)\lambda_u^nE_u+C_s(f)\lambda_s^nE_s+C_-(f)(-1)^nE_-+O\left(e^{\frac{C_u(f)\lambda_u^{n-4}}{\ell}}\right)+w_{fix},$$ and $$\begin{matrix}
\text{dist}(\varphi_{n}(f))&=&O\left(e^{\frac{C_u(f)\lambda_u^{n-3}}{\ell}}\right),\\ \text{dist}(\varphi_{n}^{l}(f))&=&O\left(e^{\frac{C_u(f)\lambda_u^{n-2}}{\ell}}\right), \\
\text{dist}(\varphi_{n}^{r}(f))&=&O\left(e^{\frac{C_u(f)\lambda_u^{n-1}}{\ell}}\right).
\end{matrix} $$ \end{prop} The rest of the section is devoted to prove this proposition. In particular we show that \begin{enumerate} \item $\lambda_s=\frac{\frac{1}{\ell}-\sqrt{\left(\frac{1}{\ell}\right)^2+\frac{4}{\ell}}}{2}\in(-1,0),$ \item $\lambda_u=\frac{\frac{1}{\ell}+\sqrt{\left(\frac{1}{\ell}\right)^2+\frac{4}{\ell}}}{2}>1,$ \item $E_u=\left(1,\frac{-\lambda_u+\ell-1}{\ell\lambda_u(1+\lambda_u)},\frac{1}{1+\lambda_u},\frac{\ell-1}{\ell\lambda_u}\right),$ \item $E_s=\left(1,\frac{-\lambda_s+\ell-1}{\ell\lambda_s(1+\lambda_s)},\frac{1}{1+\lambda_s},\frac{\ell-1}{\ell\lambda_s}\right),$ \item $E_{-}=(0,0,1,0).$ \end{enumerate}
\subsection{Asymptotics of the scaling ratio} We define the following sequence of scaling ratio which plays a main role in the rest of the paper. Let $f\in{\mathscr W}$, the scaling ratio $\alpha_n$ is
$$\alpha_n:=\frac{\left|\left[0, x_{3,n}\right]\right|}{\left|\left[0, x_{4,n}\right]\right|}=\frac{x_{3,n} }{x_{4,n}}=\frac{S_{2,n}S_{3,n}}{1-S_{2,n}+\left(1-S_{1,n}\right)S_{2,n}S_{3,n}}.$$
\begin{theo}\label{alpha} For every $f\in{\mathscr W}$, there exists $\alpha_0<1$ such that $$\alpha_n=O\left(\alpha_0^{\left(\frac{2}{\ell}\right)^{\left(\frac{n}{2}\right)}}\right).$$ \end{theo} \begin{proof} Let $f\in{\mathscr W}\subset{\mathscr L}^r$, $r\geq 2$ and $\pi:\mathbb{R}\to{\mathbb S}^1$ the natural projection with period $1-x_1(f)$. Consider the lift of $f$, $F:\mathbb{R}\to\mathbb{R}$ such that $F(0)=x_1(f)$. Then, there exists $H:\mathbb{R}\to\mathbb{R}$ which is a lift of a circle homeomorphism and satisfies the following properties: \begin{itemize} \item[-] $H(x+1-x_1)=H(x)+1-x_1$,
\item[-] $H_{|[0,1]}=id$, \item[-] $H:[x_1,0]\to [x_1,0]$ is a polynomial diffeomorphism, \item[-] $G=H\circ F\circ H^{-1}$ is a ${{\mathcal C}^r}$ map except in $\pi^{-1}(x_3(f))$ and $\pi^{-1}(x_4(f))$. \end{itemize}
Observe that close to $\pi^{-1}(x_3(f))$ and $\pi^{-1}(x_4(f))$, $G$ has the form of $x^{\ell}$ up to a ${{\mathcal C}^r}$ coordinate change. Let $g:{\mathbb S}^1\to{\mathbb S}^1$ be the projection of $G$. Then $g$ belongs to the class of circle maps studied in \cite{5aut, P1, P2} and that the theorem is valid for $g$ (see Appendix in \cite{P1}). Because $H_{|(0,1)\cup (x_1,0)}$ is a diffeomorpism and $H$ is a conjugacy between $F$ and $G$ we get ${\alpha_n(f)}/{\alpha_n(g)}\to 1$ when $n$ goes to infinity. The theorem follows. \end{proof} \begin{rem}\label{prop2} Observe that, by Proposition 2 in \cite{5aut}, for every $f\in{\mathscr W}$ and for $n\in{\mathbb N}$ large enough, $$ \frac{x_{3,n}}{x_{4,n}-x_{3,n}}\leq 2\alpha_n. $$ \end{rem} \subsection{Asymptotics of the diffeomorphisms} In this section we show that the distortion of the diffeomorphic parts of the renormalizations behave as the sequence $\alpha_n$. As a consequence, the diffeomorphisms tend to identity double exponentially fast. \begin{prop}\label{affdiffeo} Let $f\in {\mathscr W}$, then for all $n\in{\mathbb N}$, $$\begin{matrix}
\text{dist}(\varphi_{n})=O\left(\alpha_{n-2}^{\frac{1}{\ell}}\right), & \text{dist}(\varphi_{n}^{l})=O\left(\alpha_{n-1}^{\frac{1}{\ell}}\right),&
\text{dist}(\varphi_{n}^{r})=O\left(\alpha_{n}^{\frac{1}{\ell}}\right).
\end{matrix} $$ \end{prop} The following Proposition is a preparation for proving Proposition \ref{affdiffeo}. \begin{prop}\label{Koebe} Let $f\in{\mathscr W}$. There exists a constant $K>0$ such that for every $\alpha>0$ the following holds. Let $T$ and $M\subset T$ be two intervals and let $S, D$ be the left and the right component of $T\setminus M$ and $n\in{\mathbb N}$. Suppose that \begin{enumerate} \item for every $0\leq i\leq n-1$ the intervals $f^i(T)$ are pairwise disjoint, \item $f^n:T\to f^n(T)$ is a diffeomorphism,
\item ${\left|f^n(M)\right|}/{\left|f^n(S)\right|},{\left|f^n(M)\right|}/{\left|f^n(D)\right|}<\alpha$. \end{enumerate} Then $$ \text{dist}\left(f^n(M)\right)\leq K\alpha. $$ \end{prop} The proof of the previous proposition can be found in \cite{dmvs}. We are now ready to prove Proposition \ref{affdiffeo}. \begin{proof} In this proof we use the notation introduced in Subsection \ref{Fib}. We start by proving the statement for $\varphi_n^l$. We define \begin{itemize} \item[-] $T=\left[\hat x_{4,n+1},\hat x_{4,n}\right]$, \item[-] $M=\left[f(x_{3}),\hat x_{3,n}\right]=\left[0,\hat x_{3,n}\right]$, \item[-] $S=\left[\hat x_{4,n+1}, f(x_{3})\right]=\left[\hat x_{4,n+1}, 0\right]$, \item[-] $D=\left[\hat x_{3,n},\hat x_{4,n}\right]$. \end{itemize} Observe that $$\varphi_{n}^l=Z_{M}f^{q_n-1}.$$ We claim that: \begin{enumerate} \item for every $0\leq i\leq q_n-2$ the intervals $f^i(T)$ are pairwise disjoint, \item $f^{q_n-1}:T\to f^{q_n-1}(T)$ is a diffeomorphism,
\item ${\left|f^{q_n-1}(M)\right|}/{\left|f^{q_n-1}(S)\right|},{\left|f^{q_n-1}(M)\right|}/{\left|f^{q_n-1}(D)\right|}=O\left(\alpha_{n-1}^{\frac{1}{\ell}}\right)$. \end{enumerate} Points $1$ and $2$ comes from general properties of circle maps. For point $3$, observe that \begin{eqnarray*}
\frac{\left|f^{q_n-1}(M)\right|}{\left|f^{q_n-1}(S)\right|}=\frac{\left|\left[f^{q_n}(x_3),x_3\right]\right|}{\left|\left[f^{-q_{n-1}}(x_4),f^{q_n}(x_3)\right]\right|}. \end{eqnarray*} As consequence, under the image of $f$, \begin{eqnarray*}
\frac{\left|f^{q_n}(M)\right|}{\left|f^{q_n}(S)\right|}=\frac{\left|\left[\hat x_{2,n-1}, 0\right]\right|}{\left|\left[\hat x_{4,n-1},\hat x_{2,n-1}\right]\right|}=O\left(\frac{\left|\left[\hat x_{3,n-1}, 0\right]\right|}{\left|\left[\hat x_{4,n-1},\hat x_{3,n-1}\right]\right|}\right)=O\left(\alpha_{n-1}\right), \end{eqnarray*} where we also used Remark \ref{prop2}. Hence, because of the form of the map near to the boundary points of the flat interval \begin{eqnarray*}
\frac{\left|f^{q_n-1}(M)\right|}{\left|f^{q_n-1}(S)\right|}=O\left(\alpha_{n-1}^{\frac{1}{\ell}}\right). \end{eqnarray*} Observe that, for $n$ large enough, \begin{eqnarray*}
\frac{\left|f^{q_n-1}(M)\right|}{\left|f^{q_n-1}(D)\right|}=\frac{\left|\left[f^{q_n}(x_3),x_3\right]\right|}{\left|\left[x_3, x_4\right]\right|}\leq \frac{\left|f^{q_n-1}(M)\right|}{\left|f^{q_n-1}(S)\right|}=O\left(\alpha_{n-1}^{\frac{1}{\ell}}\right). \end{eqnarray*} By Proposition \ref{Koebe} we get the desired distortion estimate for $\varphi_n^l$. \\ For the distortion estimate of $\varphi_n$, it is enough to repeat the previous argument for \begin{itemize} \item[-] $T=\left[\hat x_{4,n-1},\hat x_{4,n}\right]$, \item[-] $M=\left[\hat x_{1,n}, f(x_{3})\right]=\left[\hat x_{1,n}, 0\right]$, \item[-] $S=\left[\hat x_{4,n-1}, \hat x_{1,n}\right]$, \item[-] $D=\left[f(x_{3}),\hat x_{4,n}\right]=\left[0, \hat x_{4,n}\right]$, \end{itemize} and to notice that $\varphi_n=Z_{M}f^{q_{n-1}-1}$. \\ For the distortion estimate of $\varphi_n^r$, take \begin{itemize} \item[-] $T=\left[\hat x_{3,n},\hat x_{3,n-2}\right]$, \item[-] $M=\left[\hat x_{4,n}, \hat x_{2,n-2}\right]$, \item[-] $S=\left[\hat x_{3,n}, \hat x_{4,n}\right]$, \item[-] $D=\left[\hat x_{2,n-2},\hat x_{3,n-2}\right]$, \end{itemize} and notice that $\varphi_n^r=Z_{M}f^{q_{n}-1}$. We claim that: \begin{enumerate} \item for every $0\leq i\leq q_n-2$ the intervals $f^i(T)$ are pairwise disjoint, \item $f^{q_n-1}:T\to f^{q_n-1}(T)$ is a diffeomorphism,
\item ${\left|f^{q_n-1}(M)\right|}/{\left|f^{q_n-1}(S)\right|},{\left|f^{q_n-1}(M)\right|}/{\left|f^{q_n-1}(D)\right|}=O\left(\alpha_{n}^{\frac{1}{\ell}}\right)$. \end{enumerate} Points $1$ and $2$ comes from general properties of circle maps. For point $3$, observe that \begin{eqnarray*}
\frac{\left| f^{q_n-1}(M)\right|}{\left| f^{q_n-1}(D)\right|}=\frac{\left|\left[x_4, f^{q_{n+1}}(x_3)\right]\right|}{\left|\left[f^{q_{n+1}}(x_3), f^{q_{n-1}}(x_3)\right]\right|}\leq \frac{\left|\left[x_4, f^{q_{n+1}}(x_3)\right]\right|}{\left|\left[f^{-q_{n}}(x_3), f^{-q_{n}}(x_4))\right]\right|}. \end{eqnarray*} As consequence, under the image of $f$, \begin{eqnarray*}
\frac{\left|f^{q_n}(M)\right|}{\left|f^{q_n}(D)\right|}\leq\frac{\left|\left[0,\hat x_{3,n}\right]\right|}{\left|\left[\hat x_{3,n},\hat x_{4,n}\right]\right|}=O\left(\alpha_{n}\right), \end{eqnarray*} where we also used Remark \ref{prop2}. Hence \begin{eqnarray*}
\frac{\left|f^{q_n-1}(M)\right|}{\left|f^{q_n-1}(D)\right|}=O\left(\alpha_{n}^{\frac{1}{\ell}}\right). \end{eqnarray*} Observe that, for $n$ large enough, \begin{eqnarray*}
\frac{\left|f^{q_n-1}(M)\right|}{\left|f^{q_n-1}(S)\right|}=\frac{\left|\left[x_4, f^{q_{n+1}}(x_3)\right]\right|}{\left|\left[x_3, x_4\right]\right|}\leq \frac{\left|f^{q_n-1}(M)\right|}{\left|f^{q_n-1}(D)\right|}=O\left(\alpha_{n}^{\frac{1}{\ell}}\right). \end{eqnarray*} By Proposition \ref{Koebe} we get the desired distortion estimate for $\varphi_n^r$. \end{proof}
\subsection{Asymptotic linear behaviour of renormalization} We are now ready to prove Proposition \ref{superformula}. The proof will be presented in a series of lemmas. \begin{lem}\label{prev} Let $f\in {\mathscr W}$, then for all $n\in{\mathbb N}$, $$ \begin{aligned}
&1.&S_{1,n}&=O\left(\alpha_{n+1}^{\frac{1}{\ell}}\right),\\ &2.& S_{2,n}&=O\left(\alpha_{n-1}^{\frac{1}{\ell}}\right),\\ &3.& s_{n}&=O\left(\alpha_{n}^{\frac{1}{\ell}}\right),\\ &4.&S_{1,n}S_{2,n}S_{3,n}&=O\left(\alpha_{n}\right),\\ &5.&S_{2,n}S_{3,n}&=\frac{S_{2,n-1}}{\ell}\left(1+O\left(\alpha_{n-3}^{\frac{1}{\ell}}\right)\right).
\end{aligned} $$ \end{lem} \begin{proof}
Because $\varphi^l_n$ has bounded distortion (see Proposition \ref{affdiffeo}), we have that $S_{1,n}=O\left(\varphi^l_n\left(S_{1,n}\right)\right)$. Observe that $$ \left(\varphi^l_n\left(S_{1,n}\right)\right)^{\ell}=x_{2,n+1}. $$ Hence $$ S_{1,n}=O\left(x_{2,n+1}^{\frac{1}{\ell}}\right)=O\left(x_{3,n+1}^{\frac{1}{\ell}}\right)=O\left(\alpha_{n+1}^{\frac{1}{\ell}}\right). $$ Point $1$ follow. In order to s $2$ observe that, by Proposition $2$ in \cite{5aut} and Proposition \ref{affdiffeo}, there exist two constants $K_1$ and $K_2$, such that $$
S_{2,n+2}\leq K_1\frac{\left|\left[x_{4,n+2},x_{3,n}\right]\right|}{\left|\left[x_{3,n+2},x_{3,n}\right]\right|}\leq K_1K_2\frac{\left|\left[f^{-q_{n+1}}(x_4),x_{3}\right]\right|}{\left|\left[f^{-q_{n+1}}(x_3),x_{3}\right]\right|}. $$ Moreover, $$
\left(\frac{\left|\left[f^{-q_{n+1}}(x_4),x_{3}\right]\right|}{\left|\left[f^{-q_{n+1}}(x_3),x_{3}\right]\right|}\right)^{\ell}=O\left(\frac{x_{3,n+1}}{x_{4,n+1}}\right). $$ Combining the two previous inequalities, we find
$$ S_{2,n+2}^{\ell}=O\left(\frac{x_{3,n+1}}{x_{4,n+1}}\right)=O\left(\alpha_{n+1}\right). $$ Point $2$ follows.
For proving point $3$, we use point $1$ of this lemma, point $5$ of Lemma \ref{ss} and Proposition \ref{affdiffeo}. Namely,
$$ s_{n}^{\ell-1}=S_{5,n}=S_{1,n-1}^{\ell-1}\cdot\left(\frac{\varphi_n^l\left(S_{1,n-1}\right)}{S_{1,n-1}}\right)^{\ell-1}=O\left(\alpha_{n}^{\frac{\ell-1}{\ell}}\right)\cdot\left(1+O\left(\alpha_{n-2}^{\frac{1}{\ell}}\right)\right)=O\left(\alpha_{n}^{\frac{\ell-1}{\ell}}\right). $$ Point $3$ follows. The proof of point $4$ is a consequence of Remark \ref{prop2}. Namely, $$ S_{1,n}S_{2,n}S_{3,n}=\frac{x_{3,n}-x_{2,n}}{1-x_{2,n}}\leq\frac{x_{3,n}}{x_{4,n}-x_{3,n}}=O\left(\alpha_n\right). $$ For point $5$, by points $2$ and $3$ of Lemma \ref{ss} \begin{eqnarray*} S_{2,n}S_{3,n}=\frac{S_{2,n-1}}{\ell}\left[\frac{\ell}{S_{2,n-1}}\cdot\frac{1-\left(\varphi_{n-1}^{-1}\circ q_{s_{n-1}}^{-1}\right)\left(1-S_{2,n-1}\right)}{1-\left(\varphi_{n-1}^{l}\left( S_{1,n-1}\right)\right)^{\ell}}\right]. \end{eqnarray*} Observe that by Proposition \ref{affdiffeo} we get \begin{equation}\label{1-varphiq} 1-\left(\varphi_{n-1}^{-1}\circ q_{s_{n-1}}^{-1}\right)\left(1-S_{2,n-1}\right)=\frac{S_{2,n-1}}{Dq_{s_{n-1}}\left(\theta_{n-1}\right)\left(1+O\left(\alpha_{n-3}^{\frac{1}{\ell}}\right)\right)}, \end{equation}
where $\left|\theta_n-1\right|\leq S_{2,n}$. By points $2$ and $3$ we find \begin{equation}\label{Dq} Dq_{s_{n-1}}\left(\theta_{n-1}\right)=\ell\cdot \left(1+O\left(\alpha_{n-2}^{\frac{1}{\ell}}\right)\right). \end{equation} By point $1$ and Proposition \ref{affdiffeo} we get \begin{equation}\label{1-varphil} \frac{1}{1-\left(\varphi_{n-1}^l\left(S_{1,n-1}\right)\right)^{\ell}}=\left(1+O\left(\alpha_{n}\right)\right). \end{equation} Point $5$ follows. \end{proof} \begin{prop}\label{ssn} Let $f\in {\mathscr W}$ and $R^nf=(S_{1,n}, S_{2,n}, S_{3,n}, S_{4,n}, S_{5,n}, \varphi_n,\varphi^{l}_n,\varphi^{r}_n)$. Then $$ \begin{aligned} &1.& S_{1,n+1}&= 1-\left(\frac{\ell S_{1,n}^{\ell}}{S_{2,n}}\right)\left(1+O\left(\alpha_{n-2}^{\frac{1}{\ell}}\right)\right),\\ &2.& S_{2,n+1}&=\frac{ S_{1,n}S_{2,n}S_{3,n}}{\ell S_{5,n}}\left(1+O\left(\alpha_{n-2}^{\frac{1}{\ell}}\right)\right),\\ &3.& S_{3,n+1}&=\frac{S_{5,n}}{S_{1,n}S_{3,n}}\left(1+O\left(\alpha_{n-2}^{\frac{1}{\ell}}\right)\right),\\ & 4.& S_{4,n+1}&=\frac{S_{1,n}^{\ell}}{S_{4,n}}\left(1+O\left(\alpha_{n-1}^{\frac{1}{\ell}}\right)\right),\\ &5.& S_{5,n+1}&= S_{1,n}^{\ell-1}\left(1+O\left(\alpha_{n-1}^{\frac{1}{\ell}}\right)\right). \end{aligned} $$ \end{prop} \begin{proof} Let us prove point $1$. By (\ref{1-varphiq}) and (\ref{Dq}) \begin{equation}\label{eq:varphis1overs1} 1-\left(\varphi_n^{-1}\circ q_{s_n}^{-1}\right)\left(1-S_{2,n}\right)=\frac{S_{2,n}}{\ell}\left(1+O(\alpha_{n-2}^{\frac{1}{\ell}})\right). \end{equation} Finally, by point $1$ of Lemma \ref{ss}, point $1$ of Lemma \ref{prev}, \eqref{eq:varphis1overs1} and Proposition \ref{affdiffeo} we get \begin{eqnarray*} \left[\frac{S_{2,n}}{1-\left(\varphi_n^{-1}\circ q_{s_n}^{-1}\right)\left(1- S_{2,n}\right)}\cdot\frac{\left(\varphi_n^{l}\left( S_{1,n}\right)\right)^{\ell}}{\ell S_{1,n}^{\ell}}\right]&=&\\ \ell\left(1+O\left(\alpha_{n-1}^{\frac{1}{\ell}}\right)\right)\left(1+O\left(\alpha_{n-2}^{\frac{1}{\ell}}\right)\right)\cdot\frac{1}{\ell}\left(1+O\left(\alpha_{n-1}^{\frac{1}{\ell}}\right)\right)&=& 1+O\left(\alpha_{n-2}^{\frac{1}{\ell}}\right). \end{eqnarray*} Notice that the previous estimate $$ \frac{\left(\varphi_n^{l}\left( S_{1,n}\right)\right)^{\ell}}{ S_{1,n}^{\ell}}=\left(1+O\left(\alpha_{n-1}^{\frac{1}{\ell}}\right)\right), $$ proves point $4$ and $5$ by using Lemma \ref{ss}. Let us prove point $2$. By Proposition \ref{affdiffeo} we get $$ \frac{\ell S_{5,n}\left(\varphi_n^{-1}\circ q_{s_n}^{-1}\right)\left(S_{1,n}S_{2,n}S_{3,n}\right)}{S_{1,n}S_{2,n}S_{3,n}}\leq\frac{\ell S_{5,n}}{Dq_{s_n}\left(0\right)\left(1+O\left(\alpha_{n-2}^{\frac{1}{\ell}}\right)\right)}. $$ By calculations and points $3$ of Lemma \ref{prev} we find $$ Dq_{s_n}\left(0\right)=\ell \left(1+O\left(\alpha_{n}^{\frac{1}{\ell}}\right)\right)S_{5,n}. $$ Finally, by point $2$ of Lemma \ref{ss}, by (\ref{1-varphil}) and by the previous two estimates we have \begin{eqnarray*} \left[\frac{\ell S_{5,n}\left(\varphi_n^{-1}\circ q_{s_n}^{-1}\right)\left(S_{1,n}S_{2,n}S_{3,n}\right)}{S_{1,n}S_{2,n}S_{3,n}}\cdot\frac{1}{1-\left(\varphi_n^l\left(S_{1,n}\right)\right)^{\ell}}\right]&=&\\ \left(1+O\left(\alpha_{n-2}^{\frac{1}{\ell}}\right)\right)\left(1+O\left(\alpha_{n}^{\frac{1}{\ell}}\right)\right)\cdot\left(1+O\left(\alpha_{n+1}\right)\right)&=& 1+O\left(\alpha_{n-2}^{\frac{1}{\ell}}\right). \end{eqnarray*} Notice that point $1$ and $2$ proves that \begin{eqnarray}\label{S1S2}
\frac{\ell S_{1,n}^{\ell}}{S_{2,n}}=\left(1+O\left(\alpha_{n-2}^{\frac{1}{\ell}}\right)\right). \end{eqnarray} By point $5$ of Lemma \ref{prev}, (\ref{S1S2}) and point $5$ of this proposition we get \begin{equation}\label{S1S2S3} \frac{S_{1,n}S_{2,n}S_{3,n}}{s_n^{\ell}}=S_{1,n}\left(1+O\left(\alpha_{n-3}^{\frac{1}{\ell}}\right)\right). \end{equation} We are now ready to prove point $3$. Notice that $$ \left(\varphi_{n}^{-1}\circ q_{s_{n}}^{-1}\right)\left(S_{1,n}S_{2,n}S_{3,n}\right)=\frac{S_{1,n}S_{2,n}S_{3,n}}{Dq_{s_{n}}\left(\zeta_{n}\right)\left(1+O\left(\alpha_{n-2}^{\frac{1}{\ell}}\right)\right)}, $$ where $q_{s_{n}}\left(\zeta_n\right)\leq S_{1,n}S_{2,n}S_{3,n}\left(1+O\left(\alpha_{n-2}^{\frac{1}{\ell}}\right)\right)$. In particular, because $q_{s_{n}}\left(x\right)\geq x^{\ell}$, we get \begin{equation}\label{zeta} 0<\zeta_n\leq\left(S_{1,n}S_{2,n}S_{3,n}\right)^{\frac{1}{\ell}}\left(1+O\left(\alpha_{n-2}^{\frac{1}{\ell}}\right)\right). \end{equation} By Lemma \ref{ss}, (\ref{1-varphiq}) and the previous estimate we get $$ S_{3,n+1}=\frac{S_{5,n}}{S_{1,n}S_{3,n}}\left[\frac{1}{S_{5,n}}\cdot\frac{\left(\left(1-s_n\right)\zeta_n+s_n\right)^{\ell-1}}{\left(\left(1-s_n\right)\theta_n+s_n\right)^{\ell-1}}\left(1+O\left(\alpha_{n-2}^{\frac{1}{\ell}}\right)\right)\right]. $$ Now, by point $2$ and $3$ of Lemma\ref{prev} and by the definition of $S_{5,n}$ \begin{eqnarray*} S_{3,n+1}&=&\frac{S_{5,n}}{S_{1,n}S_{3,n}}\left[\frac{\left(\left(1-s_n\right)\zeta_n+s_n\right)^{\ell-1}}{S_{5,n}}\left(1+O\left(\alpha_{n-2}^{\frac{1}{\ell}}\right)\right)\right]\\&=&\frac{S_{5,n}}{S_{1,n}S_{3,n}}\left[\left(\left(1-s_n\right)\frac{\zeta_n}{s_n}+1\right)^{\ell-1}\left(1+O\left(\alpha_{n-2}^{\frac{1}{\ell}}\right)\right)\right]. \end{eqnarray*} Finally, by (\ref{zeta}), (\ref{S1S2S3}) and point $1$ of Lemma \ref{prev}, we find \begin{eqnarray*} S_{3,n+1}&=&\frac{S_{5,n}}{S_{1,n}S_{3,n}}\left[\left(1+O\left(S_{1,n}^{\frac{1}{\ell}}\right)\right)\left(1+O\left(\alpha_{n-2}^{\frac{1}{\ell}}\right)\right)\right]\\&=&\frac{S_{5,n}}{S_{1,n}S_{3,n}}\left(1+O\left(\alpha_{n-2}^{\frac{1}{\ell}}\right)\right). \end{eqnarray*} Point $3$ follows. \end{proof} \begin{cor}\label{s1} Let $f\in {\mathscr W}$ and $R^nf=(S_{1,n}, S_{2,n}, S_{3,n}, S_{4,n}, S_{5,n}, \varphi_n,\varphi^{l}_n,\varphi^{r}_n)$. Then $$ \begin{aligned} &1.& \frac{\ell S_{1,n}^{\ell}}{S_{2,n}}&=\left(1+O\left(\alpha_{n-2}^{\frac{1}{\ell}}\right)\right),\\ &2.& \frac{S_{2,n}S_{3,n}}{s_n^{\ell}}&=\left(1+O\left(\alpha_{n-3}^{\frac{1}{\ell}}\right)\right). \end{aligned} $$ \end{cor} Let $f\in{\mathscr W}$. Recall that for all $n\in{\mathbb N}$ we defined $$w_n=\left(\begin{matrix} \log S_{2,n}\\\log S_{3,n}\\\log S_{4,n}\\\log S_{5,n} \end{matrix} \right)=\left(\begin{matrix} y_{2,n}\\ y_{3,n}\\ y_{4,n}\\ y_{5,n} \end{matrix} \right).$$ We define now $$
M=\left(\begin{matrix} 1+\frac{1}{\ell} & 1& 0& -1\\
-\frac{1}{\ell} & -1& 0& 1\\
1 & 0& -1& 0\\
1-\frac{1}{\ell} & 0& 0& 0
\end{matrix}\right), $$ and $$w^*=\left(\begin{matrix} -\left(1+\frac{1}{\ell}\right)\log\ell\\\frac{1}{\ell}\log\ell\\-\log\ell\\ -\left(1-\frac{1}{\ell}\right)\log\ell \end{matrix} \right).$$ Point $1$ of Corollary \ref{s1} allows to eliminate $S_{1,n}$ which asymptotically is determined by $S_{2,n}$. With the notations introduced above, the new estimates of Proposition \ref{ssn} obtained by the substitution of $S_{1,n}$ takes the following linear form. \begin{prop}\label{wn}
Let $f\in {\mathscr W}$, then
$$w_{n+1}=Mw_n+O\left(\alpha_{n-2}^{\frac{1}{\ell}}\right)+w^*.$$ \end{prop}
\begin{lem}\label{eighvector} The eigenvalues of $M$ are $$ \begin{matrix}
-1,& 0, &\lambda_s=\frac{\frac{1}{\ell}-\sqrt{\left(\frac{1}{\ell}\right)^2+\frac{4}{\ell}}}{2}\in(-1,0),& \lambda_u=\frac{\frac{1}{\ell}+\sqrt{\left(\frac{1}{\ell}\right)^2+\frac{4}{\ell}}}{2}>1, \end{matrix}$$ and the corresponding eigenvectors $$ \begin{aligned}
E_{-}&=&(e^-_i)_{i=2,3,4,5}&=(0,0,1,0),\\
E_0&=&(e^0_i)_{i=2,3,4,5}&=(0,1,0,1),\\
E_s&=&(e^s_i)_{i=2,3,4,5}&=\left(1,\frac{-\lambda_s+\ell-1}{\ell\lambda_s(1+\lambda_s)},\frac{1}{1+\lambda_s},\frac{\ell-1}{\ell\lambda_s}\right),\\ E_u&=&(e^u_i)_{i=2,3,4,5}&=\left(1,\frac{-\lambda_u+\ell-1}{\ell\lambda_u(1+\lambda_u)},\frac{1}{1+\lambda_u},\frac{\ell-1}{\ell\lambda_u}\right). \end{aligned} $$ \end{lem} Let $w_{fix}$ be the fixed point of the equation $Lw_{fix}+w^*=w_{fix}.$
\subsection{Asymptotics in $Y$-coordinates} \begin{lem}\label{suplemma} For every $f\in {\mathscr W}$ there exist $C_u(f)$, $C_s(f)$ and $C_{-}(f)$ such that, for all $n\in{\mathbb N}$ , \begin{equation}\label{eq:wninalphan} w_n=C_u(f)\lambda_u^nE_u+C_s(f)\lambda_s^nE_s+C_-(f)(-1)^nE_-+O\left(\left(\alpha_0^{(\frac{2}{\ell})^{\frac{n-3}{2}}}\right)^{\frac{1}{\ell}}\right)+w_{fix}. \end{equation} \end{lem} \begin{proof} For all $n\in{\mathbb N}$ let $v_n=w_n-w_{fix}$. Then by Proposition \ref{wn}, $$v_{n+1}=Mv_n+\epsilon_n,$$ where $\epsilon_n=O\left(\alpha_{n-2}^{\frac{1}{\ell}}\right)=O\left(\left(\alpha_0^{(\frac{2}{\ell})^{\frac{n-2}{2}}}\right)^{\frac{1}{\ell}}\right)$, see Theorem \ref{alpha}. By iterating this formula we get \begin{equation}\label{forvn}
v_n=M^nv_0+\sum_{k=0}^{n-1}L^{n-k-1}\epsilon_k. \end{equation} By expressing $v_0$ and $\epsilon_n$ in the eigenbasis we find the following equalities: $$v_0=C_u^0E_u+C_s^0E_s+C_-^0E_-+C_0^0E_0,$$ $$\epsilon_n=\epsilon_{u,n}E_u+\epsilon_{s,n}E_s+\epsilon_{-,n}E_-+\epsilon_{0,n}E_0.$$ We consider now the following quantities depending on $f$: \begin{eqnarray}\label{cu}
C_u(f)&=&C_u^0+\sum_{k=0}^{\infty}\frac{\epsilon_{u,k}}{\lambda_u^{k+1}},\\
\label{cs}
C_s(f)&=&C_s^0+\sum_{k=0}^{\infty}\frac{\epsilon_{s,k}}{\lambda_s^{k+1}},\\
\label{c-}
C_-(f)&=&C_-^0+\sum_{k=0}^{\infty}\frac{\epsilon_{-,k}}{(-1)^{k+1}}. \end{eqnarray} Moreover, equation (\ref{forvn}) in the coordinates becomes \begin{eqnarray*} v_n&=&\left(C_u^0+\sum_{k=0}^{n-1}\frac{\epsilon_{u,k}}{\lambda_u^{k+1}}\right)\lambda_u^nE_u+\left(C_s^0+\sum_{k=0}^{n-1}\frac{\epsilon_{s,k}}{\lambda_s^{k+1}}\right) \lambda_s^nE_s+\\ &&\left(C_-^0+\sum_{k=0}^{n-1}\frac{\epsilon_{-,k}}{(-1)^{k+1}}\right)(-1)^nE_-+ \epsilon_{0, n-1}E_0\\ &=&C_u(f)\lambda_u^nE_u+C_s(f)\lambda_s^nE_s+C_-(f)(-1)^nE_-+C_0^0E_0+ \left(\sum_{k=n}^{\infty}\frac{\epsilon_{u,k}}{\lambda_u^{k+1}}\right)\lambda_u^nE_u +\\ &&\left(\sum_{k=n}^{\infty}\frac{\epsilon_{s,k}}{\lambda_s^{k+1}}\right)\lambda_s^nE_s+ \left(\sum_{k=n}^{\infty}\frac{\epsilon_{-,k}}{(-1)^{k+1}}\right)(-1)^nE_-+ \epsilon_{0, n-1}E_0\\ &=&C_u(f)\lambda_u^nE_u+C_s(f)\lambda_s^nE_s+C_-(f)(-1)^nE_-+O\left(\left(\alpha_0^{(\frac{2}{\ell})^{\frac{n-3}{2}}}\right)^{\frac{1}{\ell}}\right). \end{eqnarray*} Notice that the three tail terms were estimated in the following way. Let us start with the tail term corresponding to $E_s$. The others are treated in a similar way. Notice that for $k$ large enough we have $$\frac{\left(\alpha_0^{(\frac{2}{\ell})^{\frac{k}{2}}}\right)^{\frac{1}{\ell}}}{\left(\alpha_0^{(\frac{2}{\ell})^{\frac{k-1}{2}}}\right)^{\frac{1}{\ell}}\lambda_s}\leq\frac{1}{{2}}.$$ As a consequence \begin{eqnarray*}
\left|\sum_{k=n}^{\infty}\frac{\epsilon_{s,k}}{\lambda_s^{k+1}}\right|\lambda_s^n&=&O\left( \sum_{k=n}^{\infty}\frac{\left(\alpha_0^{(\frac{2}{\ell})^{\frac{k-2}{2}}}\right)^{\frac{1}{\ell}}}{\lambda_s^{k}}\lambda_s^{n-1}\right)\\ &=&O\left(\frac{\left(\alpha_0^{(\frac{2}{\ell})^{\frac{n-2}{2}}}\right)^{\frac{1}{\ell}}}{\lambda_s^n}\lambda_s^{n-1}\sum_{k=0}^{\infty}\left(\frac{1}{{2}}\right)^k\right)\\ &=&O\left(\left(\alpha_0^{(\frac{2}{\ell})^{\frac{n-2}{2}}}\right)^{\frac{1}{\ell}}\right). \end{eqnarray*} Finally, observe that the largest estimation comes from the term $$\epsilon_{0,n-1}= O\left(\left(\alpha_0^{(\frac{2}{\ell})^{\frac{n-3}{2}}}\right)^{\frac{1}{\ell}}\right).$$ \end{proof}
\begin{lem}\label{asymalpha} Let $f\in {\mathscr W}$ then $$\alpha_n=O\left(e^{C_u(f)\lambda_u^{n-1}}\right),$$ and $C_u(f)<0$. \end{lem} \begin{proof} By the definition of $\alpha_n$ $$\alpha_n=\frac{S_{2,n}S_{3,n}}{1-S_{2,n}+\left(1-S_{1,n}\right)S_{2,n}S_{3,n}},$$ and by applying point $2$ and $5$ of Lemma \ref{prev}, we have $$\frac{\alpha_n}{S_{2,n-1}}=1+O\left(S_{2,n-1}\right)=1+O\left(\alpha_{n-2}^{\frac{1}{\ell}}\right).$$ From the previous estimate and from \eqref{eq:wninalphan}, we get $$\log\alpha_n=\log S_{2,n-1}+O\left(\alpha_{n-2}^{\frac{1}{\ell}}\right)=C_u(f)e^u_2\lambda_u^{n-1}+C_s(f)e^s_2\lambda_s^{n-1}+O\left(\left(\alpha_0^{(\frac{2}{\ell})^{\frac{n-3}{2}}}\right)^{\frac{1}{\ell}}\right).$$
From Lemma \ref{eighvector} and Lemma \ref{alpha} it follows that
$$\alpha_n=O\left(e^{C_u(f)\lambda_u^{n-1}}\right),$$
and $C_u(f)<0$. \end{proof} Lemma \ref{suplemma} and Lemma \ref{asymalpha} proves Proposition \ref{superformula} which was the aim of this section. \subsection{Asymptotics in $X$-coordinates} In the previous section we described the asymptotics of the renormalizations in the $S$ and $Y$-coordinates. Here we deduce the asymptotics of the renormalizations in the $X$-coordinates. This is needed in Section \ref{rigidity}. \begin{lem}\label{xs} Let $f\in{\mathscr W}^{(X)}$, then \begin{eqnarray*}
\log |x_{1,n}|&=&C_u(f)\lambda_u^n(e_2^u+e_3^u-e_4^u)+C_s(f)\lambda_s^n(e_2^s+e_3^s-e_4^s)-C_-(f)(-1)^n+\delta_{1,n}, \\
\log x_{2,n}&=&C_u(f)\lambda_u^n(e_2^u+e_3^u)+C_s(f)\lambda_s^n(e_2^s+e_3^s)+
\delta_{2,n},\\
\log x_{3,n}&=&C_u(f)\lambda_u^n(e_2^u+e_3^u)+C_s(f)\lambda_s^n(e_2^s+e_3^s)+
\delta_{3,n},\\
\log |1-x_{4,n}|&=&C_u(f)\lambda_u^n(e_2^u)+C_s(f)\lambda_s^n(e_2^s)+
\delta_{4,n}, \end{eqnarray*} where $$\delta_{1,n},\delta_{2,n},\delta_{3,n},\delta_{4,n}=O\left(e^{\frac{C_u(f)\lambda_u^{n-4}}{\ell}}\right).$$ \end{lem} \begin{proof} By Lemma \ref{xtos}, $$ x_{1,n}=-\frac{S_{3,n}(1-S_{1,n})S_{2,n}}{(1+S_{3,n}(1-S_{1,n})S_{2,n})S_{4,n}}.
$$
As consequence
$$
\log\left| x_{1,n}\right|=y_{2,n}+y_{3,n}-y_{4,n}+O\left(S_{1,n}\right)+O\left(S_{2,n}S_{3,n}\right),
$$
and by points $1$, $2$ and $5$ of Lemma \ref{prev}
$$
\log\left| x_{1,n}\right|=y_{2,n}+y_{3,n}-y_{4,n}+O\left(\alpha_{n+1}^{\frac{1}{\ell}}\right)+O\left(\alpha_{n-2}^{\frac{1}{\ell}}\right).
$$
By Proposition \ref{superformula} and by Lemma \ref{asymalpha} the esimates for $\log\left| x_{1,n}\right|$ follows. Similarly one obtains the other estimates. \end{proof}
\section{Renormalization of decomposed maps}\label{deco}
The renormalization operator in the space ${\mathscr L}$ is not differentiable. We cannot use the standard cone field method for the construction of invariant manifolds. The reason for which our renormalization operator is not differentiable is that the composition of ${{\mathcal C}^3}$ diffeomorphisms is not differentiable. In this section we introduce a space $L$ and a corresponding renormalization $R:L_0\to L$ in such a way that $R$ does not involve composition\footnote{We will abuse the notation denoting both renormalization operators by $R$}. Moreover, there is a natural projection $O:L\to{\mathscr L}$ such that $R:L_0\to L$ is a lift of $R:{\mathscr L}_0\to {\mathscr L}$. The space $L$ is called the space of decompositions. The corresponding renormalization $R$ on $L$ is still not differentiable, but it is jump-out-differentiable. This will allow us to apply Theorem \ref{InvMan} to obtain the invariant manifold formed by Fibonacci Cherry maps in the space $L$ which we will pull in ${\mathscr L}$ by using the natural projection $O$.
\subsection{The space of decompositions} To discuss the differentiability of $R$ we need the space $\text{Diff }^{3+\epsilon}([0,1])$, with $\epsilon>0$, to be a normed vector space. This is achieved by using the concept of non-linearity. \begin{defin} Let $r\geq 2$. The non linearity $\eta:\text{Diff }^r([0,1])\to\mathcal{C}^{r-2}([0,1])$ is defined as $$\varphi\to\eta_{\varphi}=D\log D\varphi.$$ \end{defin} \begin{lem} $\eta$ is a bijection. \end{lem} \begin{proof} There is an explicit inverse of $\eta$. Namely, $$\varphi(x)=\frac{\int_0^x e^{\int_0^s\eta} ds}{\int_0^1 e^{\int_0^s\eta} ds}.$$ \end{proof} We can now identify $\text{Diff }^r([0,1])$ with $\mathcal{C}^{r-2}([0,1])$ and use the Banach space structure of $\mathcal{C}^{r-2}([0,1])$ on $\text{Diff }^r([0,1])$. The norm of a diffeomorphism $\varphi\in \text{Diff }^r([0,1])$ is defined as
$$|\varphi|_r=|\eta_{\varphi}|_{\mathcal{C}^{r-2}}.$$
\begin{lem} Let $r\geq 2$. The metric defined on $\text{Diff }^r([0,1])$ by the norm $|\cdot|_r$, is equivalent to the $\mathcal{C}^{r}$ distance. Namely,
$$|\varphi_n-\varphi|_r\to 0\iff
|\varphi_n-\varphi|_ {\mathcal{C}^{r}}\to 0.$$ \end{lem}
We are now ready for the definition of decomposition. Let $$T=\left\{\frac{k}{2^n}| n\geq 0, 0<k< 2^n\right\}\subset (0,1)$$ be the dyadic rationals with his natural order. Let $\theta:T\to T$ the doubling map $$\theta:\tau\to 2\tau \mod 1.$$ \\ Let $X_r$, $r\ge 2$, be the space of decomposed diffeomorphisms
$$X_r=\left\{\underline\varphi=(\varphi_\tau)_{\tau\in T}|
\varphi_\tau\in\text{Diff }^r([0,1]), \sum |\varphi_\tau|_r<\infty \right\}.$$ We define the norm of $\underline\varphi\in X_r$ as
$$|\underline\varphi|_r=\sum |\varphi_{\tau}|_r.$$ \begin{rem} One should think of $\underline\varphi$ as a chain of diffeomorphisms labeled by $T$. Their use allows to avoid composition. When $r=3+\epsilon$ we suppress the lower index and we simply use the notation $X=X_{3+\epsilon}$ and
$|\underline\varphi|=|\underline\varphi|_{3+\epsilon}$. \end{rem} Let $O_n:X\to\text{Diff }^2([0,1])$ be the partial composition defined as $$O_n\underline\varphi=\varphi_1\circ\cdots\circ\varphi_{\frac{k}{2^n}}\circ\cdots\circ \varphi_{\frac{2}{2^n}}\circ \varphi_{\frac{1}{2^n}}.$$ \begin{prop}\label{comp}
The limit $$O\underline\varphi:=\lim O_n\underline\varphi$$ exists and it
is called the composition of $\underline\varphi$.
The composition $$O:X\to\text{Diff }^2([0,1])$$ is Lipschitz on bounded sets.
\end{prop} \begin{proof}
The proof is a minor variation of the proof of the Sandwich Lemma from \cite{M}. \end{proof} We are now ready to define the space of decompositions $L$. Recall that functions in $${\mathscr L}^{(Y, 3+\epsilon)}=\Sigma^{(Y)}\times \text{Diff }^{ 3+\epsilon}([0,1])\times \text{Diff }^{ 3+\epsilon}([0,1])\times\text{Diff }^{ 3+\epsilon}([0,1])$$ are represented in $Y-$coordinates as follows: $f=(y_1,y_2,y_3,y_4, y_5, \varphi,\varphi^{l},\varphi^{r})$ where $$
\begin{matrix}
y_1=S_1, & y_2=\log S_2, & y_3=\log S_3, & y_4=\log S_4, & y_5=\log S_5, \end{matrix} $$ and \begin{equation*}
\Sigma^{(Y)}=\{(y_1,y_2,y_3,y_4,y_5)\in\mathbb R^5 | y_5<0 \}. \end{equation*} The space of decompositions is similar to ${\mathscr L}^{(Y, 3+\epsilon)}$ except that the diffeomorphisms are replaced by decomposed diffeomorphisms. Namely, $$L=\Sigma^{(Y)}\times X\times X\times X,$$
and similarly as before, a point $\underline f\in L$ is represented by $\underline f=(y_1,y_2,y_3,y_4,y_5, \underline\varphi,\underline\varphi^{l}, \underline\varphi^{r}).$
Observe that $L\subset\mathbb{R}^5\times X\times X\times X$ which carries a norm defined by the euclidian norm of $\mathbb{R}^5$ and the non-linearity norm of $X$. Namely, if $\underline f=\left(y_1,y_2,y_3,y_4,y_5, \underline\varphi,\underline\varphi^{l}, \underline\varphi^{r}\right)\in L$ then $$\left|\underline f\right|=\sum\left| y_i\right| +\left|\underline\varphi\right| +\left|\underline\varphi^{l}\right| +\left|\underline\varphi^{r}\right|.$$ \\ Let $O: L\to{\mathscr L}^{(Y,2)}$ be the composition defined as $$O:\underline f=\left(y_1,y_2,y_3,y_4,y_5, \underline\varphi,\underline\varphi^{l}, \underline\varphi^{r}\right)\to f=(y_1,y_2,y_3,y_4,y_5, O\underline\varphi,O\underline\varphi^l, O\underline\varphi^r) $$ Observe that ${\mathscr L}^{(Y, 3+\epsilon)}$ is an open subset of $\mathbb{R}^5\times \text{Diff}^{ 3+\epsilon}([0,1])\times \text{Diff}^{ 3+\epsilon}([0,1])\times \text{Diff}^{ 3+\epsilon}([0,1])$ which carries a norm defined by the euclidian norm of $\mathbb{R}^5$ and the non-linearity norm of $\text{Diff}^{ 3+\epsilon}([0,1])$. There is a natural embedding of ${\mathscr L}^{(Y, 3+\epsilon)}$ into $L$.
\subsection{Renormalization on the space of decompositions}\label{RenonL} Recall the definition of zoom, $Z_I$, from Definition \ref{zoomop}. The following lemmas hold. \begin{lem}\label{normzoom} Let $r\ge 2$ and let $I$ be an interval in $[0,1]$. The norm of the linear operator $Z_{I}:\text{Diff }^r([0,1])\to \text{Diff }^r([0,1])$ satisfies
$$|Z_{I}|_r=|I|.$$ \end{lem} \begin{lem} Let $I$ be an interval of $[0,1]$. Then $Z_{I}\varphi$ is the diffeomorphism obtained by rescaling the restriction of $\varphi$ to $I$. \end{lem} We are now ready to define the renormalization operator on $L$. We consider the subset of renormalizable maps $L_0\subset L$,
$$L_0=\left\{\underline f\in L| 0<y_1<1\right\}.$$
Let $\tau\in T$ and let $\pi^{\tau}: X\to X$ be defined as $$(\pi^{\tau}\underline{\varphi})_{\tau'}=\left\{
\begin{aligned}
&0 & \tau'>\tau\\
&\varphi_{\tau'} & \tau'\leq\tau
\end{aligned}\right. $$
Let $\tau\in T$. To describe the orbit of a point $x\in[0,1]$ in a decomposition we define the function $\gamma_{\tau}: X\times [0,1]\to [0,1]$ as
$$\gamma_{\tau}(\underline\varphi,x)=O\circ\pi^{\tau}(\underline\varphi)(x).$$
Let $\underline f=\left({y}_1,{y}_2,{y}_3,{y}_4,{y}_5,{\underline\varphi}, {\underline\varphi}^{l},{\underline\varphi}^{r}\right)\in L_0$. The renormalization of $\underline f$ is $$R\underline f=\left(\tilde{y}_1,\tilde{y}_2,\tilde{y}_3,\tilde{y}_4,\tilde{y}_5,\tilde{\underline\varphi}, \tilde{\underline\varphi}^{l},\tilde{\underline\varphi}^{r}\right)$$ which is defined in the following way. Let $f=O\left(\underline f\right)=\left({y}_1,{y}_2,{y}_3,{y}_4,{y}_5,{O\underline\varphi}, {O\underline\varphi}^{l},{O\underline\varphi}^{r}\right)$, then $$Rf=\left(\tilde{y}_1,\tilde{y}_2,\tilde{y}_3,\tilde{y}_4,\tilde{y}_5,O\tilde{\underline\varphi}, O\tilde{\underline\varphi}^{l},O\tilde{\underline\varphi}^{r}\right).$$ This defines the $\tilde{y}_k$ of $R\underline f$. It is left to define $\tilde{\underline\varphi}$, $\tilde{\underline\varphi}^l$ and $\tilde{\underline\varphi}^r.$ The definition is the same as the definition of $\tilde{\varphi}$, $\tilde{\varphi}^l$ and $\tilde{\varphi}^r$ except that the diffeomorphisms are now decomposed diffeomorphisms, see Lemma \ref{ss}. \\ Let $\tau\in T$, then $$y_1(\tau)=\gamma_{\tau}\left(\underline\varphi^{l},y_1\right),$$ and $$\tilde{\varphi}_\tau= Z_{\left[y_1(\tau), 1\right]}\varphi^{l}_\tau.$$ \\ Let $\tau\in T$, define $$\varphi^{-1}\circ q_s^{-1}\left(1-e^{y_2}\right)(\tau)=\gamma_{\tau}\left(\underline{\varphi},\left(O\underline\varphi\right)^{-1}\circ q_s^{-1}\left(1-e^{y_2}\right)\right),$$ and $$\tilde{\varphi}^{l}_{\tau}=\left\{
\begin{aligned}
& Z_{\left[\varphi^{-1}\circ q_s^{-1}\left(1-e^{y_2}\right)(\theta\tau),1\right]}\varphi_{\theta\tau} & \tau<\frac{1}{2}\\
& Z_{\left[q_s^{-1}\left(1-e^{y_2}\right),1\right]}q_s& \tau=\frac{1}{2}\\
& \varphi^r_{\theta\tau} & \tau>\frac{1}{2}
\end{aligned} \right.$$ where $s=e^{{y_5}/{\ell-1}}.$ \\ Let $\tau\in T$, define $$\varphi^{-1}\circ q_s^{-1}\left(y_1e^{y_2}e^{y_3}\right)(\tau)=\gamma_{\tau}\left(\underline{\varphi},\left(O\underline\varphi\right)^{-1}\circ q_s^{-1}\left(y_1e^{y_2}e^{y_3}\right)\right),$$ and $$\tilde{\varphi}^{r}_{\tau}=\left\{
\begin{aligned}
& Z_{\left[0,\varphi^{-1}\circ q_s^{-1}\left(y_1e^{y_2}e^{y_3}\right)(\theta\tau)\right]}\varphi_{\theta\tau} & \tau<\frac{1}{2}\\
& Z_{\left[0,q_s^{-1}\left(y_1e^{y_2}e^{y_3}\right)\right]}q_s& \tau=\frac{1}{2}\\
& Z_{\left[0, y_1(\theta\tau)\right]}\varphi^l_{\theta\tau} & \tau>\frac{1}{2}
\end{aligned} \right.$$
The renormalization operator $R:L_0\to L$ is now defined.
\begin{defin} A map $\underline{f}\in L$ is $\infty$-renormalizable if for every $n\geq 0$, $R^n \underline{f}\in L_0$. The set of $\infty$-renormalizable functions is denoted by $W\subset L$. \end{defin}
Renormalization on $L_0$ is naturally defined to be the lift of renormalization on ${\mathscr L}_0$. Namely, \begin{lem} The renormalization operator on $L_0$ commutes with the renormalization operator on ${\mathscr L}_0$ under the composition $O$, i.e $$O\circ R=R\circ O.$$ Moreover $$W=O^{-1}({\mathscr W}).$$ \end{lem}
\section{The manifold structure of the Fibonacci class}\label{manifold1} In this section we prove that the class of Fibonacci maps is a ${{\mathcal C}^1}$ codimension one manifold. \begin{theo}\label{manifold}
${\mathscr W}$ is a ${{\mathcal C}^1}$ codimension one manifold in ${\mathscr L}^{4+\epsilon}$. \end{theo} In order to apply Theorem \ref{InvMan} we use the following notation. Let $B_1=\mathbb{R}^4\times X\times X\times X$ and $B_0=\mathbb{R}^4\times X_2\times X_2\times X_2$. Observe that, $$ R: L_0\subset \mathbb{R}\times B_1\to \mathbb{R} \times B_1\subset \mathbb{R}\times B_0. $$ The renormalization operator $R$ does not globally satisfies the hypothesis of Theorem \ref{InvMan}. However these hypothesis are satisfied on a domain $D$ introduced in the following. Let $1\geq\epsilon^*>0$, $C^*\geq 1$ and $C_u^*\geq 1$. A map $\underline f=\left({y}_1,{y}_2,{y}_3,{y}_4,{y}_5,{\underline\varphi}, {\underline\varphi}^{l},{\underline\varphi}^{r}\right)\in D_{\epsilon^*,C^*, C_u^*}$ if the following holds. \\ Express the vector $w=(y_2,y_3,y_4,y_5)$ in the eigenvectors of the matrix $M$ as defined in Lemma \ref{eighvector},
$$
w=C_uE_u+C_sE_s+C_-E_-+C_0E_0.
$$ Define $U=U_{\epsilon^*,C^*, C_u^*}\subset B_1$ by $$\left\{\begin{aligned}
&C_u< -C^*_u,\\
&|C_0|, |C_s|<C^*,\\
& 16|\underline\varphi |+8
|{\underline\varphi}^{l}|+3|{\underline\varphi}^{r}|<\epsilon^*, \end{aligned}\right. $$ and $\partial_\pm: U\to \mathbb{R}$ by $$\left\{\begin{aligned} &\partial_+(b)=\left(\frac{3}{2\ell}e^{y_2}\right)^\frac{1}{\ell},\\ &\partial_-(b)=\left(\frac{1}{4\ell}e^{y_2}\right)^\frac{1}{\ell}, \end{aligned}\right. $$ where $b=\left({y}_2,{y}_3,{y}_4,{y}_5,{\underline\varphi}, {\underline\varphi}^{l},{\underline\varphi}^{r}\right)$. The map $\underline f=\left(y_1, b\right)$ is in $D_{\epsilon^*,C^*, C_u^*}$ if $$\left\{\begin{aligned} &b\in U,\\ &\partial_-(b)\leq y_1\leq \partial_+(b). \end{aligned}\right. $$ In the sequel, we supress the indices and use the notation $D=D_{\epsilon^*,C^*, C_u^*}$.
Observe that $D\subset L_0$ when $C^*_u$ is large enough.
\subsection{Jump-out-differentiability and structure of the derivatives}\label{jumpoutdiff} \begin{prop} \label{jumpoutdiffprop}
Let $D=D_{\epsilon^*,C^*, C_u^*}$ be a domain or a closed bounded set, then the map $R:D\to L\subset\mathbb{R}\times B_1\subset \mathbb{R}\times B_0$ is jump-out-differentiable. \end{prop} \begin{proof} By Propositions \ref{Aderivative}, \ref{Bsderivative}, \ref{Blderivative}, \ref{Csderivative}, \ref{Clderivative}, \ref{Crderivative}, \ref{Dslderivative}, \ref{Dlsderivative}, \ref{Dlrderivative}, \ref{Drsderivative}, \ref{Drlderivative} we get that $$ DR_{\underline{f}}: \mathbb{R}^5\times X_2\times X_2\times X_2\to \mathbb{R}^5\times X_2\times X_2\times X_2 $$ as described by (\ref{matrix}), is bounded and it depends continuously on $\underline f$. The same propositions imply that $$
\lim_{\left| \Delta\underline f\right|\to 0}\frac{\left|R\left(\underline f+\Delta\underline f\right)-\left[R\underline f+DR_{\underline f}\left(\Delta\underline f\right)\right]\right|_2}{\left|\Delta\underline f\right|}=0. $$ This shows that $R:D\to L$ is jump-out differentiable. \end{proof}
\begin{prop} There exists $E>0$ and $0\leq\kappa<1$ such that every domain $D_{\epsilon^*,C^*, C_u^*}$ has the following property. Let $\underline f=\left({y}_1, b\right)\in D_{\epsilon^*,C^*, C_u^*}$ with $R\underline f=\left(\tilde y_1, \tilde b\right)$ then $\left(\Delta\tilde y,\Delta\tilde b\right)=DR_{\underline f}\left(\Delta y,\Delta b\right)$ satisfies, \begin{equation*} \left\{\begin{aligned} &\Delta\tilde y=\frac{E_{\underline f}}{y_1}\Delta y+O\left(\Delta b\right),\\ &&\\
&\|\Delta\tilde b \|_0=O\left(\frac{1}{\tilde y_1^{\kappa}}|\Delta y|+\|\Delta b\|_0\right), \end{aligned}\right. \end{equation*}
with ${1}/{E}<|E_{\underline f}|< E$. \end{prop} \begin{proof} Consider $D=D_{1,1,1}$. From Propositions \ref{Aderivative}, \ref{Bsderivative}, \ref{Blderivative}, \ref{Csderivative}, \ref{Clderivative}, \ref{Crderivative}, \ref{Dslderivative}, \ref{Dlsderivative}, \ref{Dlrderivative}, \ref{Drsderivative}, \ref{Drlderivative} we get that \begin{eqnarray*} \frac{\partial\tilde y_1}{\partial b}&=&O(1),\\ \frac{\partial\tilde b}{\partial b}&=&O(1), \end{eqnarray*} and from Lemma \ref{partial y1 partial yj} we get that there exists $E>0$ such that $$\frac{1}{E}\frac{1}{y_1}\leq\frac{\partial\tilde y_1}{\partial y_1}\leq E\frac{1}{y_1}.$$ By Propositions \ref{Csderivative}, \ref{Clderivative}, Lemma \ref{partial y2 partial yj}, \ref{partial y3 partial yj}, \ref{partial y4 partial yj}, \ref{partial y5 partial yj}, \ref{partial phi r min 12 partial y}, \ref{partial phi r12 partial y}, \ref{partial phi r max 12 partial y} we get $$\frac{\partial\tilde b}{\partial y_1}=O\left(\frac{1}{y_1}+S_2^{\frac{1}{\ell^2}}S_3^{\frac{1}{\ell}}S_5^{-\frac{2}{\ell-1}}\right).$$ Observe that, by point $2$ of Lemma \ref{ss} and Lemma \ref{D111} we get $$ \tilde S_1S_2^{\frac{1}{\ell^2}}S_3^{\frac{1}{\ell}}S_5^{-\frac{2}{\ell-1}}=O\left(S_2^{\frac{\ell^2+\ell+1}{\ell^2}}S_3^{\frac{\ell+1}{\ell}}S_5^{\frac{-\ell^2-2\ell+1}{\ell(\ell-1)}}\right). $$ Expressing the $S$-coordinates in terms of the coordinates in the eigenbasis of the matrix $M$, see Lemma \ref{matrix}, we obtain $$ \log\left[\tilde S_1S_2^{\frac{1}{\ell^2}}S_3^{\frac{1}{\ell}}S_5^{-\frac{2}{\ell-1}}\right]= C_u\left[\frac{\lambda_u^2\left(\ell^2+\ell+1\right)+\lambda_u\left(-2\ell+1\right)-2\ell}{\ell^2\lambda_u\left(1+\lambda_u\right)}\right]+O\left(C\right). $$ A calculation shows that the coefficient of $C_u$ is positive. A similar expression holds for $ \log\left[\tilde S_1^{\kappa}S_2^{\frac{1}{\ell^2}}S_3^{\frac{1}{\ell}}S_5^{-\frac{2}{\ell-1}}\right]$ and when $0<\kappa<1$ is close enough to $1$, the coefficient of $C_u$, which depends continuously on $\kappa$, is again positive. Because $C_u\leq -1$ we get \begin{equation}\label{kappass} \tilde S_1^{\kappa}S_2^{\frac{1}{\ell^2}}S_3^{\frac{1}{\ell}}S_5^{-\frac{2}{\ell-1}}=O(1). \end{equation} Using again point $2$ of Lemma \ref{ss} and Lemma \ref{D111} we get $$ \frac{\tilde y_1}{y_1}=O\left(S_2^{\frac{1}{\ell}} S_3 S_5^{-1}\right). $$ Expressing the $S$-coordinates in terms of the coordinates in the eigenbasis of the matrix $M$, see Lemma \ref{matrix}, and because $C_u\leq -1$ we obtain $$ \log\left[\frac{\tilde y_1}{y_1}\right]\leq -\left[\frac{\lambda_u+1-\ell}{\ell\left(1+\lambda_u\right)}\right]+O\left(C\right). $$ As before, for $0<\kappa<1$ close enough to $1$ we also have that \begin{equation}\label{kappay} \frac{1}{y_1}=O\left(\frac{1}{\tilde y_1^{\kappa}}\right). \end{equation} Finally, by (\ref{kappass}) and (\ref{kappay}) we get $$\frac{\partial\tilde b}{\partial y_1}=O\left(\frac{1}{\tilde y_1^{\kappa}}\right).$$ The uniform bounds follow from the fact that $D_{\epsilon^*,C^*, C_u^*}\subset D_{1,1,1}$. \end{proof}
\begin{prop}\label{xiexpansion} For every $\epsilon^*<1$, $C^*>1$ and $\xi>0$, if $\underline f=\left({y}_1, b\right)\in D_{\epsilon^*,C^*, C_u^*}$ with $R\underline f=\left(\tilde y_1, \tilde b\right)$, then $R$ has vertical $\xi$-expansion for $C_u^*>1$ large enough, i.e. \begin{eqnarray*} \frac{y_1}{\tilde y_1^{\kappa}}&\geq &\xi. \end{eqnarray*}
\end{prop}
\begin{proof}
Using point $2$ of Lemma \ref{ss} and Lemma \ref{D111} we get $$ \frac{\tilde y_1}{y_1}=O\left(S_2^{\frac{1}{\ell}} S_3 S_5^{-1}\right). $$ Expressing the $S$-coordinates in terms of the coordinates in the eigenbasis of the matrix $M$, see Lemma \ref{matrix}, we obtain $$ \log\left[\frac{\tilde y_1}{y_1}\right]\leq -C_u\left[\frac{\lambda_u+1-\ell}{\ell\left(1+\lambda_u\right)}\right]+O\left(C\right). $$ For $0<\kappa<1$ close enough to $1$ and $C_u^*>1$ large enough, we also have that $$ \frac{\tilde y_1^{\kappa}}{y_1}\leq\frac{1}{\xi}. $$
\end{proof}
\begin{prop}\label{etadominationg} For every $\epsilon^*<1$, $C^*>1$ and $\eta>0$, if $\underline f=\left({y}_1, b\right)\in D_{\epsilon^*,C^*, C_u^*}$ with $R\underline f=\left(\tilde y_1, \tilde b\right)$, then $R$ has $\eta$-dominating horizontal expansion for $C_u^*>1$ large enough, i.e. \begin{eqnarray*} \frac{y_1^2}{\tilde y_1^{\kappa}}&\leq &\eta. \end{eqnarray*}
\end{prop}
\begin{proof}
Using point $2$ of Lemma \ref{ss} and Lemma \ref{D111} we get $$ \frac{y_1^2}{\tilde y_1}=O\left(S_2^{\frac{\ell-1}{\ell}} S_3^{-1} S_5\right). $$ Expressing the $S$-coordinates in terms of the coordinates in the eigenbasis of the matrix $M$, see Lemma \ref{matrix}, we obtain $$ \log\left[\frac{y_1^2}{\tilde y_1}\right]\leq -C_u\left[\frac{\lambda_u(\ell-1)+2\ell-1}{\ell\left(1+\lambda_u\right)}\right]+O\left(C\right). $$ For $0<\kappa<1$ close enough to $1$ and $C_u^*>1$ large enough, we also have that $$ \frac{y_1^2}{\tilde y_1^{\kappa}}\leq\eta. $$
\end{proof}
\subsection{Topological hyperbolicity}\label{tophyp} In this section we prove that, for an appropriate choice of $\epsilon^*, C^*, C_u^*$, $R:D_{\epsilon^*,C^*, C_u^*}\to L$ is topologically hyperbolic. The choice needs some preparation.
\\ The largest eigenvalue of $$Q=\left(\begin{matrix}
0 & 1 & 0\\
\frac{1}{16} & 0 & 1\\
\frac{1}{16} & \frac{1}{16} & 0\\
\end{matrix}\right) $$
is $\frac12$ with eigenvector $\underline E=(16,8,3)$. Let $\underline f=\left({y}_1,{y}_2,{y}_3,{y}_4,{y}_5,{\underline\varphi}, {\underline\varphi}^{l},{\underline\varphi}^{r}\right)\in L_0$ and $Rf=\left(\tilde{y}_1,\tilde{y}_2,\tilde{y}_3,\tilde{y}_4,\tilde{y}_5,O\tilde{\underline\varphi}, O\tilde{\underline\varphi}^{l},O\tilde{\underline\varphi}^{r}\right)$. Define
$$
m=m(\underline{f})=(|{\underline\varphi}|,
|{\underline\varphi}^{l}|,|{\underline\varphi}^{r}|),
$$
and $\tilde{m}=m(R\underline f)$.
\begin{lem}\label{distcontraction} For every $\epsilon^*<1$ and $C^*\geq 1$ we have $$ \tilde{m}\le Qm+ \frac{\epsilon^*}{33}\left(\begin{matrix} 0\\ 1\\ 1\\ \end{matrix}\right), $$ when $C_u^*$ is large enough. In particular, $$ (E,\tilde{m})\le \frac12 (E,m) +\frac{1}{3}\epsilon^*. $$ \end{lem}
\begin{proof} We estimate the norms $\tilde{w}$, $\tilde{w}^l$ and $\tilde{w}^r$. Observe,
$$
\left|\left[y_1(\tau),1\right]\right|\le 1.
$$
This implies, using Lemma \ref{normzoom} and the definition of renormalization,
\begin{equation}\label{w}
\tilde{w}\le w^l.
\end{equation}
Moreover, for $C_u^*>1$ large enough we have
\begin{equation}\label{ZS2m}
|[ \varphi^{-1}\circ q_s^{-1}\left(1-e^{y_2}\right)(\tau),1]|=O\left(S_2(1+\epsilon^*)\right)\le \frac{1}{16}.
\end{equation}
The previous inequality, Lemma \ref{normzoom} and Lemma \ref{C2} imply,
\begin{equation}\label{wl}
\tilde{w}^l\le \frac{1}{16} w+w^r+\frac{\epsilon^*}{33}.
\end{equation} For $C_u^*>1$ large enough and Lemma \ref{s1235} we get
\begin{eqnarray}\label{ZSSm}\nonumber
\varphi^{-1}\circ q_s^{-1}\left(y_1e^{y_2}e^{y_3}\right)(\theta\tau)&=&
O\left(\frac{S_1S_2S_3}{S_5}\right)=O\left(S_2^{1+\frac{1}{\ell}}S_3 S_5^{-1}\right)\\&=& O\left(S_2^{1+\frac{1}{\ell}}S_3 S_5^{-\frac{\ell}{\ell-1}}\right)\le \frac{1}{16},
\end{eqnarray}
where we used ${\ell S_1^{\ell}}/{S_2}\leq {3}/{2}$ and $\log\left({S_1S_2S_3}/{S_5^{\frac{\ell}{\ell-1}}}\right)\leq -C_u^*\left({\lambda_u^2(\ell+1)-1}/{\ell\lambda_u(1+\lambda_u)}\right)+O(C^*)$, see Lemma \ref{eighvector}. Furthermore, for $C_u^*>1$ large enough,
\begin{equation}\label{Zy}
|[0,y_1(\theta \tau)]|=O(y_1)=O\left(S_2^{\frac{1}{\ell}}\right)\le \frac{1}{16},
\end{equation} and by Lemma \ref{C9} we obtain
\begin{equation}\label{wr}
\tilde{w}^r\le \frac{1}{16} w+\frac{1}{16}w^l+\frac{\epsilon^*}{33}.
\end{equation}
Estimates (\ref{w}), (\ref{wl}), and (\ref{wr}) conclude the proof of the lemma.
\end{proof}
Let $\underline f_\pm=(\partial_\pm(b),b)\in \partial D$ with $R\underline f_\pm=(\tilde{y}_\pm, \tilde{b}_\pm)$. The following holds.
\begin{lem}\label{overlap} For every $C^*\geq 1$
we have
$$ \tilde{y}_+\le -\frac18 \text{ and } \tilde{y}_-\ge \frac{7}{16},
$$
when $C_u^*>1$ is large enough and $\epsilon^*<1$ is small enough. \end{lem}
\begin{proof} Observe that
$$ \frac{S_2}{1-\left(\varphi^{-1}\circ q_s^{-1}\right)\left(1- S_2\right)}=\ell(1+O(\epsilon^*)),
$$
and
$$
\frac{\left(\varphi^{l}\left( S_1\right)\right)^{\ell}}{\ell S_{1}^{\ell}}=1+O(\epsilon^*).
$$
Then for $\epsilon^*$ small enough we get, by using point 1 of Lemma \ref{ss},
$$
\tilde{y}_+\le 1-\frac32\left(1-\frac14\right)=-\frac18,
$$
and
$$
\tilde{y}_-\ge 1-\frac14\left(2+\frac14\right)\ge \frac{7}{16}.
$$
\end{proof}
\begin{prop} \label{proptophyp} For $C^*>1$ large enough $R: D_{\epsilon^*, C^*,C_u^* }\to \mathbb{R}\times B_1$ is topologically hyperbolic when $C_u^*>1$ is large enough and $\epsilon^*<1$ is small enough. \end{prop}
\begin{proof} Apply Lemma \ref{ss} and Lemma \ref{squarefactors} to obtain
\begin{eqnarray*}
\tilde{C}_u&\leq &\lambda_u C_u+\log 2,\\
\tilde{C}_s&\leq&\lambda_s C_s+\log 2,\\
\tilde{C}_0&\leq&\log 2.
\end{eqnarray*}
This implies, when $C^*>1$ and $C_u^*>1$ are large enough, that
$$
\tilde{C}_u\le -C^*_u \text{ and } |\tilde{C}_0|, |\tilde{C}_s|\le C^*.
$$
From Lemma \ref{distcontraction} we get
$$
(E, \tilde{w})\le \frac12 \epsilon^*+\frac{1}{3}\epsilon^*\le \epsilon^*.
$$ Finally when $C_u^*>1$ is large enough, for every $b\in U_{\epsilon^*, C^*,C_u^* }$
$$
0\le \partial_-(b)< \partial_+(b)\le \frac14.
$$ Lemma \ref{overlap} concludes the proof that $R:D\to \mathbb{R}\times B_1$ is topologically hyperbolic.
\end{proof}
\subsection{The Fibonacci class}
In this section we prove Theorem \ref{manifold} which states that the class of Fibonacci maps in ${\mathscr L}^{4+\epsilon}$ is a ${{\mathcal C}^1}$ codimension one manifold.
Consider $B_1^{4+\epsilon}=\mathbb{R}^4\times X_{4+\epsilon}\times X_{4+\epsilon}\times X_{4+\epsilon}$ and observe that $\left(B_1^{4+\epsilon}, |\cdot |_{3+\epsilon}\right)\subset B_1$ as a normed vector space. Similarly, $\left(B_1, |\cdot |_{2}\right)$ is a normed vector space in $B_0$. Define $$L_{3+\epsilon}^{4+\epsilon}=\Sigma^{(Y)}\times X_{4+\epsilon}\times X_{4+\epsilon}\times X_{4+\epsilon}\subset L\subset\mathbb{R}\times B_1,$$ and $$L_2^{3+\epsilon}=\Sigma^{(Y)}\times X_{3+\epsilon}\times X_{3+\epsilon}\times X_{3+\epsilon}\subset \mathbb{R}\times B_0.$$ By Proposition \ref{jumpoutdiffprop}, $$R:L_{3+\epsilon}^{4+\epsilon}\to L_2^{3+\epsilon}$$ is continuously differentiable. Apply Theorem \ref{InvMan} to get $D\subset [0,1]\times U$ with $U\subset B_1$ maximal and a ${{\mathcal C}^1}$ function $\hat\omega^*:U\to\mathbb{R}$ such that the invariant set\footnote{See Lemma \ref{omegaW}}, $$\omega^*=\left\{p\in D | \forall n\in{\mathbb N}\text{ }R^n(p)\in D\right\}$$ is the graph of $\hat\omega^*$.
Consider $$W=\left\{\underline f\in L | \exists n\in{\mathbb N} \text{ s.t. } R^n\underline f\in\omega^*\right\},$$ and notice that \begin{itemize} \item[-]$\omega^*$ is a ${{\mathcal C}^1}$ graph in $W$, \item[-]for all $n\in{\mathbb N}$, $R^n\left(W\right)\subset W$. \end{itemize}
\begin{lem}\label{transversaldeformation} Given $f\in {\mathscr W}\subset{\mathscr L}^{4+\epsilon}$ there exists a family $f_t\in{\mathscr L}^{4+\epsilon}$, smooth in $t\in (-1,1)$, with $f_0=f$ and $b_0>0$ such that $$
\{f'\in {\mathscr L}^{4+\epsilon} | \text{ } \exists t\ne 0 \text{ with } |f'-f_t|_{\C0}\leq b_0\cdot t\}\cap {\mathscr W} =\emptyset. $$ \end{lem}
\begin{proof} Let $f\in{\mathscr W}$ with $x_1=x_1(f)$ and $\pi:\mathbb{R}\to [x_1,1)$ be a piece wise smooth projection with period $1-x_1$ such that the lift $F:\mathbb{R}\to \mathbb{R}$ of $f$ is smooth. The proof of Lemma \ref{alpha} assures that such a projection $\pi$ exists. Moreover, we may assume that $\pi: [0,1)\to [0,1)$ is identity and $\pi:[x_1,0)\to [x_1,0)$ is a diffeomorphism. Let $F_t:\mathbb{R}\to \mathbb{R}$ be given by an added rotation $$ F_t(x)=F(x)+t. $$ Then for every lift $G$ of a map $g\in {\mathscr L}^{4+\epsilon}$ with period $1-x_1$ and \begin{equation}\label{GFt}
|G(x)-F_t(x)|\le \frac12 t, \end{equation}
the circle map $g$ is not a Fibonacci map.
\\
Using an appropriate projection of period $1-x_1$, each map $F_t$ is the lift of a map $f_t\in{\mathscr L}^{4+\epsilon}$. This needs some preparation. Observe that $J=[x_3(f),x_4(f)]$ is the flat interval of any $F_t$. Let
$$
o(t)=F_t(J)=t, \textbf{ } x_1(t)=F_t(o(t)) \textbf{ } 1(t)=x_1(t)+1-x_1.
$$ The interval $[x_1(t), 1(t))$ is a fundamental domain for $F_t$. Let $\pi_t:\mathbb{R}\to \mathbb{R}$ be the $1-x_1$ periodic piece wise smooth map given by $$ \pi_t(x)= \left\{\begin{aligned} &\pi\left(\frac{x-o(t)}{1(t)-o(t)}\right)&\text{ when } x\in [o(t), 1(t))\\ &\pi\left(x_1(f) \frac{x-o(t)}{x_1(t)-o(t)}\right) &\text{ when } x\in [x_1(t), o(t))\\ \end{aligned}\right. $$ Consider the smooth family $f_t\in{\mathscr L}^{4+\epsilon}$ with $f_t:[x_1,1)\to [x_1,1)$ defined by $$ f_t=\pi_t\circ F_t\circ \pi_t^{-1}, $$
with $t$ in a small interval centered around $0$. For $b_0>0$ small enough we have that every $f'\in {\mathscr L}^{4+\epsilon}$ with $|f'-f_t|_{\C0}\le b_0 t$ there exists a $1-x_1$ periodic lift $F'$ with
$|F'(x)-F_t(x)|\le (1/2) t$. According to \ref{GFt}, $f'$ is not a Fibonacci map. Hence $f'\notin{\mathscr W}$. \end{proof}
Define the embedding $i:{\mathscr L}^{4+\epsilon}\to L$ as follows. If $f=\left(y_1, y_2, y_3, y_4, y_5, \varphi, \varphi^l, \varphi^r\right)\in{\mathscr L}^{4+\epsilon}$, then $$i(f)=\left(y_1, y_2, y_3, y_4, y_5, \underline\varphi, \underline\varphi^l, \underline\varphi^r\right),$$ with $$\varphi_{\frac{1}{2}}=\varphi,\text{ }\varphi^l_{\frac{1}{2}}=\varphi^l, \text{ }\varphi^r_{\frac{1}{2}}=\varphi^r,$$ and $\varphi_{\tau}=\varphi^l_{\tau}=\varphi^r_{\tau}=0$ for $\tau\neq{1}/{2}$. This is a ${{\mathcal C}^1}$ map with $O\circ i=id$.
\begin{lem}\label{transversality} Let $\underline f\in W\cap L_{3+\epsilon}^{4+\epsilon}$ such that $W$ is locally a ${{\mathcal C}^1}$ codimension 1 manifold in $L$ containing $R\underline f$, then $$DR_{\underline f}\pitchfork T_{R\underline f}W.$$ \end{lem} \begin{proof} Let $f=O(\underline f)$. Consider the family $f_t$ through $f$ from Lemma \ref{transversaldeformation}. The Lipschitz continuity of $O$, see Proposition\ref{comp}, implies that for $b>0$ small enough $$
\{\underline{f}'\in L | \text{ } \exists t\ne 0 \text{ with } |\underline{f}'-i(f_t)|_{\C0}\leq b\cdot t\}\cap W =\emptyset. $$
Moreover, for every $\underline{f}'\in L$ with $ |\underline{f}'-i(f_t)|_{\C0}\leq b\cdot t$ for some $t\ne 0$ we have \begin{equation}\label{RfnotW} R\underline{f}'\notin W. \end{equation} Let $\Delta v=\frac{\partial( i( f_t))}{\partial t}$. Then $DR_{\underline f}(\Delta v)\in \mathbb{R}^5\times X_{3+\epsilon}\times X_{3+\epsilon}\times X_{3+\epsilon}$ and because (\ref{RfnotW}) $$ DR_{\underline f}(\Delta v) \pitchfork T_{R\underline f}W. $$ \end{proof}
\begin{cor}\label{transi} Let $f\in {\mathscr W}\cap {\mathscr L}_{3+\epsilon}^{4+\epsilon}$ such that $W$ is locally a ${{\mathcal C}^1}$ codimension 1 manifold in $L$ containing $i(f)$, then $$Di_{f}\pitchfork T_{i(f)} W.$$ \end{cor}
\begin{prop}\label{WC1} Let $\underline f\in W\cap L_{3+\epsilon}^{4+\epsilon}$ then $W$ is locally a ${{\mathcal C}^1}$ codimension 1 manifold in $L$ containing $\underline f$. \end{prop} \begin{proof} By contradiction, suppose that there exists $\underline f\in W\cap L_{3+\epsilon}^{4+\epsilon}$ such that $W$ is not locally a ${{\mathcal C}^1}$ codimension 1 manifold in $L$ containing $\underline f$. Then we may assume that $W$ is locally a ${{\mathcal C}^1}$ codimension 1 manifold containing $R\underline f$. Such an $\underline f$ exists because $\omega^*$ is a ${{\mathcal C}^1}$ graph in $W$. Let $n\ge 1$ such that $R^n\underline{f}\in \omega^*$. The derivative of $R^n$ extends to a bounded operator $DR^n_{\underline{f}}: \mathbb{R}^5\times X_{2}\times X_{2}\times X_{2}\to \mathbb{R}^5\times X_{2}\times X_{2}\times X_{2}$ because $R$ is jump-out-differentiable. Hence the tangent space at $R\underline f$ to $W$, $T_{R\underline f}W$, extends to a plane in $\mathbb{R}^5\times X_{2}\times X_{2}\times X_{2}$. Namely, $T_{R\underline{f}}W=(DR^n_{\underline{f}})^{-1}(T_{R^n\underline{f}} \omega^*)$. The contradiction comes straight from Lemma \ref{transversality} and Lemma \ref{pullbacklemma}. \end{proof}
\begin{lem}\label{DegcondinL} If $f\in{\mathscr W}$, then $i(f)\in W$. \end{lem} \begin{proof}
Fix $\epsilon^*$, $C^*$ and $C_u^*$ as in Proposition \ref{proptophyp} and let $f\in{\mathscr W}$. The aim is to prove that there exists $n\in{\mathbb N}$ such that $R^n i(f)\in\omega^*$. Proposition \ref{superformula} ensures that, for $n$ large enough, $C_u\left(R^ni(f)\right)=C_u\left(R^nf\right)<-C_u^*$ and $\left|C_s\left(R^ni(f)\right)\right|=\left|C_s\left(R^nf\right)\right|<C^*$. \\
It remains to prove that, for $n$ large enough, $\left(E, w(R^n i(f))\right)\leq\epsilon^*$. For all $n\in{\mathbb N}$, denote by $ w_n=w\left(R^ni(f)\right)$, $ w^l_n=w^l\left(R^ni(f)\right)$ and $ w^r_n=w^r\left(R^ni(f)\right)$. By the definition of renormalization on $L$, see Section \ref{RenonL}, and Lemma \ref{normzoom} we get
$$|w_{n+1}|+|w^l_{n+1}|+|w^r_{n+1}|\leq |w_{n}|+|w^l_{n}|+\left|w^r_{n}\right|+\left| Z_{\left[q_s^{-1}\left(1-S_2\right),1\right]}q_s\right|+\left|Z_{\left[0,q_s^{-1}\left(S_1S_2S_3\right)\right]}q_s\right|.$$ Applying (\ref{ZS2}) and (\ref{ZSS}) we get
$$|w_{n+1}|+|w^l_{n+1}|+|w^r_{n+1}|\leq |w_{n}|+|w^l_{n}|+\left|w^r_{n}\right|+O\left(S_2\right)+O\left(S_1 S_2 S_3 S_5^{-\frac{\ell}{\ell-1}}\right).$$ Writing the affine terms in the eigenvector base, see Lemma \ref{eighvector}, we notice that they tend to zero double exponentially fast. As consequence,
$$|w_{n}|+|w^l_{n}|+\left|w^r_{n}\right|\leq O(1).$$ By the definition of renormalization on $L$, see Section \ref{RenonL}, Lemma \ref{normzoom}, (\ref{ZS2}), (\ref{ZSS}), (\ref{ZS2m}), (\ref{ZSSm}) and (\ref{Zy}) we get $$ \left\{\begin{aligned}
|w_{n+1}|\leq& |w^l_{n}|,\\
|w^l_{n+1}|\leq& O\left(S_2\right)|w_{n}|+ |w^r_{n}|+O\left(S_2\right),\\
|w^r_{n+1}|\leq& O\left(S_1 S_2 S_3 S_5^{-\frac{\ell}{\ell-1}}\right)|w_{n}|+O\left(S_2^{\frac{1}{\ell}}\right) |w^l_{n}|+O\left(S_1 S_2 S_3 S_5^{-\frac{\ell}{\ell-1}}\right).\\ \end{aligned}\right. $$ As before, the coefficients in the orders decay double exponentially fast. Iterating the above estimates three times, we obtain
$$|w_{n+1}|+|w^l_{n+1}|+|w^r_{n+1}|\leq\lambda^{n-2}\left(|w_{n-2}|+|w^l_{n-2}|+\left|w^r_{n-2}\right|\right)+\lambda^{n-2},$$ where $\lambda<1$. The lemma follows. \end{proof} By Corollary \ref{transi}, Proposition \ref{WC1}, Lemma \ref{DegcondinL} and Lemma \ref{pullbacklemma}, $i^{-1}\left( W\right)={\mathscr W}$ is a ${{\mathcal C}^1}$ codimension one manifold in ${\mathscr L}$. This concludes the proof of Theorem \ref{manifold}.
\section{Rigidity}\label{rigidity} For any function $f\in{\mathscr W}$, the non-wandering set\footnote{The non wandering set of a map $f$ is the set of the points $x$ such that for any open neighborhood $V\ni x$ there exists an integer $n>0$ such that the intersection of $V$ and $f^n(V)$ is non-empty.} of $f$ is $K_f={\mathbb S}^1\setminus\cup_{i\geq 0}f^{-i}(U_f)$, where $U_f=[x_3(f),x_4(f)]$ is the flat interval of $f$, see \cite{5aut}, \cite{M-vS-dM-M}. \begin{defin} Let $f\in{\mathscr L}$. The set $A_f$ is the attractor of $f$ if $A_f$ is the limit set of every point. \end{defin} The following lemma holds. For its proof, the reader can refer to \cite{5aut} and \cite{M-vS-dM-M}. \begin{lem} Let $f\in{\mathscr W}$. The non-wandering set $K_f$ is a Cantor set and it is the attractor of the system. \end{lem} Let $f,g\in {\mathscr W}$ and let $h$ be an homeomorphism which conjugates $f$ and $g$.
Observe that $h$ is such that $h(U_f)=U_g$ but $h$ is not uniquely defined inside $U_f$. However, $h_{|K_f}$ is uniquely defined.
Being interested in the geometry of the Cantor set $K_f$, from now on we
only discuss the quality of $h_{|K_f}$ which is simply denote by $h$. \begin{defin} Let $K\subset\mathbb{R}$ be a Cantor set. A function $h:K\to\mathbb{R}$ is differentiable if there exists a function $Dh:K\to\mathbb{R}$ such that, for every $x_0\in K$,
$$h\left(x\right)=h\left(x_0\right)+Dh\left(x_0\right)\left(x-x_0\right)+o\left(\left|x-x_0\right|\right)$$ for every $x\in K$. Moreover $h$ is $\mathcal C^{1}$ differentiable if $Dh$ is continuous and $h$ is $\mathcal C^{1+\beta}$ differentiable if $Dh$ is H\"older with exponent $\beta>0$. \end{defin} We are now ready to state the main theorem of this section. \begin{theo}\label{LM} Let $1<\ell<2$ and let $\beta=\frac{\lambda_u-1}{\lambda_u^4\ell}>0$. If $f,g\in{\mathscr W}$ with critical exponent $\ell$ and if $h$ is the topological conjugacy between $f$ and $g$, then $h$ is a H\"older homeomorphism. Moreover the following holds. \begin{eqnarray*}
h \text{ is a Lipschitz homeomorphism} &\iff& C_u(f)=C_u(g),\\
h \text{ is a } \mathcal C^1 \text{ diffeomorphism }&\iff& C_u(f)=C_u(g), C_-(f)=C_-(g),\\
h \text{ is a } \mathcal C^{1+\beta} \text{ diffeomorphism}&\iff& C_u(f)=C_u(g), C_-(f)=C_-(g), C_s(f)=C_s(g).\\ \end{eqnarray*} \end{theo}
\begin{prop}\label{nothol}
Let $f,g\in{\mathscr W}$ with different critical exponents $\ell_f\neq\ell_g$, $1<\ell_f,\ell_g<2$. Then $h$ is not H\"older. \end{prop} \begin{proof} Because $\ell_f\neq\ell_g$, then $\lambda_u(f)\neq\lambda_u(g)$. Without loose of generality we may assume $\lambda_u(g)<\lambda_u(f).$
Observe that $$\frac{\hat x_{2,n}}{-\hat x_{1,n}}=\frac{f^{q_{n+1}}(0)}{-f^{q_{n}}(0)}=\frac{x_{2,n}}{-x_{1,n}}=S_{4,n}.$$ As a consequence, by Proposition \ref{superformula},
\begin{equation}\label{asymqn}
\hat x_{2,n}= f^{q_{n+1}}(0)=\prod_{k\leq n}S_{4,k}|\hat x_{1,0}|\sim e^{C_u(f)\lambda_{u}(f)^ne^u_4(f)}.
\end{equation}
In the same way
$$g^{q_{n+1}}(0)\sim e^{C_u(g)\lambda_{u}(g)^ne^u_4(g)}.$$ Notice that, if $h$ conjugates $f$ and $g$ then $h(f(U_f))=g(U_g)$. Suppose now that $h$ is H\"older continuous with exponent
$\beta>0$ then $$\lim_{n\to\infty}\frac{g^{q_{n+1}}(0)}{\left( f^{q_{n+1}}(0)\right)^{\beta}}
\sim \lim_{n\to\infty}e^{C_u(g)\lambda_{u}(g)^ne^u_4(g)-\beta C_u(f)\lambda_{u}(f)^ne^u_4(f)}=\infty$$ where we used that $C_u(f)<0$
(see Lemma \ref{asymalpha}) and that $\lambda_u(g)<\lambda_u(f)$. Finally, $h$ cannot be H\"older. \end{proof} We would like to compare Proposition \ref{nothol} with the main Theorem in \cite{P3}. The author proves that, if $f,g\in{\mathscr W}$ correspond to smooth circle maps with different critical exponents $\ell_f\neq\ell_g$ both belonging to $(2,+\infty]$, then $h$ is a quasi-symmetric homeomorphism, in particular it is H\"older. The reason for the regularity of $h$ under such a weak condition is related with the fact that, for functions with critical exponent $\ell>2$, the sequence $\alpha_n$ is bounded away from zero (see \cite{P1}). In the setting of our paper, of functions with critical exponent $\ell<2$, the sequence $\alpha_n$ goes
to zero double exponentially fast and this causes the loss of regularity of the conjugacy. \subsection{Proof of Theorem \ref{LM}} \begin{prop}\label{lipc1}
Let $1<\ell<2$. If $f,g\in {\mathscr W}$ with critical exponent $\ell$ and if $h$ is the topological conjugacy between $f$ and $g$ then, the following holds. \begin{eqnarray*}
h \text{ is a Lipschitz homeomorphism} &\implies& C_u(f)=C_u(g),\\
h \text{ is a } \mathcal C^1 \text{ diffeomorphism }&\implies& C_u(f)=C_u(g), C_-(f)=C_-(g). \end{eqnarray*} \end{prop} \begin{proof} Without loss of generality we may assume that $C_u(g)\geq C_u(f)$.
From (\ref{asymqn}) and Proposition \ref{superformula} we get
$$D=\limsup_{n\to\infty}\frac{g^{q_{n+1}}(0)}{f^{q_{n+1}}(0)}=\limsup_{n\to\infty}e^{(C_u(g)-C_u(f))\lambda_{u}^ne^u_4+(C_-(g)-C_-(f))(-1)^n}.$$
Observe that $h(f^{q_{n+1}}(0))=(g^{q_{n+1}}(0))$. If $h$ is a Lipschitz homeomorphism then $D$ is bounded by a positive constant. Hence, $C_u(g)=C_u(f)$. If $h$ is a $\mathcal C^1$ diffeomorphism then $D=\lim_{n\to\infty}{g^{q_{n+1}}(0)}/{f^{q_{n+1}}(0)}$. Hence, $C_u(g)=C_u(f)$ and
$C_-(g)=C_-(f)$. \end{proof} For all $n\in{\mathbb N}$ we define the following intervals: \begin{itemize}
\item[-] $A_n(f)=\left[\hat x_{1,n},0 \right]$,
\item[-] $B_n(f)=\left[0, \hat x_{3,n}\right]$,
\item[-] $C_n(f)=\left[\hat x_{3,n}, \hat x_{4,n}\right]$,
\item[-] $D_n(f)=\left[\hat x_{4,n}, \hat x_{1,n-1}\right]$, \end{itemize} and their iterates \begin{itemize}
\item[-] $A_n^i(f)=f^i(A_n(f))$ for $0\leq i<q_{n-1}$,
\item[-] $B_n^i(f)=f^i(B_n(f))$ for $0\leq i<q_{n}$,
\item[-] $C_n^i(f)=f^i(C_n(f))$ for $0\leq i<q_{n}$,
\item[-] $D_n^i(f)=f^i(D_n(f))$ for $0\leq i<q_{n}$. \end{itemize}
Observe that, for all $n\in{\mathbb N}$, $$\mathscr P_n(f)=\left\{A_n^i(f), B_n^j(f), C_n^j(f), D_n^j(f)|0\leq i<q_{n-1}, 0\leq j<q_{n}\right\}$$ is a partition of the domain of $f$ and that \begin{itemize}
\item[-] $h(A_n^i(f))=A_n^i(g)$ for $0\leq i<q_{n-1}$,
\item[-] $h(B_n^i(f))=B_n^i(g)$ for $0\leq i<q_{n}$,
\item[-] $h(C_n^i(f))=C_n^i(g)$ for $0\leq i<q_{n}$,
\item[-] $h(D_n^i(f))=D_n^i(g)$ for $0\leq i<q_{n}$. \end{itemize}
For all $n\in{\mathbb N}$ we define the mesurable function $Dh_n:[0,1]\longrightarrow\mathbb{R}^+$ as $$Dh_n(x)=\frac{|h(I)|}{|I|},$$ with $I\in\mathscr P_n$ and $x\in\text{Interior}{(I)}$. Observe that, if $T=\cup I_i$ with $I_i\in\mathscr P_n$, then for all $m\geq n$, \begin{equation}\label{ht}
|h(T)|=\int_TDh_m. \end{equation} \begin{lem}\label{length} Let $1<\ell<2$. There exists $C>0$ such that, for every interval $I\in \mathscr P_n$, $I\neq C_n^i$, then $$
|I|\geq \frac{1}{C} e^{C_u(f)\lambda_u^{n}\frac{\lambda_u}{\lambda_u-1}}. $$
\end{lem} \begin{proof}
Let $J\in\mathscr P_{n-1}$ with $I\subset J$. By the construction of $\mathscr P_n$ from $\mathscr P_{n-1}$ we see that
$$\frac{|I|}{|J|}\geq\left(1+O\left(\alpha_{n-3}^{\frac{1}{\ell}}\right)\right)\min\left\{1-x_{4,n},x_{3,n}, S_{1,n-1}\right\},$$ where we used Lemma \ref{affdiffeo}.
From Lemma \ref{asymalpha}, Lemma \ref{xs}, Corollary \ref{s1} and Proposition \ref{superformula}, there exists a positive constant $K>0$ such that
\begin{enumerate}
\item $1-x_{4,n}\geq e^{C_u\lambda_u^{n}e_2^u-K\lambda_s^n},$
\item $x_{3,n}\geq e^{C_u\lambda_u^{n}(e_2^u+e^u_3)-K\lambda_s^n},$
\item $S_{1,n-1}\geq e^{C_u\lambda_u^{n-1}(\frac{e_2^u}{\ell})-K\lambda_s^n-\frac{1}{\ell}\log\ell}$.
\end{enumerate} From Proposition \ref{superformula}, Lemma \ref{eighvector} and the value of $\lambda_u$, we get that
$$\frac{|I|}{|J|}\geq e^{C_u\lambda_u^{n}-K\lambda_s^n}.$$
Finally, because $J\neq C_{n-1}^j$, we can repeat this estimates and we get
$$|I|\geq e^{\sum_{k=0}^{n} C_u\lambda_u^{k}-K\lambda_s^k}.$$ The lemma follows. \end{proof}
\begin{lem}\label{lengthup} Let $1<\ell<2$. There exists $C>0$ such that, for every interval $I\in \mathscr P_n$, $I\neq C_n^i$, then $$
|I|\leq C e^{\frac{C_u(f)\lambda_u^{n-3}}{\ell}}. $$ \end{lem} \begin{proof} By Lemma \ref{affdiffeo}, \begin{itemize}
\item[-] $\max |A_{n}^i|\leq \max |B_{n}^i|\left(1+O\left(\alpha_{n-3}^{\frac{1}{\ell}}\right)\right)$,
\item[-] $\max |B_{n}^i|\leq\max\left[ x_{3,n}\max |A_{n-1}^i|, \max |D_{n-1}^i|\right]\left(1+O\left(\alpha_{n-3}^{\frac{1}{\ell}}\right)\right)$,
\item[-] $\max |D_{n}^i|\leq\max\left[\left(1-x_{4,n}\right)\max |A_{n-1}^i|, S_{1,n-1}\max |B_{n-1}^i|\right]\left(1+O\left(\alpha_{n-3}^{\frac{1}{\ell}}\right)\right)$,
\end{itemize}
and by recursion
\begin{itemize}
\item[-] $\max |D_{n}^i|\leq O\left(\max\left[ 1-x_{4,n}, S_{1,n-1}\right]\right)$,
\item[-] $\max |B_{n}^i|\leq O\left(\max\left[ x_{3,n}, 1-x_{4,n-1}, S_{1,n-2}\right]\right)$,
\item[-] $\max |A_{n}^i|\leq O\left(\max\left[ x_{3,n-1}, 1-x_{4,n-2}, S_{1,n-3}\right]\right)$.
\end{itemize}
From Lemma \ref{xs}, Corollary \ref{s1} and Proposition \ref{superformula}
\begin{itemize}
\item[-] $1-x_{4,n-2}=O\left(e^{C_u\lambda_u^{n-2}e_2^u}\right)$,
\item[-] $x_{3,n-1}=O\left(e^{C_u\lambda_u^{n-1}\left(e_2^u+e^u_3\right)}\right)$,
\item[-] $S_{1,n-3}=O\left(e^{C_u\lambda_u^{n-3}\left(\frac{e_2^u}{\ell}\right)}\right)$.
\end{itemize} By Lemma \ref{eighvector}, we have that ${1}/{\ell}\leq\min\left[\lambda_u^2\left(1+e_3^u\right), \lambda_u \right]$ and by the fact that $C_u<0$, the lemma follows. \end{proof} \begin{lem}\label{Dh} Let $1<\ell<2$. If $f,g\in{\mathscr W}$ with critical exponent $\ell$ and $C_u=C_u(f)=C_u(g)$ then
$$\log\frac{Dh_{n+1}}{Dh_n}=O\left(\alpha_{n-2}(f)^{\frac{1}{\ell}}+\left|\left(C_s(g)-C_s(f)\right)\lambda_s^n\right|\right).$$ \end{lem} \begin{proof} The hypothesis $C_u=C_u(f)=C_u(g)$ implies that $\log\left({\alpha_n(g)}/{\alpha_n(f)}\right)=O\left(\lambda_s^{n-1}\right)$, see Lemma \ref{asymalpha}. By Lemma \ref{affdiffeo} and point $1$ of Lemma \ref{prev} we get \begin{eqnarray*}
\frac{Dh_{n+1|A^i_{n+1}(f)}}{Dh_{n|A^i_{n+1}(f)}}&=&\frac{{|A^i_{n+1}(g)|}/{|A^i_{n+1}(f)|}} {{|B^i_{n}(g)|}/{|B^i_{n}(f)|}}=
\frac{{|A^i_{n+1}(g)|}/{|B^i_{n}(g)|}}{{|A^i_{n+1}(f)|}/{|B^i_{n}(f)|}}\\&=&\frac{{|A_{n+1}(g)|}/{|B_{n}(g)|}}
{{|A_{n+1}(f)|}/{|B_{n}(f)|}}\left(1+O\left({\alpha_{n-2}}(f)^{\frac{1}{\ell}}\right)\right)\\&=&\frac{{x_{2,n}(g)}/{x_{3,n}(g)}}{{x_{2,n}(f)}/{x_{3,n}(f)}}\left(1+O\left({\alpha_{n-2}}(f)^{\frac{1}{\ell}}\right)\right)\\&=& \frac{1-S_{1,n}(g)}{1-S_{1,n}(f)}\left(1+O\left({\alpha_{n-2}}(f)^{\frac{1}{\ell}}\right)\right)\\&=&1+O\left({\alpha_{n-2}}(f)^{\frac{1}{\ell}}\right). \end{eqnarray*}
Observe that, for $i\geq q_{n-1}$, $B^i_{n+1}(f)=D^{i-q_{n-1}}_{n}(f)$ and in particular $$\frac{Dh_{n+1|B^i_{n+1}(f)}}{Dh_{n|B^i_{n+1}(f)}}=1.$$ Let $i< q_{n-1}$. With a similar calculation as before, we get \begin{eqnarray*}
\frac{Dh_{n+1|B^i_{n+1}(f)}}{Dh_{n|B^i_{n+1}(f)}}&=&\frac{{|B^i_{n+1}(g)|}/{|B^i_{n+1}(f)|}}{{|A^i_{n}(g)|}/{|A^i_{n}(f)|}}
=\frac{{|B_{n+1}(g)|}/{|A_{n}(g)|}}
{{|B_{n+1}(f)|}/{|A_{n}(f)|}}\left(1+O\left({\alpha_{n-2}}(f)^{\frac{1}{\ell}}\right)\right)\\&=&\frac{x_{3,n+1}(g)}{x_{3,n+1}(f)}\left(1+O\left({\alpha_{n-2}}(f)^{\frac{1}{\ell}}\right)\right)\\&=&
e^{\left(C_s(g)-C_s(f)\right)\lambda_s^{n+1}\left(e_2^s+e_3^s\right)+O\left(e^{\frac{C_u\lambda_u^{n-3}}{\ell}}\right)}\left(1+O\left({\alpha_{n-2}}(f)^{\frac{1}{\ell}}\right)\right)\\&=&1+O\left(\alpha_{n-2}(f)^{\frac{1}{\ell}}+\left|\left(C_s(g)-C_s(f)\right)\lambda_s^n\right|\right), \end{eqnarray*} where we used Lemma \ref{xs}. \\
Observe that, for $i\geq q_{n-1}$, $C^i_{n+1}(f)=C^{i-q_{n-1}}_{n}(f)$ and in particular $$\frac{Dh_{n+1|C^i_{n+1}(f)}}{Dh_{n|C^i_{n+1}(f)}}=1.$$ Let $i< q_{n-1}$. With a similar calculation as before, we get \begin{eqnarray*}
\frac{Dh_{n+1|C^i_{n+1}(f)}}{Dh_{n|C^i_{n+1}(f)}}&=&\frac{{|C_{n+1}(g)|}/{|C_{n+1}(f)|}}{{|A_{n}(g)|}/{|A_{n}(f)|}}\left(1+O\left({\alpha_{n-2}}(f)^{\frac{1}{\ell}}\right)\right)\\ &=&\frac{x_{4,n+1}(g)-x_{3,n+1}(g)}{x_{4,n+1}(f)-x_{3,n+1}(f)}\left(1+O\left({\alpha_{n-2}}(f)^{\frac{1}{\ell}}\right)\right)\\&=& \frac{1-[x_{3,n+1}(g)+(1-x_{4,n+1}(g))]}{1-[x_{3,n+1}(f)+(1-x_{4,n+1}(f))]}\left(1+O\left({\alpha_{n-2}}(f)^{\frac{1}{\ell}}\right)\right)\\&=&\frac{1-O\left(\alpha_{n+2}(g)\right)}{1-O\left(\alpha_{n+2}(f)\right)} \left(1+O\left({\alpha_{n-2}}(f)^{\frac{1}{\ell}}\right)\right)=1+O\left({\alpha_{n-2}}(f)^{\frac{1}{\ell}}\right), \end{eqnarray*} where we used Lemma \ref{xs} and Lemma \ref{asymalpha}. \\ We denote by $\Delta_n(f)=\left(f^{q_{n+1}}(0),f^{-q_{n}}(0)\right)=\left(\hat x_2,\hat x_3\right)$ and its iterates by $\Delta^i_n(f)=f^i\left(\Delta_n(f)\right)$ for $0\leq i<q_n$. Observe that, for $i\geq q_{n-1}$, $D^i_{n+1}(f)=\Delta^{i-q_{n-1}}_{n}(f)$ and in particular \begin{eqnarray*}
\frac{Dh_{n+1|D^i_{n+1}(f)}}{Dh_{n|D^i_{n+1}(f)}}&=&\frac{{|\Delta^{i-q_{n-1}}_{n}(g)|}/{|\Delta^{i-q_{n-1}}_{n}(f)|}}
{{|B^{i-q_{n-1}}_{n}(g)|}/{|B^{i-q_{n-1}}_{n}(f)|}}\\&=&\frac{{|\Delta_{n}(g)|}/{|\Delta_{n}(f)|}}
{{|B_{n}(g)|}/{|B_{n}(f)|}}\left(1+O\left({\alpha_{n-2}}(f)^{\frac{1}{\ell}}\right)\right)\\&=&\frac{{|\Delta_{n}(g)|}/{|B_{n}(g)|}}
{{|\Delta_{n}(f)|}/{|B_{n}(f)|}}\left(1+O\left({\alpha_{n-2}}(f)^{\frac{1}{\ell}}\right)\right)\\ &=&\frac{S_{1,n}(g)}{S_{1,n}(f)}\left(1+O\left({\alpha_{n-2}}(f)^{\frac{1}{\ell}}\right)\right)=1+O\left({\alpha_{n-2}}(f)^{\frac{1}{\ell}}\right), \end{eqnarray*} where we used point $1$ of Lemma \ref{prev}. Let $i< q_{n-1}$, then \begin{eqnarray*}
\frac{Dh_{n+1|D^i_{n+1}(f)}}{Dh_{n|D^i_{n+1}(f)}}&=&\frac{{|D_{n+1}(g)|}/{|D_{n+1}(f)|}}{{|A_{n}(g)|}/{|A_{n}(f)|}}\left(1+O\left({\alpha_{n-2}}(f)^{\frac{1}{\ell}}\right)\right)\\
&=&\frac{{|D_{n+1}(g)|}/{|A_{n}(g)|}}{{|D_{n+1}(f)|}/{|A_{n}(f)|}}\left(1+O\left({\alpha_{n-2}}(f)^{\frac{1}{\ell}}\right)\right)\\ &=&\frac{1-x_{4,n+1}(g)}{1-x_{4,n+1}(f)}\left(1+O\left({\alpha_{n-2}}(f)^{\frac{1}{\ell}}\right)\right)\\&=&
e^{\left(C_s(g)-C_s(f)\right)\lambda_s^{n+1}+O\left(e^{\frac{C_u\lambda_u^{n-3}}{\ell}}\right)}\left(1+O\left({\alpha_{n-2}}(f)^{\frac{1}{\ell}}\right)\right)\\&=&1+O\left({\alpha_{n-2}}(f)^{\frac{1}{\ell}}+\left|\left(C_s(g)-C_s(f)\right)\lambda_s^n\right|\right), \end{eqnarray*} where we used Lemma \ref{xs} and Lemma \ref{asymalpha}. \end{proof} Observe that every boundary point of an interval in $\mathscr P_{n}$ is in the orbit of the critical point $\text{Orbit}(0)$. As a consequence, if $x\in\text{Orbit}(0)$, then $Dh_{{n}}(x)$ is well defined and Lemma \ref{Dh} implies that the following limit $$D(x)=\lim_{n\to\infty} Dh_{{n}}(x),$$ exists. Observe that, \begin{equation}\label{boundD} 0<\inf_{x}D(x)<\sup_xD(x)<\infty. \end{equation} \begin{lem}\label{Dcont} Let $1<\ell<2$. If $f,g\in{\mathscr W}$ with critical exponent $\ell$ and $C_u=C_u(f)=C_u(g)$ then, the function $D:[0,1]\setminus\text{Orbit}(0)\to(0,\infty)$ is continuous.
In particular for all $n\in{\mathbb N}$, $$\left|\log\frac{D(x)}{Dh_{{{n}}}(x)}\right|=O\left(\alpha_{n-2}(f)^{\frac{1}{\ell}}+\left|\left(C_s(g)-C_s(f)\right)\lambda_s^{n}\right|\right).$$
Moreover, if $x,y\in I\in \mathscr P_{{n}}$,
$$\left|\log\frac{D(x)}{D(y)}\right|=O\left(\alpha_{n-2}(f)^{\frac{1}{\ell}}+\left|\left(C_s(g)-C_s(f)\right)\lambda_s^{n}\right|\right).$$ \end{lem} \begin{proof} Fix $n_0\in{\mathbb N}$, then by Lemma \ref{Dh} \begin{eqnarray*}
\left|\log\frac{D(x)}{Dh_{{{n_0}}}(x)}\right|&\leq&\sum_{k\geq 0}\left|\log\frac{Dh_{{{n_0+k+1}}}(x)}{Dh_{{{n_0+k}}}(x)}\right|\\
&\leq&\sum_{k\geq 0}O\left(\alpha_{k-2}(f)^{\frac{1}{\ell}}+\left|\left(C_s(g)-C_s(f)\right)\lambda_s^{k}\right|\right)\\&=&
O\left(\alpha_{n_0-2}(f)^{\frac{1}{\ell}}+\left|\left(C_s(g)-C_s(f)\right)\lambda_s^{n_0}\right|\right). \end{eqnarray*} As a consequence, if $x,y\in I\in \mathscr P_{{n_0}}$, then
$$\left|\log\frac{D(x)}{D(y)}\right|\leq \left|\log\frac{D(x)}{Dh_{{{n_0}}}(x)}\right|+\left|\log\frac{D(y)}
{Dh_{{{n_0}}}(y)}\right|=O\left(\alpha_{n_0-2}(f)^{\frac{1}{\ell}}+\left|\left(C_s(g)-C_s(f)\right)\lambda_s^{n_0}\right|\right)$$ which means that $D$ is continuous. \end{proof}
Let $T$ be an interval in $\text{Domain}(f)$, then by (\ref{ht}) we have \begin{equation}\label{hD}
|h(T)|=\lim_{n\to\infty}|h_{n}(T)|=\lim_{n\to\infty}\int_{T}Dh_{n}\leq\int_{T}D<\sup_x D(x)|T|. \end{equation} \begin{prop}\label{lipc1}
Let $1<\ell<2$ and let $f,g\in{\mathscr W}$ with critical exponent $\ell$. If $h$ is the topological conjugacy between $f$ and $g$, then the following holds. \begin{eqnarray*}
h \text{ is a Lipschitz homeomorphism} &\Leftarrow& C_u(f)=C_u(g),\\
h \text{ is a } \mathcal C^1 \text{ diffeomorphism }&\Leftarrow& C_u(f)=C_u(g), C_-(f)=C_-(g). \end{eqnarray*} \end{prop} \begin{proof} Observe that, by (\ref{hD}) and by Lemma \ref{Dcont} it follows immediately that if $C_u(f)=C_u(g)$ then $h$ is a Lipschitz homeomorphism. \\ Assume now that $C_u(f)=C_u(g)$ and $C_-(f)=C_-(g)$. We prove that, under these conditions, $D$ extends to a continuous function of the $\text{Domain}(f)$. This extension together with (\ref{hD}) implies that $h$ is differentiable with $Dh(x)=D(x)$ for all $x\in\text{Domain}(f)$ and by (\ref{boundD}), $h$ is a diffeomorphism. The only problematic points to extend $D$ could be $c_k=f^k(0)$ with $k\geq 0$. We prove that $$\lim_{x\uparrow c_k}D(x)=\lim_{x\downarrow c_k}D(x),$$ and in particular $D$ extends to a continuous function $D:\text{Domain}(f)\to\mathbb{R}$. \\ Fix $k\geq 0$ and let $n\in{\mathbb N}$ large enough, such that $q_{n-1}>k$. Define $$D_+(c_k)=\lim_{x\downarrow c_k} D(x)=
\lim_{n\to\infty}\frac{|B_{2n}^k(g)|}{|B_{2n}^k(f)|},$$ and $$D_-(c_k)=\lim_{x\uparrow c_k} D(x)=
\lim_{n\to\infty}\frac{|A_{2n}^k(g)|}{|A_{2n}^k(f)|}.$$ By Lemma \ref{affdiffeo} and Lemma \ref{xs} it follows that \begin{eqnarray*}
\frac{D_+(c_k)}{D_-(c_k)}&=&\lim_{n\to\infty}\frac{{|B_{2n}^k(g)|}/{|A_{2n}^k(g)|}}{{|B_{2n}^k(f)|}/{|A_{2n}^k(f)|}}
=\lim_{n\to\infty}\frac{{|B_{2n}(g)|}/{|A_{2n}(g)|}}{{|B_{2n}(f)|}/{|A_{2n}(f)|}}\\
&=&\lim_{n\to\infty}\frac{{x_{3,2n}(g)}/{-x_{1,2n}(g)}}{{x_{3,2n}(f)}/{-x_{1,2n}(f)}}
=\lim_{n\to\infty} e^{O\left(\lambda_s^{2n}\right)}=1. \end{eqnarray*} \end{proof} \begin{prop} Let $1<\ell<2$, $\beta=\frac{\lambda_u-1}{\lambda_u^4\ell}>0$ and $f,g\in{\mathscr W}$ with critical exponent $\ell$. If $h$ is the topological conjugacy between $f$ and $g$, then the following holds. \begin{eqnarray*} h \text{ is a } \mathcal C^{1+\beta} \text{ diffeomorphism }&\iff& C_u(f)=C_u(g), C_-(f)=C_-(g), C_s(f)=C_s(g) \end{eqnarray*} \end{prop} \begin{proof} Assume that $h$ is $\mathcal C^{1+\beta}$, then by Proposition \ref{lipc1}, $C_u(f)=C_u(g)$ and $C_-(f)=C_-(g)$. Because $C_u=C_u(f)=C_u(g)$ we have that $\log\left({\alpha_n(g)}/{\alpha_n(f)}\right)=O\left(\lambda_s^{n-1}\right)$, see Lemma \ref{asymalpha}. Observe that \begin{eqnarray*}
\frac{Dh(f^{q_{n+1}}(0))}{Dh(f^{q_{n}}(0))}&=&\frac{Dg^{q_{n-1}}(g^{q_{n}}(0))}{Df^{q_{n-1}}(f^{q_{n}}(0))}\\
&=&\frac{Dq_{s_{n}(g)}(0)\left({1-x_{2,n}(g)}/{x_{1,n}(g)}\right)}{Dq_{s_{n}(f)}(0)\left({1-x_{2,n}(f)}/{x_{1,n}(f)}\right)}\left(1+O\left(\alpha_{n-2}(f)^{\frac{1}{\ell}}\right)\right)\\
&=&\frac{S_{5,n}(g)\left(1+O\left(s_n(g)\right)\right)\left({1-x_{2,n}(g)}/{x_{1,n}(g)}\right)}{S_{5,n}(f)\left(1+O\left(s_n(f)\right)\right)\left({1-x_{2,n}(f)}{x_{1,n}(f)}\right)}\left(1+O\left(\alpha_{n-2}(f)^{\frac{1}{\ell}}\right)\right)\\
&=&\frac{{S_{5,n}(g)}/{x_{1,n}(g)}}{{S_{5,n}(f)}/{x_{1,n}(f)}}\left(1+O\left(x_{2,n}(f)\right)+O\left(s_{n}(f)\right)+O\left(\alpha_{n-2}(f)^{\frac{1}{\ell}}\right)\right)\\
&=&e^{-(C_s(g)-C_s(f))\lambda_s^n(e_2^s+e_3^s-e_4^s-e_5^s)}\left(1+O\left(e^{\frac{C_u(e^u_2\lambda_u^{n-4}}{\ell}}\right)\right),
\end{eqnarray*}
where we used Proposition \ref{superformula}, Lemma \ref{xs}, point $3$ of Lemma \ref{prev} and Lemma \ref{asymalpha}. From Lemma \ref{eighvector} it follows that $$(e_2^s+e_3^s-e_4^s-e_5^s)\neq 0.$$ Moreover, by the hypothesis that $h$ is $\mathcal C^{1+\beta}$ and by (\ref{boundD}) we get \begin{eqnarray*}
{Dh(f^{q_{n+1}+1}(U))}{Dh(f^{q_{n}+1}(U))}&=&O\left(|f^{q_{n}}(0),f^{q_{n+1}}(0)|^{\beta}\right)=O\left(|f^{q_{n}}(0),0|^{\beta}\right)\\&=&O\left(e^{\beta C_u(f)\lambda_u^n\left(e_2^u+e_3^u-e_4^u\right)}\right), \end{eqnarray*} where, by Lemma \ref{eighvector}, $$(e_2^u+e_3^u-e_4^u)\neq 0.$$ The two above estimates imply that $C_s(f)=C_s(g)$. This proves one side of the implication.
\\ Assume now that $C_u(f)=C_u(g), C_-(f)=C_-(g)$ and $C_s(f)=C_s(g)$. Let $x,y\in K_f$ and choose $n$ maximal such that, there exists $I\in\mathscr P_n$ containing $x$ and $y$. Because of the fact that $B_{n+1}^{i+q_{n-1}}=D_n^i$ and by the maximality of $n$, we have $I\neq D_n^i$. Moreover, because $x,y\in K_f$, then $I\neq C_n^i$. It remains to study two cases: either $I=A_n^i$ or $I=B_n^i$. \\ Suppose that $I=A_n^i$, then by the maximality of $n$, $x\in D_{n+1}^i$ and $y\in B_{n+1}^i$. From this, the fact that $1-x_{4,n+1},x_{3,n+1}<<1$, Lemma \ref{affdiffeo} and Lemma \ref{length} it follows that \begin{equation}\label{hol1}
|x-y|\geq\frac{1}{2}|A_n^i|\geq\frac{1}{2C}e^{C_u(f)\lambda_u^{n}\frac{\lambda_u}{\lambda_u-1}}. \end{equation} Moreover, by Lemma \ref{Dcont} and by Lemma \ref{asymalpha} we have \begin{equation}\label{hol2}
\log\frac{Dh(x)}{Dh(y)}=O\left(e^{\frac{C_u\lambda_u^{n-3}}{\ell}}\right). \end{equation} By (\ref{hol1}) and (\ref{hol2}) we find
$$\frac{|Dh(x)-Dh(y)|}{|x-y|^\beta}=O\left(e^{\frac{C_u\lambda_u^{n-3}}{\ell}-\beta C_u\lambda_u^n\frac{\lambda_u}{\lambda_u-1}}\right)=O(1),$$ for $0<\beta\leq \frac{\lambda_u-1}{\lambda_u^4\ell}$, see Lemma \ref{eighvector}. \\ Suppose now that $I=B_n^i$, then $x\in A_{n+1}^i$ and $y\in D_{n+1}^{i+q_{n-1}}$. Moreover, $x\leq f^{q_{n+1}+i}(0)\leq y$. Let $m\geq n+1$ maximal such that $x\in A_{m}^{i+q_{n+1}}$ and $y\in B_{m}^{i+q_{n+1}}$. As a consequence, using Lemma \ref{length} we have \begin{equation}\label{hol3}
|x-y|\geq\frac{1}{2}\min\left(|A_{m}^{i+q_{n+1}}|,|B_{m}^{i+q_{n+1}}|\right)\geq\frac{1}{2C}e^{C_u(f)\lambda_u^{m}\frac{\lambda_u}{\lambda_u-1}}. \end{equation} Observe now that, by Lemma \ref{Dcont} and Lemma \ref{asymalpha} we have \begin{equation}\label{hol4}
\left|\log\frac{Dh(x)}{Dh(y)}\right|=\left|\log\frac{Dh(y)}{Dh(f^{q_{n+1}+i}(0))}\right|+\left|\log\frac{Dh(x)}{Dh(f^{q_{n+1}+i}(0))}\right|=
O\left(e^{\frac{C_u\lambda_u^{m-3}}{\ell}}\right) \end{equation} By (\ref{hol3}) and (\ref{hol4}) we find
$$\frac{|Dh(x)-Dh(y)|}{|x-y|^\beta}=O\left(e^{\frac{C_u\lambda_u^{m-3}}{\ell}-\beta C_u\lambda_u^m\frac{\lambda_u}{\lambda_u-1}}\right)=O(1),$$ for $0<\beta\leq \frac{\lambda_u-1}{\lambda_u^4\ell}$, see Lemma \ref{eighvector}. \end{proof}
\begin{prop} Let $1<\ell<2$. If $f,g\in{\mathscr W}$ with critical exponent $\ell$, then the topological conjugacy between $f$ and $g$ is $\mathcal C^{\beta} $ with $\beta=\frac{C_u(g)}{C_u(f)}\frac{\lambda_u-1}{\lambda_u^4\ell}>0$. \end{prop} \begin{proof}
Let $x,y\in K_f$ and choose $n$ maximal such that there exists $I\in\mathscr P_n$ containing $x$ and $y$. Because of the fact that $B_{n+1}^{i+q_{n-1}}=D_n^i$ and by the maximality of $n$, then $I\neq D_n^i$. Moreover, because $x,y\in K_f$, then $I\neq C_n^i$. It remains to study two cases: either $I=A_n^i$ or $I=B_n^i$. \\ Suppose that $I=A_n^i$, then by the maximality of $n$, $x\in D_{n+1}^i$ and $y\in B_{n+1}^i$. From this, the fact that $1-x_{4,n+1},x_{3,n+1}<<1$, Lemma \ref{affdiffeo} and Lemma \ref{length} it follows that \begin{equation}\label{chol1}
|x-y|\geq\frac{1}{2}|A_n^i|\geq\frac{1}{2C}e^{C_u(f)\lambda_u^{n}\frac{\lambda_u}{\lambda_u-1}}. \end{equation} Observe now that, by Lemma \ref{lengthup}, \begin{equation}\label{chol2}
|h(y)-h(x)|\leq |h(I)|=O\left(e^{\frac{C_u(g)\lambda_u^{n-3}}{\ell}}\right). \end{equation} By (\ref{chol1}) and (\ref{chol2}) we find
$$\frac{|h(y)-h(x)|}{|x-y|^\beta}=O\left(e^{\frac{C_u(g)\lambda_u^{n-3}}{\ell}-\beta C_u(f)\lambda_u^n\frac{\lambda_u}{\lambda_u-1}}\right)=O(1,)$$ for $0<\beta\leq\frac{C_u(g)}{C_u(f)}\frac{\lambda_u-1}{\lambda_u^4\ell}$, see Lemma \ref{eighvector}. \\ Suppose now that $I=B_n^i$, then $x\in A_{n+1}^i$ and $y\in D_{n+1}^{i+q_{n-1}}$. Moreover, $x\leq f^{q_{n+1}+i}(0)\leq y$. Let $m\geq n+1$ maximal such that $x\in A_{m}^{i+q_{n+1}}$ and $y\in B_{m}^{i+q_{n+1}}$. As a consequence, using Lemma \ref{length} we have \begin{equation}\label{chol3}
|x-y|\geq\frac{1}{2}\min\left(|A_{m}^{i+q_{n+1}}|,|B_{m}^{i+q_{n+1}}|\right)\geq\frac{1}{2C}e^{C_u(f)\lambda_u^{m}\frac{\lambda_u}{\lambda_u-1}}. \end{equation} Observe now that, by Lemma \ref{lengthup} we have \begin{equation}\label{chol4}
|h(y)-h(x)|\leq |h(A_{m}^{i+q_{n+1}})|+|h(B_{m}^{i+q_{n+1}})|=O\left(e^{\frac{C_u(g)\lambda_u^{m-3}}{\ell}}\right). \end{equation} By (\ref{chol3}) and (\ref{chol4}) we find
$$\frac{|h(y)-h(x)|}{|x-y|^\beta}=O\left(e^{\frac{C_u(g)\lambda_u^{m-3}}{\ell}-\beta C_u(f)\lambda_u^m\frac{\lambda_u}{\lambda_u-1}}\right)=O(1),$$ for $0<\beta\leq\frac{C_u(g)}{C_u(f)}\frac{\lambda_u-1}{\lambda_u^4\ell}$, see Lemma \ref{eighvector}. \end{proof} \section{Appendix I: technical lemmas}\label{AppendixII} We collect here technical lemmas which are needed in Subsections \ref{jumpoutdiff} and \ref{tophyp}. The proofs of the lemmas are based on calculations. \\
Let $D\subset L\subset \mathbb{R}^5\times X_{3+\epsilon}\times X_{3+\epsilon}\times X_{3+\epsilon}$ as defined in Section \ref{manifold1} or a bounded closed set. Refer to Section \ref{manifold1} for the definitions of $\epsilon^*$, $C^*$ and $C_u^*$ and to Definition \ref{zoomop} for the definition of the zoom operator $Z_{I}$.
\begin{lem}\label{D111} On $D=D_{1,1,1}$ the following holds:
$$
\frac{\ell S_5 q^{-1}_s(S_1S_2S_3)}{S_1S_2S_3}=O(1).
$$
\end{lem}
\begin{proof} A calculation shows that,
$$
\frac{\ell S_5 q^{-1}_s(S_1S_2S_3)}{S_1S_2S_3}\le \frac{\ell S_5}{Dq_s(0)}=
\frac{1-s^\ell}{1-s}=O(1),
$$
where we used $\log s\leq {-{1}/{\ell\lambda_u}+{1}/{\ell\lambda_s}+{1}/{\ell-1}}$, see Lemma \ref{eighvector}.
\end{proof}
\begin{lem}\label{C2} For every $\delta>0$ and $C^*\geq 1$ we have
$
\left |Z_{\left[q_s^{-1}\left( 1-e^{y_2} \right),1\right]}q_s\right|\le \delta
$
when $C_u^*$ is large enough.
\end{lem}
\begin{proof} A calculation shows
\begin{equation}\label{ZS2}
\left|Z_{\left[q_s^{-1}\left( 1-e^{y_2} \right),1\right]}q_s\right|=
\left|Z_{\left[q_s^{-1}\left( 1-S_2 \right),1\right]}q_s\right|\le
\left|Z_{\left[1-S_2 ,1\right]}q_0\right|
\le 2(\ell-1)\frac{S_2}{1-S_2}\le \delta
\end{equation}
where we used $S_2\leq e^{C^*}e^{-C_u^*}$, see Lemma \ref{eighvector}.
\end{proof}
\begin{lem}\label{s1235} For every $\delta>0$ and $C^*\geq 1$ we have
$$ 1-2\delta \le \frac{\ell S_5 q^{-1}_s(S_1S_2S_3)}{S_1S_2S_3}\le 1+2\delta
$$
when $C_u^*$ is large enough.
\end{lem}
\begin{proof} A calculation shows for $C_u^*$ is large enough,
$$
\frac{\ell S_5 q^{-1}_s(S_1S_2S_3)}{S_1S_2S_3}\le \frac{\ell S_5}{Dq_s(0)}=
\frac{1-s^\ell}{1-s}\le 1+2\delta,
$$
where we used $\log s\leq {-C_u^*\left({1}/{\ell\lambda_u}\right)+C^*\left({1}/{\ell\lambda_s}\right)+C^*\left({1}/{\ell-1}\right)}$, see Lemma \ref{eighvector}. A calculation shows for $C_u^*$ is large enough,
$$
\frac{\ell S_5 q^{-1}_s(S_1S_2S_3)}{S_1S_2S_3}\ge
\frac{\ell S_5}{Dq_s(q^{-1}_s(S_1S_2S_3))}\ge
\left(1-\frac{S_1S_2S_3}{S_5^{\frac{\ell}{\ell-1}}}\right)^{\frac{\ell-1}{\ell}}\ge
1-\frac{S_1S_2S_3}{S_5^{\frac{\ell}{\ell-1}}}\ge 1-2\delta,
$$
where we used ${\ell S_1^{\ell}}/{S_2}\leq{3}/{2}$ and $\log\left({S_1S_2S_3}/{S_5^{\frac{\ell}{\ell-1}}}\right)\leq -C_u^*\left({\lambda_u^2(\ell+1)-1}/{\ell\lambda_u(1+\lambda_u)}\right)+O(C^*)$, see Lemma \ref{eighvector}.
\end{proof}
\begin{lem}\label{C9} For every $\delta>0$ and $C^*\geq 1$ we have
$
\left|Z_{\left[0,q_s^{-1}\left(y_1e^{y_2}e^{y_3}\right)\right]}q_s\right|\le \delta.
$
when $C_u^*$ is large enough.
\end{lem}
\begin{proof} A calculation shows for $C_u^*$ is large enough,
\begin{equation}\label{ZSS}
\left|Z_{\left[0,q_s^{-1}\left(y_1e^{y_2}e^{y_3}\right)\right]}q_s\right|=
\left|Z_{\left[0,q_s^{-1}\left(S_1 S_2 S_3\right)\right]}q_s\right|\le
2S_1 S_2 S_3 \frac{\ell-1}{\ell} S_5^{-\frac{\ell}{\ell-1}}\le \delta,
\end{equation}
where we used ${\ell S_1^{\ell}}/{S_2}\leq {3}/{2}$ and $\log\left({S_1S_2S_3}/{S_5^{\frac{\ell}{\ell-1}}}\right)\leq -C_u^*\left({\lambda_u^2(\ell+1)-1}/{\ell\lambda_u(1+\lambda_u)}\right)+O(C^*)$, see Lemma \ref{eighvector}.
\end{proof}
\begin{lem}\label{squarefactors} For every $C^*\geq 1$
we have
$$
\begin{aligned} 1.& \text{ } \frac12\le \left[\frac{\ell S_5 \left(\varphi^{-1}\circ q_s^{-1}\right)\left(S_{1}S_{2}S_{3}\right)}{S_{1}S_{2}S_{3}}\cdot\frac{1}{1-\left(\varphi^{l}\left( S_1\right)\right)^{\ell}}\right]\le 2,\\
2.& \text{ } \frac12\le \left[\frac{S_{1}S_{3} \left(1-\left(\varphi^{-1}\circ q_s^{-1}\right)\left(1-S_{2}\right)\right)}{S_5 \left(\varphi^{-1}\circ q_s^{-1}\right)\left(S_{1}S_{2}S_{3}\right)}\right]\le 2,\\ 3.& \text{ } \frac12 \le \left[\left(\frac{\varphi^{l}\left( S_1\right)}{S_{1}}\right)^{\ell}\right]\le 2,\\
4.& \text{ } \frac12\le \left[\left(\frac{\varphi^{l}\left( S_1\right)}{S_{1}}\right)^{\ell -1}\right]\le 2, \end{aligned}
$$
when $C_u^*>1$ is large enough and $\epsilon^*<1$ is small enough. \end{lem}
\begin{proof} Properties $3$ and $4$ hold when $\epsilon^*<1$ is small enough. Moreover, for $C_u^*$ large enough,
\begin{equation}\label{top2} \frac{S_{1}S_{3} \left(1-\left(\varphi^{-1}\circ q_s^{-1}\right)\left(1-S_{2}\right)\right)}{\ell S_{1}S_{2}S_{3}}=1 + O(\epsilon^*),
\end{equation}
where we used $S_2\leq e^{C^*}e^{-C_u^*}$. For $C_u^*$ large enough and Lemma \ref{s1235} we get
\begin{equation}\label{bottom2} \frac{S_{1}S_{2}S_{3}} {\ell S_5 \left(\varphi^{-1}\circ q_s^{-1}\right)\left(S_{1}S_{2}S_{3}\right)}=1 + O(\epsilon^*).
\end{equation}
The equations (\ref{top2}) and (\ref{bottom2}) imply property $2$ by taking $\epsilon^*<1$ small enough. Finally, for $C_u^*$ large enough,
\begin{equation}\label{top1} \frac{1}{1-\left(\varphi^{l}\left( S_1\right)\right)^{\ell}}=1 + O(\epsilon^*). \end{equation} The esitimates (\ref{bottom2}) and (\ref{top1}) imply property $1$ by taking $\epsilon^*<1$ small enough.
\end{proof}
The following lemma is needed in Appendix II. \begin{lem}\label{deltaqs} On $D$, $$
\begin{aligned}
\left|\frac{\partial q_s^{-1}(1-S_2)}{\partial s}\right| &=O\left(S_2\right)=O\left(1\right),\\
\left|\frac{\partial q_s^{-1}(S_1S_2S_3)}{\partial s}\right| &=O\left(S_2^{\frac{\ell+1}{\ell}}S_3 S_5^{-\frac{\ell}{\ell-1}}\right)=O\left(1\right),\\
\left| S_2S_3 D q_s^{-1}(S_1S_2S_3)\right| &=O\left(S_2^{\frac{1}{\ell^2}}S_3^{\frac{1}{\ell}}\right)=O\left(1\right),\\
\left| S_1S_2S_3 D q_s^{-1}(S_1S_2S_3)\right| |q_s|_3&=O\left(S_2^{\frac{\ell+1}{\ell}} S_3 S_5^{-\frac{\ell+1}{\ell-1}}\right)=O\left(1\right),\\
\left| S_5^{\frac{1}{\ell-1}}\frac{\partial q_s^{-1}(S_1S_2S_3)}{\partial s}\right| |q_s|_3&=O\left(S_2^{\frac{\ell+1}{\ell}} S_3 S_5^{-\frac{\ell+1}{\ell-1}}\right)=O\left(1\right).
\end{aligned} $$
\end{lem}
\begin{proof} A calculation shows
$$
\frac{\partial q_s^{-1}(y)}{\partial s}=
\frac{(1-y)(1-s)s^{\ell-1}\left[ (1-s)q^{-1}_s(y)+s \right]^{1-\ell}-1 + (1-s)q^{-1}_s(y)+s}
{(1-s)^2}.
$$
Observe, $q^{-1}_s(1-S_2)=1-O(S_2)$. This implies $\left|{\partial q_s^{-1}(1-S_2)}/{\partial s}\right|=O(S_2)$. The first statement of the lemma follows.
From Lemma \ref{s1235} we have $q_s^{-1}(S_1S_2S_3)=O({S_1S_2S_3}/{S_5})$. Using the expression for ${\partial q_s^{-1}(y)}/{\partial s}$ above we obtain
\begin{equation}\label{parts qs}
\left|\frac{\partial q_s^{-1}(S_1S_2S_3)}{\partial s}\right|=O\left(S_2^{\frac{\ell+1}{\ell}}S_3 S_5^{-\frac{\ell}{\ell-1}}\right), \end{equation} where we used that $S_1^{\ell}=O(S_2)$ and $s^{\ell-1}=S_5$. By expressing $\log S_2, \log S_3, \log S_5$ in the base of the eigenvectors of the matrix $M$, see Lemma \ref{eighvector}, we have $$ S_2^{\frac{\ell+1}{\ell}}S_3 S_5^{-\frac{\ell}{\ell-1}}=e^{O(C_u^*)}=O(1). $$ A calculation shows that $$Dq_s^{-1}\left(S_1S_2S_3\right)=O\left(\left[S_1S_2S_3 (1-s^{\ell})+ s^{\ell}\right]^{\frac{1}{\ell}-1}\right).$$ With a similar procedure as before one can show that $$ S_1S_2S_3=e^{-KC_u^*}s^{\ell}<< s^{\ell}, $$ and as consequence, \begin{equation}\label{Ds} Dq_s^{-1}\left(S_1S_2S_3\right)=O\left(s^{1-\ell}\right). \end{equation} Notice that \begin{equation}\label{qs3}
|q_s|_3=O\left(\frac{1}{s^2}\right). \end{equation} The third and fourth equations are achieved by using (\ref{Ds}), (\ref{qs3}) and the previous procedure of using coordinates corresponding to the base of the eigenvectors of $M$. For the last bound one should use (\ref{parts qs}), (\ref{qs3}) and the eigenvectors procedure.
\end{proof} \section{Appendix II: differentiability properties of renormalization}\label{Appendix} Let $D\subset L\subset \mathbb{R}^5\times X_{3+\epsilon}\times X_{3+\epsilon}\times X_{3+\epsilon}$ as defined in Section \ref{manifold1} or a bounded closed set. We show that $$ R:D\to \mathbb{R}^5\times X_2\times X_2\times X_2, $$ is differentiable and the derivative in a point ${\underline{f}}$ extends to a bounded operator $$ DR_{\underline{f}}: \mathbb{R}^5\times X_2\times X_2\times X_2\to \mathbb{R}^5\times X_2\times X_2\times X_2, $$ and ${\underline{f}}\mapsto DR_{\underline{f}}$ is continuous. Let us start introducing some notation. \\ We denote by $\underline f=\left({y}_1,{y}_2,{y}_3,{y}_4,{y}_5,{\underline\varphi}, {\underline\varphi}^{l},{\underline\varphi}^{r}\right)=\left(\underline{y}, {\underline\varphi}, {\underline\varphi}^{l},{\underline\varphi}^{r}\right)\in L_0$, the composition by $f=O(\underline f)=\left({y}_1,{y}_2,{y}_3,{y}_4,{y}_5,{O(\underline\varphi}), O({\underline\varphi}^{l}),O({\underline\varphi}^{r})\right)=\left(y, \varphi, \varphi^{l},\varphi^{r}\right)\in {\mathscr L}_0$ and the renormalization by $R\underline f=\left(\tilde{y}_1,\tilde{y}_2,\tilde{y}_3,\tilde{y}_4,\tilde{y}_5,\tilde{\underline\varphi}, \tilde{\underline\varphi}^{l},\tilde{\underline\varphi}^{r}\right)=\left(\underline{\tilde{y}},\tilde{\underline\varphi}, \tilde{\underline\varphi}^{l},\tilde{\underline\varphi}^{r}\right)$. The partial derivatives are denoted accordingly. \begin{equation}\label{matrix}DR_{\underline{f}}=\left(\begin{matrix}
A & B_s & B_l & B_r\\
C_s & D_{ss} & D_{sl} &D_{sr}\\
C_l & D_{ls} & D_{ll} &D_{lr}\\
C_r & D_{rs} & D_{rl} &D_{rr}\\
\end{matrix}\right)=\left(\begin{matrix}
A & B_s & B_l & 0\\
C_s & 0 & D_{sl} &0\\
C_l & D_{ls} & 0 &D_{lr}\\
C_r & D_{rs} & D_{rl} &0\\
\end{matrix}\right) \end{equation} where, for example, $$ \begin{aligned} A&=\frac{\partial \tilde{y}}{\partial y}: \mathbb{R}^5\to \mathbb{R}^5,\\ B_l&=\frac{\partial \tilde{y}}{\partial \underline\varphi^{l}}: X_2\to \mathbb{R}^5,\\ C_s&=\frac{\partial \tilde{\underline\varphi}}{\partial y}: \mathbb{R}^5\to X_2 ,\\ D_{lr}&=\frac{\partial \tilde{\underline\varphi}^{l}}{\partial \underline{\varphi}^r}: X_2\to X_2. \end{aligned} $$
Observe, $B_r=0$, $D_{ss}=D_{sr}=D_{ll}=D_{rr}=0$.
\subsection{The partial derivative $A$}\label{A} The partial derivative $A$ can be calculated explicitly by using Lemma \ref{ss}. In particular, $A$ depends continuously on $\underline{f}$. The operator $A$ does not cause that $R:L_0\to L$ is not differentiable. In this subsection, the expressions for the partial derivatives follow from short explicit calculations which are left to the reader. The bounds on the norms follow from the definition of the domain $D$, Lemma \ref{s1235} and Lemma \ref{deltaqs}.
First observe that $$ \frac{\partial \tilde{y}_1}{\partial y_3}= \frac{\partial \tilde{y}_1}{\partial y_4}= \frac{\partial \tilde{y}_2}{\partial y_4}= \frac{\partial \tilde{y}_3}{\partial y_4}=0, $$ $$ \frac{\partial \tilde{y}_4}{\partial y_j}=0 \text{ when } j=2,3,5 $$ and $$ \frac{\partial \tilde{y}_5}{\partial y_j}=0 \text{ when } j=2,3,4,5. $$ \begin{lem}\label{partial y1 partial yj} Let $\underline{f}\in D$. The partial derivatives of $$ \Delta y_j\mapsto \tilde{y}_1(\underline{f}+\Delta y_j), $$
$i=1,2,5$, are given by $$ \begin{aligned} \frac{\partial \tilde{y}_1}{\partial y_1}&= -\frac{\ell}{S_1}\cdot\left[ \frac{D\varphi^{l}(S_1)\varphi^l(S_1)^{\ell-1}S_1}{1-\varphi^{-1}\circ q_s^{-1}(1-S_2)}\right],\\ \frac{\partial \tilde{y}_1}{\partial y_2}&= \left[ \frac{D(\varphi^{-1}\circ q_s^{-1})(1-S_2)\varphi^l(S_1)^{\ell} S_2} {\left(1-\varphi^{-1}\circ q_s^{-1}(1-S_2)\right)^2} \right],\\ \frac{\partial \tilde{y}_1}{\partial y_5}&=-1\cdot\left[\frac{1}{\ell-1}\frac{D\varphi^{-1}(q_s^{-1}(1-S_2)) \varphi^l(S_1)^{\ell} S_5^{\frac{1}{\ell-1}}} {\left(1-\varphi^{-1}\circ q_s^{-1}(1-S_2)\right)^2}\cdot\frac{\partial q^{-1}_s(1-S_2)}{\partial s} \right],\\ \end{aligned} $$ with norms $$ \begin{aligned}
\left|\frac{\partial \tilde{y}_1}{\partial y_1}\right|&=O\left(\frac{1}{S_1}\right),\\
\left|\frac{\partial \tilde{y}_1}{\partial y_2}\right|&=O\left(1\right),\\
\left|\frac{\partial \tilde{y}_1}{\partial y_5}\right|&=O\left(S_5^{\frac{1}{\ell-1}}\right),\\ \end{aligned} $$ and there exists $E>0$ such that $$
\left|\frac{\partial \tilde{y}_1}{\partial y_1}\right|>E\frac{1}{S_1}.$$ The dependance $\underline{f}\mapsto {\partial \tilde{y}_1}/{\partial y_j}$ is continuous. \end{lem}
\begin{lem}\label{partial y2 partial yj} Let $\underline{f}\in D$. The partial derivatives of $$ \Delta y_j\mapsto \tilde{y}_2(\underline{f}+\Delta y_j), $$
$i=1,2,3,5$, are given by $$ \begin{aligned} \frac{\partial \tilde{y}_2}{\partial y_1}&=\frac{1}{S_1}\cdot \left[ \frac{D(\varphi^{-1}\circ q^{-1}_s)(S_1S_2S_3)S_1S_2S_3}{\varphi^{-1}\circ q^{-1}_s(S_1S_2S_3)}+\frac{\ell \varphi^l(S_1)^{\ell-1} D\varphi^l(S_1) S_1}{1-\varphi^l(S_1)^\ell}\right],\\ \frac{\partial \tilde{y}_2}{\partial y_2}&= 1\cdot \left[ \frac{D(\varphi^{-1}\circ q^{-1}_s)(S_1S_2S_3)S_1S_2S_3}{\varphi^{-1}\circ q^{-1}_s(S_1S_2S_3)}\right],\\ \frac{\partial \tilde{y}_2}{\partial y_3}&= 1\cdot \left[ \frac{D(\varphi^{-1}\circ q^{-1}_s)(S_1S_2S_3)S_1S_2S_3}{\varphi^{-1}\circ q^{-1}_s(S_1S_2S_3)}\right],\\ \frac{\partial \tilde{y}_2}{\partial y_5}&=-1\cdot \left[ \frac{1}{1-\ell}\frac{D\varphi^{-1}(q_s^{-1}(S_1S_2S_3)S_5^{\frac{1}{\ell-1}}} {\varphi^{-1}\circ q_s^{-1}(S_1S_2S_3)} \cdot \frac{\partial q_s^{-1}(S_1S_2S_3)}{\partial s}\right] , \\ \end{aligned} $$ with norms $$ \begin{aligned}
\left|\frac{\partial \tilde{y}_2}{\partial y_1}\right|&=O\left(\frac{1}{S_1}\right),\\
\left|\frac{\partial \tilde{y}_2}{\partial y_2}\right|&=O\left(1\right),\\
\left|\frac{\partial \tilde{y}_2}{\partial y_3}\right|&=O\left(1\right),\\
\left|\frac{\partial \tilde{y}_2}{\partial y_5}\right|&=O\left(1\right).\\ \end{aligned} $$
The dependance $\underline{f}\mapsto {\partial \tilde{y}_1}/{\partial y_j}$ is continuous. \end{lem}
\begin{lem}\label{partial y3 partial yj} Let $\underline{f}\in D$. The partial derivatives of $$ \Delta y_j\mapsto \tilde{y}_3(\underline{f}+\Delta y_j), $$
$i=1,2,3,5$, are given by $$ \begin{aligned} \frac{\partial \tilde{y}_3}{\partial y_1}&= -\frac{1}{S_1}\cdot \left[ \frac{D(\varphi^{-1}\circ q^{-1}_s)(S_1S_2S_3)S_1S_2S_3}{ \varphi^{-1}\circ q^{-1}_s(S_1S_2S_3)}\right],\\ \frac{\partial \tilde{y}_3}{\partial y_2}&= \left[ \frac{D(\varphi^{-1}\circ q_s^{-1})(1-S_2) S_2} {1-\varphi^{-1}\circ q_s^{-1}(1-S_2)} -\frac{D(\varphi^{-1}\circ q^{-1}_s)(S_1S_2S_3)S_1S_2S_3}{\varphi^{-1}\circ q^{-1}_s(S_1S_2S_3)}\right],\\ \frac{\partial \tilde{y}_3}{\partial y_3}&= -1\cdot \left[ \frac{D(\varphi^{-1}\circ q^{-1}_s)(S_1S_2S_3)S_1S_2S_3}{ \varphi^{-1}\circ q^{-1}_s(S_1S_2S_3)} \right],\\ \frac{\partial \tilde{y}_3}{\partial y_5}&= 1\cdot \left[ \frac{1}{1-\ell}\left\{ \frac{D\varphi^{-1}(q_s^{-1}(S_1S_2S_3)S_5^{\frac{1}{\ell-1}}} {\varphi^{-1}\circ q_s^{-1}(S_1S_2S_3)} \cdot \frac{\partial q_s^{-1}(S_1S_2S_3)}{\partial s}\right\}\right]+\\ &+ 1\cdot\left[ \frac{1}{1-\ell}\left\{\frac{D\varphi^{-1}(q_s^{-1}(1-S_2)) S_5^{\frac{1}{\ell-1}}} {1-\varphi^{-1}\circ q_s^{-1}(1-S_2)}\cdot\frac{\partial q^{-1}_s(1-S_2)} {\partial s} \right\}\right], \end{aligned} $$ with norms $$ \begin{aligned}
\left|\frac{\partial \tilde{y}_3}{\partial y_1}\right|&=O\left(\frac{1}{S_1}\right),\\
\left|\frac{\partial \tilde{y}_3}{\partial y_2}\right|&=O\left(1\right),\\
\left|\frac{\partial \tilde{y}_3}{\partial y_3}\right|&=O\left(1\right),\\
\left|\frac{\partial \tilde{y}_3}{\partial y_5}\right|&=O\left(1\right).\\ \end{aligned} $$ The dependance $\underline{f}\mapsto {\partial \tilde{y}_1}/{\partial y_j}$ is continuous. \end{lem}
\begin{lem}\label{partial y4 partial yj} Let $\underline{f}\in D$. The partial derivatives of $$ \Delta y_j\mapsto \tilde{y}_4(\underline{f}+\Delta y_j), $$
$j=1,4$, are given by $$ \begin{aligned} \frac{\partial \tilde{y}_4}{\partial y_1}&= \frac{\ell}{S_1}\cdot \left[\frac{D\varphi^l(S_1)}{\varphi^l(S_1)}\cdot S_1\right],\\ \frac{\partial \tilde{y}_4}{\partial y_4}&=-1,\\ \end{aligned} $$ with norms $$ \begin{aligned}
\left|\frac{\partial \tilde{y}_4}{\partial y_1}\right|&=O\left(\frac{1}{S_1}\right),\\
\left|\frac{\partial \tilde{y}_4}{\partial y_4}\right|&=O\left(1\right).\\ \end{aligned} $$ The dependance $\underline{f}\mapsto {\partial \tilde{y}_4}/{\partial y_j}$, $j=1,4$, is continuous. \end{lem}
\begin{lem}\label{partial y5 partial yj} Let $\underline{f}\in D$. The partial derivative of $$ \Delta y_1\mapsto \tilde{y}_5(\underline{f}+\Delta y_1), $$ is given by $$ \frac{\partial \tilde{y}_5}{\partial y_1}= \frac{\ell-1}{S_1}\cdot \left[\frac{D\varphi^l(S_1)}{\varphi^l(S_1)}\cdot S_1\right], $$ with norm $$ \begin{aligned}
\left|\frac{\partial \tilde{y}_5}{\partial y_1}\right|&=O\left(\frac{1}{S_1}\right).\\ \end{aligned} $$ The dependance $\underline{f}\mapsto {\partial \tilde{y}_5}/{\partial y_1}$ is continuous. \end{lem}
Let $A=A(\underline{f})=\left({\partial \tilde{y}_i}/{\partial y_j}\right): \mathbb{R}^5\to \mathbb{R}^5$.
\begin{prop}\label{Aderivative} The dependance $\underline{f}\mapsto A(\underline{f})$ is continuous and $$
\left| \tilde{y}(\underline{f}+\Delta y)-
\left[ \tilde{y}(\underline{f})+A \Delta y\right]\right|=O_{\underline{f}}(|\Delta y |^2). $$ \end{prop}
\subsection{The partial derivative $B_l$}\label{Bl}
Recall the identification of $\text{Diff }^r([0,1])$ with $\mathcal{C}^{r-2}([0,1])$, $r\ge 2$. Given a non-linearity $\eta\in \mathcal{C}^{r-2}([0,1])$ denote the corresponding diffeomorphism with $$\varphi_\eta(x)=\frac{\int_0^x e^{\int_0^s\eta} ds}{\int_0^1 e^{\int_0^s\eta} ds}.$$ The following lemma is obtained by a straight forward calculation. The proof is left to the reader.
\begin{lem}\label{evaluation} Let $x\in [0,1]$. The evaluation operator $E: \text{Diff }^2([0,1])=\mathcal{C}^{0}([0,1])\to \mathbb{R}$ $$ E: \varphi\mapsto \varphi(x) $$ is differentiable with derivative ${\partial \varphi(x)}/{\partial \varphi}: \mathcal{C}^{0}([0,1])\to \mathbb{R}$ given by $$ \frac{\partial \varphi(x)}{\partial \varphi}(\Delta \eta)= \left( \frac{\int_0^x[\int_0^s \Delta \eta] e^{\int_0^s \eta } ds}{\int_0^x e^{\int_0^s \eta } ds}- \frac{\int_0^1[\int_0^s \Delta \eta] e^{\int_0^s \eta } ds}{\int_0^1 e^{\int_0^s \eta } ds}\right)\varphi(x), $$ where $\varphi=\varphi_\eta$. The norm is bounded by $$
\left| \frac{\partial \varphi(x)}{\partial \varphi}\right|\le 2\min\left\{\varphi(x), 1-\varphi(x)\right\}. $$ Moreover\footnote{ We identify $\Delta \varphi$ with $\Delta \eta$}, $$
\left| E(\varphi+\Delta \varphi)-\left[E(\varphi)+\frac{\partial \varphi(x)}{\partial \varphi}(\Delta \varphi)\right]\right|=O(|\Delta \varphi|_2^2). $$ \end{lem}
\begin{cor}\label{corphil} Let $\psi^+,\psi^-\in \text{Diff }^2([0,1])$ and $x\in [0,1]$. The evaluation operator $E^{\psi^+,\psi^-}: \text{Diff }^2([0,1])=\mathcal{C}^{0}([0,1])\to \mathbb{R},$ $$ E^{\psi^+,\psi^-}: \varphi\mapsto \psi^+\circ \varphi\circ \psi^-(x) $$ is differentiable with derivative $ {\partial (\psi^+\circ \varphi\circ \psi^-(x))}/{\partial \varphi}: \mathcal{C}^{0}([0,1])\to \mathbb{R}$ given by $$ \frac{\partial (\psi^+\circ \varphi\circ \psi^-(x))}{\partial \varphi}=D\psi^+(\varphi\circ \psi^-(x))\frac{\partial \varphi(\psi^-(x))}{\partial \varphi}, $$ with norm $$
\left| \frac{\partial (\psi^+\circ \varphi\circ \psi^-(x))}{\partial \varphi}\right|\le 2 \max \left\{D\psi^+\right\} \cdot \varphi(\psi^-(x)). $$ Moreover, $$
\left| E^{\psi^+,\psi^-}(\varphi+\Delta \varphi)-\left[E^{\psi^+,\psi^-}(\varphi)+\frac{\partial (\psi^+\circ \varphi\circ \psi^-(x))}{\partial \varphi}(\Delta \varphi)\right]\right|=O(|\Delta \varphi|_2^2). $$ \end{cor}
Let $\tau\in T$ and recall that $\pi^{\tau}: X_3\to X_3$ is defined as $$\pi^{\tau}\underline{\varphi}_{\tau'}=\left\{
\begin{matrix}
0 & \tau'>\tau\\
\varphi_{\tau'} & \tau'\leq\tau
\end{matrix}\right. $$ Also define $\pi_{\tau}: X_3\to X_3$ is defined as $$\pi_{\tau}\underline{\varphi}_{\tau'}=\left\{
\begin{matrix}
\varphi_{\tau'} & \tau'>\tau \\
0 & \tau'\leq \tau\\
\end{matrix}\right.
$$
Let
$$
\varphi^l_{+,\tau}=O\circ \pi_\tau(\underline{\varphi}^l) \text{ and }
\varphi^l_{-,\tau}=O\circ \pi^\tau(\underline{\varphi}^l) .
$$
Observe, $O\underline{\varphi}^l=\varphi^l_{+,\tau}\circ \varphi^l_\tau\circ \varphi^l_{-,\tau}$. The proof of the following lemma is a straight forward calculation of derivatives using Lemma \ref{ss}. The estimates on the norms follow from Corollary \ref{corphil}.
\begin{lem}\label{partial ypartial phil} Let $\underline{f}\in D$ and $\tau\in T$. The maps $$ \text{Diff }^{3+\epsilon}([0,1])\ni \Delta \varphi^l_\tau \mapsto \tilde{y}_i(\underline{f}+\Delta \varphi^l_\tau)\in \mathbb{R} $$ are differentiable, $i=1,2,3,4,5$. The derivatives are given by $$ \begin{aligned} \frac{\partial \tilde{y}_1}{\partial \varphi^l_\tau}&= D\varphi^l_{+,\tau}(\varphi^l_\tau\circ \varphi^l_{-,\tau}(S_1))\cdot \frac{\ell \varphi^l(S_1)^{\ell-1}}{1-\varphi^{-1}\circ q_s^{-1}(1-S_2)}\cdot \frac{\partial \varphi^l_\tau(\varphi^l_{-,\tau}(S_1))}{\partial \varphi^l_\tau},\\ \frac{\partial \tilde{y}_2}{\partial \varphi^l_\tau}&=D\varphi^l_{+,\tau}(\varphi^l_\tau\circ \varphi^l_{-,\tau}(S_1))\cdot \frac{\ell \varphi^l(S_1)^{\ell-1}}{1-\left(\varphi^l(S_1)\right)^\ell}\cdot \frac{\partial \varphi^l_\tau(\varphi^l_{-,\tau}(S_1))}{\partial \varphi^l_\tau},\\ \frac{\partial \tilde{y}_3}{\partial \varphi^l_\tau}&=0,\\ \frac{\partial \tilde{y}_4}{\partial \varphi^l_\tau}&= D\varphi^l_{+,\tau}(\varphi^l_\tau\circ \varphi^l_{-,\tau}(S_1))\cdot \frac{\ell }{\varphi^l(S_1)}\cdot \frac{\partial \varphi^l_\tau(\varphi^l_{-,\tau}(S_1))}{\partial \varphi^l_\tau},\\ \frac{\partial \tilde{y}_5}{\partial \varphi^l_\tau}&=D\varphi^l_{+,\tau}(\varphi^l_\tau\circ \varphi^l_{-,\tau}(S_1))\cdot \frac{\ell -1}{\varphi^l(S_1)}\cdot \frac{\partial \varphi^l_\tau(\varphi^l_{-,\tau}(S_1))}{\partial \varphi^l_\tau}.\\ \end{aligned} $$ In particular, the partial derivatives extend to bounded functionals on $\text{Diff }^2([0,1])$ with norms $$ \begin{aligned}
\left|\frac{\partial \tilde{y}_1}{\partial \varphi^l_\tau}\right|&= O(1),\\
\left|\frac{\partial \tilde{y}_2}{\partial \varphi^l_\tau}\right|&=O(S_1^{\ell}),\\
\left|\frac{\partial \tilde{y}_3}{\partial \varphi^l_\tau}\right|&= 0,\\
\left|\frac{\partial \tilde{y}_4}{\partial \varphi^l_\tau}\right|&= O(1),\\
\left|\frac{\partial \tilde{y}_5}{\partial \varphi^l_\tau}\right|&= O(1).\\ \end{aligned} $$ The dependance $\underline{f}\mapsto {\partial \tilde{y}_i}/{\partial \varphi^l_\tau}$ is continuous and $$
\left| \tilde{y}_i(\underline{f}+\Delta \varphi^l_\tau)-
\left[ \tilde{y}_i(\underline{f})+\frac{\partial \tilde{y}_i}{\partial \varphi^l_\tau}\Delta \varphi^l_\tau\right]\right|=O_{\underline{f}}(|\Delta \varphi^l_\tau |_{3+\epsilon}^2). $$ \end{lem} The functionals ${\partial \tilde{y}_i}/{\partial \varphi^l_\tau}$ define the operator ${\partial \tilde{y}}/{\partial \varphi^l_\tau}: \text{Diff }^2([0,1])\to \mathbb{R}^5$. Let the operator $B_l:X_2 \to \mathbb{R}^5$ be defined by $$ B_l: \Delta \underline{\varphi}^l\mapsto \sum_{\tau\in T} \frac{\partial \tilde{y}}{\partial \varphi^l_\tau} \Delta \varphi^l_\tau. $$ This is a uniformly bounded operator. This follows from Lemma \ref{partial ypartial phil}. We obtain the following.
\begin{prop}\label{Blderivative} The dependance $\underline{f}\mapsto B_l(\underline{f})$ is continuous and $$
\left| \tilde{y}(\underline{f}+\Delta \underline{\varphi}^l)-
\left[ \tilde{y}(\underline{f})+B_l \Delta \underline{\varphi}^l\right]\right|=O_{\underline{f}}(|\Delta \underline{\varphi}^l |_{3+\epsilon}^2). $$ Moreover, the partial derivative extends to a uniformly bounded operator $B_l:X_2 \to \mathbb{R}^5$. \end{prop}
\subsection{The partial derivative $B_s$}\label{Bs}
The following lemma is obtained by a straight forward calculation. The proof is left to the reader.
\begin{lem}\label{evaluationinv} Let $x\in [0,1]$. The evaluation operator $E_-: \text{Diff }^2([0,1])=\mathcal{C}^{0}([0,1])\to \mathbb{R}$, $$ E_-: \varphi\mapsto \varphi^{-1}(x) $$ is differentiable with derivative ${\partial \varphi^{-1}(x)}/{\partial \varphi}: \mathcal{C}^{0}([0,1])\to \mathbb{R}$ given by $$ \frac{\partial \varphi^{-1}(x)}{\partial \varphi}= -\frac{1}{D\varphi(\varphi^{-1}(x))}\cdot \frac{\partial \varphi(\varphi^{-1}(x))}{\partial \varphi}. $$
The norm is bounded by $$
\left| \frac{\partial \varphi^{-1}(x)}{\partial \varphi}\right|\le 2 \max \left\{\frac{1}{D\varphi}\right\} \cdot \min\left\{x, 1-x\right\}. $$ Moreover, $$
\left| E_-(\varphi+\Delta \varphi)-\left[E_-(\varphi)+\frac{\partial \varphi^{-1}(x)}{\partial \varphi}(\Delta \varphi)\right]\right|=O(|\Delta \varphi|_2^2). $$ \end{lem}
\begin{cor}\label{corphilinv} Let $\psi^+,\psi^-\in \text{Diff }^2([0,1])$ and $x\in [0,1]$. The evaluation operator $E^{\psi^+,\psi^-}_-: \text{Diff }^2([0,1])=\mathcal{C}^{0}([0,1])\to \mathbb{R},$ $$ E^{\psi^+,\psi^-}_-: \varphi\mapsto \psi^+\circ \varphi^{-1}\circ \psi^-(x) $$ is differentiable with derivative $ {\partial (\psi^+\circ \varphi^{-1}\circ \psi^-(x))}/{\partial \varphi}: \mathcal{C}^{0}([0,1])\to \mathbb{R}$ given by $$ \frac{\partial (\psi^+\circ \varphi^{-1}\circ \psi^-(x))}{\partial \varphi}=D\psi^+(\varphi^{-1}\circ \psi^-(x))\frac{\partial \varphi^{-1}(\psi^-(x))}{\partial \varphi}, $$ with norm $$
\left| \frac{\partial (\psi^+\circ \varphi^{-1}\circ \psi^-(x))}{\partial \varphi}\right|\le 2 \frac{\max D\psi^+}{\min D\varphi}\cdot \min\left\{\psi^-(x), 1-\psi^-(x)\right\}. $$ Moreover, $$
\left| E^{\psi^+,\psi^-}_-(\varphi+\Delta \varphi)-\left[E^{\psi^+,\psi^-}_-(\varphi)+\frac{\partial (\psi^+\circ \varphi^{-1}\circ \psi^-(x))}{\partial \varphi}(\Delta \varphi)\right]\right|=O(|\Delta \varphi|_2^2). $$ \end{cor}
Let
$$
\varphi_{+,\tau}=O\circ \pi_\tau(\underline{\varphi}) \text{ and }
\varphi_{-,\tau}-=O\circ \pi^\tau(\underline{\varphi}) .
$$
Observe, $O\underline{\varphi}=\varphi_{+,\tau}\circ \varphi_\tau\circ \varphi_{-,\tau}$. The proof of the following lemma is a straight forward calculation of derivatives using Lemma \ref{ss}. The estimates on the norms follow from Corollary \ref{corphilinv}.
\begin{lem}\label{partial ypartial philinv} Let $\underline{f}\in D$ and $\tau\in T$. The maps $$ \text{Diff }^3([0,1])\ni \Delta \varphi_\tau \mapsto \tilde{y}_i(\underline{f}+\Delta \varphi_\tau)\in \mathbb{R} $$ are differentiable, $i=1,2,3,4,5$. The derivatives are given by $$ \begin{aligned} \frac{\partial \tilde{y}_1}{\partial \varphi_\tau}&= D\varphi_{-,\tau}^{-1}( \varphi^{-1}_\tau(\varphi_{+,\tau}^{-1} (q_s^{-1}(1-S_2))) )\cdot \frac{\varphi^l(S_1)^{\ell}}{\left[1-\varphi^{-1}(q_s^{-1}(1-S_2))\right]^2}\cdot \frac{\partial \varphi_\tau(\varphi_{+,\tau}^{-1} (q_s^{-1}(1-S_2)))}{\partial \varphi_\tau},\\ \frac{\partial \tilde{y}_2}{\partial \varphi_\tau}&= D\varphi_{-,\tau}^{-1}( \varphi^{-1}_\tau(\varphi_{+,\tau}^{-1} (q_s^{-1}(S_1S_2S_3))))\cdot \frac{1}{\varphi^{-1}(q_s^{-1}(S_1S_2S_3))}\cdot \frac{\partial \varphi_\tau(\varphi_{+,\tau}^{-1} (q_s^{-1}(S_1S_2S_3)))}{\partial \varphi_\tau},\\ \frac{\partial \tilde{y}_3}{\partial \varphi_\tau}&=D\varphi_{-,\tau}^{-1}( \varphi^{-1}_\tau(\varphi_{+,\tau}^{-1} (q_s^{-1}(1-S_2))) )\cdot \frac{\varphi^l(S_1)^{\ell}}{1-\varphi^{-1}(q_s^{-1}(1-S_2))}\cdot \frac{\partial \varphi_\tau(\varphi_{+,\tau}^{-1} (q_s^{-1}(1-S_2)))}{\partial \varphi_\tau}+\\ &\text{ }-D\varphi_{-,\tau}^{-1}( \varphi^{-1}_\tau(\varphi_{+,\tau}^{-1} (q_s^{-1}(S_1S_2S_3))))\cdot \frac{1}{\varphi^{-1}(q_s^{-1}(S_1S_2S_3))}\cdot \frac{\partial \varphi_\tau(\varphi_{+,\tau}^{-1} (q_s^{-1}(S_1S_2S_3)))}{\partial \varphi_\tau},\\ \frac{\partial \tilde{y}_4}{\partial \varphi_\tau}&=0,\\ \frac{\partial \tilde{y}_5}{\partial \varphi_\tau}&=0.\\ \end{aligned} $$ In particular, the partial derivatives extend to bounded functionals on $\text{Diff }^2([0,1])$ with norms $$ \begin{aligned}
\left|\frac{\partial \tilde{y}_1}{\partial \varphi_\tau}\right|&= O(1),\\
\left|\frac{\partial \tilde{y}_2}{\partial \varphi_\tau}\right|&=O(1),\\
\left|\frac{\partial \tilde{y}_3}{\partial \varphi_\tau}\right|&=O(1),\\
\left|\frac{\partial \tilde{y}_4}{\partial \varphi_\tau}\right|&=0,\\
\left|\frac{\partial \tilde{y}_5}{\partial \varphi_\tau}\right|&=0.\\ \end{aligned} $$ The dependance $\underline{f}\mapsto {\partial \tilde{y}_i}/{\partial \varphi_\tau}$ is continuous and $$
\left| \tilde{y}_i(\underline{f}+\Delta \varphi_\tau)-
\left[ \tilde{y}_i(\underline{f})+\frac{\partial \tilde{y}_i}{\partial \varphi_\tau}\Delta \varphi_\tau\right]\right|=O_{\underline{f}}(|\Delta \varphi_\tau |_{3+\epsilon}^2). $$ \end{lem}
The functionals ${\partial \tilde{y}_i}/{\partial \varphi_\tau}$ define the operator ${\partial \tilde{y}}/{\partial \varphi_\tau}: \text{Diff }^2([0,1])\to \mathbb{R}^5$. Define the operator $B_s:X_2 \to \mathbb{R}^5$ by $$ B_s: \Delta \underline{\varphi}\mapsto \sum_{\tau\in T} \frac{\partial \tilde{y}}{\partial \varphi_\tau} \Delta \varphi_\tau. $$ This is a uniformly bounded operator, see Lemma \ref{partial ypartial philinv}. We obtain the following.
\begin{prop}\label{Bsderivative} The dependance $\underline{f}\mapsto B_s(\underline{f})$ is continuous and $$
\left| \tilde{y}(\underline{f}+\Delta \underline{\varphi})-
\left[ \tilde{y}(\underline{f})+B_s \Delta \underline{\varphi}\right]\right|=O_{\underline{f}}(|\Delta \underline{\varphi}|_{3+\epsilon}^2) $$ Moreover, the partial derivative extends to a uniformly bounded operator $B_s:X_2 \to \mathbb{R}^5$. \end{prop}
\subsection{The partial derivatives $C_s$}\label{Cs}
The proof of the following lemma is a straightforward calculation and it is left to the reader. Recall that we identify $\text{Diff }^r([0,1])$ with $C^{r-2}([0,1])$, diffeomorphisms with their non-linearities.
\begin{lem}\label{zoompartials} Let $\varphi\in \text{Diff }^{3+\epsilon}([0,1]).$ The zoom curve $Z:[0,1]^2 \ni (a,b)\mapsto Z_{[a,b]}\varphi\in \text{Diff }^2([0,1])$ is differentiable with partial derivatives $$ \frac{\partial Z_{[a,b]}\varphi}{\partial a} = (b-a)(1-x)D\eta((b-a)x+a)-\eta((b-a)x+a), $$ and $$ \frac{\partial Z_{[a,b]}\varphi}{\partial b} = (b-a)xD\eta((b-a)x+a)+\eta((b-a)x+a). $$ The norms are bounded by $$
\left| \frac{\partial Z_{[a,b]}\varphi}{\partial a} \right|_{2},
\left| \frac{\partial Z_{[a,b]}\varphi}{\partial b} \right|_{2}\le 2 |\varphi |_{3}, $$ and $$
\left| Z_{[a+\Delta a, b+\Delta b]}\varphi- \left[ Z_{[a,b]}\varphi+\frac{\partial Z_{[a,b]}\varphi}{\partial a}\Delta a + \frac{\partial Z_{[a,b]}\varphi}{\partial b} \Delta b\right]
\right| $$ $$
\le |\varphi |_{3+\epsilon}
\left( | \Delta a|^{1+\epsilon}+| \Delta b|^{1+\epsilon}|\right). $$ \end{lem}
Observe, ${\partial \tilde{\varphi}_\tau}/{\partial y_i}=0$ when $i=2,3,4,5$.
\begin{lem}\label{partial phi partial y} Let $\underline{f}\in D$ and $\tau\in T$. The map $$ \mathbb{R} \ni \Delta y_1 \mapsto \tilde{\varphi}_\tau(\underline{f}+\Delta y_1)\in \text{Diff}^2([0,1]) $$ is differentiable. The derivative is given by $$ \frac{\partial \tilde{\varphi}_\tau}{\partial y_1}=D(O\circ \pi^\tau(\underline{\varphi}^l))(y_1)\cdot \frac{\partial Z_{[y_1(\tau,1]}\varphi^l_\tau}{\partial y_1(\tau)}, $$ with norm $$
\left| \frac{\partial \tilde{\varphi}_\tau}{\partial y_1}\right|_2\le
2e^{\frac{\delta}{8}} | \varphi^l_\tau|_{3}=O(1). $$ The dependance $\underline{f}\mapsto {\partial \tilde{\varphi}_\tau}/{\partial y_1}$ is continuous and $$
\left| \tilde{\varphi}_\tau(\underline{f}+\Delta y_1)-
\left[\tilde{\varphi}_\tau(\underline{f})+\frac{\partial \tilde{\varphi}_\tau}{\partial y_1}\Delta y_1\right]\right|_2\le O_{\underline{f}}\left( | \varphi^l_\tau|_{3+\epsilon} |\Delta y_1|^{1+\epsilon}\right). $$ \end{lem}
Define the operator $C_s:\mathbb{R}^5 \to X_2$ by $$ C_s(\Delta \underline{y})_\tau= \frac{\partial \tilde{\varphi}_\tau}{\partial y_1}\Delta y_1. $$ This is a uniformly bounded operator, see Lemma \ref{partial phi partial y}. We obtain the following.
\begin{prop}\label{Csderivative} The dependance $\underline{f}\mapsto C_s(\underline{f})$ is continuous and $$
\left| \tilde{\varphi}(\underline{f}+\Delta \underline{y})-
\left[\tilde{\varphi}(\underline{f})+C_s \Delta \underline{y}\right]\right|_2
=O_{\underline{f}}(|\Delta \underline{y}|^{1+\epsilon}). $$ Moreover, the partial derivative extends to a uniformly bounded operator $C_s:\mathbb{R}^5 \to X_2$. \end{prop}
\subsection{The partial derivatives $C_l$}\label{Cl}
Observe, ${\partial \tilde{\varphi}^l_\tau}/{\partial y_i}=0$ when $i=1,3,4$ and $\tau\in T$. Moreover, ${\partial \tilde{\varphi}^l_\tau}/{\partial y_i}=0$ when $i=1,2,3,4,5$ and $\tau> 1/2$. The following lemmas describe the nonzero partial derivatives. The bounds on the norms of the derivatives follow from Lemma \ref{deltaqs}. \begin{lem}\label{partial phi l min 12 partial y} Let $\underline{f}\in D$. The maps $$ \mathbb{R} \ni \Delta y_i \mapsto \tilde{\varphi}^l_\tau(\underline{f}+\Delta y_i)\in \text{Diff }^2([0,1]) $$ are differentiable, $i=2,5$ and $\tau<1/2$. The derivatives are given by $$ \begin{aligned} \frac{\partial \tilde{\varphi}^l_\tau}{\partial y_2}&= -S_2 D(\varphi_{+,\theta \tau}^{-1}\circ q_s^{-1})(1-S_2)\cdot \frac{\partial Z_{[\varphi^{-1}\circ q^{-1}_s(1-S_2)(\theta \tau),1]}\varphi_{\theta \tau}} {\partial \varphi^{-1}\circ q^{-1}_s(1-S_2)(\theta \tau)}, \\ \frac{\partial \tilde{\varphi}^l_\tau}{\partial y_5}&=\frac{S_5^\frac{1}{\ell-1}}{\ell-1} D\varphi_{+,\theta \tau}^{-1}(q_s^{-1}(1-S_2)) \frac{\partial q^{-1}_s(1-S_2)}{\partial s} \cdot \frac{\partial Z_{[\varphi^{-1}\circ q^{-1}_s(1-S_2)(\theta \tau),1]}\varphi_{\theta \tau}} {\partial \varphi^{-1}\circ q^{-1}_s(1-S_2)(\theta \tau)}, \\ \end{aligned} $$ with norms $$ \begin{aligned}
\left|\frac{\partial \tilde{\varphi}^l_\tau}{\partial y_2}\right|_{2}&\le 2S_2 D(\varphi_{+,\theta \tau}^{-1}\circ q_s^{-1})(1-S_2)|\varphi_{\theta \tau}|_3=O(1),\\
\left|\frac{\partial \tilde{\varphi}^l_\tau}{\partial y_5}\right|_{2}&\le 2\frac{S_5^\frac{1}{\ell-1}}{\ell-1} D\varphi_{+,\theta \tau}^{-1}(q_s^{-1}(1-S_2))
\frac{\partial q^{-1}_s(1-S_2)}{\partial s} |\varphi_{\theta \tau}|_3=O(1). \\ \end{aligned} $$ The dependance $\underline{f}\mapsto {\partial \tilde{\varphi}^l_{\tau}}/{\partial y_i}$ is continuous and $$
\left| \tilde{\varphi}^l_\tau(\underline{f}+\Delta \underline{y})-
\left[\tilde{\varphi}^l_\tau(\underline{f})+\frac{\partial \tilde{\varphi}^l_\tau}{\partial y_1}\Delta y_1+\frac{\partial \tilde{\varphi}^l_\tau}{\partial y_5}\Delta y_5\right]\right|_2\le $$ $$
\le O_{\underline{f}}\left( |\varphi_{\theta \tau}|_{3+\epsilon}\left\{|\Delta y_2|^{1+\epsilon}+|\Delta y_5|^{1+\epsilon}\right\}\right). $$ \end{lem}
\begin{lem}\label{partial phi l12 partial y} Let $\underline{f}\in D$. The maps $$ \mathbb{R} \ni \Delta y_i \mapsto \tilde{\varphi}^l_{\frac12}(\underline{f}+\Delta y_i)\in \text{Diff }^2([0,1]) $$ are differentiable, $i=2,5$. The derivatives are given by $$ \begin{aligned} \frac{\partial \tilde{\varphi}^l_{\frac12}}{\partial y_2}&= -S_2 Dq^{-1}_s(1-S_2)\cdot\frac{\partial Z_{[q^{-1}_s(1-S_2),1]}q_s}{\partial q^{-1}_s(1-S_2)}, \\ \frac{\partial \tilde{\varphi}^l_\frac12}{\partial y_5}&=\frac{S_5^\frac{1}{\ell-1}}{\ell-1} \frac{\partial q^{-1}_s(1-S_2)}{\partial s} \cdot \frac{\partial Z_{[q^{-1}_s(1-S_2),1]}q_s}{\partial q^{-1}_s(1-S_2)}, \\ \end{aligned} $$ with norms $$ \begin{aligned}
\left|\frac{\partial \tilde{\varphi}^l_\frac12}{\partial y_2}\right|_{2}&= 2S_2 Dq^{-1}_s(1-S_2) 2\ell\le 4S_2=O(1),\\
\left|\frac{\partial \tilde{\varphi}^l_\frac12}{\partial y_5}\right|_{2}&=\frac{4\ell}{\ell-1} \frac{\partial q^{-1}_s(1-S_2)}{\partial s} S_5^\frac{1}{\ell-1}=O(1). \\ \end{aligned} $$ The dependance $\underline{f}\mapsto {\partial \tilde{\varphi}^l_\frac12}/{\partial y_i}$ is continuous and $$
\left| \tilde{\varphi}^l_\frac12(\underline{f}+\Delta \underline{y})-
\left[\tilde{\varphi}^l_\frac12(\underline{f})+\frac{\partial \tilde{\varphi}^l_\frac12}{\partial y_2}\Delta y_2+\frac{\partial \tilde{\varphi}^l_\frac12}{\partial y_5}\Delta y_5\right]\right|_2\le $$ $$
\le O_{\underline{f}}\left( |\Delta y_2|^{1+\epsilon}+|\Delta y_5|^{1+\epsilon}\right). $$ \end{lem} Define the operator $C_l:\mathbb{R}^5 \to X_2$ by $$ C_l(\Delta \underline{y})_\tau= \frac{\partial \tilde{\varphi}^l_\tau}{\partial y_2}\Delta y_2+ \frac{\partial \tilde{\varphi}^l_\tau}{\partial y_5}\Delta y_5. $$ This is a uniformly bounded operator, see Lemma \ref{partial phi l min 12 partial y}, Lemma \ref{partial phi l12 partial y}. We obtain the following.
\begin{prop}\label{Clderivative} The dependance $\underline{f}\mapsto C_l(\underline{f})$ is continuous and $$
\left| \tilde{\varphi}^l(\underline{f}+\Delta \underline{y})-
\left[\tilde{\varphi}^l(\underline{f})+C_l\Delta \underline{y}\right]\right|_2
=O_{\underline{f}}(|\Delta \underline{y}|^{1+\epsilon}). $$ Moreover, the partial derivative extends to a uniformly bounded operator $C_l:\mathbb{R}^5 \to X_2$. \end{prop}
\subsection{The partial derivatives $C_r$}\label{Cr} Observe that, ${\partial \tilde{\varphi}^l_\tau}/{\partial y_4}=0$ when $\tau\in T$. Moreover, ${\partial \tilde{\varphi}^l_\tau}/{\partial y_i}=0$ when $i=2,3, 5$ and $\tau> 1/2$. The following lemmas describe the nonzero partial derivatives. The bounds on the norms of the derivatives follow from Lemma \ref{deltaqs}.
\begin{lem}\label{partial phi r min 12 partial y} Let $\underline{f}\in D$. The maps $$ \mathbb{R} \ni \Delta y_i \mapsto \tilde{\varphi}^r_\tau(\underline{f}+\Delta y_i)\in \text{Diff }^2([0,1]) $$ are differentiable, $i=1,2,3,5$ and $\tau<1/2$. The derivatives are given by $$ \begin{aligned} \frac{\partial \tilde{\varphi}^r_\tau}{\partial y_1}&= S_2S_3 D(\varphi_{+,\theta \tau}^{-1}\circ q_s^{-1})(S_1S_2S_2)\cdot \frac{\partial Z_{[0,\varphi^{-1}\circ q^{-1}_s(S_1S_2S_3)(\theta \tau)]}\varphi_{\theta \tau}} {\partial \varphi^{-1}\circ q^{-1}_s(S_1S_2S_3)(\theta \tau)}, \\ \frac{\partial \tilde{\varphi}^r_\tau}{\partial y_2}&= S_1S_2S_3 D(\varphi_{+,\theta \tau}^{-1}\circ q_s^{-1})(S_1S_2S_2)\cdot \frac{\partial Z_{[0,\varphi^{-1}\circ q^{-1}_s(S_1S_2S_3)(\theta \tau)]}\varphi_{\theta \tau}} {\partial \varphi^{-1}\circ q^{-1}_s(S_1S_2S_3)(\theta \tau)}, \\ \frac{\partial \tilde{\varphi}^r_\tau}{\partial y_3}&= S_1S_2S_3 D(\varphi_{+,\theta \tau}^{-1}\circ q_s^{-1})(S_1S_2S_2)\cdot \frac{\partial Z_{[0,\varphi^{-1}\circ q^{-1}_s(S_1S_2S_3)(\theta \tau)]}\varphi_{\theta \tau}} {\partial \varphi^{-1}\circ q^{-1}_s(S_1S_2S_3)(\theta \tau)}, \\ \frac{\partial \tilde{\varphi}^r_\tau}{\partial y_5}&=\frac{S_5^\frac{1}{\ell-1}}{\ell-1} D\varphi_{+,\theta \tau}^{-1}(q_s^{-1}(S_1S_2S_3)) \frac{\partial q^{-1}_s(S_1S_2S_3)}{\partial s} \cdot \frac{\partial Z_{[0,\varphi^{-1}\circ q^{-1}_s(S_1S_2S_3)(\theta \tau)]}\varphi_{\theta \tau}} {\partial \varphi^{-1}\circ q^{-1}_s(S_1S_2S_3)(\theta \tau)}, \\ \end{aligned} $$ with norms $$ \begin{aligned}
\left|\frac{\partial \tilde{\varphi}^r_\tau}{\partial y_1}\right|_{2}&\le 2S_2S_3 D(\varphi_{+,\theta \tau}^{-1}\circ q_s^{-1})(S_1S_2S_3)|\varphi_{\theta \tau}|_3=O\left(|\varphi_{\theta \tau}|_3\right),\\
\left|\frac{\partial \tilde{\varphi}^r_\tau}{\partial y_2}\right|_{2}&\le 2S_1S_2S_3 D(\varphi_{+,\theta \tau}^{-1}\circ q_s^{-1})(S_1S_2S_3)|\varphi_{\theta \tau}|_3=O\left(|\varphi_{\theta \tau}|_3\right),\\
\left|\frac{\partial \tilde{\varphi}^r_\tau}{\partial y_3}\right|_{2}&\le 2S_1S_2S_3 D(\varphi_{+,\theta \tau}^{-1}\circ q_s^{-1})(S_1S_2S_3)|\varphi_{\theta \tau}|_3=O\left(|\varphi_{\theta \tau}|_3\right),\\
\left|\frac{\partial \tilde{\varphi}^r_\tau}{\partial y_5}\right|_{2}&\le 2\frac{S_5^\frac{1}{\ell-1}}{\ell-1} D\varphi_{+,\theta \tau}^{-1}(q_s^{-1}(S_1S_2S_3))
\frac{\partial q^{-1}_s(S_1S_2S_3)}{\partial s} |\varphi_{\theta \tau}|_3=O\left(|\varphi_{\theta \tau}|_3\right). \\ \end{aligned} $$ The dependance $\underline{f}\mapsto {\partial \tilde{\varphi}^r(\tau)}/{\partial y_i}$ is continuous and $$
\left| \tilde{\varphi}^r_\tau(\underline{f}+\Delta \underline{y})-
\left[\tilde{\varphi}^r_\tau(\underline{f})+\frac{\partial \tilde{\varphi}^r_\tau}{\partial y_1}\Delta y_1+\frac{\partial \tilde{\varphi}^r_\tau}{\partial y_2}\Delta y_2+\frac{\partial \tilde{\varphi}^r_\tau}{\partial y_3}\Delta y_3+\frac{\partial \tilde{\varphi}^r_\tau}{\partial y_5}\Delta y_5\right]\right|_2\le $$ $$
\le O_{\underline{f}}\left( |\varphi_{\theta \tau}|_{3+\epsilon}\left\{|\Delta y_1|^{1+\epsilon}+|\Delta y_2|^{1+\epsilon}+|\Delta y_3|^{1+\epsilon}+|\Delta y_5|^{1+\epsilon}\right\}\right). $$ \end{lem} \begin{lem}\label{partial phi r12 partial y} Let $\underline{f}\in D$. The maps $$ \mathbb{R} \ni \Delta y_i \mapsto \tilde{\varphi}^r_{\frac12}(\underline{f}+\Delta y_i)\in \text{Diff }^2([0,1]) $$ are differentiable, $i=1,2,3,5$. The derivatives are given by $$ \begin{aligned} \frac{\partial \tilde{\varphi}^r_{\frac12}}{\partial y_1}&= S_2S_3 Dq^{-1}_s(S_1S_2S_3)\cdot\frac{\partial Z_{[0,q^{-1}_s(S_1S_2S_3)]}q_s}{\partial q^{-1}_s(S_1S_2S_3)}, \\ \frac{\partial \tilde{\varphi}^r_{\frac12}}{\partial y_2}&= S_1S_2S_3 Dq^{-1}_s(S_1S_2S_3)\cdot\frac{\partial Z_{[0,q^{-1}_s(S_1S_2S_3)]}q_s}{\partial q^{-1}_s(S_1S_2S_3)}, \\ \frac{\partial \tilde{\varphi}^r_{\frac12}}{\partial y_3}&= S_1S_2S_3 Dq^{-1}_s(S_1S_2S_3)\cdot\frac{\partial Z_{[0,q^{-1}_s(S_1S_2S_3)]}q_s}{\partial q^{-1}_s(S_1S_2S_3)}, \\ \frac{\partial \tilde{\varphi}^l_\frac12}{\partial y_5}&=\frac{S_5^\frac{1}{\ell-1}}{\ell-1} \frac{\partial q^{-1}_s(S_1S_2S_3)}{\partial s} \cdot \frac{\partial Z_{[0,q^{-1}_s(S_1S_2S_3)]}q_s}{\partial q^{-1}_s(S_1S_2S_3)}, \\ \end{aligned} $$ with norms $$ \begin{aligned}
\left|\frac{\partial \tilde{\varphi}^r_\frac12}{\partial y_1}\right|_{2}&=
2S_2S_3 Dq^{-1}_s(S_1S_2S_3) |q_s|_3,\\
\left|\frac{\partial \tilde{\varphi}^r_\frac12}{\partial y_2}\right|_{2}&=
2S_1S_2S_3 Dq^{-1}_s(S_1S_2S_3) |q_s|_3=O(1),\\
\left|\frac{\partial \tilde{\varphi}^r_\frac12}{\partial y_3}\right|_{2}&=
2S_1S_2S_3 Dq^{-1}_s(S_1S_2S_3) |q_s|_3=O(1),\\
\left|\frac{\partial \tilde{\varphi}^r_\frac12}{\partial y_5}\right|_{2}&=2\frac{S_5^\frac{1}{\ell-1}}{\ell-1}
\frac{\partial q^{-1}_s(S_1S_2S_3)}{\partial s} |q_s|_3=O(1). \\ \end{aligned} $$ The dependance $\underline{f}\mapsto {\partial \tilde{\varphi}^r_\frac12}/{\partial y_i}$ is continuous and $$
\left| \tilde{\varphi}^r_\frac12(\underline{f}+\Delta \underline{y})-
\left[\tilde{\varphi}^r_\frac12(\underline{f})+\frac{\partial \tilde{\varphi}^r_\frac12}{\partial y_1}\Delta y_1+\frac{\partial \tilde{\varphi}^r_\frac12}{\partial y_2}\Delta y_2+\frac{\partial \tilde{\varphi}^r_\frac12}{\partial y_3}\Delta y_3+\frac{\partial \tilde{\varphi}^r_\frac12}{\partial y_5}\Delta y_5\right]\right|_2\le $$ $$
\le O_{\underline{f}}\left( |\Delta y_1|^{1+\epsilon}+|\Delta y_2|^{1+\epsilon}+|\Delta y_3|^{1+\epsilon}+|\Delta y_5|^{1+\epsilon}\right). $$ \end{lem} \begin{lem}\label{partial phi r max 12 partial y} Let $\underline{f}\in D$. The maps $$ \mathbb{R} \ni \Delta y_1\mapsto \tilde{\varphi}^r_\tau(\underline{f}+\Delta y_1)\in \text{Diff }^2([0,1]) $$ are differentiable, $\tau>1/2$. The derivative is given by $$ \begin{aligned} \frac{\partial \tilde{\varphi}^r_\tau}{\partial y_1}&= D\left(\varphi^l_{-,\theta \tau}(y_1)\right)\cdot \frac{\partial Z_{[0, y_1(\theta \tau)]}\varphi^l_{\theta \tau}} {\partial y_1(\theta \tau)}, \end{aligned} $$ with norm $$ \begin{aligned}
\left|\frac{\partial \tilde{\varphi}^r_\tau}{\partial y_1}\right|_{2}&\le
2 D\left(\varphi^l_{-,\theta \tau}(y_1)\right) |\varphi^l_{\theta \tau}|_3 \end{aligned}. $$ The dependance $\underline{f}\mapsto {\partial \tilde{\varphi}^r(\tau)}/{\partial y_1}$ is continuous and $$
\left| \tilde{\varphi}^r_\tau(\underline{f}+\Delta \underline{y})-
\left[\tilde{\varphi}^r_\tau(\underline{f})+\frac{\partial \tilde{\varphi}^r_\tau}{\partial y_1}{\Delta y_1}\right]\right|_2\le $$ $$
\le O_{\underline{f}}\left( |\varphi_{\theta \tau}^l|_{3+\epsilon}\left\{|\Delta y_1|^{1+\epsilon}\right\}\right). $$ \end{lem} Define the operator $C_r:\mathbb{R}^5 \to X_2$ by $$ C_r(\Delta \underline{y})_\tau= \frac{\partial \tilde{\varphi}^r_\tau}{\partial y_1}\Delta y_1+ \frac{\partial \tilde{\varphi}^r_\tau}{\partial y_2}\Delta y_2+ \frac{\partial \tilde{\varphi}^r_\tau}{\partial y_3}\Delta y_3+ \frac{\partial \tilde{\varphi}^r_\tau}{\partial y_5}\Delta y_5. $$
This is a bounded operator, see Lemma \ref{partial phi r min 12 partial y}, Lemma \ref{partial phi r12 partial y} and Lemma \ref{partial phi r max 12 partial y} . We obtain the following.
\begin{prop}\label{Crderivative} The dependance $\underline{f}\mapsto C_r(\underline{f})$ is continuous and $$
\left| \tilde{\varphi}^r(\underline{f}+\Delta \underline{y})-
\left[\tilde{\varphi}^r(\underline{f})+C_r\Delta \underline{y}\right]\right|_2
=O_{\underline{f}}(|\Delta \underline{y}|^{1+\epsilon}). $$ The operator $C_r:\mathbb{R}^5 \to X_2$ is bounded. Moreover the operator $C_r:\mathbb{R}^4 \to X_2$ with $$ C_r(\Delta \underline{y})_\tau= \frac{\partial \tilde{\varphi}^r_\tau}{\partial y_2}\Delta y_2+ \frac{\partial \tilde{\varphi}^r_\tau}{\partial y_3}\Delta y_3+ \frac{\partial \tilde{\varphi}^r_\tau}{\partial y_5}\Delta y_5 $$ is uniformly bounded. \end{prop}
\subsection{The partial derivatives $D_{sl}$}\label{Dsl} Observe that, ${\partial \tilde{\varphi}_\tau}/{\partial {\varphi}^l_{\tau'}}=0$ when $\tau'>\tau$. The following lemmas describe the nonzero partial derivatives.
\begin{lem}\label{partial phi tau=tau' partial phil} Let $\underline{f}\in D$. The maps $$ \mathbb{R} \ni \Delta{\varphi}^l_\tau \mapsto \tilde{\varphi}_\tau(\underline{f}+\Delta {\varphi}^l_\tau)\in \text{Diff }^2([0,1]) $$ are linear. Namely, $$ \tilde{\varphi}_\tau(\underline{f}+\Delta {\varphi}^l_\tau)=\tilde{\varphi}_\tau(\underline{f})+\frac{\partial \tilde{\varphi}_\tau}{\partial {\varphi}^l_{\tau}}\Delta{\varphi}^l_\tau, $$ where $$ \frac{\partial \tilde{\varphi}_\tau}{\partial {\varphi}^l_{\tau}}=Z_{\left[y_1(\tau),1\right]}, $$ with norm $$ \begin{aligned}
\left|\frac{\partial \tilde{\varphi}_\tau}{\partial {\varphi}^l_{\tau}}\right|_{2}&\le 1. \end{aligned} $$ \end{lem} Let $\tau>\tau'$. For $\underline\varphi\in X_3$, we define $$ \underline\varphi_{[\tau,\tau')}=O\circ\pi_{\tau'}\circ\pi^{\tau}\left(\underline\varphi\right). $$ By Lemma \ref{zoompartials} and Lemma \ref{evaluation} we get the following. \begin{lem}\label{partial phi tau>tau' partial phil} Let $\underline{f}\in D$ and let $\tau>\tau'$. The maps $$ \mathbb{R} \ni \Delta{\varphi}^l_{\tau'} \mapsto \tilde{\varphi}_\tau(\underline{f}+\Delta {\varphi}^l_{\tau'})\in \text{Diff }^2([0,1]) $$ are differentiable. The derivatives are given by $$ \begin{aligned} \frac{\partial \tilde{\varphi}_\tau}{\partial {\varphi}^l_{\tau'}}&=D\underline\varphi_{[\tau,\tau')}^l\left(\varphi^l_{\tau'}\left(y_1(\tau')\right)\right)\cdot\frac{\partial \tilde{\varphi}_\tau}{\partial y_1({\tau})}\circ\frac{\partial {\varphi}^l_{\tau'}\left(y_1({\tau'})\right)}{\partial {\varphi}^l_{\tau'}},\\ \end{aligned} $$ with norms $$ \begin{aligned}
\left|\frac{\partial \tilde{\varphi}_\tau}{\partial {\varphi}^l_{\tau'}}\right|_{2}&=O\left(\left|\varphi_\tau^l\right|_3S_1\right). \end{aligned} $$ The dependance $\underline{f}\mapsto {\partial \tilde{\varphi}_\tau}/{\partial {\varphi}^l_{\tau'}}$ is continuous and $$
\left| \tilde{\varphi}_\tau(\underline{f}+\Delta{\varphi}_{\tau'}^l)-
\left[\tilde{\varphi}_\tau(\underline{f})+\frac{\partial \tilde{\varphi}_\tau}{\partial {\varphi}^l_{\tau'}}\Delta {\varphi}^l_{\tau'} \right]\right|_2
= O_{\underline{f}}\left( \left|{\varphi}^l_{\tau}\right|_{3+\epsilon}\left|\Delta{\varphi}_{\tau'}^l\right|_{2}^{1+\epsilon}\right). $$ \end{lem} Define the operator $D_{sl}: X_2 \to X_2$ by $$ \left(D_{sl}\left(\Delta \underline{\varphi}^l\right)\right)_\tau=\sum_{\tau'\leq\tau}\frac{\partial \tilde{\varphi}_\tau}{\partial {\varphi}^l_{\tau'}}\Delta\varphi^l_{\tau'}. $$
This is a uniformly bounded operator, see Lemma \ref{partial phi tau=tau' partial phil} and Lemma \ref{partial phi tau>tau' partial phil} . We obtain the following.
\begin{prop}\label{Dslderivative} The dependance $\underline{f}\mapsto D_{sl}(\underline{f})$ is continuous and $$
\left| \tilde{\varphi}(\underline{f}+\Delta \underline{\varphi}^l)-
\left[\tilde{\varphi}(\underline{f})+D_{sl}\Delta \underline{\varphi}^l\right]\right|_2
=O_{\underline{f}}(|\Delta \underline{\varphi}^l|_2^{1+\epsilon}). $$ Moreover, the partial derivative extends to a uniformly bounded operator $D_{sl}: X_2 \to X_2$. \end{prop} \subsection{The partial derivatives $D_{lr}$}\label{Dlr} Observe that, ${\partial \tilde{\varphi}^l_\tau}/{\partial {\varphi}^r_{\tau'}}=0$ when $\tau\leq\frac{1}{2}$ and when $\tau>\frac{1}{2}, \theta\tau\neq\tau'$. The following lemma describe the nonzero partial derivatives.
\begin{lem}\label{partial phil tau=thetatau' partial phir} Let $\underline{f}\in D$, let $\tau>\frac{1}{2}$ and $\theta\tau=\tau'$. The maps $$ \mathbb{R} \ni \Delta{\varphi}^r_{\tau'} \mapsto \tilde{\varphi}^l_\tau(\underline{f}+\Delta {\varphi}^r_{\tau'})\in \text{Diff }^2([0,1]) $$ are linear. Namely, $$ \tilde{\varphi}^l_\tau(\underline{f}+\Delta {\varphi}^r_{\tau'})=\tilde{\varphi}^l_\tau(\underline{f})+\frac{\partial \tilde{\varphi}^l_\tau}{\partial {\varphi}^r_{\tau'}}\Delta{\varphi}^r_{\tau'}, $$ where $$ \frac{\partial \tilde{\varphi}^l_\tau}{\partial {\varphi}^r_{\tau'}}=\text{Id}, $$ with norms $$ \begin{aligned}
\left|\frac{\partial \tilde{\varphi}^l_\tau}{\partial {\varphi}^r_{\tau'}}\right|_{2}&=1. \end{aligned} $$ \end{lem} Define the operator $D_{lr}: X_2 \to X_2$ by $$ \left(D_{lr}\left(\Delta \underline{\varphi}^r\right)\right)_\tau=\Delta\varphi^r_{\theta\tau}, $$ where $\tau>\frac{1}{2}$.
This is a uniformly bounded operator, see Lemma \ref{partial phil tau=thetatau' partial phir}. We obtain the following.
\begin{prop}\label{Dlrderivative} The dependance $\underline{f}\mapsto D_{lr}(\underline{f})$ is continuous and $$
\left| \tilde{\varphi}^l(\underline{f}+\Delta \underline{\varphi}^r)-
\left[\tilde{\varphi}^l(\underline{f})+D_{lr}\Delta \underline{\varphi}^r\right]\right|_2 =0. $$ Moreover, the partial derivative extends to a uniformly bounded operator $D_{lr}: X_2 \to X_2$. \end{prop} \subsection{The partial derivatives $D_{ls}$}\label{Dls} Observe that, ${\partial \tilde{\varphi}^l_\tau}/{\partial {\varphi}_{\tau'}}=0$ when $\tau\geq\frac{1}{2}$ and when $\tau<\frac{1}{2}, \tau'<\theta\tau$. The following lemmas describe the nonzero partial derivatives.
\begin{lem}\label{partial phil thetatau=tau' partial phi} Let $\underline{f}\in D$ and let $\tau<\frac{1}{2}$. The maps $$ \mathbb{R} \ni \Delta{\varphi}_{\theta\tau} \mapsto \tilde{\varphi}^l_\tau(\underline{f}+\Delta {\varphi}_{\theta\tau})\in \text{Diff }^2([0,1]) $$ are differentiable. The derivatives are given by $$ \begin{aligned} \frac{\partial \tilde{\varphi}^l_\tau}{\partial {\varphi}_{\theta\tau}}&=Z_{\left[\varphi^{-1}\circ q_s^{-1}(1-S_2)(\theta\tau),1\right]}+\\&+\frac{\partial \tilde{\varphi}^l_\tau}{\partial \varphi^{-1}\circ q_s^{-1}(1-S_2)(\theta\tau)}\circ\frac{\partial {\varphi}_{\theta\tau}^{-1}\left(\varphi_{\theta\tau}\left(\varphi^{-1}\circ q_s^{-1}(1-S_2)(\theta\tau)\right)\right)}{\partial {\varphi}_{\theta\tau}} \end{aligned}, $$ with norms $$ \begin{aligned}
\left|\frac{\partial \tilde{\varphi}^l_\tau}{\partial {\varphi}_{\theta\tau}}\right|_{2}&=O(S_2). \end{aligned} $$ The dependance $\underline{f}\mapsto {\partial \tilde{\varphi}^l_\tau}/{\partial {\varphi}_{\theta\tau}}$ is continuous and $$
\left| \tilde{\varphi}^l_\tau(\underline{f}+\Delta{\varphi}_{\theta\tau})-
\left[\tilde{\varphi}^l_\tau(\underline{f})+\frac{\partial \tilde{\varphi}^l_\tau}{\partial {\varphi}_{\theta\tau}}\Delta {\varphi}_{\theta\tau} \right]\right|_2
= O_{\underline{f}}\left( \left|{\varphi}_{\theta\tau}\right|_{3+\epsilon}\left|\Delta\underline{\varphi}\right|_{2}^{1+\epsilon}\right). $$ \end{lem}
By Lemma \ref{zoompartials} and Lemma \ref{evaluationinv} we get the following. \begin{lem}\label{partial phil thetatau<tau' partial phi} Let $\underline{f}\in D$, $\tau<\frac{1}{2}$ and $\theta\tau<\tau'$. The maps $$ \mathbb{R} \ni \Delta{\varphi}_{\tau'} \mapsto \tilde{\varphi}^l_\tau(\underline{f}+\Delta {\varphi}_{\tau'})\in \text{Diff }^2([0,1]) $$ are differentiable. The derivatives are given by $$ \begin{aligned} \frac{\partial \tilde{\varphi}^l_\tau}{\partial {\varphi}_{\tau'}}&=\frac{1}{D\underline\varphi_{[\tau',\theta\tau)}\left(\varphi_{\theta\tau}\left(\varphi^{-1}\circ q_s^{-1}(1-S_2)(\theta\tau)\right)\right)}\cdot\\&\cdot\frac{\partial \tilde{\varphi}^l_\tau}{\partial \varphi^{-1}\circ q_s^{-1}(1-S_2)(\theta\tau)}\circ\frac{\partial {\varphi}_{\tau'}^{-1}\left(\varphi_{\tau'}\left(\varphi^{-1}\circ q_s^{-1}(1-S_2)(\tau')\right)\right)}{\partial {\varphi}_{\tau'}},\\ \end{aligned} $$ with norms $$ \begin{aligned}
\left|\frac{\partial \tilde{\varphi}^l_\tau}{\partial {\varphi}_{\tau'}}\right|_{2}&=O\left(\left|\varphi_{\theta\tau}\right|_3S_2\right). \end{aligned} $$ The dependance $\underline{f}\mapsto {\partial \tilde{\varphi}^l_\tau}/{\partial {\varphi}_{\tau'}}$ is continuous and $$
\left| \tilde{\varphi}^l_\tau(\underline{f}+\Delta{\varphi}_{\tau'})-
\left[\tilde{\varphi}^l_\tau(\underline{f})+\frac{\partial \tilde{\varphi}^l_\tau}{\partial {\varphi}_{\tau'}}\Delta {\varphi}_{\tau'} \right]\right|_2
= O_{\underline{f}}\left( \left|{\varphi}_{\theta\tau}\right|_{3+\epsilon}\left|\Delta{\varphi}_{\tau'}\right|_{2}^{1+\epsilon}\right). $$ \end{lem} Define the operator $D_{ls}: X_2 \to X_2$ by $$ \left(D_{ls}\left(\Delta \underline{\varphi}\right)\right)_\tau=\sum_{\tau'>\theta\tau,\\\tau<\frac{1}{2}}\frac{\partial \tilde{\varphi}^l_\tau}{\partial {\varphi}_{\tau'}}\Delta\varphi_{\tau'}. $$
This is a uniformly bounded operator, see Lemma \ref{partial phil thetatau=tau' partial phi} and Lemma \ref{partial phil thetatau<tau' partial phi} . We obtain the following.
\begin{prop}\label{Dlsderivative} The dependance $\underline{f}\mapsto D_{ls}(\underline{f})$ is continuous and $$
\left| \tilde{\varphi}^l(\underline{f}+\Delta \underline{\varphi})-
\left[\tilde{\varphi}^l(\underline{f})+D_{ls}\Delta \underline{\varphi}\right]\right|_2
=O_{\underline{f}}(|\Delta \underline{\varphi}|_2^{1+\epsilon}). $$ Moreover, the partial derivative extends to a uniformly bounded operator $D_{ls}: X_2 \to X_2$. \end{prop} \subsection{The partial derivatives $D_{rs}$}\label{Drs} Observe that, ${\partial \tilde{\varphi}^r_\tau}/{\partial {\varphi}_{\tau'}}=0$ when $\tau\geq\frac{1}{2}$ and when $\tau<{1}/{2}, \tau'<\theta\tau$. The following lemmas describe the nonzero partial derivatives.
\begin{lem}\label{partial phir thetatau=tau' partial phi} Let $\underline{f}\in D$ and let $\tau<\frac{1}{2}$. The maps $$ \mathbb{R} \ni \Delta{\varphi}_{\theta\tau} \mapsto \tilde{\varphi}^r_\tau(\underline{f}+\Delta {\varphi}_{\theta\tau})\in \text{Diff }^2([0,1]) $$ are differentiable. The derivatives are given by $$ \begin{aligned} \frac{\partial \tilde{\varphi}^r_\tau}{\partial {\varphi}_{\theta\tau}}&=Z_{\left[0,\varphi^{-1}\circ q_s^{-1}(S_1S_2S_3)(\theta\tau)\right]}+\\&+\frac{\partial \tilde{\varphi}^r_\tau}{\partial \varphi^{-1}\circ q_s^{-1}(S_1S_2S_3)(\theta\tau)}\circ\frac{\partial {\varphi}_{\theta\tau}^{-1}\left(\varphi_{\theta\tau}\left(\varphi^{-1}\circ q_s^{-1}(S_1S_2S_3)(\theta\tau)\right)\right)}{\partial {\varphi}_{\theta\tau}}, \end{aligned} $$ with norms $$ \begin{aligned}
\left|\frac{\partial \tilde{\varphi}^r_\tau}{\partial {\varphi}_{\theta\tau}}\right|_{2}&=O(q_s^{-1}\left(S_1S_2S_3\right)). \end{aligned} $$ The dependance $\underline{f}\mapsto {\partial \tilde{\varphi}^r_\tau}/{\partial {\varphi}_{\theta\tau}}$ is continuous and $$
\left| \tilde{\varphi}^r_\tau(\underline{f}+\Delta{\varphi}_{\theta\tau})-
\left[\tilde{\varphi}^r_\tau(\underline{f})+\frac{\partial \tilde{\varphi}^r_\tau}{\partial {\varphi}_{\theta\tau}}\Delta {\varphi}_{\theta\tau} \right]\right|_2
= O_{\underline{f}}\left( \left|{\varphi}_{\theta\tau}\right|_{3+\epsilon}\left|\Delta\underline{\varphi}\right|_{2}^{1+\epsilon}\right). $$ \end{lem}
By Lemma \ref{zoompartials} and Lemma \ref{evaluationinv} we get the following. \begin{lem}\label{partial phir thetatau<tau' partial phi} Let $\underline{f}\in D$, let $\tau<\frac{1}{2}$ and $\theta\tau<\tau'$. The maps $$ \mathbb{R} \ni \Delta{\varphi}_{\tau'} \mapsto \tilde{\varphi}^r_\tau(\underline{f}+\Delta {\varphi}_{\tau'})\in \text{Diff }^2([0,1]) $$ are differentiable. The derivatives are given by $$ \begin{aligned} \frac{\partial \tilde{\varphi}^r_\tau}{\partial {\varphi}_{\tau'}}&=\frac{1}{D\underline\varphi_{[\tau',\theta\tau)}\left(\varphi_{\theta\tau}\left(\varphi^{-1}\circ q_s^{-1}(S_1S_2S_3)(\theta\tau)\right)\right)}\cdot\\&\cdot\frac{\partial \tilde{\varphi}^r_\tau}{\partial \varphi^{-1}\circ q_s^{-1}(S_1S_2S_3)(\theta\tau)}\circ\frac{\partial {\varphi}_{\tau'}^{-1}\left(\varphi_{\tau'}\left(\varphi^{-1}\circ q_s^{-1}(S_1S_2S_3)(\tau')\right)\right)}{\partial {\varphi}_{\tau'}},\\ \end{aligned} $$ with norms $$ \begin{aligned}
\left|\frac{\partial \tilde{\varphi}^r_\tau}{\partial {\varphi}_{\tau'}}\right|_{2}&=O\left(\left|\varphi_{\theta\tau}\right|_3q_s^{-1}\left(S_1S_2S_3\right)\right). \end{aligned} $$ The dependance $\underline{f}\mapsto {\partial \tilde{\varphi}^r_\tau}/{\partial {\varphi}_{\tau'}}$ is continuous and $$
\left| \tilde{\varphi}^r_\tau(\underline{f}+\Delta{\varphi}_{\tau'})-
\left[\tilde{\varphi}^r_\tau(\underline{f})+\frac{\partial \tilde{\varphi}^r_\tau}{\partial {\varphi}_{\tau'}}\Delta {\varphi}_{\tau'} \right]\right|_2
= O_{\underline{f}}\left( \left|{\varphi}_{\theta\tau}\right|_{3+\epsilon}\left|\Delta{\varphi}_{\tau'}\right|_{2}^{1+\epsilon}\right). $$ \end{lem} Define the operator $D_{rs}: X_2 \to X_2$ by $$ \left(D_{rs}\left(\Delta \underline{\varphi}\right)\right)_\tau=\sum_{\tau'>\theta\tau,\\\tau<\frac{1}{2}}\frac{\partial \tilde{\varphi}^r_\tau}{\partial {\varphi}_{\tau'}}\Delta\varphi_{\tau'}. $$
This is a uniformly bounded operator, see Lemma \ref{partial phir thetatau=tau' partial phi} and Lemma \ref{partial phir thetatau<tau' partial phi} . We obtain the following.
\begin{prop}\label{Drsderivative} The dependance $\underline{f}\mapsto D_{rs}(\underline{f})$ is continuous and $$
\left| \tilde{\varphi}^r(\underline{f}+\Delta \underline{\varphi})-
\left[\tilde{\varphi}^r(\underline{f})+D_{rs}\Delta \underline{\varphi}\right]\right|_2
=O_{\underline{f}}(|\Delta \underline{\varphi}|_2^{1+\epsilon}). $$ Moreover, the partial derivative extends to a uniformly bounded operator $D_{rs}: X_2 \to X_2$. \end{prop} \subsection{The partial derivatives $D_{rl}$}\label{Drl} Observe that, ${\partial \tilde{\varphi}^r_\tau}/{\partial {\varphi}^l_{\tau'}}=0$ when $\tau\leq{1}/{2}$ and when $\tau>{1}/{2}$ with $\tau'>\theta\tau$. The following lemmas describe the nonzero partial derivatives.
\begin{lem}\label{partial phir thetatau=tau' partial phil} Let $\underline{f}\in D$ and let $\tau>\frac{1}{2}$. The maps $$ \mathbb{R} \ni \Delta{\varphi}^l_{\theta\tau} \mapsto \tilde{\varphi}^r_\tau(\underline{f}+\Delta {\varphi}^l_{\theta\tau})\in \text{Diff }^2([0,1]) $$ are linear. Namely, $$ \tilde{\varphi}^r_\tau(\underline{f}+\Delta {\varphi}^l_{\theta\tau})=\tilde{\varphi}^r_\tau(\underline{f})+\frac{\partial \tilde{\varphi}^r_\tau}{\partial {\varphi}^l_{\theta\tau}}\Delta{\varphi}^l_{\theta\tau}, $$ where $$ \frac{\partial \tilde{\varphi}^r_\tau}{\partial {\varphi}^l_{\theta\tau}}=Z_{\left[0,y_1(\theta\tau)\right]}, $$ with norms $$ \begin{aligned}
\left|\frac{\partial \tilde{\varphi}^r_\tau}{\partial {\varphi}^l_{\theta\tau}}\right|_{2}&=O(S_1). \end{aligned} $$ \end{lem} By Lemma \ref{zoompartials} and Lemma \ref{evaluation} we get the following. \begin{lem}\label{partial phir thetatau>tau' partial phil} Let $\underline{f}\in D$ and let $\tau>\frac{1}{2}$ with $\theta\tau>\tau'$. The maps $$ \mathbb{R} \ni \Delta{\varphi}^l_{\tau'} \mapsto \tilde{\varphi}^r_\tau(\underline{f}+\Delta {\varphi}^l_{\tau'})\in \text{Diff }^2([0,1]) $$ are differentiable. The derivatives are given by $$ \begin{aligned} \frac{\partial \tilde{\varphi}^r_\tau}{\partial {\varphi}^l_{\tau'}}&=D\underline\varphi_{[\theta\tau,\tau')}^l\left(\varphi^l_{\tau'}\left(y_1(\tau')\right)\right)\cdot\frac{\partial \tilde{\varphi}^r_\tau}{\partial y_1({\theta\tau})}\circ\frac{\partial {\varphi}^l_{\tau'}\left(y_1({\tau'})\right)}{\partial {\varphi}^l_{\tau'}},\\ \end{aligned} $$ with norms $$ \begin{aligned}
\left|\frac{\partial \tilde{\varphi}^r_\tau}{\partial {\varphi}^l_{\tau'}}\right|_{2}&=O\left(\left|\varphi_{\theta\tau}^l\right|_3S_1\right). \end{aligned} $$ The dependance $\underline{f}\mapsto {\partial \tilde{\varphi}^r_\tau}/{\partial {\varphi}^l_{\tau'}}$ is continuous and $$
\left| \tilde{\varphi}^r_\tau(\underline{f}+\Delta{\varphi}_{\tau'}^l)-
\left[\tilde{\varphi}^r_\tau(\underline{f})+\frac{\partial \tilde{\varphi}^r_\tau}{\partial {\varphi}^l_{\tau'}}\Delta {\varphi}^l_{\tau'} \right]\right|_2
= O_{\underline{f}}\left( \left|{\varphi}^l_{\theta\tau}\right|_{3+\epsilon}\left|\Delta{\varphi}_{\tau'}^l\right|_{2}^{1+\epsilon}\right). $$ \end{lem} Define the operator $D_{rl}: X_2 \to X_2$ by $$ \left(D_{rl}\left(\Delta \underline{\varphi}^l\right)\right)_\tau=\sum_{\tau'\leq\theta\tau}\frac{\partial \tilde{\varphi}^r_\tau}{\partial {\varphi}^l_{\tau'}}\Delta\varphi^l_{\tau'}. $$
This is a uniformly bounded operator, see Lemma \ref{partial phir thetatau=tau' partial phil}, and Lemma \ref{partial phir thetatau>tau' partial phil} . We obtain the following.
\begin{prop}\label{Drlderivative} The dependance $\underline{f}\mapsto D_{rl}(\underline{f})$ is continuous and $$
\left| \tilde{\varphi}^r(\underline{f}+\Delta \underline{\varphi}^l)-
\left[\tilde{\varphi}^r(\underline{f})+D_{rl}\Delta \underline{\varphi}^l\right]\right|_2
=O_{\underline{f}}(|\Delta \underline{\varphi}^l|_2^{1+\epsilon}). $$ Moreover, the partial derivative extends to a uniformly bounded operator $D_{rl}: X_2 \to X_2$. \end{prop}
\end{document} | arXiv |
\begin{document}
\keywords{Fractional Sobolev--Poincar\'e inequality, fractional Sobolev inequality, fractional Hardy inequality, unbounded John domain} \subjclass[2010]{26D10 (46E35)}
\begin{abstract} We prove fractional Sobolev--Poincar\'e inequalities in unbounded John domains and we characterize fractional Hardy inequalities there. \end{abstract}
\maketitle
\markboth{\textsc{R. Hurri-Syrj\"anen and A. V. V\"ah\"akangas}} {\textsc{Fractional Sobolev--Poincar\'e and fractional Hardy inequalities}}
\section{Introduction}
Let $D$ be a bounded $c$-John domain in $\R^n$, $n\ge 2$. Let numbers $\delta,\tau \in (0,1)$ and exponents $p,q\in [1,\infty)$ be given such that $1/p-1/q = \delta/n$. Then there is a constant $C=C(\delta,\tau,p,n,c)$ such that the fractional Sobolev--Poincar\'e inequality \begin{equation}\label{fractionalqp} \int_D\vert u(x)-u_D\vert ^q\,dx \le C \biggl(\int_D\int_{B^n(x,\tau \qopname\relax o{dist}(x,\partial D))}\frac{\vert u(x)-u(y)\vert ^p}{\vert x-y\vert ^{n+\delta p}}\,dy\,dx \biggr)^{q/p} \end{equation} holds for all functions $u \in L^1(D)$. For a proof we refer the reader to \cite[Theorem 4.10]{H-SV} when $1<p<n/\delta$ and to \cite{Dyda3} when $p=1$.
We prove the inequality corresponding to \eqref{fractionalqp} in unbounded John domains, Theorem \ref{t.main}. The classical Sobolev--Poincar\'e inequality for an unbounded $c$-John domain $D$ has been proved in \cite[Theorem 4.1]{MR1190332}: there is a finite constant $C(n,p,c)$ such that the inequality \[ \inf_{a\in\R}Ê\int_D \lvert u(x)-a\rvert^{np/(n-p)}\,dx \le C(n,p,c)\bigg(\int_D \lvert \nabla u(x)\rvert^p\,dx\bigg)^{n/(n-p)} \] holds for all $u\in L^1_p(D)=\{u\in\mathscr{D}'(D)\,:\,\nabla u\in L^p(D)\}$; here $1\le p<n$. We obtain the fractional Sobolev inequalities \eqref{frac_Sobolev_inequality} in unbounded John domains too, Theorem \ref{t.main_emb}.
As an application of the fractional Sobolev inequalities we characterize the fractional Hardy inequalities \[ \int_{D} \frac{\lvert u(x)\rvert^q}{\qopname\relax o{dist}(x,\partial D)^{q(\delta+n(1/q-1/p))}}\,dx \le C\bigg(\int_{D} \int_{D} \frac{\lvert u(x)-u(y)\rvert ^p}{\lvert x-y\rvert ^{n+\delta p}}\,dy\,dx\bigg)^{q/p} \] in unbounded John domains $D$ whenever $\delta\in (0,1)$ and exponents $p,q\in [1,\infty)$ are given such that $p<n/\delta$ and $0\le 1/p-1/q\le \delta/n$ and the constant $C$ does not depend on $u\in C_0(D)$, Theorem \ref{t.hardy_c}. We also give sufficient geometric conditions for the fractional Hardy inequalities in Corollary \ref{t.cor}.
\section{Notation and preliminaries}\label{s.notation}
Throughout the paper we assume that $D$ is a domain and $G$ is an open set in the Euclidean $n$-space $\R^n$, $n\geq 2$. The open ball centered at $x\in \R^n$ and with radius $r>0$ is $B^n(x,r)$. The Euclidean distance from $x\in G$ to the boundary of $G$ is written as $\qopname\relax o{dist}(x,\partial G)$. The diameter of a set $A$ in $\R^n$ is $\mathrm{diam}(A)$. The Lebesgue $n$-measure of a measurable set $A$ is denoted by $\vert A\vert.$ For a measurable set $A$ with finite and positive measure and for an integrable function $u$ on $A$ the integral average is written as \[ u_A=\frac{1}{\lvert A \rvert} \int_{A}u(x)\,dx\,. \] We write $\chi_A$ for the characteristic function of a set $A$. For a proper open set $G$ in $\R^n$ we fix a Whitney decomposition $\mathcal{W}(G)$. The construction and the properties of Whitney cubes can be found in \cite[VI 1]{S}. The family $C_0(G)$ consists of all continuous functions $u:G\to \R$ with a compact support in $G$. We let $C(\ast,\dotsb,\ast)$ denote a constant which depends on the quantities appearing in the parentheses only.
We define the $c$-John domains so that unbounded domains are allowed, too. For other equivalent definitions we refer the reader to \cite{MR1246886} and \cite{MR1190332}.
\begin{defn}\label{sjohn} A domain $D$ in $\R^n$ with $n\ge 2$ is a {\em $c$-John domain}, $c\ge 1$, if each pair of points $x_1,x_2\in D$ can be joined by a rectifiable curve $\gamma:[0,\ell]\to D$ parametrized by its arc length such that
$\qopname\relax o{dist}(\gamma(t),\partial D)\ge \min\{t,\ell -t\}/c$ for every $t\in [0,\ell]$. \end{defn}
\noindent Examples of unbounded John domains are the Euclidean $n$-space $\R^n$ and the infinite cone \[ \big\{(x',x_n)\in\R^n \,:\, x_n>\lVert x' \rVert\big\}\,. \]
For more examples we refer the reader to \cite[4.3 Examples]{MR1190332}.
We recall a useful property of bounded John domains from \cite[Theorem 3.6]{MR1246886}.
\begin{lem}\label{t.equi}
Let $D$ in $\R^n$ be a bounded $c$-John domain, $n\ge 2$.
Then there exists a central point $x_0\in D$ such that every
point $x$ in $D$ can be joined to $x_0$ by a rectifiable curve
$\gamma:[0,\ell]\to D$, parametrized by its arc length, with
$\gamma(0)=x$, $\gamma(\ell)=x_0$, and $\qopname\relax o{dist}(\gamma(t),\partial D)\ge t/4c^2$ for each $t\in [0,\ell]$. \end{lem}
The following engulfing property is in \cite[Theorem 4.6]{MR1246886}.
\begin{lem}\label{t.engulfing} A $c$-John domain $D$ in $\R^n$ can be written as the union of domains $D_1,D_2,\ldots$ such that \begin{itemize} \item[(1)]Ê$\overline{D_i}$ is compact in $D_{i+1}$ for each $i=1,2,\ldots$, \item[(2)]Ê$D_i$ is a $c_1$-John domain for each $i=1,2,\ldots$ with $c_1=c_1(c,n)$. \end{itemize} \end{lem}
We define the upper and lower Assouad dimension of a given set $E\not=\emptyset$ in $\R^n$. The upper Assouad dimension measures how thin a given set is and the lower Assouad dimension measures its fatness. For further discussion on these dimensions we refer to \cite[\S1]{KLV}.
\begin{defn}\label{def:uAssouad} The upper Assouad dimension of $E$, written as $\overline{\mathrm{dim}}_A(E)$, is defined as the infimum of all numbers $\lambda\ge 0$ as follows: There exists a constant $C=C(E,\lambda)> 0$ such that for every $x\in E$ and for all $0<r<R<2\mathrm{diam}(E)$ the set $E\cap B^n(x,R)$ can be covered by at most $C(R/r)^\lambda$ balls that are centered in $E$ and have radius $r$. \end{defn}
\begin{defn} The lower Assouad dimension of $E$, written as $\underline{\mathrm{dim}}_{A}(E)$, is defined as the supremum of all numbers $\lambda\ge 0$ as follows: There exists a constant $C=C(E,\lambda)> 0$ such that for every $x\in E$ and for all $0<r<R<2\diam(E)$ at least $C(R/r)^\lambda$ balls centered in $E$ and with radius $r$ are needed to cover the set $B^n(x,R)\cap E$. \end{defn}
Let $G$ be an open set in $\R^n$. Let $0< p<\infty$ and $0<\tau ,\delta<1$ be given. We write \[ \lvert u \rvert_{W^{\delta,p}(G)} = \bigg( \int_G\int_{G}\frac{\lvert u(x)-u(y)\rvert^p}{\lvert x-y\rvert^{n+\delta p}}\, dy\,dx\,\bigg)^{1/p} \] and \[ \lvert u \rvert_{W^{\delta,p}_\tau(G)} = \bigg( \int_G\int_{B^n(x,\tau \mathrm{dist}(x,\partial G))}\frac{\lvert u(x)-u(y)\rvert^p}{\lvert x-y\rvert^{n+\delta p}}\, dy\,dx\,\bigg)^{1/p} \] for appropriate measurable functions $u$ on $G$. When $G=\R^n$ both of the integrals in the latter form are taken over the whole space. The homogeneous fractional Sobolev space $\dot{W}^{\delta,p}_\tau(G)$ consists of all measurable functions $u:G\to \R$ with $\lvert u\rvert_{W^{\delta,p}_\tau(G)}<\infty$.
The following lemma tells that the functions $u\in \dot{W}^{\delta,p}_\tau(G)$ are locally $L^p$-integrable in $G$, that is $u\in L^p_{\textup{loc}}(G)$. We improve this for John domains in Corollary \ref{embed}.
\begin{lem}\label{l.integrability} Suppose that $G$ is an open set in $\R^n$. Let $0< p<\infty$ and $0<\tau,\delta<1$ be given. Let $K$ be a compact set in $G$. If $u\in \dot{W}^{\delta,p}_\tau(G)$ then $u\in L^p(K)$. \end{lem}
\begin{proof} We may assume that $G\not=\R^n$. If $G=\R^n$, then we just remove one point from $G\setminus K$. By covering $K$ with a finite number of balls $B$ such that $\overline{B}\subset G$ we may assume that $K$ is the closure of such a ball. Let us fix $\varepsilon>0$ such that $\varepsilon\tau/(1-\varepsilon \tau)<\tau$. We obtain \begin{equation} \label{base_fin} \begin{split} \int_K\int_{K\cap B^n(z,\tau \mathrm{dist}(z,\partial G))}& \lvert u(z)-u(y)\rvert^p\,dy\,dz \\&\le \mathrm{diam}(K)^{n+\delta p}\int_K\int_{K\cap B^n(z,\tau \mathrm{dist}(z,\partial G))} \frac{\lvert u(z)-u(y)\rvert^p}{\lvert z-y\rvert^{n+\delta p}}\,dy\,dz\\&\le \diam(K)^{n+\delta p}\lvert u\rvert_{W^{\delta,p}_\tau(G)}^p<\infty\,. \end{split} \end{equation} Let us fix $x\in K$ and $0<r_x<\varepsilon \tau \qopname\relax o{dist}(x,\partial G)$. Since $K$ is the closure of some ball, we have the inequality $\lvert K\cap B^n(x,r_x)\rvert>0$. By our estimates in \eqref{base_fin} there is a point $z_x\in K\cap B^n(x,r_x)$ so that \begin{equation}\label{finite} \int_{K\cap B^n(z_x,\tau \mathrm{dist}(z_x,\partial G))} \lvert u(z_x)-u(y)\rvert^p\,dy < \infty\,. \end{equation}
By the choice of $\varepsilon$ we have $x\in B^n(z_x,\tau \qopname\relax o{dist}(z_x,\partial G))$ for each $x\in K$. Thus, \[ K\subset \bigcup_{x\in K} B^n(z_x,\tau \qopname\relax o{dist}(z_x,\partial G))\,. \] By the compactness of the set $K$ there are points $x_1,\ldots,x_N$ in $K$ such that $K$ is contained in the union of the balls $B^n(z_i,\tau \qopname\relax o{dist}(z_i,\partial G))$, where $z_i=z_{x_i}$ for each $i$. Hence, by inequality \eqref{finite} we obtain \begin{align*} \int_K \lvert u(y)\rvert^p\,dy &\le \sum_{i=1}^N \int_{K\cap B^n(z_i,\tau \qopname\relax o{dist}(z_i,\partial G))} \lvert u(y)\rvert^p\,dy \\ &\le 2^p\sum_{i=1}^N \int_{K\cap B^n(z_i,\tau \qopname\relax o{dist}(z_i,\partial G))} \lvert u(z_i)\rvert^p + \lvert u(z_i)-u(y)\rvert^p\,dy<\infty\,. \end{align*} This concludes the proof. \end{proof}
The following definition is from \cite[\S1]{MR2317850}. It arises from generalized Poincar\'e inequalities that are studied in \cite[\S 7]{MR1609261}. Let us fix $\kappa \ge 1$ and an open set $G$ in $\R^n$. For $\delta\in [0,1]$, $0<p\le \infty$, and $u\in L^1_{\textup{loc}}(G)$ we write \[ \lvert u\rvert_{A^{\delta,p}_\kappa(G)} = \sup_{\mathcal{Q}_\kappa(G)} \Bigg\lVert \sum_{Q\in\mathcal{Q}_\kappa(G)} \bigg(\frac{1}{\lvert Q\rvert^{1+\delta/n}}\int_Q \lvert u(x)-u_Q\rvert\,dx \bigg)\chi_Q\Bigg\lVert_{L^p(G)}\,, \] where the supremum is taken over all families of cubes $\mathcal{Q}_\kappa(G)$ such that $\kappa Q\subset G$ for every $Q\in\mathcal{Q}_\kappa(G)$ and $Q\cap R = \emptyset$ if $Q$ and $R$ belong to $\mathcal{Q}_\kappa(G)$ and $Q\not=R$.
\begin{lem}\label{embed_l} Suppose that $G$ is an open set in $\R^n$. Let $0<\tau,\delta<1$ and $1\le p<\infty$ be given. Then there is a constant $\kappa=\kappa(n,\tau)\ge 1$ such that inequality \begin{equation}\label{e.embed} \lvert u\rvert_{A^{\delta,p}_\kappa(G)}\le (\sqrt n)^{n/p+\delta}\lvert u\rvert_{W^{\delta,p}_\tau(G)} \end{equation} holds for every $u\in L^1(G)$. \end{lem}
\begin{proof} Let us choose $\kappa=\kappa(n,\tau)\ge 1$ such that $Q\subset B^n(x,\tau \qopname\relax o{dist}(x,\partial G))$ whenever $x\in Q\in\mathcal{Q}_\kappa(G)$. Then we fix a family of cubes $\mathcal{Q} :=\mathcal{Q}_\kappa(G)$. By Jensen's inequality we obtain \begin{align*}
\sum_{Q\in\mathcal{Q}} \lvert Q\rvert \bigg(\frac{1}{\lvert Q\rvert^{1+\delta/n}}\int_Q \lvert u(x)-u_Q\rvert\,dx\bigg)^p \le \sum_{Q\in\mathcal{Q}} \lvert Q\rvert^{-\delta p/n}\int_Q \lvert u(x)-u_Q\rvert^p\,dx\,. \end{align*} By using Jensen's inequality again \begin{align*} &\sum_{Q\in\mathcal{Q}} \lvert Q\rvert^{-\delta p/n}\int_Q \lvert u(x)-u_Q\rvert^p\,dx\\&\le (\sqrt n)^{n+\delta p}\sum_{Q\in\mathcal{Q}} \int_Q\int_Q \frac{\lvert u(x)-u(y)\rvert^p}{\lvert x-y\rvert^{n+\delta p}}\,dy\,dx \le (\sqrt n)^{n+\delta p}\lvert u\rvert_{W^{\delta,p}_\tau(G)}^p\,. \end{align*} Taking supremum over all families $\mathcal{Q}_k(G)$ gives inequality \eqref{e.embed}. \end{proof}
\section{Inequalities in bounded John domains}\label{s.bounded}
We give the following fractional Sobolev--Poincar\'e inequality in bounded John domains. The inequality for $p>1$ is already in \cite[Theorem 4.10]{H-SV}, but we need a better control over the dependencies of the constant $C$.
\begin{thm}\label{t.sub} Suppose that $D$ is a bounded $c$-John domain in $\R^n$, $n\ge 2$. Let $\tau,\delta\in (0,1)$ and $1\le p< n/\delta$ be given. Then there is a constant $C=C(\delta,\tau,p,n,c)>0$ such that the fractional Sobolev--Poincar\'e inequality \begin{equation}\label{fractionalq1} \int_D\vert u(x)-u_D\vert ^{np/(n-\delta p)}\,dx \le C \lvert u \rvert_{W^{\delta,p}_\tau(D)}^{np/(n-\delta p)} \end{equation} holds for every $u\in L^1(D)$. \end{thm}
Theorem \ref{t.sub} follows from Proposition \ref{WeakEquivToStrong} and Proposition \ref{p.weak}. The following result from \cite{Dyda3}, based upon the Maz'ya truncation method \cite{Maz} adapted to the fractional setting, shows that it is enough to prove a weak fractional Sobolev--Poincar\'e inequality.
\begin{prop}\label{WeakEquivToStrong} Suppose that $G$ is an open set in $\R^n$ with $\lvert G\rvert<\infty$. Let $0<\delta,\tau <1$ and $0<p\le q<\infty$ be given. Then the following conditions are equivalent. \begin{itemize} \item[(A)] There is a constant $C_1>0$ such that inequality \begin{align*}
&\quad\inf_{a\in\R}\sup_{t>0}\lvert \{x\in G:\,|u(x)-a|>t\}\rvert t^{q}
\\&\qquad\qquad\qquad \le C_1 \biggl(\int_{G}\int_{B^n(y,\tau \qopname\relax o{dist}(y,\partial G))}\frac{\vert u(y)-u(z)\vert^p}{\vert y-z\vert ^{n+\delta p}} \,dz\,dy\biggr)^{q/p} \end{align*} holds for every $u\in L^\infty(G)$. \item[(B)] There is a constant $C_2>0$ such that inequality \begin{align*} \quad \inf_{a\in\R} \int_G\vert u(x)-a\vert ^{q}\,dx \le C_2 \bigg(\int_G\int_{B^n(y,\tau \qopname\relax o{dist}(y,\partial G))}\frac{\vert u(y)-u(z)\vert^p}{\vert y-z\vert ^{n+\delta p}}\,dz \,dy\bigg)^{q/p} \end{align*} holds for every $u\in L^1(G)$. \end{itemize} In the implication from (A) to (B) $C_2=C(p,q)C_1$ and from (B) to (A)
$C_1=C_2$. \end{prop}
The weak fractional Sobolev--Poincar\'e inequalities hold in bounded John domains by the following proposition.
\begin{prop}\label{p.weak} Suppose that $D$ is a bounded $c$-John domain in $\R^n$. Let $\tau,\delta\in (0,1)$ and $1\le p< n/\delta$ be given. Then there is a constant $C=C(\delta,\tau,p,n,c)>0$ such that the weak fractional Sobolev--Poincar\'e inequality \begin{equation*} \inf_{a\in\R} \sup_{t>0} \vert \{x\in D\,:\, \lvert u(x)-a\rvert > t \}\rvert t^{np/(n-\delta p)}\le C \lvert u \rvert_{W^{\delta,p}_\tau(D)}^{np/(n-\delta p)} \end{equation*} holds for every $u\in L^\infty(D)$. \end{prop}
For a simple proof of Proposition \ref{p.weak} we refer to \cite[Theorem 4.10]{H-SV}. The dependencies of the constants appearing in \cite[Theorem 4.10]{H-SV} can be tracked more explicitly in order to obtain Proposition \ref{p.weak}. In the present paper, we give a more general argument that might be of independent interest.
The following Theorem \ref{iso} is the key result for proving Proposition \ref{p.weak}.
\begin{thm}\label{suff_thm}\label{iso} Suppose that $D$ is a bounded $c$-John domain in $\R^n$. Let $\kappa\ge 1$ be fixed. Let $\delta\in [0,1]$ and $1\le p<n/\delta$ be given. Then there exists a constant $C=C(n,\kappa,p,\delta,c)$ such that the inequality \begin{equation}\label{fract} \inf_{a\in\R} \sup_{t>0}Ê\lvert \{x\in D\,:\, \lvert u(x)-a\rvert > t\}\rvert t^{np/(n-\delta p)} \le C\lvert u\rvert_{A^{\delta,p}_\kappa(D)}^{np/(n-\delta p)} \end{equation} holds for every $u\in L^1(D)$. \end{thm}
We give the proof of Theorem \ref{iso} in Section \ref{s.iso_proof}. By using Theorem \ref{iso} the claim of Proposition \ref{p.weak} follows easily.
\begin{proof}[Proof of Proposition \ref{p.weak}]
By Lemma~\ref{embed_l} it is enough to prove that there is a constant $C=C(\delta , \tau , p, n, c)$ such that the inequality \[ \inf_{a\in\R} \sup_{t>0} \vert \{x\in D\,:\, \lvert u(x)-a\rvert >t \}\rvert t^{np/(n-\delta p)}\le C\lvert u\rvert_{A^{\delta,p}_{\kappa(n,\tau)}(D)}^{np/(n-\delta p)} \] holds for all $u\in L^\infty(D)$. This inequality follows from Theorem \ref{iso}. \end{proof}
\section{Proof of Theorem \ref{iso}}\label{s.iso_proof} We start to build up the proof for Theorem \ref{iso} by giving auxiliary results. The following lemma gives local inequalities. Similar results are known in metric measure spaces, \cite[Theorem 4.1]{MR2317850}.
\begin{lem}\label{cube_lem} Let $1\le p,q<\infty$ be given such that $1/p-1/q=\delta/n$ with $\delta\in [0,1]$. Then there is a constant $C=C(n,p,\delta)>0$ such that inequality \begin{equation}\label{assertion} \sup_{t>0}\lvert \{x\in Q\,:\, \lvert u(x)-u_Q\rvert >t\}\rvert t^q\le C \lvert u\rvert_{A^{\delta,p}_1(Q)}^q \end{equation} holds for every cube $Q$ in $\R^n$ and for all $u\in L^1_{\textup{loc}}(\R^n)$. \end{lem}
\begin{proof} Let us fix $u\in L^1_{\textup{loc}}(\R^n)$. We write for cubes $Q$ in $\R^n$ \begin{align*} a(Q) &= \lvert u\rvert_{A^{\delta,p}_1(Q)}\cdot \lvert Q\rvert^{-1/q} \\&= \bigg\{ \lvert Q\rvert^{-p/q} \cdot \sup_{\mathcal{Q}_1(Q)} \sum_{R\in \mathcal{Q}_1(Q)} \lvert R\rvert^{1-\delta p/n} \bigg(\frac{1}{\lvert R\rvert}Ê\int_R \lvert u(x)-u_R\rvert\,dx\bigg)^p \bigg\}^{1/p}\,. \end{align*} Inequality \eqref{assertion} follows from the generalized Poincar\'e inequality theorem \cite[Theorem 7.2(a)]{MR1609261} as soon as we prove inequalities \eqref{f_control} and \eqref{a_control}. The inequality \begin{equation}\label{f_control} \frac{1}{\lvert Q\rvert}\int_Q \lvert u(x)-u_Q\rvert\,dx \le a(Q) \end{equation} holds for every cube $Q$ in $\R^n$. Namely, \begin{align*} \frac{1}{\lvert Q\rvert}\int_Q \lvert u(x)-u_Q\rvert\,dx &= \bigg\{Ê\lvert Q\rvert^{-p/q} \cdot\lvert Q\rvert^{1-\delta p/n} \bigg(\frac{1}{\lvert Q\rvert}\int_Q \lvert u(x)-u_Q\rvert\,dx\bigg)^{p}\bigg\}^{1/p}\\&\le a(Q)\,, \end{align*} because $1-p/q-\delta p/n=0$. We need to show that the inequality \begin{equation}\label{a_control} \sum_{P\in \mathcal{Q}_1(Q)} a(P)^q \lvert P\rvert \le 2^{q/p} a(Q)^q\lvert Q\rvert \end{equation} holds for all cubes $Q$ in $\R^n$ and all families $\mathcal{Q}_1(Q)$ of pairwise disjoint cubes inside $Q$. In order to prove inequality \eqref{a_control} let us fix a cube $Q$ and its family $\mathcal{Q}_1(Q)$. For each $P\in\mathcal{Q}_1(Q)$ we fix its family $\mathcal{Q}_1(P)$ such that \[ \lvert u\rvert_{A^{\delta,p}_1(P)}^p\le 2\sum_{R\in \mathcal{Q}_1(P)} \lvert R\rvert^{1-\delta p/n} \bigg(\frac{1}{\lvert R\rvert}Ê\int_R \lvert u(x)-u_R\rvert\,dx\bigg)^p\,. \] Since $q/p\ge 1$, \begin{align*} \sum_{P\in\mathcal{Q}_1(Q)} a(P)^q \lvert P\rvert \le 2^{q/p}\bigg\{ \sum_{P\in\mathcal{Q}_1(Q)} \sum_{R\in \mathcal{Q}_1(P)} \lvert R\rvert^{1-\delta p/n} \bigg(\frac{1}{\lvert R\rvert}Ê\int_R \lvert u(x)-u_R\rvert\,dx\bigg)^p \bigg\}^{q/p}\,. \end{align*}
Then writing
$\mathcal{Q}:=\cup_{P\in\mathcal{Q}_1(Q)}Ê\mathcal{Q}_1(P)$ allows us to estimate
\begin{align*}
&\sum_{P\in\mathcal{Q}_1(Q)} a(P)^q \lvert P\rvert \\&\le 2^{q/p}\bigg\{ \sum_{R\in \mathcal{Q}} \lvert R\rvert^{1-\delta p/n} \bigg(\frac{1}{\lvert R\rvert}Ê\int_R \lvert u(x)-u_R\rvert\,dx\bigg)^p \bigg\}^{q/p} \le 2^{q/p} a(Q)^q\lvert Q\rvert\,.
\end{align*} This implies inequality \eqref{a_control}. \end{proof}
For a bounded $c$-John domain $D$ we let $\mathcal{W}^\kappa(D)$ be its modified Whitney decomposition with a fixed $\kappa\geq 1$ such that $\kappa Q^*=\kappa \tfrac 98 Q\subset D$ for each $Q\in\mathcal{W}^\kappa(D)$. This decomposition is obtained by dividing each Whitney cube $Q\in\mathcal{W}(D)$ into sufficiently small dyadic subcubes, their number depending on $\kappa$ and $n$ only. The family of cubes in $\mathcal{W}^\kappa(D)$ with side length $2^{-j}$, $j\in\mathbf{Z}$, is written as $\mathcal{W}_{j}^\kappa(D)$.
Let $Q$ be in $\mathcal{W}^\kappa(D)$. Let us suppose that we are given a chain $\mathcal{C}(Q)\subset\mathcal{W}^\kappa(D)$ of cubes \[\mathcal{C}(Q) = (Q_0,\ldots,Q_k)\,,\] joining a fixed cube $Q_0\in\mathcal{W}^\kappa(D)$ to $Q_k=Q$ such that there exists a constant $C(n,\kappa )$ so that the inequality \[ \lvert u_{Q^*} - u_{Q_0^*}\rvert \le C(n,\kappa)\sum_{R\in\mathcal{C}(Q)} \frac{1}{\lvert R^*\rvert}\int_{R^*} \lvert u(x)-u_{R^*}\rvert\,dx \] holds whenever $u\in L^1_{\textup{loc}}(D)$. The family $\{\mathcal{C}(Q)\,:\,Q\in\mathcal{W}^\kappa(D)\}$ of chains of cubes is called a chain decomposition of $D$. The shadow of a given cube $Q\in \mathcal{W}^\kappa(D)$ is the family \[\mathcal{S}(R) = \{ Q\in\mathcal{W}^\kappa(D)\,:\, R\in\mathcal{C}(Q) \}\,.\]
The following key lemma is a straightforward modification of \cite[Proposition 2.5]{H-SMV} once we have Lemma \ref{t.equi}.
\begin{lem}\label{chain} Let $D$ be a bounded $c$-John domain in $\R^n$. Let $\kappa\ge 1$ and $1\le q<\infty$ be given. Then there exist a chain decomposition of $D$ and constants $\sigma, \rho\in\mathbf{N}$ such that \begin{itemize} \item[(1)] $\ell(Q)\le 2^{\rho}\ell(R)$ for each $R\in\mathcal{C}(Q)$
and $Q\in\mathcal{W}^\kappa(D)$, \item[(2)] $\sharp \{R\in\mathcal{W}^\kappa_j(D):\ R\in\mathcal{C}(Q)\}\le
2^ \rho$ for each $Q\in \mathcal{W}^\kappa(D)$ and $j\in \mathbf{Z}$, \item[(3)] the inequality \begin{equation*} \sup_{j\in \mathbf{Z}}\sup_{R\in\mathcal{W}^\kappa_j(D)} \frac{1}{\lvert R\lvert} \sum_{k=j-\rho}^\infty \sum_{\substack{Q\in \mathcal{W}^\kappa_k(D) \\ Q\in\mathcal{S}(R) }} \lvert Q\rvert (\rho+1+k-j)^{q} < \sigma \end{equation*} holds. \end{itemize} The constants $\sigma$ and $\rho$ depend on $\kappa$, $n$, $q$, and the John constant $c$ only. \end{lem}
We are ready for the proof of Theorem \ref{suff_thm}.
\begin{proof}[Proof of Theorem \ref{suff_thm}] Let us denote $q=np/(n-\delta p)$. We need to show that there is a constant $C(n,\kappa,p,\delta,c)$ such that the inequality \[
\inf_{a\in\R}\sup_{t>0} \lvert \{x\in D:\,|u(x)-a|>t\}\rvert t^q \le C(n,\kappa,p,\delta,c) \lvert u\rvert_{A^{\delta,p}_\kappa(D)}^q \] holds for each $u\in L^1(D)$. Let $Q_0$ be the fixed cube in the chain decomposition of $D$ given by Lemma \ref{chain}. By the triangle inequality we obtain \begin{align*}
\lvert u(x)-u_{Q_0^*}\rvert &\le \left|u(x)-\sum_{Q\in\mathcal{W}^\kappa(D)} u_{Q^*}\chi_{Q}(x)\right|+
\left|\sum_{Q\in\mathcal{W}^\kappa(D)} u_{Q^*} \chi_Q(x) - u_{Q_0^*}\right| \end{align*} for almost every $x\in D$. We write \[
\left|u(x)-\sum_{Q\in\mathcal{W}^\kappa(D)} u_{Q^*}\chi_{Q}(x)\right|
=: g_1(x) \] and \[
\left|\sum_{Q\in\mathcal{W}^\kappa(D)} u_{Q^*} \chi_Q(x) - u_{Q_0^*}\right|
=: g_2(x) \] for $x\in D$. For a fixed $t>0$ we estimate \begin{align*} & t^q \lvert \{x\in D:\ \lvert
u(x)-u_{Q_0^*}\rvert>t\}\rvert \\&\qquad\le
t^q\left\lvert \{x\in D:\ g_1(x)>t/2\}\right\lvert +
t^q\left\lvert \{x\in D:\ g_2(x)>t/2\}\right\lvert \,.
\end{align*} The local term $g_1$ is estimated by Lemma \ref{cube_lem} and the inequality $p\le q$: \begin{align*}
t^q\left\lvert \{x\in D:\ g_1(x)>t/2\}\right\lvert & = \sum_{Q\in\mathcal{W}^\kappa(D)} t^q \lvert
\{x\in \textup{int}(Q)\,:\, \lvert u(x)-u_{Q^*}\rvert >t/2\}\rvert
\\& \le C 2^q\bigg(\sum_{Q\in\mathcal{W}^\kappa(D)} \lvert u\rvert_{A^{\delta,p}_1(Q^*)}^p\bigg)^{q/p}\,. \end{align*} Let us note that $\kappa R \subset \kappa Q^*\subset D$ if $R\in \mathcal{Q}_1(Q^*)$ and $Q\in\mathcal{W}^\kappa(D)$. We divide the family $\{Q^*\,:\,Q\in\mathcal{W}^\kappa(D)\}$ of cubes into $C(n,\kappa)$ families so that each of them consists of pairwise disjoint cubes. As in the proof of Lemma \ref{cube_lem} we obtain \[
t^q \lvert \{x\in D: \ g_1(x)>t/2\}\rvert \le C\lvert u\rvert^q_{A^{\delta,p}_\kappa(D)}\,. \]
We start to estimate the chaining term $g_2$: \begin{align*}
t^q\left\lvert \{x\in D:\ g_2(x)>t/2\}\right\lvert & = t^q\sum_{Q\in\mathcal{W}^\kappa(D)} \lvert
\{ x\in \textup{int}(Q)\,:\, \lvert u_{Q^*}-u_{Q_0^*}\rvert >t/ 2\}\rvert \\ &\le
2^q \sum_{Q\in\mathcal{W}^\kappa(D)} \lvert Q\rvert\lvert
u_{Q^*}-u_{Q_0^*}\rvert^q=:\Sigma\,. \end{align*} By property (1) of the chain decomposition in Lemma \ref{chain} we obtain \begin{align*} \Sigma &\le C\sum_{k=-\infty}^\infty \sum_{Q\in
\mathcal{W}^\kappa_k(D)} \lvert Q\rvert\Bigg(\sum_{j=-\infty}^{k+\rho} \underbrace{ \sum_{\substack{R\in\mathcal{W}^\kappa_j(D) \\ R\in \mathcal{C}(Q)}}
\frac{1}{\lvert R^*\rvert}\int_{R^*} \lvert u(x)-u_{R^*}\rvert\,dx}_{=:\Sigma_{j,Q}}
\Bigg)^q\\ &=C\sum_{k=-\infty}^\infty \sum_{Q\in
\mathcal{W}^\kappa_k(D)} \lvert Q\rvert \Bigg(\sum_{j=-\infty}^{k+\rho}
\underbrace{(\rho+1+k-j)^{-1}(\rho+1+k-j)}_{=1}\Sigma_{j,Q}\Bigg)^q\,. \end{align*} Property (2) in Lemma \ref{chain} and the equation $1/p-1/q=\delta/n$ give
\begin{align*}
\Sigma_{j,Q}^q & = \Bigg(\sum_{\substack{R\in\mathcal{W}^\kappa_j(D) \\ R\in \mathcal{C}(Q)}}
\frac{1}{\lvert R^*\rvert}\int_{R^*} \lvert u(x)-u_{R^*}\rvert\,dx\Bigg)^q\\ & \le C\sum_{\substack{R\in\mathcal{W}^\kappa_j(D) \\ R\in \mathcal{C}(Q)}} \Bigg( \frac{1}{\lvert R^*\rvert}\int_{R^*} \lvert u(x)-u_{R^*}\rvert\,dx\Bigg)^q \le C\sum_{\substack{R\in\mathcal{W}^\kappa_j(D) \\ R\in \mathcal{C}(Q)}} \frac{\lvert u\rvert_{A^{\delta,p}_1(R^*)}^{q}}{\lvert R^*\rvert}\,.
\end{align*}
Thus, H\"older's inequality and property (3) in Lemma \ref{chain} imply that \begin{align*} \Sigma& \leq C\sum_{k=-\infty}^\infty \sum_{Q\in
\mathcal{W}^\kappa_k(D)} \lvert Q\rvert\sum_{j=-\infty}^{k+\rho} (\rho+1+k-j)^q
\sum_{\substack{R\in\mathcal{W}^\kappa_j(D) \\ R\in
\mathcal{C}(Q)}}\frac{\lvert u\rvert_{A_1^{\delta,p}(R^*)}^{q}}{\lvert
R^*\rvert} \\
& = C\sum_{j=-\infty}^\infty \sum_{R\in \mathcal{W}^\kappa_j(D)}
\lvert u\rvert_{A_1^{\delta,p}(R^*)}^{q} \cdot \frac{1}{\lvert R\rvert} \sum_{k=j-\rho}^\infty
\sum_{\substack{Q\in\mathcal{W}^\kappa_k(D) \\ Q\in \mathcal{S}(R)}}
\lvert Q\rvert(\rho+1+k-j)^q \\
&
\leq C\bigg(\sum_{j=-\infty}^\infty \sum_{R\in \mathcal{W}^\kappa_j(D)}
\lvert u\rvert_{A^{\delta,p}_1(R^*)}^{p}\bigg)^{q/p} \leq C\lvert u\rvert_{A^{\delta,p}_\kappa(D)}^{q}\,. \end{align*} The theorem is proved. \end{proof}
\section{Sobolev--Poincar\'e inequalities in unbounded John domains}\label{s.fractional}
We prove a fractional Sobolev--Poincar\'e inequality in unbounded John domains.
\begin{thm}\label{t.main} Suppose that $D$ in $\R^n$ is an unbounded $c$-John domain and that $\tau,\delta\in (0,1)$ are given. Let $1\le p< n/\delta$. Then there is a constant $C=C(\delta,\tau,p,n,c)>0$ such that the fractional Sobolev--Poincar\'e inequality \begin{equation*} \inf_{a\in\R}\int_D\vert u(x)-a\vert ^{np/(n-\delta p)}\,dx \le C \lvert u \rvert_{W^{\delta,p}_\tau(D)}^{np/(n-\delta p)} \end{equation*} holds for each $u\in \dot{W}^{\delta,p}_\tau(D)$. \end{thm}
The proof is similar to the proof of \cite[Theorem 4.1]{MR1190332} where the classical Sobolev--Poincar\'e inequality has been proved in unbounded domains which have an engulfing property. The proof is based on an idea from \cite{MR802488}.
\begin{proof}[Proof of Theorem \ref{t.main}] By Lemma \ref{t.engulfing} the $c$-John domain $D$ has an engulfing property. That is, there are bounded $c_1$-John domains $D_i$ with $c_1=c_1(c,n)$ such that \begin{equation*} D_i\subset \overline{D_i}\subset D_{i+1}\,,\qquad i=1,2,\dots\,, \end{equation*} and \[ D=\bigcup_{i=1}^{\infty}D_i\,. \] Let us fix $u\in \dot{W}^{\delta,p}_\tau (D)$. By Lemma \ref{l.integrability} with $K=\overline{D_i}$ we obtain that $u\in L^p(D_i)$ and hence $u\in L^1(D_i)$ for each $i$. Therefore we may write \begin{equation*} u_i=u_{D_i}=\frac{1}{\vert D_i\vert}\int_{D_i}u(x)\,dx\,,\qquad i=1,2,\dots. \end{equation*} The sequence $(u_i)$ is bounded. Namely, by the triangle inequality \begin{equation*} \vert u_i\vert = \frac{1}{\vert D_1\vert}\int_{D_1}\vert u_i\vert\,dx \le \frac{1}{\vert D_1\vert}\biggl( \int_{D_1}\vert u(x)-u_i\vert\,dx +\int_{D_1}\vert u(x)\vert \,dx\biggl)\,. \end{equation*} By H\"older's inequality with exponents $(np/(np-n+\delta p), np/(n-\delta p))$ and by Theorem \ref{t.sub} applied in a bounded $c_1$-John domain $D_i$ we obtain \begin{align*} \int_{D_1}\vert u(x)-u_i\vert \,dx &\le \vert D_1\vert^{1-1/p+\delta /n} \vert\vert u-u_{D_i}\vert\vert_{L^{np/(n-\delta p)}(D_1)}\\ & \le \vert D_1\vert^{1-1/p+\delta /n} \vert\vert u-u_{D_i}\vert\vert_{L^{np/(n-\delta p)}(D_i)} \le\vert D_1\vert^{1-1/p+\delta /n} C \lvert u \rvert_{W^{\delta,p}_\tau(D)}<\infty \end{align*} with a constant $C=C(\delta,\tau,p,n,c_1)$.
The bounded sequence $(u_i)$ has a convergent subsequence $(u_{i_j})$ and hence there is a constant $a\in\R$ such that $\lim_{j\to\infty} u_{i_j}=a$. By Fatou's lemma and Theorem \ref{t.sub} applied with the function $u\in L^p(D_j)$ we obtain \begin{align*} \int_D\vert u(x)- a\vert ^{np/(n-\delta p)}\,dx &=\int_D\liminf_{j\to\infty}\chi_{D_{i_j}}(x)\vert u(x)-u_{i_j}\vert^{np/(n-\delta p)}\,dx\\ &\le\liminf_{j\to\infty}\int_{D_{i_j}}\vert u(x)-u_{i_j}\vert^{np/(n-\delta p)}\,dx\\ &\le C\liminf_{j\to\infty}\lvert u \rvert_{W^{\delta,p}_\tau(D_{i_j})}^{np/(n-\delta p)} \le C\lvert u \rvert_{W^{\delta,p}_\tau(D)}^{np/(n-\delta p)}\,, \end{align*} where $C=C(\delta,\tau,p,n,c_1)$. The claim follows. \end{proof}
A fractional Sobolev inequality holds in unbounded John domains.
\begin{thm}\label{t.main_emb} Suppose that $D$ in $\R^n$ is an unbounded $c$-John domain and that $\tau,\delta\in (0,1)$ are given. Let $1\le p< n/\delta$. Then there is a constant $C=C(\delta,\tau,p,n,c)>0$ such that the fractional Sobolev inequality \begin{equation}\label{frac_Sobolev_inequality} \int_D\vert u(x)\vert ^{np/(n-\delta p)}\,dx \le C \lvert u \rvert_{W^{\delta,p}_\tau(D)}^{np/(n-\delta p)} \end{equation} holds for each $u\in \dot{W}^{\delta,p}_\tau(D)$ with a compact support in $D$. \end{thm}
\begin{proof} We write $D=\cup_{i=1}^\infty D_i$ as in the proof of Theorem \ref{t.main}. By Lemma \ref{l.integrability} and the fact that the numbers $\lvert D_i\rvert$ converge to $ \lvert D\rvert=\infty$ as $i\to \infty$ we obtain that \[ \lvert u_i\rvert= \lvert u_{D_i}\rvert \le \bigg( \frac{1}{\lvert D_i\rvert}\int_{D_i}Ê\lvert u(x)\rvert^p\,dx\bigg)^{1/p}\le \lvert D_i\rvert^{-1/p}Ê\lVert u\rVert_{L^p(D)}\xrightarrow{i\to \infty}Ê0 \] for $u\in \dot{W}^{\delta,p}_\tau(D)$ with a compact support in $D$. Therefore, the proof follows as the proof of Theorem \ref{t.main} with
$a=0$. \end{proof}
The following is a corollary of Theorem \ref{t.main} and Theorem \ref{t.main_emb}. It shows that $\dot W^{\delta,p}_\tau(D)$ is embedded to $L^{np/(n-\delta p)}(D)$ if we identify any two functions in $\dot W^{\delta,p}_\tau(D)$ whose difference is a constant almost everywhere. This identification is usually included already in the definition of homogeneous spaces of smoothness $0<\delta <1$.
\begin{cor}\label{embed} Suppose that $D$ in $\R^n$ is a $c$-John domain and that $\tau,\delta\in (0,1)$ are given. Let $1\le p< n/\delta$ and $q=np/(n-\delta p)$. Then there is a nonlinear bounded operator \[ \mathbf{E}:\dot W^{\delta,p}_{\tau}(D)\to L^q(D)\,,\qquad \mathbf{E}(u)= u-a_u\,, \] whose norm is bounded by a constant $C=C(\delta,\tau,p,n,c)$; here $a_u\in \R$ for each $u\in \dot W^{\delta,p}_{\tau}(D)$. If $D$ is an unbounded $c$-John domain, then $\mathbf{E}(u)=u$ for each $u\in \dot W^{\delta,p}_{\tau}(D)$ whose support is a compact set in $D$. \end{cor}
\section{Fractional Hardy inequalities in unbounded John domains}
We characterize certain fractional Hardy inequalities in unbounded John domains as an application of Theorem \ref{t.main_emb}. The following definition is motivated by the fractional Hardy inequalities from \cite{EH-SV}. The classical $(q,p)$-Hardy inequalities are studied in \cite{EH-S}.
We say that a fractional $(\delta,q,p)$-Hardy inequality with $0<\delta<1$ and $0< p,q<\infty$ holds in a proper open set $G$ in $\R^n$, if there is a constant $C>0$ such that the inequality \begin{equation}\label{e.hardy} \int_{G} \frac{\lvert u(x)\rvert^q}{\qopname\relax o{dist}(x,\partial G)^{q(\delta+n(1/q-1/p))}}\,dx \le C\bigg(\int_{G} \int_{G} \frac{\lvert u(x)-u(y)\rvert ^p}{\lvert x-y\rvert ^{n+\delta p}}\,dy\,dx\bigg)^{q/p} \end{equation} holds for all functions $u\in C_0(G)$. The fractional Sobolev inequality \eqref{frac_Sobolev_inequality} is obtained when $1/p-1/q=\delta/n$. The usual fractional $(\delta,p,p)$-Hardy inequality is obtained when $q=p$.
Our characterization of fractional Hardy inequalities is given in terms of Whitney cubes from $\mathcal{W}(G)$ and the $(\delta,p)$-capacities \[ \mathrm{cap}_{\delta,p}(K,G) = \inf_u \lvert u\rvert_{W^{\delta,p}(G)}^p \] of compact sets $K$ in $G$, where the infimum is taken over all $u\in C_0(G)$ such that $u(x)\ge 1$ for each point $x\in K$.
\begin{thm}\label{t.hardy_c} Let $D$ be an unbounded $c$-John domain in $\R^n$, $D\not=\R^n$. Let $\delta\in (0,1)$ and $1\le p,q<\infty$ be given such that $p<n/\delta$ and $0\le 1/p-1/q\le \delta/n$. Then the following conditions are equivalent. \begin{itemize} \item[(A)] A fractional $(\delta,q,p)$-Hardy inequality holds in $D$. \item[(B)] There exists a positive constant $N>0$ such that inequality \begin{equation*} \bigg(\sum_{Q\in\mathcal{W}(D)} \mathrm{cap}_{\delta,p}(K\cap Q,D)^{q/p}\bigg)^{p/q} \le N\,\mathrm{cap}_{\delta,p}(K,D) \end{equation*} holds for every compact set $K$ in $D$. \end{itemize} \end{thm}
The proof of Theorem \ref{t.hardy_c} is based on the fractional Sobolev inequalities and the Maz'ya type characterization for the validity of a fractional $(\delta,q,p)$-Hardy inequality, Theorem \ref{t.maz'ya}. Before the proof of Theorem \ref{t.hardy_c} we give some remarks, corollaries and auxiliary results.
\begin{rem} There exist unbounded John domains where a fractional $(\delta,p,p)$-Hardy inequality fails for some values of $\delta$ and $p$. As an example let us define $D=\R^2\setminus L$, where $L$ is a closed line-segment in $\R^2$. Then, the fractional $(\delta,p,p)$-Hardy inequality fails whenever $1< p<\infty$ and $\delta = 1/p$. This example is a modification of \cite[Theorem 9]{Dyda2}. \end{rem}
Sufficient geometric conditions for the fractional Hardy inequalities are given in the following corollary. For the relevant notation we refer to Section \ref{s.notation}.
\begin{cor}\label{t.cor} Let $D$ be an unbounded $c$-John domain in $\R^n$, $D\not=\R^n$. Let $0<\delta<1$ and $1\le p,q<\infty$ be given such that $p<n/\delta$ and $0\le 1/p-1/q\le \delta/n$. Then the fractional $(\delta,q,p)$-Hardy inequality \eqref{e.hardy} holds in $D$ if either condition (A) or condition (B) holds. \begin{itemize} \item[(A)] $\overline{\mathrm{dim}}_A(\partial D) < n-\delta p$; \item[(B)] $\underline{\mathrm{dim}}_A(\partial D)> n-\delta p$ and $\partial D$ is unbounded. \end{itemize} \end{cor}
\begin{proof} By Theorem \ref{t.hardy_c} it is enough to prove a $(\delta,p,p)$-Hardy inequality which is a consequence of \cite[Theorem 2]{Dyda1}. The plumpness condition required there follows from the John condition in Definition \ref{sjohn}. \end{proof}
Now we start to build up our proof for Theorem \ref{t.hardy_c}. First we give a characterization which is an extension of \cite[Proposition 5]{Dyda2} where the special case of $p=q$ is considered. This type of characterizations go back to Vl. Maz'ya, \cite{Maz}.
\begin{thm}\label{p.maz'ya} Suppose that $G$ is an open set in $\R^n$ and $\omega:G\to [0,\infty)$ is measurable. Then the following conditions are equivalent whenever $0<\delta<1$ and $0<p\le q<\infty$. \begin{itemize} \item[(A)] There is a constant $C_1>0$ such that the inequality \[\int_G \lvert u(x)\rvert^q\,\omega(x)\,dx \le C_1 \lvert u\rvert_{W^{\delta,p}(G)}^q\] holds for every $u\in C_0(G)$. \item[(B)] There is a constant $C_2>0$ such that the inequality \[ \int_K \omega(x)\,dx \le C_2\, \mathrm{cap}_{\delta,p}(K,G)^{q/p}\ \] holds for every compact set $K$ in $G$. \end{itemize} In the implication from (A) to (B) $C_2=C_1$ and from (B) to (A) $C_1 = \frac{C_2 2^{3q+2q/p}}{ (1-2^{-p})^{q/p}}$. \end{thm}
As a corollary of Theorem \ref{p.maz'ya} we obtain Theorem \ref{t.maz'ya} when we choose
\[ \omega=\mathrm{dist}(\cdot,\partial G)^{-q(\delta+n(1/q-1/p))}\,. \]
\begin{thm}\label{t.maz'ya} Let $0<\delta<1$ and $0<p\le q<\infty$. Then a $(\delta,q,p)$-Hardy inequality \eqref{e.hardy} holds in a proper open set $G$ in $\R^n$ if and only if there is a constant $C>0$ such that the inequality \begin{equation}\label{e.maz'ya} \int_K \mathrm{dist}(x,\partial G)^{-q(\delta+n(1/q-1/p))}\,dx \le C\,\mathrm{cap}_{\delta,p}(K,G)^{q/p} \end{equation} holds for every compact set $K$ in $G$. \end{thm}
The proof for Theorem \ref{p.maz'ya} is a simple modification of the proof of \cite[Proposition 5]{Dyda2}, but we give the proof in the general case for the convenience of the reader.
\begin{proof}[Proof of Theorem \ref{p.maz'ya}] Let us first assume that condition (A) holds. Let $u\in C_0(G)$ be such that $u(x)\geq 1$ for every point $x\in K$. By condition (A) we obtain \[
\int_K \omega(x)\,dx \leq \int_G \lvert u(x)\rvert^q\,\omega(x)\,dx \leq C_1 \bigg(\int_G
\int_G \frac{\lvert u(x)-u(y)\rvert^p}{\lvert x-y\rvert^{n+\delta p}}\, dy\, dx\bigg)^{q/p}\,. \] Taking infimum over all such functions $u$ we obtain condition (B) with $C_2=C_1$.
Now let us assume that condition (B) holds and let $u\in C_0(G)$. We write \[
E_k = \{x\in G \,:\, \lvert u(x)\rvert > 2^k \} \quad \text{and} \quad A_k = E_k \setminus E_{k+1}\,, k\in \Z\,. \] Let us note that \begin{equation}\label{e.decomp} G= \{x\in G\,:\, 0\le \lvert u(x)\rvert <\infty\} =\{x\in G \,:\, u(x)=0\}\cup \bigcup_{i\in \Z} A_i\,. \end{equation} By condition (B) \begin{equation*} \begin{split}
\int_G \lvert u(x)\rvert^q \omega(x)\,dx &\leq
\sum_{k\in \Z} 2^{(k+2)q} \int_{A_{k+1}} \omega(x)\,dx \\&\leq C_2 2^{2q} \sum_{k\in \Z} 2^{kq} \mathrm{cap}_{\delta ,p}(\overline{A}_{k+1}, G)^{q/p} \,. \end{split} \end{equation*} Let us define $u_k:G\to [0,1]$ by \[ u_k(x) = \begin{cases} 1, & \text{if $\lvert u(x)\rvert \geq 2^{k+1}$\,,}\\ \frac{\lvert u(x)\rvert}{2^k}-1 &\text{if $2^k < \lvert u(x)\rvert < 2^{k+1}$\,,}\\ 0, & \text{if $\lvert u(x)\rvert \leq 2^k$\,.} \end{cases} \] Then $u_k \in C_0(G)$ and $u_k(x)=1$ if $x\in \overline{E}_{k+1}$. We note that $\overline{A}_{k+1} \subset\overline{E}_{k+1}$. Thus we may take $u_k$ as a test function for the capacity. Let us write $F=\{x\in G \,:\, u(x)=0\}$. By \eqref{e.decomp}, \begin{align*} &\mathrm{cap}_{\delta ,p}(\overline{A}_{k+1}, G) \leq \int_G \int_G \frac{\lvert u_k(x)-u_k(y)\rvert^p}{\lvert x-y\rvert^{n+\delta p}}\,dy\,dx \nonumber\\ &\leq
2 \sum_{i\leq k} \sum_{j\geq k} \int_{A_i} \int_{A_j} \frac{\lvert u_k(x)-u_k(y)\rvert ^p}{\lvert x-y\rvert^{n+\delta p}}\,dy\,dx + 2 \sum_{j\geq k} \int_{F} \int_{A_j} \frac{\lvert u_k(x)-u_k(y)\rvert ^p}{\lvert x-y\rvert^{n+\delta p}}\,dy\,dx\,. \end{align*} The inequality \begin{equation*} \lvert u_k(x)-u_k(y)\rvert \leq 2\cdot 2^{-j} \lvert u(x)-u(y)\rvert \end{equation*} holds whenever $(x,y)\in A_i\times A_j$ and $i\leq k \leq j$. Namely: $\lvert u_k(x)-u_k(y)\rvert \leq 2^{-k} \lvert u(x)-u(y)\rvert$ if $x,y\in G$. If $x\in A_i$ and $y\in A_j$ with $i+2\leq j$, then $ \lvert u(x)-u(y)\rvert \geq \lvert u(y)\rvert -\lvert u(x)\rvert \geq 2^{j-1}$. Hence $\lvert u_k(x) - u_k(y)\rvert \leq 1 \leq 2\cdot 2^{-j} \lvert u(x)-u(y)\rvert$. \noindent Thus, since $q\ge p$, \begin{align*} \sum_{k\in\Z} 2^{kq}\bigg( \sum_{i\leq k} &\sum_{j\geq k} \int_{A_i} \int_{A_j} \frac{\lvert u_k(x)-u_k(y)\rvert ^p}{\lvert x-y\rvert^{n+\delta p}}\,dy\,dx\bigg)^{q/p} \\ &\le 2^q\bigg( \sum_{k\in\Z} \sum_{i\leq k} \sum_{j\geq k} 2^{(k-j)p} \int_{A_i} \int_{A_j} \frac{\lvert u(x)-u(y)\rvert ^p}{\lvert x-y\rvert^{n+\delta p}}\,dy\,dx\bigg)\,.
\end{align*} By proceeding in a similar way as before we obtain that \begin{align*} \sum_{k\in\Z} 2^{kq}\bigg( &\sum_{j\geq k} \int_{F} \int_{A_j} \frac{\lvert u_k(x)-u_k(y)\rvert ^p}{\lvert x-y\rvert^{n+\delta p}}\,dy\,dx\bigg)^{q/p} \\ &\le 2^q\bigg( \sum_{k\in\Z} \sum_{j\geq k} 2^{(k-j)p} \int_{F} \int_{A_j} \frac{\lvert u(x)-u(y)\rvert ^p}{\lvert x-y\rvert^{n+\delta p}}\,dy\,dx\bigg)^{q/p}\,. \end{align*}
Using the sum of the geometric series $\sum_{k=i}^j 2^{(k-j)p} < \sum_{k=-\infty}^j 2^{(k-j)p} \le \frac{1}{1-2^{-p}}$ and changing the order of summations gives \[ \int_G \lvert u(x)\rvert^q\omega(x)\,dx
\leq \frac{C_2 2^{3q+2q/p}}{(1-2^{-p})^{q/p}} \bigg(\int_G \int_G \frac{\lvert u(x)-u(y)\rvert^p}{\lvert x-y\rvert^{n+\delta p}}\, dy\, dx\bigg)^{q/p}\,. \] Thus condition (A) is true with $C_1 = C_2 2^{3q+2q/p} (1-2^{-p})^{-q/p}$. \end{proof}
The following lemma is an extension of \cite[Proposition 6]{Dyda2}.
\begin{lem}\label{p.necessary} Let $0<\delta <1$ and $0< p\le q<\infty$ be given. Suppose that the fractional $(\delta ,q,p)$-Hardy inequality \eqref{e.hardy} holds in a proper open set $G$ in $\R^n$ with a constant $C_1>0$. Then there is a constant $N=N(C_1,n,\delta,q,p)>0$ such that the inequality \begin{equation}\label{e.wanted} \sum_{Q\in\mathcal{W}(G)} \mathrm{cap}_{\delta ,p}(K\cap Q,G)^{q/p}\le N^{q/p}\, \mathrm{cap}_{\delta,p}(K,G)^{q/p} \end{equation} holds for every compact set $K$ in $G$. \end{lem}
\begin{proof} If $Q\in\mathcal{W}(G)$ we write $\widehat{Q}=\tfrac {17}{16}Q$ and $Q^*= \tfrac{9}{8}Q$. We recall that the side lengths of these cubes are comparable to their distances from $\partial G$.
Let us fix a compact set $K$ in $G$ and $u\in C_0(G)$ such that $u(x)\ge 1$ for each $x\in K$. For every $Q\in\mathcal{W}(G)$ we let $\varphi_Q$ be a smooth function such that $\lvert \nabla \varphi_Q\rvert \le C\ell(Q)^{-1}$ and $\chi_{Q}\le \varphi_Q\le \chi_{\widehat{Q}}$. Then, $u_Q:= u\varphi_Q$ is an admissible test function for $\mathrm{cap}_{\delta ,p}(K\cap Q,G)$. Hence, we can estimate the left hand side of inequality \eqref{e.wanted} by \begin{align*} &\sum_{Q\in\mathcal{W}(G)} \bigg(\int_G \int_G \frac{\lvert u_Q(x)-u_Q(y)\rvert^{p}}{\lvert x-y\rvert^{n+\delta p}}\,dy\,dx\bigg)^{q/p}\\ &\le C \sum_{Q\in\mathcal{W}(G)}\bigg( \int_{\widehat{Q}} \frac{\lvert u_Q(x)\rvert^p}{\mathrm{dist}(x,\partial G)^{\delta p}}\,dx+ \int_{Q^*} \int_{Q^*}\frac{\lvert u_Q(x)-u_Q(y)\rvert^{p}}{\lvert x-y\rvert^{n+\delta p}}\,dy\,dx
\bigg)^{q/p}\\
&\le C\sum_{Q\in\mathcal{W}(G)} \bigg\{ \bigg( \int_{\widehat{Q}} \frac{\lvert u_Q(x)\rvert^p}{\mathrm{dist}(x,\partial G)^{\delta p}}\,dx\bigg)^{q/p}
+\bigg(\int_{Q^*} \int_{Q^*}\frac{\lvert u_Q(x)-u_Q(y)\rvert^{p}}{\lvert x-y\rvert^{n+\delta p}}\,dy\,dx
\bigg)^{q/p} \bigg\}\,. \end{align*} Since $\lvert u_Q\rvert \le \lvert u\rvert$ and $\sum_{Q} \chi_{\widehat{Q}}\le C$, we may apply H\"older's inequality with $(q/p,q/(q-p))$ and the $(\delta ,q,p)$-Hardy inequality \eqref{e.hardy} to obtain \begin{equation}\label{e.etehd} \begin{split} \sum_{Q\in\mathcal{W}(G)}\bigg(\int_{\widehat{Q}} \frac{\lvert u_Q(x)\rvert^p}{\mathrm{dist}(x,\partial G)^{\delta p}}\,dx\bigg)^{q/p}
&\le C \sum_{Q\in\mathcal{W}(G)}\int_{\widehat{Q}} \frac{\lvert u(x)\rvert^q}{\mathrm{dist}(x,\partial G)^{q(\delta +n(1/q-1/p))}}\,dx\\ &\le C \bigg(\int_G\int_G \frac{\lvert u(x)-u(y)\rvert^p}{\lvert x-y\rvert^{n+\delta p}}\,dy\,dx\bigg)^{q/p} =C \lvert u\rvert_{W^{\delta,p}(G)}^{q}\,. \end{split} \end{equation} We fix $x,y\in G$ to estimate the remaining series. The following pointwise estimates will be useful, \begin{align*} \lvert u_Q(x)-u_Q(y)\rvert &\le \lvert u(x)\rvert \lvert \varphi_Q(x)-\varphi_Q(y)\rvert + \lvert u(x)-u(y)\rvert \varphi_Q(y)\\ &\le C\cdot \ell(Q)^{-1}\cdot\lvert u(x)\rvert\cdot \lvert x-y\rvert + \lvert u(x)-u(y)\rvert\,. \end{align*} Namely, since $\sum_{Q\in\mathcal{W}(G)} \chi_{Q^*}\le C\chi_G$ and $q\ge p$, we find that \begin{align*}
\sum_{Q\in\mathcal{W}(G)} \bigg(\int_{Q^*} \int_{Q^*}\frac{\lvert u(x)-u(y)\rvert^{p}}{\lvert x-y\rvert^{n+\delta p}}\,dy\,dx\bigg)^{q/p} \le C \lvert u\rvert_{W^{\delta,p}(G)}^{q}\,. \end{align*} By recalling that $0<\delta <1$ and by estimating as in \eqref{e.etehd} we obtain \begin{align*} &\sum_{Q\in\mathcal{W}(G)} \bigg( \ell(Q)^{-p}\int_{Q^*} \lvert u(x)\rvert ^p \int_{Q^*}\frac{\lvert x-y\rvert^p}{\lvert x-y\rvert^{n+\delta p}}\,dy\,dx\bigg)^{q/p}\\ &\le C\sum_{Q\in\mathcal{W}(G)} \bigg(\ell(Q)^{-\delta p}\int_{Q^*} \lvert u(x)\rvert ^p \,dx\bigg)^{q/p}\\ &\le C\sum_{Q\in\mathcal{W}(G)} \bigg(\int_{Q^*} \frac{\lvert u(x)\rvert ^p}{\mathrm{dist}(x,\partial G)^{\delta p}}\, dx\bigg)^{q/p}\le C \lvert u\rvert_{W^{\delta,p}(G)}^q\,. \end{align*} By collecting the estimates and taking the infimum over all admissible $u$ for $\mathrm{cap}_{\delta,p}(K,G)$ we complete the proof. \end{proof}
Now we are able to give the proof for Theorem \ref{t.hardy_c}.
\begin{proof}[Proof of Theorem \ref{t.hardy_c}] The implication from (A) to (B) is a consequence of Lemma \ref{p.necessary}. Let us then assume that condition (B) holds. In order to have
inequality \eqref{e.hardy} in $D$, by Theorem~\ref{t.maz'ya} it is enough to prove that there is a constant $C=C(\delta,p,n,c,N)>0$ such that inequality \begin{equation}\label{e.maz'ya_app} \int_K \mathrm{dist}(x,\partial D)^{-q(\delta+n(1/q-1/p))}\,dx \le C\,\mathrm{cap}_{\delta,p}(K,D)^{q/p} \end{equation} holds for every compact set $K$ in $D$. Let us fix a compact set $K$ in $D$. We consider a variation of inequality \eqref{e.maz'ya_app}: there is a constant $C=C(\delta,p,n,c)>0$ such that the inequality
\begin{equation}\label{e.cap_est} \bigg( \int_{K\cap Q}\qopname\relax o{dist}(x,\partial D)^{-q(\delta+n(1/q-1/p))}\,dx\bigg)^{1/q} \le C\, \mathrm{cap}_{\delta,p}(K\cap Q,D)^{1/p}
\end{equation} holds for every Whitney cube $Q\in\mathcal{W}(D)$. To prove inequality \eqref{e.cap_est} we let $u\in C_0(D)$ be a test function such that $u(x)\ge 1$ for every $x\in K\cap Q$. By the properties of Whitney cubes and Theorem \ref{t.main_emb} we estimate the left hand side of inequality \eqref{e.cap_est} \begin{align*}
C \lvert K\cap Q\rvert^{1/q} \lvert Q\rvert^{-(\delta+n(1/q-1/p))/n} &\le C \lvert K\cap Q\rvert^{1/q-(\delta+n(1/q-1/p))/n}\\ &\le C\lVert u\rVert_{L^{np/(n-\delta p)}(D)} \\&\le C\lvert u\rvert_{W^{\delta,p}(D)}\,. \end{align*} Inequality \eqref{e.cap_est} follows when we take the infimum over all admissible functions $u$ for the capacity $\mathrm{cap}_{\delta,p}(K\cap Q,D)$.
We may now finish the proof by using inequality \eqref{e.cap_est} and condition (B) \begin{align*} \int_K \mathrm{dist}(x,\partial D)^{-q(\delta+n(1/q-1/p))}\,dx &=\sum_{Q\in\mathcal{W}(D)} \int_{K\cap Q} \mathrm{dist}(x,\partial D)^{-q(\delta+n(1/q-1/p))}\,dx \\&\le C \sum_{Q\in\mathcal{W}(D)} \mathrm{cap}_{\delta,p}(K\cap Q,D)^{q/p} \\& \le CN^{q/p} \mathrm{cap}_{\delta,p}( K,D)^{q/p}\,, \end{align*} where $C=C(\delta,p,n,c)$. The proof is complete. \end{proof}
\end{document} | arXiv |
Fitting's theorem
Fitting's theorem is a mathematical theorem proved by Hans Fitting. It can be stated as follows:
If M and N are nilpotent normal subgroups of a group G, then their product MN is also a nilpotent normal subgroup of G; if, moreover, M is nilpotent of class m and N is nilpotent of class n, then MN is nilpotent of class at most m + n.
By induction it follows also that the subgroup generated by a finite collection of nilpotent normal subgroups is nilpotent. This can be used to show that the Fitting subgroup of certain types of groups (including all finite groups) is nilpotent. However, a subgroup generated by an infinite collection of nilpotent normal subgroups need not be nilpotent.
Order-theoretic statement
In terms of order theory, (part of) Fitting's theorem can be stated as:
The set of nilpotent normal subgroups form a lattice of subgroups.
Thus the nilpotent normal subgroups of a finite group also form a bounded lattice, and have a top element, the Fitting subgroup.
However, nilpotent normal subgroups do not in general form a complete lattice, as a subgroup generated by an infinite collection of nilpotent normal subgroups need not be nilpotent, though it will be normal. The join of all nilpotent normal subgroups is still defined as the Fitting subgroup, but it need not be nilpotent.
External links
• Fitting's Theorem at PlanetMath.
| Wikipedia |
\begin{definition}[Definition:Minkowski Functional]
Let $E$ be a vector space over $\R$.
A functional $p: E \to \R$ is called a '''Minkowski functional''' {{{iff}} it satisfies:
{{begin-axiom}}
{{axiom | n = 1
| q = \forall x \in E, \forall \lambda \in \R_{>0}
| ml= \map p {\lambda x}
| mo= =
| mr= \lambda \map p x
| rc= that is, $p$ is positive homogeneous
}}
{{axiom | n = 2
| q = \forall x, y \in E
| ml= \map p {x + y}
| mo= \le
| mr= \map p x + \map p y
| rc= that is, $p$ is sub-additive
}}
{{end-axiom}}
{{NamedforDef|Hermann Minkowski|cat = Minkowski}}
Category:Definitions/Functional Analysis
Category:Definitions/Topology
\end{definition} | ProofWiki |
Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.
Genetically encoded cell-death indicators (GEDI) to detect an early irreversible commitment to neurodegeneration
Jeremy W. Linsley ORCID: orcid.org/0000-0001-6464-87281,
Kevan Shah ORCID: orcid.org/0000-0002-6284-20301,
Nicholas Castello1,
Michelle Chan1,
Dominik Haddad2,
Zak Doric ORCID: orcid.org/0000-0003-2608-41962,3,
Shijie Wang1,
Wiktoria Leks1,
Jay Mancini1,
Viral Oza1,
Ashkan Javaherian1,
Ken Nakamura ORCID: orcid.org/0000-0002-9192-182X2,4,5,
David Kokel6,7 &
Steven Finkbeiner ORCID: orcid.org/0000-0002-3480-394X1,8
Nature Communications volume 12, Article number: 5284 (2021) Cite this article
Cell death is a critical process that occurs normally in health and disease. However, its study is limited due to available technologies that only detect very late stages in the process or specific death mechanisms. Here, we report the development of a family of fluorescent biosensors called genetically encoded death indicators (GEDIs). GEDIs specifically detect an intracellular Ca2+ level that cells achieve early in the cell death process and that marks a stage at which cells are irreversibly committed to die. The time-resolved nature of a GEDI delineates a binary demarcation of cell life and death in real time, reformulating the definition of cell death. We demonstrate that GEDIs acutely and accurately report death of rodent and human neurons in vitro, and show that GEDIs enable an automated imaging platform for single cell detection of neuronal death in vivo in zebrafish larvae. With a quantitative pseudo-ratiometric signal, GEDIs facilitate high-throughput analysis of cell death in time-lapse imaging analysis, providing the necessary resolution and scale to identify early factors leading to cell death in studies of neurodegeneration.
Neurodegenerative diseases such as Parkinson's disease (PD)1,2, Huntington's disease (HD)3,4,5,6,7, frontotemporal dementia (FTD), Alzheimer's disease (AD), and amyotrophic lateral sclerosis (ALS)8,9 are characterized by progressive neuronal dysfunction and death, leading to a deterioration of cognitive, behavioral or motor functions. In some cases, neuronal death itself is a better correlate of clinical symptoms than other pathological hallmarks of disease such as Lewy bodies in PD10, or β-Amyloid in AD11, and can be used to effectively characterize the relationship of an associated disease phenotype with degenerative pathology12,13. Using neuronal death as a consistent and unequivocal endpoint, longitudinal single-cell analysis can be performed on model systems to reveal the antecedents and forestallments of cell death14,15. Together with statistical tools used in clinical trials that account for variability, stochasticity, and asynchronicity amongst individuals within a cohort, it is possible to regress premortem phenotypic markers of neurodegeneration16, determine which are beneficial, pathological, or incidental to degeneration, and quantify the magnitude of their contribution to fate. For instance, although inclusion bodies are a hallmark of disease in HD and Tar DNA binding protein 43 (TDP43) ALS, their presence appears to be more consistent with a coping mechanism rather than a causative factor, suggesting clinical intervention to inhibit inclusion body formation could be a misguided approach3,8. While there is an ongoing debate about the relative contribution of neuronal dysfunction prior to neuronal death to the clinical deficits that patients exhibit, it is clear that neuronal death marks an irreversible step in neurodegenerative disease. Thus, neuronal death is an important, disease-relevant phenotypic endpoint that is important to understanding neurodegeneration, characterizing the mechanisms of neurodegenerative disease, and developing novel therapeutics.
Nevertheless, precisely determining whether a particular neuron is alive, dead, or dying can be challenging, particularly in live-imaging studies. Vital dyes, stains, and indicators have been developed to selectively label live or dead cells and neurons in culture17, but the onset of these signals may be delayed until long after a neuron has shown obvious signs of degeneration12. Additionally, long-term exposure to exogenous dyes can increase the risk of accumulated exposure toxicity, negating their ability to noninvasively provide information on cell death. Many assays can distinguish between cell death pathways such as apoptosis or necrosis13 and are conducive to longitudinal imaging18,19, but these typically require a priori knowledge of which cell death pathway is relevant, limiting their utility in neurodegenerative disease, which often involves a spectrum of neuronal death mechanisms20. Moreover, there is often high interconnectivity of signaling molecules across cell death pathways, and cells that begin to die by one cell death pathway may resort to a different one if the original pathway is blocked21, which can confound analyses. Furthermore, the interpretation and accuracy of apoptotic markers can vary based on the specific cellular system, and some are associated with reversible processes, meaning multiple assays must be used in parallel to unambiguously characterize the precise extent of death within a sample13,21. For these reasons, an early and sensitive cell-death pathway agnostic marker is often necessary to give a complete and unbiased readout of cell death. In live-imaging experiments, the loss of fluorescence of transfected neurons, indicating the rupture of the plasma membrane, has been shown to clearly mark neuronal death1,3, but fluorescent debris often persists for days after initial morphological signs of death and decay occur, limiting the ability to identify the precise time of death or introducing human error in the scoring of neuronal death by morphology12. In summary, without a strict criterion of what constitutes a "point of no return" at which a neuron's fate is unambiguously sealed, investigation of the causative factors that precede cell death remains challenging.
Although dyes have been used to detect neuronal death in vivo22,23,24, the permeability of dyes throughout tissue is inconsistent, making quantification difficult. Genetically encoded fluorescent proteins have greatly facilitated the ability to track single neurons within culture3 and in tissue25 over time. Furthermore, genetic targeting allows labeling of specific cell subtypes, as well as simultaneous expression of other biosensors, perturbagens, or activators15. Some of the most commonly used biosensors in neuroscience are the genetically-encoded Ca2+ indicators (GECIs), including the yellow cameleons and GCaMPs/pericams26,27. Based on the fusion of circularly permuted fluorescent proteins such as GFP with the Ca2+ binding M13-calmodulin domain, GECIs are relatively bright biosensors with low toxicity in neurons28. GECIs are used to detect either relative or absolute Ca2+ levels or neuronal circuit activity within neurons in culture29, in tissue30, within immobilized animals in virtual environments31,32, and even within freely moving animals33,34. The category of GECIs has been diversified and further optimized through the use of alternate fusion proteins to the M13-calmodulin domain as well as targeted mutagenesis35,36. Recently, Suzuki et al. engineered endoplasmic reticulum (ER)-targeted calcium-measuring organelle-entrapped protein indicators (CEPIAer) variants that emit either green, red, or blue/green fluorescence to specifically detect Ca2+ release events from the ER36. In contrast to previously developed ER Ca2+ indicator dyes, CEPIAs and recently engineered derivatives37 can be genetically and subcellularly targeted and are capable of long-term imaging over the lifetime of the neuron, enabling measurement of the full range of calcium dynamics within single neurons over time.
Here, we introduce a class of GECIs for the detection of cell death in neurons that we call genetically encoded death indicators (GEDIs). We show that GEDIs can robustly indicate the moment when a neuron's ability to maintain Ca2+ homeostasis is lost and cannot be restored, providing an earlier and more acute demarcation of the moment of death in a degenerating neuron than previously possible. In combination with a second fluorescent protein fused with a self-cleaving P2a peptide, pseudo-ratiometric GEDIs are easily quantifiable in high-throughput, give a highly reproducible signal, and are amenable to long-term imaging. GEDIs can also be targeted to specific neuronal subtypes for imaging in vivo. These data establish GEDIs as important tools for studying the time course of neurodegeneration, providing previously unobtainable delimitation and clarity to the time course of cell death.
Development of genetically encoded death indicators
GECIs such as GCaMP6f have been engineered to increase fluorescence in response to fluctuations in the range of cytosolic Ca2+ concentrations that occur during neuronal firing (Fig. 1A). The CEPIA GECIs have been engineered with elevated Kd to detect Ca2+ transients in organelles such as the ER or mitochondria that contain higher Ca2+ (Fig. 1A)36. We reasoned that removing the ER retention signals and allowing CEPIA variants to localize to the cytosol would render a GECI that was not responsive to activity-based Ca2+ transients, but that would increase in fluorescence intensity when cytosolic Ca2+ levels approached those of intracellular organelles or the extracellular milieu, which would constitute a catastrophic event for the neuron. We named these reengineered indicators GEDIs. In rat cortical primary neurons, brief electrical field stimulation (3 s at 30 Hz) increased fluorescence within cells expressing GCaMP6f, but not in those expressing the red GEDI variant (RGEDI) (Fig. 1B–D). The addition of NaN3, to a known cytotoxin which induces neuronal death, caused increased fluorescence in cells expressing either GCaMP6f or RGEDI (Fig. 1B–D). The peak GCaMP6f fluorescence response after stimulation was nearly identical to the fluorescence response to NaN3 treatment; in contrast, the ratio of stimulation to death response in RGEDI expressing cells was close to 0, indicating that the RGEDI sensor preferentially responds to death (Fig. 1E). Removal of extracellular Ca2+ abrogated the fluorescence response to NaN3 treatment, indicating the primary increase in cytosolic Ca2+ required influx from the extracellular space (Supplementary Fig. 1). To further optimize the RGEDI construct, we appended sequence encoding a porcine teschovirus-1 2a (P2a) "self-cleaving peptide" and EGFP, allowing normalization of the RGEDI signal to EGFP expression (GEDI ratio). This facilitated simple detection of the moment of neuronal death based on a cell's color change when the green and red channels are overlaid (Fig. 1F, G).
Fig. 1: GEDI detects the death of neurons.
A Relative fluorescence of GECI's GCaMP6f79 and RCEPIA36/RGEDI across Ca2+ concentrations present in the cytosol, endoplasmic reticulum, and extracellular milieu modeled from previously reported Hill coefficients and Kd values. B Representative fluorescence images of rat primary cortical neurons transfected with GCaMP6f or RGEDI at baseline, during 30 Hz × 3 s field stimulation, or after 2% NaN3 treatment, scale = 10 μm. The experiment was repeated six times for each condition with similar results. C, D Representative trace of the time course of standardized ΔF/F fluorescence after 30 Hz × 3 s stimulation (open arrowhead), and after NaN3 treatment (black arrowhead) of GCaMP6f (C) and RGEDI (D) expressing neurons. E To determine whether RGEDI signals were specific for cell death and not confounded by physiological Ca2+ transients, the ratio of the maximum signal from electrical stimulation to toxin treatment per neuron are shown, demonstrating that GEDI does not respond to physiological Ca2+ transients (n = 6 cells/group, 1 cell per coverslip, *** Unpaired, two-tailed T-test, p < 0.0001). Error bars represent SEM. F Design of RGEDI-P2a-EGFP cassette for pseudo-ratiometric expression in neurons. G Illustration of color change in red:green image overlay expected in a live versus dead neuron. Live neurons have EGFP (green) and basal RGEDI (red) fluorescence (overlaid as yellow) within the soma of the neuron, surrounded by green fluorescence that expands through the neurites. Dead neurons display yellow fluorescence throughout, with edges of red fluorescence around the soma and throughout degenerating neurites as extracellular Ca2+ permeates the membrane (arrows). H Time course images of rat primary cortical neurons expressing RGEDI-P2a-EGFP at 24–39 h post transfection (hpt). Neuron 1 shows characteristic morphology features of life through 27 hpt at which time it also shows elevated GEDI ratio (yellow asterisk), followed by loss of fluorescence at 39 hpt. Neuron 2 remains alive through time course, scale = 20 μm. Color scales are annotated in arbitrary units for each color channel. I Quantification of change in GEDI ratio of Neurons 1 and 2 in (H). J Quantification of GEDI ratio in rat primary cortical neurons before (blue dots) and from a separate well 5 min after NaN3-induced neuronal death (green dots = live, red dots = dead). The Black dotted line represents the calculated GEDI threshold. (*** One-tailed T-test p < 0.0001). K Quantification and classification of death in neuronal cultures at 24 and 48 h after co-transfection of RGEDI-P2a-EGFP and an N-terminal exon 1 fragment of Huntingtin, the protein that causes Huntington's disease, with a disease-associated expansion of the polyglutamine stretch (HttEx1Q97) to induce neuronal death using the derived GEDI threshold from (J) to define death. Independently, neurons were scored as dead (red), or live (green) by eye using EGFP at 24, 48, and 72 post transfection. Neurons that were classified as live by eye but above the GEDI threshold and classified as dead at the subsequent time point were called human errors (black).
To assess the ability of RGEDI to detect death in live-cell imaging, we used automated longitudinal microscopy3 to image individual neurons transfected with hSyn1:RGEDI-P2a-EGFP repeatedly at 3 h intervals (Fig. 1H, I). Neurons that died over the course of imaging were marked by clear fragmentation of morphology in the EGFP channel, followed by the disappearance of the debris3,38,39. Once a cell died, the increase in GEDI ratio remained stable over the course of imaging until the disappearance of the debris (Fig. 1H, I). In some cases, an increase in the GEDI ratio preceded obvious morphology changes (Fig. 1H). GEDI signal correlated with and often preceded standard markers for neuronal death such as TUNEL40, ethidium homodimer D1 (EthD1)41, propidium iodide (PI)42, or the human curation of the morphology3 (Supplementary Fig. 2). GEDI signal also preceded markers of apoptotic death such as Caspase3/7 signal (Supplementary Fig. 3). Due to the large separation of the GEDI ratio between live and dead neurons, we established a formal threshold of death for the GEDI ratio that could be used to quantify the amount of cell death in high throughput. Rat primary cortical neurons were transfected with hSyn1:RGEDI-P2a-EGFP in a 96-well plate and the GEDI ratio was derived from each well before and after a subset was exposed NaN3 (Fig. 1J). After 5 min, all neurons exposed to NaN3 showed an increased GEDI ratio compared to neurons before treatment and those not exposed to NaN3 (Fig. 1J). From these data, a GEDI ratio corresponding to the threshold of death (GEDI threshold) was calculated according to Eq. 1 (see Methods).
Automated microscopy was then performed at 24 h intervals for four days on the remaining 94 wells of the plate, and the previously derived GEDI threshold was used to assess the spontaneous neuronal death of hSyn1:RGEDI-P2a-EGFP-transfected neurons (Fig. 1K). In parallel, neuronal death was assessed by manual curation based on the abrupt loss of neuronal fluorescence over time. All neurons identified as live by manual curation had a GEDI ratio below the GEDI threshold, and most neurons identified as dead by manual curation had a GEDI ratio above the GEDI threshold (Fig. 1K). The few neurons classified as live by manual curation that contained a GEDI ratio above the threshold were recognized in hindsight to be difficult to classify based on morphology alone. Furthermore, each of these neurons could later be unequivocally classified as dead due to loss or fragmented pattern of fluorescence at the next imaged time point (Supplementary Fig. 4A′, A″), suggesting an error or inability of humans to correctly classify, and demonstrating GEDI is a more acute and accurate means of classifying neuronal death than manual curation. To examine the possibility that a temporarily increased GEDI ratio signal may give a false positive death indication, a large longitudinal data set of time-lapse GEDI images containing 94,106 tracked neurons across multiple imaging conditions was generated (Supplementary Table 1). Across all 32 longitudinal experiments, a consistent GEDI threshold indicating cell death could be established (Supplementary Fig. 4B, C). Only 0.28% of all neurons exhibited a GEDI ratio above the threshold at a one-time point that subsequently decreased below the death threshold (Supplementary Fig. 4D). Upon closer examination of those 304 neurons, each was subsequently found to have an automated segmentation artifact, which distorted the ratio of RGEDI to GFP or a tracking artifact that confused objects, rather than decreased RGEDI signal in relation to GFP (Supplementary Fig. 4E′, E″). No neurons were found to "die twice," as would be indicated by two fluctuations in the GEDI ratio above the death threshold. These data show that the GEDI signal is unlikely to increase above the GEDI threshold in a neuron that is not dead. Maintaining Ca2+ homeostasis is thought to be important for all cell types, and GEDI signal also differentiated live and dead cells when expressed in HEK293 cells (Supplementary Fig. 5). Therefore, we conclude that the GEDI biosensor specifically signals an early and irreversible commitment to degeneration and death and can serve as ground truth for quantifying cell death.
Automated identification of toxin-resistant and sensitive subpopulations of neurons
We predicted that by combining GEDI and time-lapse imaging, we would be able to monitor a heterogeneous death process and identify time-resolved subpopulations of neurons with differing sensitivities. Glutamate is the most common neurotransmitter in the brain43, but glutamate excess occurs in neurodegenerative disease and has been shown to be toxic to specific subpopulations of neurons44. Glutamate toxicity induces either apoptosis44 or necrosis45, and a reliable death sensor capable of detecting either death type can facilitate an unbiased accounting of toxicity. In principle, RGEDI should be able to detect all cell death events, as it detects loss of Ca2+ homeostasis rather than a specific substrate of a cell death pathway. To assess glutamate toxicity, rat primary neurons were transfected with hSyn1:RGEDI-P2a-EGFP and followed with automated microscopy after exposure to different levels of glutamate (Fig. 2A–C). A GEDI threshold was used to define dead neurons, and Kaplan–Meier survival curves were generated for the time course of imaging (Fig. 2D). While over 90% of neurons died within 3 h of exposure to 0.1 mM or 1 mM glutamate, recognizable by the stark change in composite GEDI color of images of wells (Fig. 2A, B), sparse neurons resistant to those treatments could be identified by their low GEDI ratio that remained alive in the culture long after the initial wave of death. Some neurons remained alive to the end of the 108 h imaging window in the presence of glutamate (Fig. 2C, D).
Fig. 2: Detection and characterization of subpopulations of neurons resistant or sensitive to glutamate treatment using GEDI.
A Representative two-color overlay image from a well of rat primary cortical neurons expressing RGEDI-P2a-EGFP before treatment with 0.1 mM glutamate (Scale = 400 μm) and zoom-in of two individual neurons within the yellow box (Scale = 100 μm). Live neurons appear green/yellow, dead neurons appear yellow/red. B Representative two-color overlay image from neurons 3 h after treatment with 1 mM glutamate and enlargement of the same two neurons within the yellow box in (A). Scale = 100 μm. C Time-course images of a neuron before and after exposure to 0.1 mM glutamate that survives until 96 h post-treatment (Scale = 50 μm). Experiments A–C were repeated on 16 wells with similar results. D Kaplan–Meier plot of neuron survival after exposure to 1, 0.1, 0.01 mM, and 0 glutamate (n = 699, 585, 527, 476). E Linear regressions of decay of EGFP (left) and RGEDI (right) after neuronal death marked by GEDI signal above the threshold. F Slopes of decay of EGFP and RGEDI signals are not different (Mann–Whitney, two-tailed, ns not significant, p = 0.93, n = 5513). Horizontal lines in the violin plots represent quartiles and median.
It is known that GFP and RFP are differentially sensitive to lysosomal hydrolases due to the difference in pH stability of GFP and RFP, and that differential sensitivity has been exploited to develop tandem tag biosensors to measure autophagy46,47. Since Ca2+ is known to activate certain proteases48, we wondered if differential degradation rates of RGEDI and EGFP fluorescence signals in dead neurons could cause the GEDI ratio to fluctuate and under- or overrepresent death. To investigate, we characterized the decay rate of EGFP and RGEDI signals after rapid death from glutamate toxicity. In dead neurons, the relative fluorescence of each protein decayed at equivalent rates (t1/2 = 20.45 h RGEDI, 20.73 h EGFP), indicating the activated GEDI ratio is a stable indicator of death across long time intervals of imaging during which neuronal debris remains present (Fig. 2E, F). These data suggest that a GEDI is a powerful tool to accurately identify live neurons within a culture in which extensive death has occurred.
Automated survival analysis of multiple neurodegenerative disease models in different species with GEDI
Neurodegenerative disease-related neuronal death is associated with a spectrum of death mechanisms including apoptosis, necrosis, excitotoxicity, and autophagic cell death20,49, necessitating the use of a death indicator of all types of cell death for effective and unbiased detection of total death. To test the effectiveness of GEDI across neurodegenerative disease models, hSyn1:RGEDI-P2a-EGFP was cotransfected into rat cortical primary neurons with pGW1:HttEx1-Q25 or pGW1:HttEx1-Q97, pCAGGs:α-synuclein, or pGW1:TDP43 to generate previously characterized overexpression models of HD3, PD39, and ALS or FTD50, respectively (Fig. 3A). Each model has been associated with multiple types of cell death to varying degrees40,51,52,53. In each model, neurons with characteristic yellow overlays of the RGEDI and EGFP channel could be detected at 24 h (Fig. 3A–A″). GEDI ratios for neurons in each model and control at each time point were quantified, and a GEDI threshold was calculated using a subset of neurons designated dead or live by manual curation (Fig. 3A, B). GEDI ratios from dead neurons in disease models were lower than GEDI ratios from their respective controls, likely due to the combined effects of reduced total exogenous protein expression observed in each disease model compared to control, and lower separation of GEDI ratio between live and dead neurons at lower RGEDI-P2a-EGFP expression levels (Supplementary Fig. 6). High expression of RGEDI-P2a-EGFP likely correlates with high expression of co-transfected disease-causing protein, leading neurons with high exogenous protein expression levels to die and disappear sooner, causing underrepresentation of high expression of RGEDI-P2a-EGFP in disease models (Supplementary Fig. 6). Nevertheless, a clear separation of populations of live and dead neurons could still be observed in each case (Fig. 3B–B″). Using the labels generated from the GEDI threshold a cumulative risk-of-death (CRD), a statistical measure of survival used in clinical studies54, was generated showing significant toxicity of each model compared to controls as previously reported (Fig. 3C). This showed that GEDI can be used to report toxicity over time across a variety of neurodegenerative disease models.
Fig. 3: Detection of death in neurodegenerative disease models with GEDI.
A Representative two-color overlay images of rat primary cortical neuron at 24 and 48 h after transfection co-expressing RGEDI-P2a-EGFP and HttEx1Q97 (A′), α-synuclein (A″), or TDP43 (A″′). The GEDI ratio identifies each neuron as live at 24, but dead at 48 h post transfection. Scale = 25 μm. B Quantification of GEDI ratio during longitudinal imaging across 168 h of live culture of neurons expressing HttEx1Q97 (B), α-synuclein (B′), or TDP43 (B″) with GEDI thresholds at 0.05 for each data set. Dots are color-coded for time post imaging. C Cumulative risk-of-death of HttEx1Q97 (HR = 1.83, 95% CI = 1.67–2.01, Cox proportional hazard (CPH) *p < 0.001), HttEx1Q25 (HR = 1.07, 95% CI = 0.99–1.15, ns not significant, p = 0.08), α-synuclein (HR = 1.73, 95% CI = 1.58–1.89, CPH *p < 0.001), TDP43 (HR = 1.77, 95% CI = 1.6–1.94, CPH *p < 0.001), and RGEDI-P2a-EGFP alone (control) generated from GEDI ratio quantification and classification against the GEDI threshold. The number of neurons in Control = 1670, HttEx1Q25-CFP = 1333, HttEx1Q97-CFP = 668, TDP43 = 610, and α-synuclein = 743. D Representative time-lapse imaging of a control iPSC motor neuron expressing RGEDI-P2a-EGFP that survives throughout imaging. Scale = 25 μm. E Representative time-lapse imaging of a SOD1 D90A iPSC motor neuron that is dead by GEDI signal at 84 h of after transfection. Scale = 25 μm. F Quantification of the GEDI ratio of Control and SOD1 D90A neurons and the derived GEDI threshold at 0.05. G CRD plot of SOD1-D90A and Control (95% CI = 1.11–1.44, CPH p < 0.0001, number of neurons in Control = 714, and SOD1 D90A = 363).
Cell-based overexpression models of neurodegeneration can be difficult to interpret because protein expression above physiological levels can introduce artifacts, which could also affect GEDI quantification. Neurons derived from induced pluripotent stem cells (iPSC) have the advantage of maintaining the genomic variants of the patients from whom the cells came, facilitating the modeling of neurodegenerative diseases55. To test whether GEDI can detect death in an iPSC-derived model of the neurodegeneration in which the endogenous disease-causing protein is expressed at physiologically relevant levels, motor neurons (MNs) were derived from iPSCs from normal control patients and compared to neurons derived from patients with a D90A SOD1 mutation, which has been shown to cause ALS56. iPSC MNs generated from patients' fibroblasts that carry the D90A SOD1 mutation have been previously shown to model key pathologies associated with ALS, such as neurofilament-containing inclusions and axonal degeneration, though a clear survival phenotype using a CRD to evaluate toxicity over time has not been established57. Neurons were transfected with hSyn1:RGEDI-P2a-EGFP after 19 days of differentiation and imaged every 12 h with automated microscopy (Fig. 3D–F). The GEDI ratio was quantified and a GEDI threshold was derived and used to generate a CRD plot, which showed that SOD1-D90A–containing neurons exhibited an increased risk of death compared to controls, with a CRD of 1.26 (Fig. 3G). These data show that GEDI can be used to automatically detect neuronal death and derive CRDs from human neurons.
Development of an expanded family of GEDI sensors
To expand the applications of the GEDI biosensor, we tested other GEDI variants with alternative characteristics that could be useful in different experimental contexts. For example, reliance on green and red emission spectra for death detection with RGEDI-P2a-EGFP restricts the ability to concurrently image other biosensors whose spectra overlap, limiting the opportunity to investigate other co-variates of disease14. Accordingly, we engineered RGEDI-P2a-3xBFP, so that death can be reported during simultaneous imaging of green biosensors. RGEDI-P2a-3xBFP showed a significant increase in signal after death compared to RFP-P2a-EGFP and the rate of signal increase following exposure to NaN3 was not different between RGEDI-P2a-3xBFP and RGEDI-P2a-EGFP or GCaMP6f-P2a-mRuby58 (GEDI ratio = GCaMP6f/mRuby) (Fig. 4A–D ANOVA p = 0.44).
Fig. 4: Comparison of GEDI variants to detect neuronal death.
A–C Representative images of rat primary cortical neurons expressing RGEDI-P2a-3xBFP (A), mRuby-P2a-GCaMP6f (B), or RGEDI-P2a-EGFP (C), before, 5 min and 10 min after exposure to NaN3. D Quantification of the peak signal and response rate of signal increase (τ) from fitted nonlinear regressions of increases in fluorescence signals over time from variants of the GEDI biosensor: RGEDI-P2a-3xBFP (n = 23), mRuby-P2a-GCaMP6f (n = 52), RGEDI-P2a-EGFP (n = 41), GC150-P2a-mApple, RGEDI-NLS-P2a-EGFP-NLS, and GC150-NLS-P2a-mApple-NLS (n = 40), compared to EGFP-P2a-mApple (n = 18). Error bars represent SE, ANOVA Tukey's ****p < 0.0001; ns not significant, n values represent individual neurons sampled across at least three independent wells. E Relative fluorescence of GECI's GCaMP6f79 and RCEPIA36/RGEDI and GCaMP150ER37/GC150 across Ca2+ concentrations modeled from previously reported Hill coefficients and Kd values. F Representative images of rat primary cortical neurons expressing GC150-P2a-mApple, G GC150-NLS-P2a-mApple-NLS, and H RGEDI-NLS-P2a-EGFP-NLS before, 5 min, and 10 min after exposure to NaN3. Error bars represent SEM. Scale = 25 μm. Experiments in A–C and F–H were repeated at least 18 times with similar results.
Next, we tested a recently engineered ER Ca2+ sensor based on the GCaMP GECI called GCaMP6-150, which was recently reported to have an excellent dynamic range37 and higher affinity for Ca2+ than RGEDI (Fig. 4E). We generated a GEDI sensor based on the GCaMP6-150 template by removing the ER signaling peptides from the GCaMP6-150 cassette and combining it with a P2a peptide and mApple to generate GC150-P2a-mApple. As expected, cells expressing GC150-P2a-mApple showed a significantly increased GEDI ratio (GEDI ratio = GC150/mApple) after NaN3-induced death (Fig. 4D–F). Similar to RGEDI-P2a-EGFP, GC150-P2a-mApple did not increase in signal in response to field stimulation of 30 Hz (Supplementary Fig. 7), indicating GC150 signal only increases in response to irreversible high levels of Ca2+ but not normal Ca2+ transients that occur within neurons.
GECIs targeted to the nucleus have been shown to increase the resolution and duration of signal in whole-brain studies of zebrafish59. Therefore, we generated GEDI constructs with nuclear localization signals (NLS): GC150-NLS-P2a-mApple-NLS and RGEDI-NLS-P2a-EGFP-NLS (Fig. 4F, G). Each nuclear-localized GEDI showed a similar significant increase in GEDI ratio upon NaN3 treatment, corresponding to the death of neurons (Fig. 4D, G, H). The kinetics of the responses were not different between any of the GEDI variants (ANOVA p = 0.65), indicating a similar ability to detect death across imaging situations in which different versions of GEDI are needed (Fig. 4D). These data demonstrate that the GEDI approach offers an acute, versatile, and quantitative method to detect neuron death in time-lapse imaging.
GCaMP acutely reports death in vivo
Many zebrafish larvae models of neurodegeneration have been developed, in part to take advantage of their unique characteristics, including translucent skin and the ability to be immobilized for long periods of time, that make them amenable to live imaging60,61. However, it has not been possible to acutely detect neuronal death and characterize the preceding events in vivo with time-lapse imaging in these models, limiting the characterization of neuronal death to static snapshots of single time points62,63,64.
We sought to develop a platform to visualize neuronal death longitudinally in vivo with GEDI by adapting our automated four-dimensional (4D) longitudinal single-cell tracking microscopy platform25 to in toto longitudinal imaging of live zebrafish larvae over multiple days (Supplementary Fig. 8). Larvae at 72 h postfertilization were anesthetized in tricaine, immobilized in low melting point agarose in 96 well optical ZFplates (Diagnocine), where they can typically remain alive for 120 h as assayed by heartbeat. Automated confocal microscopy was used to repeatedly image each fish in three dimensions at specified intervals, generating 4D images of each fish in an array (Supplementary Fig. 8). To evoke neuronal death, the inducible cell ablation protein nitroreductase (NTR)65 was expressed in MNs using the mnx1 promoter66, and 10 μM of metronidazole (MTZ), a harmless prodrug that is activated by NTR, was added to the zebrafish media. By 24 h after the addition of MTZ, some MN axons became clumped and MN cell bodies looked fragmented compared to DMSO, yet no difference in MN axon area was detected (Supplementary Fig. 9A–C). By 48 h after the addition of MTZ, MN axons appeared to retract, and a reduction in axon area could be detected, yet somas and/or debris from the MNs remained in the spinal cord (Supplementary Fig. 9A, B). In contrast, non-immobilized mnx:Gal4; UAS:NTR-mCherry;UAS:EGFP zebrafish larvae became immotile upon incubation with MTZ for 24 h, swimming no more than larvae in which the neuromuscular junction has been blocked with botulinum toxin (UAS:BoTx-EGFP)67 (Supplementary Fig. 9D–F), indicating that 24 h incubation in MTZ is sufficient to functionally ablate MNs. Neuronal death at 24 h post MTZ was also confirmed with the use of PhiPhiLux G1D2a live fluorescent reporter of caspase-3-like activity23,68 (Supplementary Fig. 10A–C). These data suggest that an acute marker for neuronal death is required to monitor neurodegeneration in vivo that more accurately distinguishes live neurons from functionally ablated neurons.
GCaMP is commonly and widely used in zebrafish for studies of neuronal activity and functionality69, and transgenic lines with GCaMP expression in the nervous system are widely available, making it easy to apply to studies of neuronal death. GCaMP signal due to endogenous Ca2+ transients in MNs is not distinguishable from signal due to loss of membrane integrity associated with neuronal death (see Fig. 1). However, we found that GCaMP can be made to function as a GEDI within the zebrafish by blocking endogenous neuronal activity with the use of tricaine, an anesthetic that blocks voltage-gated channels in the nervous system70 even when the motor swimming circuit was activated by application of 0.1% acetic acid (AcOH), which stimulates swimming71 (Supplementary Fig. 11A–C). In contrast, with the use of a muscle contraction blocker 4-methyl-N-(phenylmethyl)benzenesulfonamide (BTS), which immobilizes larvae but does not block neuronal activity72, GCaMP7 calcium transients are still present (Supplementary Fig. 11C–E). GCaMP7 signal significantly increased from baseline in MNs after 24 h of MTZ application compared to those incubated in DMSO alone or those not expressing NTR (Supplemental 9G–I and Supplementary Fig. 12A, B). Similarly, cacnb1−/− mutants, which are immobile due to loss of skeletal muscle function but maintain normal MN activity73, also showed increased GCaMP7 signal in response to MTZ treatment without tricaine immobilization (Supplementary Fig. 12A–C). Thus, GCaMP7 can be used as a GEDI and an accurate measure of neuronal death in immobilized zebrafish.
Although GCaMP is effective at labeling neuronal death within tricaine-anesthetized larvae, tricaine application can have adverse effects on physiology74, The dampening of neuronal activity within the zebrafish can potentially complicate the interpretation of neurodegenerative disease models, especially those in which hyperexcitability is thought to be a disease-associated phenotype, such as AD, FTD, and ALS75,76. Thus, a true GEDI would be preferable to GCaMP because it would eliminate the need for immobilization in zebrafish imaging preparations in which CNS activity is preserved, such as fictive swimming assays72,73,77. We first tested the ability of RGEDI-P2a-EGFP to detect neuronal death in vivo by co-injecting DNA encoding neuroD:NTR-BFP and neuroD:RGEDI-P2a-EGFP at the 1-cell stage, and then using in toto live longitudinal imaging to track fluorescence of co-expressing neurons within the larval spinal cord after NTR-MTZ–mediated ablation beginning at 72 hpf (Fig. 5A). After 24 h of incubation in 10 μM MTZ, the morphology of neurons showed signs of degeneration including neurite retraction and loss of fluorescence, yet neurons co-expressing NTR-BFP and RGEDI-P2a-EGFP did not show increases in GEDI ratio (Fig. 5B, C), indicating RGEDI signal cannot distinguish live from dead neurons in this system.
Fig. 5: Single-cell tracking and specific detection of death within live zebrafish larvae with GC150 but not RGEDI.
A Cartoon schematic of zebrafish larvae showing the approximate location of sparsely labeled clusters of neurons. B Neurons in the zebrafish larvae spinal cord co-expressing NTR-BFP, EGFP, and RGEDI at 0, 24, and 48 h after mounting for automated imaging in MTZ. White arrows indicate neurons co-expressing NTR-BFP and EGFP. Scale = 50 μm. C Quantification of GEDI ratio in neurons co-expressing NTR-BFP and RGEDI-P2a-EGFP exposed to MTZ (n = 7) or DMSO (n = 5) showing no increase in GEDI signal (ANOVA Sidak's, ns not significant). D Neurons in the zebrafish larvae spinal cord co-expressing NTR-BFP, mApple, and GC150 at 0, 24, and 48 h after mounting for automated imaging in DMSO. White arrows indicate neurons with the fluorescence of BFP and mApple, asterisks indicate autofluorescence from pigment in larvae skin, scale = 20 μm. E Quantification of average GEDI ratio of neurons from zebrafish larvae incubated in DMSO (n = 7) or 10 μM MTZ (n = 17) over time showing an increase in GEDI ratio in neurons indicating neuronal death after 24 h in MTZ (ANOVA Tukey's ****p < 0.0001, ***p < 0.0005). F Neurons in zebrafish larvae spinal cord co-expressing NTR-BFP, mApple, and GC150 at 0, 24, and 48 h after mounting for automated imaging in 10 μM MTZ. White arrows indicate neurons with the fluorescence of BFP and mApple, yellow arrows indicate neurons with the fluorescence of BFP, mApple, and GC150, indicating neuronal death. Asterisks indicate autofluorescence from pigment in larvae skin, scale = 20 μm. Error bars represent SEM. The experiment was on 12 larvae with similar results.
We hypothesized that extracellular Ca2+ levels in vivo within the zebrafish larvae could be too low to reach the concentration required for RGEDI to optimally fluoresce. Therefore, we next tested if GC150, which has a higher binding affinity for Ca2+ (Fig. 4E), could better report neuronal death in vivo. Sporadic expression of GC150-P2a-mApple was generated by co-injection of DNA encoding neuroD: GC150-P2a-mApple with neuroD:NTR-BFP at the 1-cell stage. Live longitudinal imaging was performed on larvae incubated in either DMSO or 10 μM MTZ, and individual neurons within the brain expressing both mApple and BFP were tracked in 4D within the whole larvae (Fig. 5D–F). Larvae exposed to 10 μM MTZ showed increased GC150 signal by 24 h after MTZ extending to 48 h, while larvae exposed to DMSO alone did not show signs of neuronal death or increases in GC150 (Fig. 5D–F). With increased binding affinity compared to RGEDI, GC150 could potentially be more susceptible to detecting physiological Ca2+ transients within neurons, similar to GCaMP, which could confound its utility as a GEDI. To test whether GC150 increases in fluorescence during Ca2+ transients, GC150 was targeted to MNs by injection of mnx1:GC150-P2a-mApple. Larvae were immobilized in BTS, and no response of GC150 was detected after activation of the motor circuit, indicating GC150 signal does not increase in response to normal calcium transients within neurons (Supplementary Fig. 11F, G). These data indicate GC150 is suitable for in vivo detection of neuronal death in an un-anesthetized animal.
The study of neurodegenerative diseases has been hampered by an inability to distinguish populations of neurons destined to die from those that have already perished, which precludes the investigation of mechanisms that drive selective degeneration. Here, we characterize a biosensor family, GEDI, which is specifically tuned to detect neuronal death in longitudinal imaging studies, facilitating analysis of neurons in time points leading up to neuronal death. Using automated microscopy, we show that the analysis of GEDI is compatible with fully automated, single-cell survival analysis, which has been previously shown to be 100–1000 times more sensitive than population-based studies that rely on a single snapshot in time15. We believe this tool will lead to more precision in the discovery of the mechanisms of neurodegeneration and increase the throughput of quantitative studies to discover novel therapeutics.
Similar to other death-pathway agnostic indicators of cell death or viability, GEDI detects the loss of membrane integrity as a readout of cell death78. However, by uniquely measuring the Ca2+ permeability of the membrane, GEDI holds several key advantages for longitudinal imaging. For one, unlike death indicators that rely on DNA intercalation, such as EthD1, PI, or 4ʹ,6-diamidino-2-phenylindole (DAPI)/Höechst, GEDIs are nontoxic. Additionally, intercalating agents require a breach of both the plasma membrane and the nuclear membrane, which can be problematic in the stochastic process of degeneration and could result in delayed signal (Supplementary Fig. 2). In contrast, each indicator we used to engineer GEDIs binds Ca2+ in less than a second36,37,79, which is over 500× faster than the time course of death after NaN3 exposure (Fig. 4). GEDIs can also be combined with complementary labels specific for cell death pathways (Supplementary Figs. 2, 3), which could help resolve ambiguity in cell death pathway crosstalk such as when a cell resorts to a different cell death pathway when the primary pathway is blocked21, by providing more direct temporal linkage between death pathway signal and death. The stability of the fluorescence tags used in the GEDIs (Fig. 2D) enables the signal to be sampled at long time intervals such as every 24 h, a particularly important property during long-term imaging studies to minimize phototoxicity. These properties allowed us to use GEDIs to empirically determine the level of cytosolic Ca2+ associated with an irreversible fate (Fig. 1 and Supplementary Fig. 4), a property no other death indicator is capable of, to our knowledge. Furthermore, with the increased time resolution of death possible using GEDI, subpopulations of death-resistant neurons and the spread of neuronal death can now be imaged. For instance, reports of subpopulations of neurons displaying resistance to glutamate have been described in culture and in vivo80,81,82,83, yet each report has relied on dyes to characterize death at static time points, limiting the ability to resolve the time course and cell-to-cell transmission of excitotoxic injury on a single-cell level now possible with GEDI (Fig. 2A–D). While this study focused on the use of GEDI in degenerating neurons, it should be noted that most if not all cells maintain a concentration gradient of Ca2+, suggesting that these death indicators could be adapted for use to report cell death and spread of cell death in other cell or tissue types84.
Although much of what we know about the etiology of neurodegenerative disease has come from a two-dimensional culture of neurons, it is becoming increasingly clear that the progression of neurodegenerative disease is dependent on the relationship of neurons to their surrounding tissue. For instance, multiple neurodegenerative diseases are linked to abnormal circuitry75,76,85,86, and evidence suggests that propagation of neurodegenerative disease throughout the brain may proceed via the spread of pathogenic proteins87,88,89. To study such cell nonautonomous phenomena of neurodegeneration, an approach that integrates the three-dimensionality of tissue is required. Previously, we used organotypic slice culture to create a more tissue-like environment in which to study neurodegeneration over time, with an automated 4D imaging platform90. Here, in combination with GEDI, we apply this technology to a zebrafish model of neurodegeneration to study neuronal death in vivo. In contrast to organotypic slice culture models, which expose neurons to the stresses of brain dissection and culturing, in vivo 4D imaging of zebrafish larvae fully preserves the architecture of the brain. Zebrafish larvae studies can also be scaled up for high-throughput imaging screens91, and to our knowledge, our platform represents the first optimized for 4D longitudinal imaging of immobilized fish. This can be especially useful, as zebrafish larvae are a well-characterized behavioral model92,93, and 4D imaging can be used in parallel with behavioral analysis, providing an important behavioral correlate of neurodegeneration over time.
Our use of 4D imaging of zebrafish generated several important findings. First, we show that the loss of fluorescence from neurons expressing mCherry is not an acute indicator of neuronal death in vivo (Supplementary Fig. 9). One implication of this finding is that studies of neurodegeneration using fluorescent proteins in zebrafish could underestimate the time course of degeneration in the model, and behavioral characterization may be a more acute indicator of degeneration. Time-lapse imaging of zebrafish neurodegeneration is also complicated by the species' high neuronal regeneration capacity94 and the presence of scavenger cells such as microglia62, which could alter the apparent rate of death overtime, underscoring the necessity of an acute indicator to track neurons as they die over time. Second, our successful application of GCaMP7 to detect neuronal death in tricaine-immobilized zebrafish larvae means that commonly used GECIs, under neuronal paralytic conditions, can act as GEDIs. While GECIs prove to be a convenient tool for the study of neurodegeneration, this finding should also raise caution in the interpretation of GECI systems under conditions in which neuronal death occurs, as a chronic increase in GECI signal after death could be confused with normal Ca2+ transient activity. Finally, we showed that although the RGEDI construct could not be used to detect death in vivo, GC150 and GCaMP7 could. Due to the differences in Ca2+ binding affinity between the three indicators, these data suggest that free Ca2+ in the extracellular spaces in the brain of the zebrafish larvae is somewhat limited and may be insufficient to induce the RGEDI fluorescence upon cell death. Thus, the difference in the functionality of RGEDI in cultured neurons (Figs. 1–4) and in vivo in the zebrafish brain (Fig. 5 and Supplementary Fig. 9), could be due to the virtually unlimited supply of Ca2+ in culture medium compared to the brain, where extracellular Ca2+ can be limiting in times of high activity95 and in dense synaptic areas96. Interestingly, the limited extracellular Ca2+ in our in vivo assays raises the intriguing possibility that the sequestration of free calcium in dying neurons could be a previously unexplored mechanism of cognitive decline in neurodegeneration. Future studies using 4D modeling will be key to address this question.
With few disease-modifying therapies for neurodegenerative diseases available, there is a great need to understand disease mechanisms and etiology to develop new therapeutic targets12. Our studies using neurodegenerative disease models of PD, ALS/FTD, HD, glutamate toxicity, and in vivo neuronal ablation demonstrate the ability of GEDIs to acutely identify the moment of death in time-lapse imaging studies, allowing a unique time-resolved view of neurodegeneration. We believe the use of GEDIs will aid longitudinal single-cell analysis of neurons to complete our understanding of the underlying causes of neurodegeneration, and provide assays to help generate much-needed therapeutics.
Animals and culturing
All animal experiments complied with UCSF Institutional Animal Care and Use Committee (IACUC, protocols AN183829-02 and AN189188-01) regulations. Animals are housed in approved facilities with humidity regulated between 30 and 70%, temperature between 68 and 79 °F, and 12 h light/dark cycles. Primary mouse (C57BL/6) and rat (Long-Evans) cortical neurons were prepared at embryonic days 20–21. Brain cortices were dissected in dissociation medium (DM-81.8 mM Na2SO4, 30 mM K2SO4, 5.8 mM MgCl2, 0.25 mM CaCl2, 1 mM HEPES, 20 mM glucose, 0.001% phenol red, and 0.16 mM NaOH) with kynurenic acid (1 mM final) (DM/KY). KY solution was diluted from stock 10x KY solution (10 mM KY, 0.0025% phenol red, 5 mM HEPES, and 100 mM MgCl2). For cell disassociation, the cortices were treated with papain (100 U, Worthington Biochemical) for 10 min, followed by treatment with trypsin inhibitor solution (15 mg/mL trypsin inhibitor, Sigma) for 10 min. Both solutions were made up in DM/KY, sterile filtered, and kept in a 37 °C water bath. The cortices were then gently triturated to dissociate single neurons in Opti-MEM (Thermo Fisher Scientific) and glucose medium (20 mM). Neurons were plated into 96-well plates at a density of 100,000 cells/mL. Two hours after plating, the plating medium was replaced with a Neurobasal growth medium with 100X GlutaMAX, pen/strep, and B27 supplement (NB medium). Zebrafish embryos raised to 48 hpf were enzymatically dechorionated using 2 mg/ml Pronase (Protease, Type XIV, Sigma) for 20 min. For behavioral analysis, embryos at 72 and 96 hpf were lightly tapped on the tail with #2 forceps while recording a 10 s movie and analyzed for movement within the dish by binarizing the image and quantifying movement97. A list of all zebrafish lines used in this study are available in Supplementary Table 2.
Plasmids, transfections, toxins, dyes, injections, and transgenics
The mammalian expression constructs phSyn1:RGEDI-P2a-EGFP, phSyn1:RGEDI-P2a-3xTagBFP2, phSyn1:RGEDI-NLS-P2a-EGFP-NLS, and phSyn1:TDP43, phSyn1:empty were generated by synthesizing the insert into a pBluescript Sk + backbone. All constructs were verified by sequencing. At 4–5 days in vitro (DIV), rat cortical neurons were transfected with plasmids and Lipofectamine 2000 to achieve sparse labeling of neurons within each well. For survival analysis, each well of a 96-well plate containing primary rat cortical neurons was cotransfected with 0.15 μg of DNA of phSyn1:RGEDI-P2a-EGFP, phSyn1:RGEDI-P2a-3xTagBFP2, phSyn1:RGEDI-NLS-P2a-EGFP-NLS, and phSyn1:mRuby-P2a- GCaMP6f58, 0.1 μg of DNA of phsyn1:empty, pGW1-HttEx1Q97-mCerulean, pGW1- HttEx1Q25-mCerulean3, or 0.075 μg of DNA pCAGGS–α-synuclein39 or phSyn1:TDP43. l-Glutamic acid monosodium salt was diluted in NB media with 0.5% DMSO. 2% NaN3 (Sigma) was dissolved in NB media, or in PBS with or without Ca2+. For TUNEL staining, the Alexa647 Click-iT Assay for in situ apoptosis detection was used (Life Technologies). For Caspase detection, PhiPhiLux (Gentaur) and NucView 405 Caspase Substrate (Biotium) were used according to directions23,68,98. For testing other cell death indicators, mouse primary cortical neurons were isolated from embryonic 17-day pups and at 3 DIV, neurons were transfected with either 0.02 ug of hSyn1:RGEDI-P2a-EGFP or pGW1-EGFP. At 6 DIV, neurons transfected with RGEDI were treated with neural basal media while neurons transfected with EGFP were treated with 1 uM ethidium homodimer-1 Life Technologies) or 0.5 uM propidium iodide (Life Technologies). Dyes were allowed to incubate for 30 min before a pretreatment timepoint was taken. Neurons were exposed to 90 s UV light with a custom-built LED light box to induce cell death and cells were imaged every 4 h to track the signal of death indicators.
For experiments using electrical field stimulation, primary mouse neurons were cultured for 4 days before transfection with phSyn1:RGEDI-P2a-EGFP, phSyn1:mRuby-P2a-GCaMP6f, or phSyn1:GC150-P2a-mApple and then imaged at 8 DIV in Tyrode's medium (pH 7.4; in mm: 127 NaCl, 10 HEPES-NaOH, 2.5 KCl, 2 MgCl2, and 2 CaCl2, 30 mm glucose and 10 mm pyruvate) using a Nikon CFI Plan Apo ×40/0.95 numerical aperture air objective on a Nikon Ti-E inverted microscope with an iXon EMCCD camera (Andor Technology). Field stimulation (3 s at 30 Hz) was done using an A385 current isolator and an SYS-A310 Accupulser signal generator (World Precision Instruments). NaN3 was then directly injected into the imaging chamber to achieve rapid mixing. GCaMP6f and RGEDI fluorescence images were obtained using the following filters [490/20(ex),535/50 (em) and 543/22 (ex), 617/73 (em) respectively, Chroma] and regions of interest were drawn over cell bodies. The fold change (ΔF/F0) in fluorescence was calculated for each time point after background subtraction.
To generate zebrafish constructs, gateway recombination-based cloning was performed (Life Technologies) using the Tol2kit99. A pME entry clone was generated for RGEDI-P2a3xBFP and RGEDI-P2a-EGFP by subcloning from phSyn1:RGEDI-P2a3xBFP, phSyn1:RGEDI-P2a-EGFP, or phSyn1:GCaMP6f-P2a-mRuby using the pCR8/GW/TOPO TA cloning kit (Life Technologies). pME:GC150-P2a-mApple and pME-NTR-BFP were synthesized (Genscript) and cloned into pET-30a. Each was combined with p5E-neuroD100, P3E polyA and pDestTol2pA99, to generate neuroD: RGEDI-P2a-EGFP, neuroD: GC150-P2a-mApple, and neuroD:NTR-BFP. Embryos were injected at the 1-cell stage into the yolk with ~20 nl containing 100 ng/ul of each DNA. For stable transgenic neuroD:GC150-P2a-mApple, embryos were co-injected with neuroD:GC150-P2a-mApple and T7 in vitro transcribed tol2 mRNA (100 ng/ul) (Bio-Synthesis, Inc) into the cell at the 1-cell stage, and then raised to adulthood. Progeny were then screened by outcrossing to identify founder lines that were propagated and used for subsequent experiments. A list of all constructs used in this study are available in Supplementary Table 3.
In toto automated 4D high-content imaging of immobilized zebrafish
For imaging experiments, zebrafish embryos were dechorionated at 24 hpf, sorted for fluorescence, and put into 200 μM 1-phenyl 2-thiourea (PTU) E2 medium at 28.5 °C to inhibit melanogenesis101. At 72 hpf embryos were immobilized in 0.05% tricaine methanesulfonate (Sigma) or 50 μM BTS (Tocris) for an hour, then were loaded into wells of a ZFplate (Diagnocine) in 100 ul of media with paralytic using wide-bore tips and allowed to settle into the slits. Using a multichannel pipette, 100 ul of molten 1.5% low melting point agarose was loaded into each nozzle hole, and then 100 ul of agarose/paralytic mixture was immediately removed and agarose was allowed to solidify. Finally, liquid media was added to the top of each well containing paralytic with DMSO and/or with 10 uM MTZ (Sigma).
The automated spinning disk confocal imaging system was previously described25. Briefly, a custom system was used combining a Nikon microscope base (Nikon Ti-E), a Yokogawa spinning disk confocal (Yokogawa CSU-W1), an automated three-axis stage (Applied Scientific Instrumentation, MS-2500-Ti and PZ-2300), and modified custom software controlling Micromanager (version 2.0 gamma) allowing an automated return to the same location on the imaging plate for continual imaging of the same location in 3D space. Automation of the system was performed with Green Button Go (Biosero, Freemont). To accommodate zebrafish imaging at 28.5 °C rather than at the enclosure temperature of 37 °C for mammalian cell culture, a custom-made homeostatic Peltier cooling lid was designed and constructed (Physiotemp) to sit on top of the ZF plate.
Data analysis and quantification
Quantification of GEDI and morphology channel fluorescence intensity from 2D cultures was done using files obtained by automated imaging as previously described3,14,102. Files were processed using custom scripts running within a custom-built image processing Galaxy bioinformatics cluster25,103 that background subtracts, montages, fine-tunes alignment across time points of imaging, segments individual neurons, tracks segmented neurons over time, and then extracts intensity and feature information from each neuron into a csv file. Background subtraction was performed by subtracting the median intensity of each image and was required for the calculation of a translatable GEDI signal across data sets. Segmentation of neurons was targeted towards detection of the brightest area of the morphology of the neuron, usually the soma, that was larger than a minimal size threshold of 100 pixels. The GEDI ratio is sensitive to extent of segmentation of neurons, and errors in segmentation that include background can reduce the precision of the GEDI ratio obtained. A tight segmentation of neurons around the soma is desirable as segmentation of neurites frequently results in segmentation of multiple neurons at a time because of overlapping projections, and the dimmer and inconsistent morphology fluorescence signal extension throughout neurites. Tracking of neurons was performed by labeling an object as the same object at the next time point based on the proximity of the coordinates of the segmented mask to a segmented mask at the previous time point. Survival analysis was performed by defining the time of death as the point at which the GEDI ratio of a longitudinally imaged neuron exceeds the empirically calculated GEDI threshold, or the point at which point a tracked segmentation label is lost, which was performed using custom scripts written in R. The GEDI threshold was determined using the following equation:
$${{{{{\mathrm{GEDI}}}}}}\,{{{{{\mathrm{ratio}}}}}}\,{{{{{\mathrm{threshold}}}}}}= \; ([({{{{{\mathrm{mean}}}}}}\,{{{{{\mathrm{GEDI}}}}}}\,{{{{{\mathrm{ratio}}}}}}\,{{{{{\mathrm{dead}}}}}})-({{{{{\mathrm{mean}}}}}}\,{{{{{\mathrm{GEDI}}}}}}\,{{{{{\mathrm{ratio}}}}}}\,{{{{{\mathrm{live}}}}}})]\ast 0.25)\\ +[{{{{{\mathrm{mean}}}}}}\,{{{{{\mathrm{GEDI}}}}}}\,{{{{{\mathrm{ratio}}}}}}\,{{{{{\mathrm{live}}}}}}]$$
The survival package for R statistical software was used to construct Kaplan–Meier curves from the survival data based on the GEDI ratio, and survival functions were fit to these curves to derive cumulative survival and risk-of-death curves that describe the instantaneous risk of death for individual neurons as previously described38. Linear regressions of log decay and nonlinear regressions of GEDI signal increase were calculated in Prism using plateau followed by one-phase association kinetics.
For zebrafish motor axon area quantification, in toto z-stacks of immobilized fish were maximum projected, stitched together, background subtracted, and binarized. The spinal cord soma, brain, and eye fluorescence typical of mnx1 transgenics66 was manually masked out using FIJI (version 1.53c), leaving only the motor axon projections and the total area of the signal was quantified per fish and standardized to the initial time point for that fish. For zebrafish analysis of GEDI signal, in toto z-stacks of immobilized fish were stitched together, background subtracted per z plane, and all imaging channels over time were combined into a 5D composite hyperstacks (x, y, z, color, and time dimensions). Due to a spherical aberration commonly present in confocal 3D imaging and present on this platform25,104, maximum projections across z planes created blurry images that limited precision in quantification, so individual z planes were used for all fluorescence quantifications of GCaMP 7 and GEDI. In Tg:mnx1:GCaMP7/NTR-mCherry experiments, GEDI ratio was calculated per hemi-segment because individual neurons could not be easily resolved. In neuroD:NTR-BFP experiments, individual neurons were located by the co-expression of the GEDI morphology channel with the NTR-BFP while blinded to the GEDI fluorescence.
iPSC differentiation to MNs
iPSC line derived from a healthy control individual (KW4) was obtained from the Yamanaka lab96. A line containing the SOD1 D90A mutation was acquired from the iPSC repository at the Packard Center at Johns Hopkins. Reprogramming and characterization of the SOD1 D90 iPSC line were previously reported105. Healthy and SOD1 D90A iPSCs were found to be karyotypically normal and were differentiated into MNs using a modified dual-SMAD inhibition protocol106 (http://neurolincs.org/pdf/diMN-protocol.pdf). At day 18 of differentiation, iPSC-derived MNs were dissociated using trypsin (Thermo Fisher), embedded in diluted Matrigel (Corning) to limit cell motility, and plated onto Matrigel-coated 96-well plates. From day 20–35, the neurons underwent a medium change every 2–3 days.
Reporting Summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
Raw data acquired by robotic microscopy used in this study are too large to post online but the raw and processed data that support the findings of this study can be obtained from the corresponding author upon reasonable request. Additional representative data, measurements, and analysis scripts are available at: https://doi.org/10.5281/zenodo.5107973.
The code is copyright protected, and its use in performing the described research is patented (US Patent 7,139,415 and US Patent Application 14/737,325). Code is available upon request to the corresponding author. Access to and use of the code is subject to a nonexclusive, revocable, nontransferable, and limited right to use the code for the exclusive purpose of undertaking academic, governmental, or not-for-profit research. Use of the code or any part thereof for commercial or clinical purposes is strictly prohibited in the absence of a Commercial License Agreement from The J. David Gladstone Institutes.
Skibinski, G. et al. Nrf2 mitigates LRRK2- and α-synuclein–induced neurodegeneration by modulating proteostasis. Proc. Natl Acad. Sci. USA 114, 1165–1170 (2016).
Skibinski, G., Nakamura, K., Cookson, M. R. & Finkbeiner, S. Mutant LRRK2 toxicity in neurons depends on LRRK2 levels and synuclein but not kinase activity or inclusion bodies. J. Neurosci. 34, 418–433 (2014).
Arrasate, M., Mitra, S., Schweitzer, E. S., Segal, M. R. & Finkbeiner, S. Inclusion body formation reduces levels of mutant huntingtin and the risk of neuronal death. Nature 431, 805–810 (2004).
ADS CAS PubMed Article PubMed Central Google Scholar
Miller, J. et al. Identifying polyglutamine protein species in situ that best predict neurodegeneration. Nat. Chem. Biol. 7, 925–934 (2011).
Miller, J. et al. Quantitative relationships between huntingtin levels, polyglutamine length, inclusion body formation, and neuronal death provide novel insight into huntington's disease molecular pathogenesis. J. Neurosci. 30, 10541–10550 (2010).
Tsvetkov, A. S. et al. Proteostasis of polyglutamine varies among neurons and predicts neurodegeneration. Nat. Chem. Biol. 9, 586–592 (2013).
Tsvetkov, A. S. et al. A small-molecule scaffold induces autophagy in primary neurons and protects against toxicity in a Huntington disease model. Proc. Natl Acad. Sci. USA 107, 16982–16987 (2010).
ADS CAS PubMed PubMed Central Article Google Scholar
Barmada, S. J. et al. Autophagy induction enhances TDP43 turnover and survival in neuronal ALS models. Nat. Chem. Biol. 10, 677–685 (2014).
Barmada, S. J. et al. Cytoplasmic mislocalization of TDP-43 is toxic to neurons and enhanced by a mutation associated with familial amyotrophic lateral sclerosis. J. Neurosci. 30, 639–649 (2010).
Surmeier, D. J., Obeso, J. A. & Halliday, G. M. Selective neuronal vulnerability in Parkinson disease. Nat. Rev. Neurosci. 18, 101–113 (2017).
Rodrigue, K. M. et al. Risk factors for beta-amyloid deposition in healthy aging: vascular and genetic effects. JAMA Neurol. 70, 600–606 (2013).
Linsley, J. W., Reisine, T. & Finkbeiner, S. Cell death assays for neurodegenerative disease drug discovery. Expert Opin. Drug Discov. 14, 901–913 (2019).
Kepp, O., Galluzzi, L., Lipinski, M., Yuan, J. & Kroemer, G. Cell death assays for drug discovery. Nat. Rev. Drug Discov. 10, 221–237 (2011).
Arrasate, M. & Finkbeiner, S. Automated microscope system for determining factors that predict neuronal fate. Proc. Natl Acad. Sci. USA 102, 3840–3845 (2005).
Finkbeiner, S., Frumkin, M. & Kassner, P. D. Cell-based screening: extracting meaning from complex data. Neuron 86, 160–174 (2015).
Guo, F., Liu, X., Cai, H. & Le, W. Autophagy in neurodegenerative diseases: pathogenesis and therapy. Brain Pathol. 28, 3–13 (2018).
Riss, T. L. et al. Cell Viability Assays. In Eli Lilly & Company and the National Center for Advancing Translational Sciences. (eds Markossian, S.) 317–342 (Assay Guidance Manual, Bethesda, MD, 2004).
van Ham, T. J., Mapes, J., Kokel, D. & Peterson, R. T. Live imaging of apoptotic cells in zebrafish. FASEB J. 24, 4336–4342 (2010).
Zhang, J. et al. Visualization of caspase-3-like activity in cells using a genetically encoded fluorescent biosensor activated by protein cleavage. Nat. Commun. 4, 2157 (2013).
ADS PubMed Article CAS PubMed Central Google Scholar
Gorman, A. M. Neuronal cell death in neurodegenerative diseases: recurring themes around protein handling. J. Cell. Mol. Med. 12, 2263–2280 (2008).
Galluzzi, L. et al. Molecular mechanisms of cell death: recommendations of the Nomenclature Committee on Cell Death 2018. Cell Death Differ. 25, 486–541 (2018).
Tucker, B. & Lardelli, M. A rapid apoptosis assay measuring relative acridine orange fluorescence in zebrafish embryos. Zebrafish 4, 113–116 (2007).
Horstick, E. J., Tabor, K. M., Jordan, D. C. & Burgess, H. A. Genetic ablation, sensitization, and isolation of neurons using nitroreductase and tetrodotoxin-insensitive channels. Methods Mol. Biol. 1451, 355–366 (2016).
Abdelilah, S. et al. Mutations affecting neural survival in the zebrafish Danio rerio. Dev. 123, 217–227 (1996).
Linsley, J. W. et al. Automated four-dimensional long term imaging enables single cell tracking within organotypic brain slices to study neurodevelopment and degeneration. Commun. Biol. 2, 155 (2019).
Miyawaki, A. et al. Fluorescent indicators for Ca2+ based on green fluorescent proteins and calmodulin. Nature 388, 882–887 (1997).
Nagai, T., Sawano, A., Park, E. S. & Miyawaki, A. Circularly permuted green fluorescent proteins engineered to sense Ca2+. Proc. Natl Acad. Sci. USA 98, 3197–3202 (2001).
Rose, T., Goltstein, P. M., Portugues, R. & Griesbeck, O. Putting a finishing touch on GECIs. Front. Mol. Neurosci. 7, 88 (2014).
Lin, M. Z. & Schnitzer, M. J. Genetically encoded indicators of neuronal activity. Nat. Neurosci. 19, 1142–1153 (2016).
Tominaga, K., Geusz, M. E., Michel, S. & Inouye, S. T. Calcium imaging in organotypic cultures of the rat suprachiasmatic nucleus. Neuroreport 5, 1901–1905 (1994).
Ahrens, M. B., Huang, K. H., Narayan, S., Mensh, B. D. & Engert, F. Two-photon calcium imaging during fictive navigation in virtual environments. Front. Neural Circuits 7, 104 (2013).
Huang K., et al. Predictive neural processing in adult zebrafish depends on shank3b. Preprint at bioRXiv. 2019.
Sheintuch, L. et al. Tracking the same neurons across multiple days in Ca(2+) imaging data. Cell Rep. 21, 1102–1115 (2017).
Kim, D. H. et al. Pan-neuronal calcium imaging with cellular resolution in freely swimming zebrafish. Nat. Methods 14, 1107–1114 (2017).
Zhao, Y. et al. An expanded palette of genetically encoded Ca(2)(+) indicators. Science 333, 1888–1891 (2011).
Suzuki, J. et al. Imaging intraorganellar Ca2+ at subcellular resolution using CEPIA. Nat. Commun. 5, 4153 (2014).
ADS CAS PubMed Article Google Scholar
de Juan-Sanz, J. et al. Axonal endoplasmic reticulum Ca(2+) content controls release probability in CNS nerve terminals. Neuron 93, 867–881.e866 (2017).
Skibinski, G. et al. Nrf2 mitigates LRRK2- and alpha-synuclein-induced neurodegeneration by modulating proteostasis. Proc. Natl Acad. Sci. USA 114, 1165–1170 (2017).
Nakamura, K. et al. Direct membrane association drives mitochondrial fission by the Parkinson disease-associated protein alpha-synuclein. J. Biol. Chem. 286, 20710–20726 (2011).
Venderova, K. & Park, D. S. Programmed cell death in Parkinson's disease. Cold Spring Harb. Perspect. Med. 2, a009365 (2012).
Merrilees, M. J., Beaumont, B. W. & Scott, L. J. Fluoroprobe quantification of viable and non-viable cells in human coronary and internal thoracic arteries sampled at autopsy. J. Vasc. Res. 32, 371–377 (1995).
Crissman, H. A. & Steinkamp, J. A. Rapid, simultaneous measurement of DNA, protein, and cell volume in single cells from large mammalian cell populations. J. Cell Biol. 59, 766–771 (1973).
Zhou, Y. & Danbolt, N. C. Glutamate as a neurotransmitter in the healthy brain. J. Neural Transm. 121, 799–817 (2014).
Lewerenz, J. & Maher, P. Chronic glutamate toxicity in neurodegenerative diseases-What is the evidence? Front. Neurosci. 9, 469 (2015).
Fujikawa, D. G. The role of excitotoxic programmed necrosis in acute brain injury. Comput. Struct. Biotechnol. J. 13, 212–221 (2015).
Katayama, H., Kogure, T., Mizushima, N., Yoshimori, T. & Miyawaki, A. A sensitive and quantitative technique for detecting autophagic events based on lysosomal delivery. Chem. Biol. 18, 1042–1052 (2011).
Kimura, S., Noda, T. & Yoshimori, T. Dissection of the autophagosome maturation process by a novel reporter protein, tandem fluorescent-tagged LC3. Autophagy 3, 452–460 (2007).
Huang, C. J. et al. Calcium-activated calpain-2 is a mediator of beta cell dysfunction and apoptosis in type 2 diabetes. J. Biol. Chem. 285, 339–348 (2010).
Chi, H., Chang, H. Y. & Sang, T. K. Neuronal cell death mechanisms in major neurodegenerative diseases. Int. J. Mol. Sci. 19, 3082 (2018).
Barmada, S. J. et al. Amelioration of toxicity in neuronal models of amyotrophic lateral sclerosis by hUPF1. Proc. Natl Acad. Sci. USA 112, 7821–7826 (2015).
Sawa, A., Tomoda, T. & Bae, B. I. Mechanisms of neuronal cell death in Huntington's disease. Cytogenetic Genome Res. 100, 287–295 (2003).
Morrice, J. R., Gregory-Evans, C. Y. & Shaw, C. A. Necroptosis in amyotrophic lateral sclerosis and other neurological disorders. Biochim. Biophys. Acta Mol. Basis Dis. 1863, 347–353 (2017).
Kostrzewa, R. M. Review of apoptosis vs. necrosis of substantia nigra pars compacta in Parkinson's disease. Neurotox. Res. 2, 239–250 (2000).
Jager, K. J., van Dijk, P. C., Zoccali, C. & Dekker, F. W. The analysis of survival data: the Kaplan-Meier method. Kidney Int. 74, 560–565 (2008).
Haston, K. M. & Finkbeiner, S. Clinical trials in a dish: the potential of pluripotent stem cells to develop therapies for neurodegenerative diseases. Annu. Rev. Pharm. Toxicol. 56, 489–510 (2016).
Luigetti, M. et al. Heterozygous SOD1 D90A mutation presenting as slowly progressive predominant upper motor neuron amyotrophic lateral sclerosis. Neurol. Sci. 30, 517–520 (2009).
Chen, H. et al. Modeling ALS with iPSCs reveals that mutant SOD1 misregulates neurofilament balance in motor neurons. Cell Stem Cell 14, 796–809 (2014).
Rose, T., Jaepel, J., Hubener, M. & Bonhoeffer, T. Cell-specific restoration of stimulus preference after monocular deprivation in the visual cortex. Science. 352, 1319–1322 (2016).
Kim, C. K. et al. Prolonged, brain-wide expression of nuclear-localized GCaMP3 for functional circuit mapping. Front. Neural Circuits 8, 138 (2014).
Martin-Jimenez, R., Campanella, M. & Russell, C. New zebrafish models of neurodegeneration. Curr. Neurol. Neurosci. Rep. 15, 33 (2015).
PubMed Article CAS Google Scholar
Saleem, S. & Kannan, R. R. Zebrafish: an emerging real-time model system to study Alzheimer's disease and neurospecific drug discovery. Cell Death Discov. 4, 45 (2018).
Xu, J., Wang, T., Wu, Y., Jin, W. & Wen, Z. Microglia colonization of developing zebrafish midbrain is promoted by apoptotic neuron and lysophosphatidylcholine. Dev. Cell 38, 214–222 (2016).
Fett, M. E. et al. Parkin is protective against proteotoxic stress in a transgenic zebrafish model. PLoS ONE 5, e11783 (2010).
ADS PubMed PubMed Central Article CAS Google Scholar
Pyati, U. J., Look, A. T. & Hammerschmidt, M. Zebrafish as a powerful vertebrate model system for in vivo studies of cell death. Semin. Cancer Biol. 17, 154–165 (2007).
Curado, S., Stainier, D. Y. & Anderson, R. M. Nitroreductase-mediated cell/tissue ablation in zebrafish: a spatially and temporally controlled ablation method with applications in developmental and regeneration studies. Nat. Protoc. 3, 948–954 (2008).
Zelenchuk, T. A. & Bruses, J. L. In vivo labeling of zebrafish motor neurons using an mnx1 enhancer and Gal4/UAS. Genes. 49, 546–554 (2011).
Sternberg, J. R. et al. Optimization of a neurotoxin to investigate the contribution of excitatory interneurons to speed modulation in vivo. Curr. Biol. 26, 2319–2328 (2016).
Tabor, K. M. et al. Direct activation of the Mauthner cell by electric field pulses drives ultrarapid escape responses. J. Neurophysiol. 112, 834–844 (2014).
Muto, A. et al. Genetic visualization with an improved GCaMP calcium indicator reveals spatiotemporal activation of the spinal motor neurons in zebrafish. Proc. Natl Acad. Sci. USA 108, 5425–5430 (2011).
Attili, S. & Hughes, S. M. Anaesthetic tricaine acts preferentially on neural voltage-gated sodium channels and fails to block directly evoked muscle contraction. PLoS ONE 9, e103751 (2014).
Lopez-Luna, J., Al-Jubouri, Q., Al-Nuaimy, W. & Sneddon, L. U. Impact of stress, fear and anxiety on the nociceptive responses of larval zebrafish. PLoS ONE 12, e0181010 (2017).
Cheung, A. et al. A small-molecule inhibitor of skeletal muscle myosin II. Nat. Cell Biol. 4, 83–88 (2002).
Zhou, W. et al. Non-sense mutations in the dihydropyridine receptor beta1 gene, CACNB1, paralyze zebrafish relaxed mutants. Cell Calcium 39, 227–236 (2006).
Zahl, I. H., Samuelsen, O. & Kiessling, A. Anaesthesia of farmed fish: implications for welfare. Fish. Physiol. Biochem. 38, 201–218 (2012).
Kanai, K. et al. Motor axonal excitability properties are strong predictors for survival in amyotrophic lateral sclerosis. J. Neurol. Neurosurg. Psychiatry 83, 734–738 (2012).
Vossel, K. A. et al. Seizures and epileptiform activity in the early stages of Alzheimer disease. JAMA Neurol. 70, 1158 (2013).
Naumann, E. A., Kampff, A. R., Prober, D. A., Schier, A. F. & Engert, F. Monitoring neural activity with bioluminescence during natural behavior. Nat. Neurosci. 13, 513–520 (2010).
Zhang, Y., Chen, X., Gueydan, C. & Han, J. Plasma membrane changes during programmed cell deaths. Cell Res. 28, 9–21 (2018).
Chen, T. W. et al. Ultrasensitive fluorescent proteins for imaging neuronal activity. Nature 499, 295–300 (2013).
Reyes, R. C., Brennan, A. M., Shen, Y., Baldwin, Y. & Swanson, R. A. Activation of neuronal NMDA receptors induces superoxide-mediated oxidative stress in neighboring neurons and astrocytes. J. Neurosci. 32, 12973–12978 (2012).
Minnella, A. M. et al. Excitotoxic superoxide production and neuronal death require both ionotropic and non-ionotropic NMDA receptor signaling. Sci. Rep. 8, 17522 (2018).
Brennan-Minnella, A. M., Won, S. J. & Swanson, R. A. NADPH oxidase-2: linking glucose, acidosis, and excitotoxicity in stroke. Antioxid. Redox Signal. 22, 161–174 (2015).
Brennan, A. M. et al. NADPH oxidase is the primary source of superoxide induced by NMDA receptor activation. Nat. Neurosci. 12, 857–863 (2009).
Berridge, M. J., Lipp, P. & Bootman, M. D. The versatility and universality of calcium signalling. Nat. Rev. Mol. cell Biol. 1, 11–21 (2000).
Kanai, K. et al. Altered axonal excitability properties in amyotrophic lateral sclerosis: Impaired potassium channel function related to disease stage. Brain 129, 953–962 (2006).
Poudel, G. R. et al. Abnormal synchrony of resting state networks in premanifest and symptomatic Huntington disease: the IMAGE-HD study. J. Psychiatry Neurosci. 39, 87–96 (2014).
Jucker, M. & Walker, L. C. Self-propagation of pathogenic protein aggregates in neurodegenerative diseases. Nature 501, 45–51 (2013).
Masnata, M. & Cicchetti, F. The evidence for the spread and seeding capacities of the mutant Huntingtin protein in in vitro systems and their therapeutic implications. Front. Neurosci. 11, 647 (2017).
Jucker, M. & Walker, L. C. Pathogenic protein seeding in Alzheimer disease and other neurodegenerative disorders. Ann. Neurol. 70, 532–540 (2011).
Abdelfattah, A. S. et al. A bright and fast red fluorescent protein voltage indicator that reports neuronal activity in organotypic brain slices. J. Neurosci. 36, 2458–2472 (2016).
Pardo-Martin, C. et al. High-throughput in vivo vertebrate screening. Nat. Methods 7, 634–636 (2010).
Saint-Amant, L. & Drapeau, P. Time course of the development of motor behaviors in the zebrafish embryo. J. Neurobiol. 37, 622–632 (1998).
Kalueff, A. V. et al. Towards a comprehensive catalog of zebrafish behavior 1.0 and beyond. Zebrafish 10, 70–86 (2013).
Ghosh, S. & Hui, S. P. Regeneration of zebrafish CNS: adult neurogenesis. Neural Plast. 2016, 5815439 (2016).
Nicholson, C., ten Bruggencate, G., Stockle, H. & Steinberg, R. Calcium and potassium changes in extracellular microenvironment of cat cerebellar cortex. J. Neurophysiol. 41, 1026–1039 (1978).
Egelman, D. M. & Montague, P. R. Calcium dynamics in the extracellular space of mammalian neural tissue. Biophys. J. 76, 1856–1867 (1999).
Linsley, J. W. et al. Congenital myopathy results from misregulation of a muscle Ca2+ channel by mutant Stac3. Proc. Natl Acad. Sci. USA 114, E228–e236 (2017).
Cen, H., Mao, F., Aronchik, I., Fuentes, R. J. & Firestone, G. L. DEVD-NucView488: a novel class of enzyme substrates for real-time detection of caspase-3 activity in live cells. FASEB J. 22, 2243–2252 (2008).
Kwan, K. M. et al. The Tol2kit: a multisite gateway-based construction kit for Tol2 transposon transgenesis constructs. Dev. Dyn. 236, 3088–3099 (2007).
McGraw, H. F., Snelson, C. D., Prendergast, A., Suli, A. & Raible, D. W. Postembryonic neuronal addition in zebrafish dorsal root ganglia is regulated by Notch signaling. Neural Dev. 7, 23 (2012).
Karlsson, J., von Hofsten, J. & Olsson, P. E. Generating transparent zebrafish: a refined method to improve detection of gene expression during embryonic development. Mar. Biotechnol. 3, 522–527 (2001).
Daub, A., Sharma, P. & Finkbeiner, S. High-content screening of primary neurons: ready for prime time. Curr. Opin. Neurobiol. 19, 537–543 (2009).
Afgan, E. et al. The Galaxy platform for accessible, reproducible and collaborative biomedical analyses: 2016 update. Nucleic Acids Res. 44, W3–w10 (2016).
Caster, A. H. & Kahn, R. A. Computational method for calculating fluorescence intensities within three-dimensional structures in cells. Cell. Logist. 2, 176–188 (2012).
Li, Y. et al. A comprehensive library of familial human amyotrophic lateral sclerosis induced pluripotent stem cells. PLoS ONE 10, e0118266 (2015).
Chambers, S. M. et al. Highly efficient neural conversion of human ES and iPS cells by dual inhibition of SMAD signaling. Nat. Biotechnol. 27, 275–280 (2009).
This work was supported by grants from the NIH (U54 NS191046, NS101996, RF1 AG058476, RF1 AG056151, RF1 AG058447, P01 AG054407, U01 MH115747, R01 LM013617 and RF1 AG064170), as well as support from the Koret Foundation Artificial Intelligence Program for Biomedical Research and the Taube/Koret Center for Neurodegenerative Disease Research (SF). The Gladstone Institutes received support from a National Center for Research Resources Grant RR18928. Z.D. was supported by a Hillblom Graduate Fellowship. We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan Xp GPU used for this research. Kathryn Claiborn provided editorial assistance, Kelley Nelson and Gayane Abramova administrative assistance, Caitlyn Bonilla helped troubleshoot the microscope, and David Cahill provided lab maintenance and organization. Elliot Mount provided key assistance in configuring and troubleshooting the microscope to image live zebrafish. Mnx1 plasmid and transgenic zebrafish line were a kind gift from E. Isakoff (UC-Berkeley). Jack Taylor, Matthew Mccarroll, Louie Ramos, and Ethan Fertsch (UCSF) and Claire Quinata (Gladstone Institutes) provided zebrafish expertize and colony maintenance. Sean Low (University of Michigan) provided zebrafish expertize, mentoring, guidance, and calcium imaging assistance.
Gladstone Center for Systems and Therapeutics, San Francisco, CA, USA
Jeremy W. Linsley, Kevan Shah, Nicholas Castello, Michelle Chan, Shijie Wang, Wiktoria Leks, Jay Mancini, Viral Oza, Ashkan Javaherian & Steven Finkbeiner
Gladstone Institute of Neurologic Disease, San Francisco, CA, USA
Dominik Haddad, Zak Doric & Ken Nakamura
Neuroscience Graduate Program, University of California, San Francisco, CA, USA
Zak Doric
Biomedical Sciences and Neuroscience Graduate Program, University of California, San Francisco, CA, USA
Ken Nakamura
Department of Neurology, University of California, San Francisco, CA, USA
Department of Physiology, University of California, San Francisco, CA, USA
David Kokel
Institute for Neurodegenerative Disease, University of California, San Francisco, CA, USA
Taube/Koret Center for Neurodegenerative Disease, Gladstone Institutes, San Francisco, CA, USA
Steven Finkbeiner
Jeremy W. Linsley
Kevan Shah
Nicholas Castello
Dominik Haddad
Shijie Wang
Wiktoria Leks
Jay Mancini
Viral Oza
Ashkan Javaherian
J.W.L. and S.F. wrote the manuscript. J.W.L., K.S., N.C., K.N., A.J., and S.F. designed the automatic microscopy experiments. Molecular biology, biosensor design, and rodent primary neuron culturing and imaging performed by J.W.L. J.W.L. and D.K. designed zebrafish experiments. J.W.L., V.O., and W.L. performed zebrafish microinjections and imaging experiments, J.W.L. and J.M. performed zebrafish behavior experiments. N.C., M.C., S.W., D.H., and Z.D. carried out death delay imaging experiments with stimulation, dyes, stains, and GEDI. K.S. performed iPS differentiation, culturing, transfection, and imaging experiments. Custom scripts for analysis of imaging experiments done by J.L. and K.S. All authors reviewed the manuscript.
Correspondence to Steven Finkbeiner.
The authors declare no competing interests, but the following competing financial interests: S.F. is the inventor of Robotic Microscopy Systems, US Patent 7,139,415 and Automated Robotic Microscopy Systems, US Patent Application 14/737,325, both assigned to the J. David Gladstone Institutes. A provisional US and EPO patent for the GEDI biosensor (inventors J.W.L., K.S., and S.F.) assigned to the J. David Gladstone Institutes has been placed GL2016-815, May 2019.
Peer review information Nature Communications thanks Takeharu Nagai, Fumihito Ono and the other, anonymous reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Peer Review File
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
Linsley, J.W., Shah, K., Castello, N. et al. Genetically encoded cell-death indicators (GEDI) to detect an early irreversible commitment to neurodegeneration. Nat Commun 12, 5284 (2021). https://doi.org/10.1038/s41467-021-25549-9
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.
Editorial Values Statement
Journal Impact
Editors' Highlights
Top 50 Articles
Search articles by subject, keyword or author
Show results from All journals This journal
Explore articles by subject
Nature Communications (Nat Commun) ISSN 2041-1723 (online)
nature.com sitemap
Protocol Exchange
Nature portfolio policies
Author & Researcher services
Nature Masterclasses
Nature Research Academies
Libraries & institutions
Librarian service & tools
Partnerships & Services
Nature Careers
Nature Conferences
Nature Africa
Nature China
Nature India
Nature Italy
Nature Middle East
© 2022 Springer Nature Limited
Close banner Close
Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.
I agree my information will be processed in accordance with the Nature and Springer Nature Limited Privacy Policy.
Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing | CommonCrawl |
Maclaurin Series of Combinations Of Functions
Recall from the Frequently Used Maclaurin Series page the following common Maclaurin series:
The Geometric Series: $\frac{1}{1 - x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + ...$, for $-1 < x < 1$
The Derivative of the Geometric Series: $\frac{1}{(1 - x)^2} = \sum_{n=0}^{\infty} nx^{n-1} = x + 2x + 3x^2 + ...$, for $-1 < x < 1$.
The Antiderivative of the Geometric Series: $-\ln (1 - x) = \sum_{n=0}^{\infty} \frac{x^{n+1}}{n+1} = x + \frac{x^2}{2} + \frac{x^3}{3} + ...$, for $-1 ≤ x < 1$.
Inverse Tangent Function: $\tan ^{-1} x = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} x^{2n+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - ...$, for $-1 ≤ x ≤ 1$.
Euler Exponential Function: $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + \frac{x}{1!} + \frac{x^2}{2!} + ...$, for $(-\infty, \infty)$.
Natural Logarithm: $\ln (1 + x) = \sum_{n=0}^{\infty} (-1)^{n+1} \frac{x^n}{n}$.
Sine Function: $\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - ...$, for $(-\infty, \infty)$.
Cosine Function: $\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n} = 1 - \frac{x^2}{2} + \frac{x^4}{4} - ...$, for $(-\infty, \infty)$.
Using some substitutions, we can easily obtain a bunch of other Maclaurin series for some somewhat more complicated functions as we demonstrate below.
Find a Maclaurin series representation of the function $e^{2x^2/5}$.
We have that the Maclaurin series for $e^x$ is given for all $x \in \mathbb{R}$ by:
\begin{align} \quad e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + \frac{x}{1!} + \frac{x^2}{2!} + .. \end{align}
Substituting in $\frac{2x^2}{5}$ for $x$ gives us a Maclaurin series for $e^{\frac{2x^2}{5}}$:
\begin{align} \quad e^{\frac{2x^2}{5}} = \sum_{n=0}^{\infty} \frac{\left ( \frac{2x^2}{5} \right )^n}{n!} = \sum_{n=0}^{\infty} \frac{2^n x^{2n}}{5^n n!} \end{align}
Find a Maclaurin series representation for the function $x \tan^{-1} (7x)$.
We have that the Maclaurin series for the inverse tangent function is given for $-1 ≤ x < 1$ by:
\begin{align} \quad \tan ^{-1} x = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} x^{2n+1} \end{align}
Substituting $7x$ for $x$ gives us that:
\begin{align} \quad \tan^{-1} (7x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} (7x)^{2n+1} = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} 7^{n+1}x^{2n+1} \end{align}
Now multiplying the equation above by $x$ gives us:
\begin{align} \quad x \tan^{-1} (7x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} 7^{n+1}x^{2n+2} \end{align}
Note that this Maclaurin series converges for $\mid 7x \mid < 1$, that is $\mid x \mid < \frac{1}{7}$, which is a relatively small interval. | CommonCrawl |
Torsion abelian group
In abstract algebra, a torsion abelian group is an abelian group in which every element has finite order.[1] For example, the torsion subgroup of an abelian group is a torsion abelian group.
See also
• Betti number
References
1. Dummit, David; Foote, Richard. Abstract Algebra, ISBN 978-0471433347, pp. 369
| Wikipedia |
I have 2 questions regarding the naming of secondary structure elements ($\alpha$-helix and $\beta$-sheets), like helix C or sheet 2, which are often used in publications.
Who assigns the characters and numbers to the helices and sheets? The authors of the paper where the structure is published (considering possible homologues or canonical folds)? Some institution?
Is there a database/website where one can easily look up the right naming? I could not find any such information on RCSB webpage's entries. Or is it always necessary to look at the publication?
I read that there would be a 'canonical P450 fold' - but I could not find any naming conventions.
Browse other questions tagged biochemistry molecular-biology pdb xray-crystallography or ask your own question.
How to predict a mRNA secondary structure with a large sequence?
Can pymol show cartoon (secondary structure) for a pdb of multiple frames?
How to confirm secondary structure formation of Precursor miRNA on gel? | CommonCrawl |
\begin{definition}[Definition:Constructed Semantics]
Let $\LL$ be a formal language.
A '''constructed semantics''' for $\LL$ is a formal semantics which is invented ''solely'' for proving a property about $\LL$ or other entities related to $\LL$.
{{expand}}
\end{definition} | ProofWiki |
System Of Differential Equations Examples
The notes begin with a study of well-posedness of initial value problems for a first- order differential equations and systems of such equations. For example: equation 2x 16 = 10 has a solution x =3, as 23 16 =6 16 = 10. A differential equation which does not depend on the variable, say x is known as an autonomous differential equation. An integrating factor, I (x), is found for the linear differential equation (1 + x 2) d y d x + x y = 0, and the equation is rewritten as d d x (I (x) y) = 0. Introduction. However, since we are beginners, we will mainly limit ourselves to 2×2 systems. The negative eigenenergies of the Hamiltonian are sought as a solution, because these represent the bound states of the atom. SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS 3 Example 4. It is therefore important to learn the theory of ordinary differential equation, an important tool for mathematical modeling and a basic language of science. 3) Asdescribed above, welookfor asolutionto (2. x 5 0 xy2. The ultimate test is this: does it satisfy the equation?. Partial Differential Equations For Scientists And Engineers This book list for those who looking for to read and enjoy the Partial Differential Equations For Scientists And Engineers, you can read or download Pdf/ePub books and don't forget to give credit to the trailblazing authors. More Examples Here are more examples of how to solve systems of equations in Algebra Calculator. Solve the following system non-linear first order Lokta Volterra equations with boundary conditions x0 = 10, y0 = 5. In partial differential equations, they may depend on more than one variable. Reduce Differential Order of DAE System. A "system" of linear equations means that all of the equations are true at the same time. Here we present a collection of examples of general systems of linear differential equations and some applications in Physics and the Technical Sciences. Partial Differential Equations For Scientists And Engineers. In the past 30 years, however, macroeconomics has seen. of a System of ODEs. Systems of Differential Equations Matrix Methods Characteristic Equation Cayley-Hamilton - Cayley-Hamilton Theorem - An Example - The Cayley-Hamilton-Ziebur Method for ~u0= A~u - A Working Rule for Solving ~u0= A~u Solving 2 2~u0= A~u - Finding ~d 1 and ~d 2 - A Matrix Method for Finding ~d 1 and ~d 2 Other Representations of the. In addition, we investigate our model in more depth. Bus Suspension System An Example to Show How to Reduce Coupled Differential Equations to a Set of First Order Differential Equations. Example 13: System of non-linear first order differential equations. Differential equations are a special type of integration problem. such equations as an equivalent system of first-order differential equations in terms of a vector y and its first derivative. Differential Equations Calculator. From the digital control schematic, we can see that a difference equation shows the relationship between an input signal e(k) and an output signal u(k) at discrete intervals of time where k represents the index of the sample. A stochastic differential equation (SDE) is an equation in which the unknown quantity is a stochastic process and the equation involves some known stochastic processes, for example, the Wiener process in the case of diffusion equations. The Summer Program will consider the consequences of overdeterminacy and partial differential equations of finite type. 1 by taking h = 0. A differential operator is an operator defined as a function of the differentiation operator. The ideas rely on computing the eigenvalues and eigenvectors of the coefficient matrix. 1 (Modelling with differential equations). The differential equation is linear. x =location along the beam (in) E =Young's modulus of elasticity of the beam (psi) I =second moment of area (in4) q =uniform loading intensity (lb/in). We include a review of fundamental con- cepts, a description of elementary numerical methods and the concepts of convergence and order for stochastic di erential equation solvers. To use TEMATH's System of Differential Equations Solver, Select System Diff Eq from the Graph menu. Consider, for example, the system of linear differential equations. 1 A simple example system Here's a simple example of a system of differential equations: solve the coupled equations dy 1 dt =−2y 1 +y2 dy2 dt =y 1 −2y2 (1) for y 1 (t)and y2 (t)given some initial values y 1 (0)and y2 (0). 4), we should only use equation (and no other environment) to produce a single equation. Pick one of our Differential Equations practice tests now and begin!. Since the functions f (x,y) and g(x,y) do not depend on the variable t, changes in the initial value t 0 only have the effect of horizontally shifting the graphs. Example (initial value problem). This is an example of an initial value problem, where the initial position and the initial velocity are used to determine the solution. Sketch Question: Give An Example Of A System Of Differential Equations For Which (t, 1) Is A Solution. Context: System of 2x2 homogeneneous differential equations: x' = Ax (A is a 2x2 matrix with real elements). 2 Equilibria of flrst order equations 129 5. Introduction and Motivation; Second Order Equations and Systems; Euler's Method for Systems; Qualitative Analysis ; Linear Systems. 3 A nonlinear pendulum 128 5. In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. Once you represent the equation in this way, you can code it as an ODE M-file. Solve a System of Differential Equations Solve a system of several ordinary differential equations in several variables by using the dsolve function, with or without initial conditions. They can be divided into several types. DSolve can solve ordinary differential equations (ODEs), partial differential equations (PDEs), differential algebraic equations (DAEs), delay differential equations (DDEs), integral equations, integro-differential equations, and hybrid differential equations. Solve the system of ODEs. 4), we should only use equation (and no other environment) to produce a single equation. The function desolve solves systems of linear ordinary differential equations using Laplace transform. In the above six examples eqn 6. The particular solution functions x(t) and y(t) to the system of differential equations satisfying the given initial values will be graphed in blue (for x(t)) and green (for y(t)). Context: System of 2x2 homogeneneous differential equations: x' = Ax (A is a 2x2 matrix with real elements). Differential Equations Calculator. First, some may ask why would do we care that we can convert a 3rd order or higher ODE into a system of equations? Well there are quite a few reasons. In the above six examples eqn 6. Solve a System of Differential Equations Solve a system of several ordinary differential equations in several variables by using the dsolve function, with or without initial conditions. differentiable" N ×N autonomous system of differential equations. The RLC Circuit The RLC circuit is the electrical circuit consisting of a resistor of resistance R, a coil of inductance L, a capacitor of capacitance C and a voltage source arranged in series. Analysis of a System of Linear Delay Differential Equations A new analytic approach to obtain the complete solution for systems of delay differential equations (DDE) based on the concept of Lambert functions is presented. When this is not the case the system is commonly known as being differential algebraic and this 1this may be subject to debate since the non-autonomous case can have special features 1. A differential story — Peter D Lax wins the 2005 Abel Prize for his work on differential equations. Differential Equations and Separation of Variables A differential equation is basically any equation that has a. Parabolic equations: exempli ed by solutions of the di usion equation. This might introduce extra solutions. 3, the initial condition y 0 =5 and the following differential equation. Substitution method. concentration of species A) with respect to an independent variable (e. This is a constant time factor so it's not the biggest deal, but I feel that we can improve some applications by reducing common latency here. 1 Matrices and Linear Systems 264 5. Remember also that the derivative term y'(t) describes the rate of change in y(t). In particular, phase portraits for such systems can be classified according to types of eigenvalues which appear (the sign of the real part, zero or nonzero imaginary parts) and the dimensions of the generalized eigenspaces. Consider the system. pdf - Example 1 Solve the following differential equation 0:5 d2 y d3y dy = x 2y 3 4 2 dx3 dx dx 0 00 y(1 = 4 y(1 = 3 y(1 = 2 a Using the rev_ivp. This is a constant time factor so it's not the biggest deal, but I feel that we can improve some applications by reducing common latency here. In general, the number of equations will be equal to the number of dependent variables i. An couple of examples would be Example 1: dx1 dt = 0. I have solved such a system once before, but that was using an adiabatic approximation, e. It may not be immediately obvious for Maxwell's equations unless you write out the divergence and curl in terms of partial derivatives. ME 563 Mechanical Vibrations Fall 2010 1-2 1 Introduction to Mechanical Vibrations 1. Under reasonable conditions on φ, such an equation has a solution and the corresponding initial value problem has a unique solution. We know means the slope of y with respect to x. MTH 244 - Matrix Method for ODE 1 MTH 244 - Additional Information for Chapter 3 Section 1 (Merino) and section 3 (Dobrushkin) - March 2003 1 Linear Systems of Differential Equations of Order One. 2 Crystal growth{a case study 137 5. Systems of Equations: Graphical Method In these lessons, we will learn how to solve systems of equations or simultaneous equations by graphing. The following are examples of ordinary differential equations: In these, y stands for the function, and either t or x is the independent variable. differential equation (1). Cramer's rule says that if the determinant of a coefficient matrix |A| is not 0, then the solutions to a system of linear equations can. System of differential equations, ex1 Differential operator notation, system of linear differential equations, solve system of differential equations by elimination, supreme hoodie ss17. Differential equations involve the derivatives of a function or a set of functions. The differential equation is linear. A calculator for solving differential equations. 6 is non-homogeneous where as the first five equations are homogeneous. discusses two-point boundary value problems: one-dimensional systems of differential equations in which the solution is a function of a single variable and the value of the solution is known at two points. 6 Linearization of Nonlinear Systems In this section we show how to perform linearization of systems described by nonlinear differential equations. example4 a mixture problem a tank contains 50 gallons of a solution. We include a review of fundamental con- cepts, a description of elementary numerical methods and the concepts of convergence and order for stochastic di erential equation solvers. But if the homogenous part of the solution has the same root, you would try multiplying it by t to get a linearly independent set. In this case, the Microsoft Excel 5. A couple of examples may help to give the flavor. I give only one example, which shows how the trigonometric functions may emerge in the solution of a system of two simultaneous linear equations, which, as we saw above, is equivalent to a second-order equation. DESJARDINS and R´emi VAILLANCOURT Notes for the course MAT 2384 3X Spring 2011 D´epartement de math´ematiques et de statistique Department of Mathematics and Statistics Universit´e d'Ottawa / University of Ottawa Ottawa, ON, Canada K1N 6N5. As in the above example, the solution of a system of linear equations can be a single ordered pair. Let's see some examples of first order, first degree DEs. The solution is given by the equations x1(t) = c1 +(c2 −2c3)e−t/3 cos(t/6) +(2c2 +c3)e−t/3 sin(t/6), x2(t) = 1 2 c1 +(−2c2 −c3)e−t/3 cos(t/6) +(c2 −2c3)e−t/3 sin(t/6),. The equations are said to be "coupled" if output variables (e. Consider, for example, the system of linear differential equations. saying that one of the differential equations was approximately zero on the timescale at which the others change. Solve a System of Differential Equations Solve a system of several ordinary differential equations in several variables by using the dsolve function, with or without initial conditions. A Differential Equation is a n equation with a function and one or more of its derivatives: Example: an equation with the function y and its derivative dy dx. The laws of the Natural and Physical world are usually written and modeled in the form of differential equations. Cain and Angela M. System of differential equations. Capable of finding both exact solutions and numerical approximations, Maple can solve ordinary differential equations (ODEs), boundary value problems (BVPs), and even differential algebraic equations (DAEs). Ramsay, Department of Psychology, 1205 Dr. I give only one example, which shows how the trigonometric functions may emerge in the solution of a system of two simultaneous linear equations, which, as we saw above, is equivalent to a second-order equation. The above problem can be. Solving Systems of Differential Equations. Simple Control Systems 4. In this course, I will mainly focus on, but not limited to, two important classes of mathematical models by ordinary differential equations: population dynamics in biology. Jun 17, 2017 · A system of differential equations is a set of two or more equations where there exists coupling between the equations. Ordinary differential equation examples by Duane Q. Using Mathcad to Solve Systems of Differential Equations Charles Nippert Getting Started Systems of differential equations are quite common in dynamic simulations. It is the same concept when solving differential equations - find general solution first, then substitute given numbers to find particular solutions. i use matlab commands 'ode23' and 'ode45' for solving systems of differential. 2 Equilibria of flrst order equations 129 5. of the system, emphasizing that the system of equations is a model of the physical behavior of the objects of the simulation. We just saw that there is a general method to solve any linear 1st order ODE. That is the main idea behind solving this system using the model in Figure 1. This example shows you how to convert a second-order differential equation into a system of differential equations that can be solved using the numerical solver ode45 of MATLAB®. The differential equation with input f(t) and output y(t) can represent many different systems. Calculus is required as specialized advanced topics not usually found in elementary differential equations courses are included, such as exploring the world of discrete dynamical systems and describing chaotic. This results in the differential equation. The differential equations we consider in most of the book are of the form Y′(t) = f(t,Y(t)), where Y(t) is an unknown function that is being sought. A differential equation is an equation that relates a function with one or more of its derivatives. The ultimate test is this: does it satisfy the equation?. $$\frac{dy(t)}{dt} = -k \; y(t)$$ The Python code first imports the needed Numpy, Scipy, and Matplotlib packages. In Section 4. Example: t y″ + 4 y′ = t 2 The standard form is y t t. 524 Systems of Differential Equations analysis, the recycled cascade is modeled by the non-triangular system x′ 1 = − 1 6 x1 + 1 6 x3, x′ 2= 1 6 x1 − 1 3 x , x′ 3= 1 3 x2 − 1 6 x. The examples ddex1, ddex2, ddex3, ddex4, and ddex5 form a mini tutorial on using these solvers. I made up the third equation to be able to get a solution. Let's see some examples of first order, first degree DEs. As a consequence, the analysis of nonlinear systems of differential equations is much more accessible than it once was. To solve a single differential equation, see Solve Differential Equation. 5, solution is AJ0. 4) In other words, p{m) is obtained from p{D) by replacing D by m. The solutions of such systems require much linear algebra (Math 220). Assembly of the single linear differential equation for a diagram com-. 0 Modeling a first order differential equation Let us understand how to simulate an ordinary differential equation (continuous time system) in Simulink through the following example from chemical engineering: "A mass balance for a chemical in a completely mixed reactor can be mathematically modeled as the differential equation 8 × Ö × ç. The effects of $\xi, \, \epsilon,\, \theta_1 $ and $\theta_2$ rates on the devices that moved from latent to recovered nodes are investigated. For this linear differential equation system, the origin is a stable node because any trajectory proceeds to the origin over time. A differential equation which does not depend on the variable, say x is known as an autonomous differential equation. Simultaneous equations can help us solve many real-world problems. A system of two first order linear differential equations is two dimensional because the state space of the solutions is two dimensional affine vector space. The ddex1 example shows how to solve the system of differential equations. The functional dependence of x_1, , x_n on an independent variable, for instance x, must be explicitly indicated in the variables and its derivatives. The particular solution functions x(t) and y(t) to the system of differential equations satisfying the given initial values will be graphed in blue (for x(t)) and green (for y(t)). 1 (Modelling with differential equations). A basic example showing how to solve systems of differential equations. Learn more about differential equations. How to Solve Differential Equations. Here, you can see both approaches to solving differential equations. The theory of systems of linear differential equations resembles the theory of higher order differential equations. Mar 28, 2018 · You are welcome, you have two systems of ODE with three unknown quantities (I1, I2 and v ). ) We show by a number of examples how they may. In the above six examples eqn 6. This article assumes that the reader understands basic calculus, single differential equations, and linear algebra. It's possible that a differential equation has no solutions. 1 First-Order Systems and Applications 228 4. is, those differential equations that have only one independent variable. Jul 25, 2016 · An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance. Here we will consider a few variations on this classic. This issue will be used to track common interface option handling. We will also learn about systems of linear differential equations, including the very important normal modes problem. An application: linear systems of differential equations We use the eigenvalues and diagonalization of the coefficient matrix of a linear system of differential equations to solve it. Introduction Differential equations are a convenient way to express mathematically a change of a dependent variable (e. Oct 25, 2019 · Solving second order ordinary differential equations is much more complex than solving first order ODEs. I am trying to find the equilibrium points by hand but it seems like it is not possible without the help of a numerical method. Once you represent the equation in this way, you can code it as an ODE M-file. Find the particular solution given that `y(0)=3`. Maple is the world leader when it comes to solving differential equations, finding closed-form solutions to problems no other system can handle. Mathcad Professional includes a variety of additional, more specialized functions for solving differential equations. 4 solving differential equations using simulink the Gain value to "4. It may not be immediately obvious for Maxwell's equations unless you write out the divergence and curl in terms of partial derivatives. For example: equation 2x 16 = 10 has a solution x =3, as 23 16 =6 16 = 10. This second edition of Noonburg's best-selling textbook includes two new chapters on partial differential equations, making the book usable for a two-semester sequence in differential equations. Basically, one simply replaces the higher order terms with new variables and includes the equations that define the new variables to form a set of first order simultaneous differential. a system of difference equations x((m+1)T+) = g(x(mT+)). ,where 4 is the time constant In this case we want to pass 0 and * as parameters, to make it easy to be able to change values for these parameters We set * = 1 We set initial condition +(0) = 1 and 4 = 5. In this example, I will show you the process of converting two higher order linear differential equation into a sinble matrix equation. Ordinary differential equation examples by Duane Q. Elliptic equations: weak and strong minimum and maximum principles; Green's functions. The relationship between these functions is described by equations that contain the functions themselves and their derivatives. 1 Systems of differential equations Find the general solution to the following system: 8 <: x0 1 (t) = 1(t) x 2)+3 3) x0 2 (t) = x 1(t)+x 2(t) x 3(t) x0 3 (t) = x 1(t) x 2(t)+3x 3(t) First re-write the system in matrix form: x0= Ax Where: x = 2 4 x 1(t) x 2(t) x 3(t) 3 5 A= 2 4 1 1 3 1 1 1 1 1 3 3 5 1. An integrating factor, I (x), is found for the linear differential equation (1 + x 2) d y d x + x y = 0, and the equation is rewritten as d d x (I (x) y) = 0. Exact differential equation examples and solutions download exact differential equation examples and solutions free and unlimited. Denoting this known solution by y 1 , substitute y = y 1 v = xv into the given differential equation and solve for v. Ordinary Differential Equations (ODES) There are many situations in science and engineering in which one encounters ordinary differential equations. Systems of differential equations Handout Peyam Tabrizian Friday, November 18th, 2011 This handout is meant to give you a couple more example of all the techniques discussed in chapter 9, to counterbalance all the dry theory and complicated ap-plications in the differential equations book! Enjoy! :) Note: Make sure to read this carefully!. In our study of chaos, we will need to expand the definitions of linear and nonlinear to include differential equations. To solve a system with higher-order derivatives, you will first write a cascading system of simple first-order equations then use them in your differential function. We'll start by attempting to solve a couple of very simple equations of such type. An example: dx1 dt = 2x1x2 +x2 dx2 dt = x1 −t2x2. For example, much can be said about equations of the form ˙y = φ(t,y) where φ is a function of the two variables t and y. ( x0(t) = x2 +1, y0(t) = x(y −1). The first tank starts with 40 pounds of salt dissolved in it, and the second tank starts with 60 pounds of salt. The following graphic outlines the method of solution. I don't really have such information now. In this example, I will show you the process of converting two higher order linear differential equation into a sinble matrix equation. Flashcards. Second Order Differential Equations 19. McKinley October 22, 2013 In these notes, which replace the material in your textbook, we will learn a modern view of analyzing systems of differential equations. The above integral equation, however,. " Then, using the Sum component, these terms are added, or subtracted, and fed into the integrator. It is not possible to solve for three variables given two equations. In most applications, the functions represent physical quantities, the derivatives represent their. Homogeneous linear differential equations produce exponential solutions. Find a solution of the differential equation from the previous example that satisfies the condition y(0) = 2. Differential equations involve the derivatives of a function or a set of functions. NDSolve solves a wide range of ordinary differential equations as well as many partial differential equations. Linear Ordinary Differential Equations If differential equations can be written as the linear combinations of the derivatives of y, then it is known as linear ordinary differential equations. Example: t y″ + 4 y′ = t 2 The standard form is y t t. As matter of fact, the explicit solution method does not exist for the general class of linear equations with variable coefficients. Such systems arise when a model involves two and more variable. such equations as an equivalent system of first-order differential equations in terms of a vector y and its first derivative. Over the last hundred years, many techniques have been developed for the solution of ordinary differential equations and partial differential equations. It provides qualitative physical explanation of mathematical results while maint. ) We show by a number of examples how they may. , position or voltage) appear in more than one equation. System of differential equations. Since they feature homogeneous functions in one or the other form, it is crucial that we understand what are homogeneous functions first. An couple of examples would be Example 1: dx1 dt = 0. logo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science. Then in the five sections that follow we learn how to solve linear higher-order differential equations. These worked examples begin with two basic separable differential equations. pdf - Example 1 Solve the following differential equation 0:5 d2 y d3y dy = x 2y 3 4 2 dx3 dx dx 0 00 y(1 = 4 y(1 = 3 y(1 = 2 a Using the rev_ivp. Exact differential equation examples and solutions download exact differential equation examples and solutions free and unlimited. After introducing each class of differential equations we consider finite difference methods for the numerical solution of equations in the class. Typically a complex system will have several differential equations. Parameter Estimation for Differential Equations: A Gen-eralized Smoothing Approach J. When working with differential equations, MATLAB provides two different approaches: numerical and symbolic. discusses two-point boundary value problems: one-dimensional systems of differential equations in which the solution is a function of a single variable and the value of the solution is known at two points. Oct 19, 2011 · Solving ordinary differential equations on the GPU. Elliptic equations: weak and strong minimum and maximum principles; Green's functions. Solve the differential equation for the spring, d2y dt2 = − k m y, if the mass were displaced by a distance y0 and then released. 3) a Nonlinear SystemofDifferentialEquations. Differential equations are a special type of integration problem. Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. Featured on Meta An apology to our community, and next steps. 4 System of two linear differential equations 51 to each other. So the equation is called an Ordinary Differential Equations (ODE) b) When the unknown function y depends on several independent variables r, s, t, etc. dsolve can't solve this system. In this section we will examine some of the underlying theory of linear DEs. ca The research was supported by Grant 320 from the Natural Science and Engineering. Here follows the continuation of a collection of examples from Calculus 4c-1, Systems of differential systems. Liberal use of examples and homework problems aids the student in the study of the topics presented and applying them to numerous applications in the real scientific world. The contents of the tank are kept. differentiable" N ×N autonomous system of differential equations. This is a system of differential equations which describes the changing positions of n bodies with mass interacting with each other under the influence of gravity. Elementary Differential Equations with Boundary Value Problems is written for students in science, en-gineering,and mathematics whohave completed calculus throughpartialdifferentiation. Solve a System of Differential Equations Solve a system of several ordinary differential equations in several variables by using the dsolve function, with or without initial conditions. Find the general solution for the differential equation `dy + 7x dx = 0` b. Do they approach the origin or are they repelled from it? We can graph the system by plotting direction arrows. Toggle Main Navigation. view of analyzing systems of differential equations. From the way these forms were constructed, it is clear that a three dimensional surface in the seven dimensional space with coordinates x, y, t, a, b, c, u which solves Pfaff's problem and can be parameterized by x, y, t corresponds to the graph of a solution to the system of differential equations, and hence to a solution of the wave equation. Clearly the trivial solution (\(x = 0\) and \(y = 0\)) is a solution, which is called a node for this system. After introducing each class of differential equations we consider finite difference methods for the numerical solution of equations in the class. The solution is given by the equations x1(t) = c1 +(c2 −2c3)e−t/3 cos(t/6) +(2c2 +c3)e−t/3 sin(t/6), x2(t) = 1 2 c1 +(−2c2 −c3)e−t/3 cos(t/6) +(c2 −2c3)e−t/3 sin(t/6),. pdf - Example 1 Solve the following differential equation 0:5 d2 y d3y dy = x 2y 3 4 2 dx3 dx dx 0 00 y(1 = 4 y(1 = 3 y(1 = 2 a Using the rev_ivp. The functional dependence of x_1, , x_n on an independent variable, for instance x, must be explicitly indicated in the variables and its derivatives. The laws of the Natural and Physical world are usually written and modeled in the form of differential equations. differential equation. Differential equations have a remarkable ability to predict the world around us. That is, we can solve the equation x t 4 separately from the equation u t 0. The theory is very deep, and so we will only be able to scratch the surface. acterises these differential equations as so-called dynamical systems. When writing a. of differential equations and view the results graphically are widely available. If you are solving several similar systems of ordinary differential equations in a matrix form, create your own solver for these systems, and then use it as a shortcut. Chapter 1 Differential and Difference Equations In this chapter we give a brief introduction to PDEs. For generality, let us consider the partial differential equation of the form [Sneddon, 1957] in a two-dimensional domain. If the highest-order derivative present in a differential equation is the first derivative, the equation is a first-order differential equation. The equations of a system are dependent if ALL the solutions of one equation are also solutions of the other equation. Systems of Differential Equations. Delay Differential Equations. To solve a system with higher-order derivatives, you will first write a cascading system of simple first-order equations then use them in your differential file. Differential equations. First-Order Linear ODE. The same equations describe a variety of mechanical and electrical systems. Aug 07, 2012 · Modeling with ordinary differential equations (ODEs) Simple examples of solving a system of ODEs Create a System of ODE's To run a fit, your system has to be written as a definition. View Videos or join the Second-order Differential Equation discussion. There are many areas where differential equations are used as a model for the problem at hand. This is just an overview of the techniques; MATLAB provides a rich set of functions to work with differential equations. 3 A nonlinear pendulum 128 5. Mathcad Professional includes a variety of additional, more specialized functions for solving differential equations. Homogeneous Differential Equations are of prime importance in physical applications of mathematics due to their simple structure and useful solutions. It is firstorder since only the first derivative of x appears in the equation. Here is a simple example of a real-world problem modeled by a differential equation involving a parameter (the constant rate H). Flexural vibration of beamsandheatconductionarestudiedasexamplesof application. Image: Second order ordinary differential equation (ODE) integrated in Xcos As you can see, both methods give the same results. Then in the five sections that follow we learn how to solve linear higher-order differential equations. Cain and Angela M. Linear Ordinary Differential Equations If differential equations can be written as the linear combinations of the derivatives of y, then it is known as linear ordinary differential equations. We focus in particular on the linear differential equations of second order of variable coefficients, although the amount of examples is far from exhausting. At the end of these lessons, we have a systems of equations calculator that can solve systems of equations graphically and algebraically. Now we have two differential equations for two mass (component of the system) and let's just combine the two equations into a system equations (simultaenous equations) as shown below. In addition to analytic and numerical methods, graphic methods are also used for the approximate solution of differential equations. System of differential equations, ex1 Differential operator notation, system of linear differential equations, solve system of differential equations by elimination, supreme hoodie ss17. Introduction and First Definitions; Vector. Nonlinear Differential Equations and The Beauty of Chaos 2 Examples of nonlinear equations 2 ( ) kx t dt d x t m =− Simple harmonic oscillator (linear ODE) More complicated motion (nonlinear ODE) ( )(1 ()) 2 ( ) kx t x t dt d x t m =− −α Other examples: weather patters, the turbulent motion of fluids Most natural phenomena are. A differential equation which does not depend on the variable, say x is known as an autonomous differential equation. Flashcards. Incontrast, a set of m equations with l unknown functions is called a system of m equations. 0 Modeling a first order differential equation Let us understand how to simulate an ordinary differential equation (continuous time system) in Simulink through the following example from chemical engineering: "A mass balance for a chemical in a completely mixed reactor can be mathematically modeled as the differential equation 8 × Ö × ç. 1 A simple example system Here's a simple example of a system of differential equations: solve the coupled equations dy 1 dt =−2y 1 +y2 dy2 dt =y 1 −2y2 (1) for y 1 (t)and y2 (t)given some initial values y 1 (0)and y2 (0). 3 A nonlinear pendulum 128 5. Example: Solve the system of equations by the substitution method. The diagram represents the classical brine tank problem of Figure 1. 98 CHAPTER 3 Higher-Order Differential Equations 3. When writing a. I The Navier-Stokes equations are a set of coupled, non-linea r, partial differential equations. Substitution method. A system (1) is called a strictly hyperbolic system if all roots of the characteristic equation are distinct for any non-zero vector. The following are examples of ordinary differential equations: In these, y stands for the function, and either t or x is the independent variable. You can also plot slope and direction fields with interactive implementations of Euler and Runge-Kutta methods. To solve a single differential equation, see Solve Differential Equation. Use * for multiplication a^2 is a 2. Such systems occur as the general form of (systems of) differential equations for vector-valued functions x in one independent variable t,. 2 Crystal growth{a case study 137 5. In this case, we speak of systems of differential equations. The equations of a system are independent if they do not share ALL solutions. 2 together with the y-axis. Count-abel even if not solve-able — The 2004 Abel Prize goes to Sir Michael Atiyah and Isadore Singer for their work on how to solve systems of equations. Find the general solution for the differential equation `dy + 7x dx = 0` b. The equations are said to be "coupled" if output variables (e. Mathematica 9 leverages the extensive numerical differential equation solving capabilities of Mathematica to provide functions that make working with parametric differential equations conceptually simple. Nykamp is licensed under a Creative Commons Attribution-Noncommercial-ShareAlike 4. 3)byintroducingthecharacteristicequations, dx ds = a dt ds = 1 dz ds = 0: (2. Remember also that the derivative term y'(t) describes the rate of change in y(t). In particular, MATLAB offers several solvers to handle ordinary differential equations of first order. Checking this solution in the differential equation shows that. Find a solution of the differential equation from the previous example that satisfies the condition y(0) = 2. A sin-gle difierential equation of second and higher order can also be converted into a system of flrst-order difierential. | CommonCrawl |
Simplify: $3!(2^3+\sqrt{9})\div 2$.
Simplify according to the order of operations. \begin{align*}
3!(2^3+\sqrt{9})\div 2 &= 6(8+3)\div 2 \\
&=6(11)\div 2 \\
&=66\div 2\\
&=\boxed{33}.
\end{align*} | Math Dataset |
\begin{definition}[Definition:Type]
Let $\MM$ be an $\LL$-structure.
Let $A$ be a subset of the universe of $\MM$.
Let $\LL_A$ be the language consisting of $\LL$ along with constant symbols for each element of $A$.
Viewing $\MM$ as an $\LL_A$-structure by interpreting each new constant as the element for which it is named, let $\map {\operatorname {Th}_A} \MM$ be the set of $\LL_A$-sentences satisfied by $\MM$.
An '''$n$-type over $A$''' is a set $p$ of $\LL_A$-formulas in $n$ free variables such that $p \cup \map {\operatorname {Th}_A} \MM$ is satisfiable by some $\LL_A$-structure.
{{Disambiguate|Definition:Logical Formula}}
\end{definition} | ProofWiki |
Lester's theorem
In Euclidean plane geometry, Lester's theorem states that in any scalene triangle, the two Fermat points, the nine-point center, and the circumcenter lie on the same circle. The result is named after June Lester, who published it in 1997,[1] and the circle through these points was called the Lester circle by Clark Kimberling.[2] Lester proved the result by using the properties of complex numbers; subsequent authors have given elementary proofs[3][4][5][6], proofs using vector arithmetic,[7] and computerized proofs.[8]
See also
• Parry circle
• Shape § Similarity classes
• van Lamoen circle
References
1. Lester, June A. (1997), "Triangles. III. Complex triangle functions", Aequationes Mathematicae, 53 (1–2): 4–35, doi:10.1007/BF02215963, MR 1436263, S2CID 119667124
2. Kimberling, Clark (1996), "Lester circle", The Mathematics Teacher, 89 (1): 26, JSTOR 27969621
3. Shail, Ron (2001), "A proof of Lester's theorem", The Mathematical Gazette, 85 (503): 226–232, doi:10.2307/3622007, JSTOR 3622007, S2CID 125392368
4. Rigby, John (2003), "A simple proof of Lester's theorem", The Mathematical Gazette, 87 (510): 444–452, doi:10.1017/S0025557200173620, JSTOR 3621279, S2CID 125214460
5. Scott, J. A. (2003), "Two more proofs of Lester's theorem", The Mathematical Gazette, 87 (510): 553–566, doi:10.1017/S0025557200173917, JSTOR 3621308, S2CID 125997675
6. Duff, Michael (2005), "A short projective proof of Lester's theorem", The Mathematical Gazette, 89 (516): 505–506, doi:10.1017/S0025557200178581, S2CID 125894605
7. Dolan, Stan (2007), "Man versus computer", The Mathematical Gazette, 91 (522): 469–480, doi:10.1017/S0025557200182117, JSTOR 40378420, S2CID 126161757
8. Trott, Michael (1997), "Applying GroebnerBasis to three problems in geometry", Mathematica in Education and Research, 6 (1): 15–28
External links
• Weisstein, Eric W. "Lester Circle". MathWorld.
| Wikipedia |
Differential Distribution and Expression of Panton-Valentine Leucocidin among Community-Acquired Methicillin-Resistant Staphylococcus aureus Strains
Battouli Saïd-Salim, Barun Mathema, Kevin Braughton, Stacy Davis, Daniel Sinsimer, William Eisner, Yekaterina Likhoshvay, Frank R. DeLeo, Barry N. Kreiswirth
Battouli Saïd-Salim
TB Center, Public Health Research Institute, Newark, New Jersey 07103
Barun Mathema
Kevin Braughton
Laboratory of Human Bacterial Pathogenesis, Rocky Mountain Laboratories, and National Institute of Allergy and Infectious Diseases, National Institutes of Health, Hamilton, Montana 59840
Stacy Davis
Daniel Sinsimer
TB Center, Public Health Research Institute, Newark, New Jersey 07103and Department of Microbiology and Molecular Genetics, University of Medicine and Dentistry of New Jersey, Newark, New Jersey 07103
William Eisner
Yekaterina Likhoshvay
Frank R. DeLeo
Barry N. Kreiswirth
For correspondence: [email protected]
DOI: 10.1128/JCM.43.7.3373-3379.2005
Community-acquired methicillin-resistant Staphylococcus aureus (CA-MRSA) is an emerging threat worldwide. CA-MRSA strains differ from hospital-acquired MRSA strains in their antibiotic susceptibilities and genetic backgrounds. Using several genotyping methods, we clearly define CA-MRSA at the genetic level and demonstrate that the prototypic CA-MRSA strain, MW2, has spread as a homogeneous clonal strain family that is distinct from other CA-MRSA strains. The Panton-Valentine leucocidin (PVL)-encoding genes, lukF and lukS, are prevalent among CA-MRSA strains and have previously been associated with CA-MRSA infections. To better elucidate the role of PVL in the pathogenesis of CA-MRSA, we first analyzed the distribution and expression of PVL among different CA-MRSA strains. Our data demonstrate that PVL genes are differentially distributed among CA-MRSA strains and, when they are present, are always transcribed, albeit with strain-to-strain variability of transcript levels. To directly test whether PVL is critical for the pathogenesis of CA-MRSA, we evaluated the lysis of human polymorphonuclear leukocytes (PMNs) during phagocytic interaction with PVL-positive and PVL-negative CA-MRSA strains. Unexpectedly, there was no correlation between PVL expression and PMN lysis, suggesting that additional virulence factors underlie leukotoxicity and, thus, the pathogenesis of CA-MRSA.
Staphylococcus aureus is an important human pathogen capable of causing diseases in the hospital and community settings (23). The increased incidence of multidrug-resistant S. aureus strains among nosocomial (or hospital-acquired [HA]) infections has added a challenging dimension to the S. aureus problem (5). These strains are typically labeled HA methicillin-resistant Staphylococcus aureus (MRSA) strains or simply MRSA strains. Several risk factors, such as recent hospitalization or exposure to a health care setting, residence in long-term-care facilities, invasive or surgical procedures, and injection drug use, predispose a patient to MRSA acquisition.
Methicillin resistance in S. aureus is conferred by the mecA gene, which encodes an altered penicillin binding protein (PBP 2′) (22). The mecA gene is harbored in a large mobile genetic element (referred to as the staphylococcal chromosomal cassette mec [SCCmec]) that has a unique chromosomal integration locus (14). Sequence analyses defined three major SCCmec types (SCCmec types I, II, and III) among nosocomial MRSA strains. SCCmec types are distinguished on the basis of their sizes, which range from 26 and 67 kb, and genetic compositions, in which their genomes include recombinases and antibiotic resistance genes (16).
In the community, the majority of S. aureus infections, which include skin and soft tissue infections (6, 8, 13, 23), are caused by methicillin-susceptible S. aureus (MSSA) strains. However, since 1991 there have been increasing reports of MRSA infections in the community and in patients with and without risk factors for MRSA infection (4, 5). These MRSA strains, commonly referred to as community-acquired MRSA (CA-MRSA) strains (7, 8, 11-13, 18, 20, 30), differ from nosocomial MRSA strains on the basis of their genetic backgrounds and antibiograms. CA-MRSA strains harbor the recently described SCCmec type IV element in their genomes. Compared with other SCCmec types, the SCCmec type IV element is distinguished by its small size, the presence of functional recombinases, and the absence of antibiotic resistance markers (25). This distinction is consistent with the observation that CA-MRSA strains are more susceptible to antibiotics than hospital-acquired MRSA strains. In fact, many CA-MRSA strains are highly susceptible and show resistance only to β-lactam antibiotics, an observation that clearly distinguishes them from the multidrug-resistant nosocomial MRSA strains.
The genome of prototypic CA-MRSA strain, MW2, was fully sequenced (1) and revealed the presence of the SCCmec type IV element described above (15). Additionally, the sequencing results for this strain revealed unique CA-MRSA-specific virulence factors. For example, MW2 harbors the Panton-Valentine leucocidin (PVL), encoded by two contiguous and cotranscribed genes (lukF and lukS); staphylococcal enterotoxin H (seh); and staphylococcal enterotoxin C (sec). PVL, a bicomponent, secreted leucocidin found in <5% of S. aureus isolates (21), is cytotoxic to human and rabbit monocytes, macrophages, and human polymorphonuclear leukocytes (PMNs) (10). The protein forms nonspecific pores in leukocyte plasma membranes, which result in increased permeability and eventual host cell lysis (32). PVL has gained much attention as one of the key virulence determinants present in CA-MRSA strains. Lina and colleagues (21) found that 50 to 93% of the S. aureus strains causing primary skin infections produce PVL. Furthermore, a study by Gillet et al. (10) demonstrated a strong association between PVL and necrotizing pneumonia in healthy children and young adults. From these studies, the authors hypothesized that the propensity of many CA-MRSA strains to cause severe skin and soft tissue infections and, occasionally, necrotizing pneumonia is due to their ability to produce PVL (10, 21).
Previous studies have defined CA-MRSA strains as isolates recovered within 48 h after hospital admission and/or in patients without known risk factors for MRSA acquisition. Given the fluidity between community and health care settings, i.e., with hospital strains being transferred into the community and vice versa (29), we opted to utilize a genetic approach based on the presence of SCCmec type IV to define CA-MRSA. The presumed role of PVL in the pathogenesis of CA-MRSA led us to investigate the distribution of PVL among CA-MRSA strains in our collection and evaluate its expression. Furthermore, to gain insight into the role of PVL during infection, we assessed PMN lysis during phagocytic interaction with PVL-negative and PVL-positive S. aureus strains of closely related genetic backgrounds.
Strain collection.One hundred twenty-one strains were grouped as CA-MRSA based on the presence of the SCCmec type IV element in their genomes. Eighty-three strains were sent to our laboratory by clinical institutions for genotyping. These strains were suspected of being CA-MRSA based on their reduced drug resistance profile; and SCCmec typing, indeed, classified them as SCCmec type IV (see below). These strains were susceptible to two or all three of the following antibiotics: fluoroquinolones, clindamycin, and erythromycin. As certain genetic backgrounds based on spa type (spa types 1, 7, and 17) appear to be prevalent among CA-MRSA strains, we selected from our database additional strains with these genotypes for SCCmec typing. Strains that were classified as SCCmec type IV were included in this study. In a previous study, we used several genotyping techniques, including a side-by-side pulsed-field gel electrophoresis (PFGE) comparison, to demonstrate that an MSSA strain known as MnCop was the parental strain of MW2 (8). The spa type 35 of MnCop is closely related to MW2 spa type 131, as they both share the same multilocus sequence typing (MLST) profile (data not shown). For this reason, we included several MSSA strains from our collection of spa type 35 strains in this study.
SCCmec typing.Multiplex PCR analysis was performed as described previously (26) to distinguish the four genetic elements for SCCmec, with one modification; that is, the pls gene was amplified by using the following primers from the sequence with GenBank accession number AF115379: primer PlsF (GGGGTGGTTAATGGTATGAATAAA) and primer PlsR (CGGAATGTTGCTCTTGGTTGTGCGTTTTC).
spa typing.The method of spa typing was developed in our laboratory, and its accuracy and discriminatory power for determination of the subspecies of S. aureus strains is superior to that of MLST analysis (19, 27). Spa typing is a DNA sequencing-based method that distinguishes strains based upon the makeup of the variable number of tandem repeats in the 3′ region of the protein A gene, which is both unique and conserved among S. aureus isolates. Currently, the Public Health Research Institute database contains 625 different spa types among a collection of over 2,500 S. aureus isolates.
PFGE.PFGE was performed and the results were interpreted as described previously (33). Briefly, the organisms were embedded in agarose, and the intact chromosome was digested with SmaI. DNA fragments were resolved for 22 h with a Bio-Rad CHEF DR-II PFGE unit (Bio-Rad, Hercules, CA).
agr typing.the S. aureus strains were analyzed for their agr types by the method of Jarraud et al. (17). The agr types were identified by PCR amplification of the hypervariable domain of the agr locus by using oligonucleotide primers specific for each of the four major agr types, as described by Shopsin et al. (31).
Detection of PVL genes.The presence of genes encoding PVL was determined by Southern blot analysis, as described below, and by PCR with the following primers: primer LukS-PV (GGCCTTTCCAATACAATATTGG) and primer LukF-PV (CCCAATCAACTTCATAAATTG). Thermal cycling was performed in a GeneAmp 9600 instrument (Perkin-Elmer Corporation, Applied Biosystems, Foster City, CA); and the parameters consisted of initial heating at 95°C for 5 min, followed by 35 cycles of denaturation (1 min at 94°C), annealing (30 s at 57°C), and extension (1 min at 72°C).
The presence of PVL was validated by molecular beacon analysis with the lukF component of pvl. The beacon experiment was carried out by using the following beacon and primers: primer lukF beacon [5′-6-FAM d(CGCGAAGAATTTATTGGTGTCCTATCTCGATCGCG)-DABCYL-3′, where FAM is 6-carboxyfluorescein and DABCYL is 4-(4′-dimethylaminophenylazo)benzoic acid], primer LukF F (5′-GCCAGTGTTATCCAGAGG-3′), and primer LukF R (CTATCCAGTTGAAGTTGATCC-3′).
The quantitative real-time PCR mixture contained 1× I.Q. supermix (Bio-Rad), 0.1 μM of each molecular beacon, 0.5 μM of each primer, and DNA template. The thermal cycling program consisted of 10 min on a spectrofluorometric thermal cycler (iCycler; Bio-Rad) at 95°C, followed by 45 cycles of 30 s at 95°C, 30 s at 50°C, and 30 s at 72°C.
Southern blot analysis.Chromosomal DNA was digested with the ClaI restriction enzyme, and Southern blot hybridization was performed as described elsewhere (28).
RNA isolation.S. aureus strains were cultured to postexponential phase, and RNA isolation and detection were performed as described previously (28).
Reverse transcription-PCR with lukF beacon.The first-strand cDNA for the lukF gene was synthesized by using a Omniscript RT PCR kit (QIAGEN) with the lukF reverse primer described above. Samples incubated in the absence of reverse transcriptase served as controls. Quantitative real-time PCR was performed as described above. A molecular beacon probe for a Staphylococcus genome-specific region of the 16S rRNA gene (SG16S) was used as a control. The sequences of the SG16S forward and reverse primers, as well as the molecular beacon sequence, are as follows: primer SG16S-F, 5′-TGGAGCATGTGGTTTAATTCGA-3′; primer SG16S-R, 5′-TGCGGGACTTAACCCAACA-3′; and probe SG16S-MB, 5′-[HEX]-CGCTGACTTACCAAATCTTGACATCCTTCAGCG-[DABCYL]-3′, where HEX is hexachlorofluorescein.
Standard curves were generated to calculate the copy numbers of each gene in the reaction. Briefly, this was accomplished by taking the cycle threshold (CT) value for each sample and applying the following formula: $$mathtex$$\[\mathrm{copy\ number}\ {=}\ 10^{[(\mathrm{C_{T}}\ {-}\ \mathrm{y\ intercept})/\mathrm{slope}]}\]$$mathtex$$ where the slope and y-intercept values were calculated from the standard curve by using Stratagene Mx4000 software (Stratagene Corporation, La Jolla, CA). Each sample was assayed in triplicate, and the average number of pvl copies was divided by the average number of SG16S copies to obtain a normalization value. The average standard deviation in the cycle thresholds for both pvl and SG16S probes was less than 1 cycle.
Isolation of human PMNs and assay for PMN lysis (release of lactate dehydrogenase [LDH]).Human PMNs were isolated from the venous blood of healthy individuals, in accordance with a protocol approved by the Institutional Review Board for Human Subjects, National Institute of Allergy and Infectious Diseases. Human PMNs were purified by the method described by Boyum (3), but with modifications. Briefly, whole human blood was mixed 1:1 with 0.9% NaCl (Irrigation USP; Abbott Laboratories, North Chicago, Ill.) containing 3% dextran 500 (Amersham Biosciences, Piscataway, NJ) for 20 min to sediment the erythrocytes. The leukocyte-rich supernatant was transferred to a new tube and was centrifuged at ∼670 × g for 10 min. The cells were resuspended in 35 ml 0.9% NaCl, and the suspension was underlaid with 10 ml Hypaque-Ficoll-Paque Plus (1.077 g/liter; Amersham). The cells were centrifuged at 350 × g for 25 min at room temperature to separate the peripheral blood mononuclear cells (PBMCs) from the PMNs and erythrocytes. The PBMC layer was removed by aspiration, and the cell pellet containing the PMNs and erythrocytes was resuspended in water (Irrigation USP; Abbott Laboratories) for 15 to 30 s. Isotonicity was restored by adding an equal volume of 1.7% NaCl. PMNs were centrifuged again, resuspended in RPMI 1640 medium (Invitrogen Corporation, Carlsbad, CA) buffered with 10 mM HEPES (RPMI/H), and enumerated with a hemacytometer. The purities of the PMN preparations and cell viability were routinely assessed by flow cytometry (FACSCalibur; BD Biosciences, San Jose, CA). Cell preparations contained 98 to 99% PMNs, and all reagents used contained <25 pg/ml endotoxin.
Strains of S. aureus were cultured to mid-exponential phase of growth (optical density at 600 nm = 0.75) and resuspended in RPMI/H. Bacteria (107) were combined on ice with human PMNs (106), and phagocytosis was synchronized by centrifugation at 350 × g for 8 min at 4°C. The culture plates were incubated at 37°C with 5% CO2 for the indicated times, and the release of cytosolic PMN LDH (cell lysis) was evaluated with a cytotoxicity detection kit (Roche Applied Sciences, Indianapolis, IN), according to the manufacturer's instructions. The assay is based on the conversion of lactate to pyruvate by LDH, whereby NAD+ is reduced to NADH/H+. Diaphorase utilizes H/H+ from NADH/H+ to reduce 2-(4-iodophenyl)-3-(4-nitrophenyl)-5-phenyltetrazolium chloride to formazan. Colorimetric detection of LDH activity was performed with triplicate wells by using a SpectraMax Plus384 microplate spectrophotometer (Molecular Devices, Sunnyvale, CA) at 490 nm and a reference λ of 600 nm. The percent cell lysis induced by S. aureus was determined with the measured absorbance readings in the equation {[(absorbance for S. aureus and PMN mixture − absorbance for S. aureus alone) − absorbance for time-matched, untreated human PMNs (spontaneous release of LDH)]/[absorbance for PMNs treated with 2% Triton X-100 (maximum releasable LDH activity) − absorbance for time-matched, untreated human PMNs (spontaneous release of LDH)]} × 100.
Molecular characterization of CA-MRSA strains. (i) CA-MRSA spa types.One hundred twenty-one geographically diverse isolates in our strain collection (clustered based on SCCmec type IV) were genotyped by sequencing the polymorphic repeat region of the gene encoding protein A (spa) and by PFGE. From this analysis the CA-MRSA strains were divided into five major groups defined by their spa types (Table 1). The first group consisted of strains with the same spa type as prototypic CA-MRSA strain MW2 (spa type 131; MLST type 1-1-1-1-1-1-1). The second group included strains with spa type 1 isolated from human immunodeficiency virus (HIV)-positive as well as HIV-negative patients in Los Angeles and New York City. Group 3 consisted of strains of spa type 7. We note that these spa types differ by a single nucleotide and that their MLST types differ by a single allele; spa type 1 is MLST 3-3-1-1-4-4, and spa type 7 is MLST 3-3-1-1-4-16. The fourth group consisted of spa type 17 strains, and the fifth group had various spa types and is listed as miscellaneous (Table 1). These results suggest that although CA-MRSA strains share certain molecular characteristics, they are not a single clonal type but, rather, are derived from a number of genetic backgrounds.
Representatives of the CA-MRSA and parental MSSA strains used in this studya
(ii) CA-MRSA PFGE.Using a more discriminatory genotyping tool, PFGE, we were able to differentiate CA-MRSA strains of the same spa type. PFGE patterns were indistinguishable among spa type 131 isolates, indicating that spa type 131 strains define a clone (Fig. 1). The data for spa type 35 strains indicate that these spa types are closely related to each other (at least at the genetic level) and to spa type 131 strains. Furthermore, among the miscellaneous spa types, spa type 194 had a spa repeat pattern similar to that of spa type 131 (Table 1). Of interest, three spa type 194 strains in our collection were isolated from the Midwest (United States), as was our spa type 131 reference strain, MW2. These strains have subtle PFGE pattern differences and are thus closely related to each other and to spa type 131 (Fig. 1). On the other hand, strains of spa types 1, 7, and 17 had significant PFGE pattern differences and deviated significantly from spa type 131 strains. These data indicate that although these strains share the same SCCmec type, they are not clonal (Fig. 1) and their genetic background is quite distinct from that of the MW2 clone.
PFGE of strains with different spa types. STD, molecular size standard.
(iii) CA-MRSA agr types.Four major agr types (agr types I, II, III, and IV) have been identified among S. aureus strains. Two recent studies demonstrated that CA-MRSA strains fall into the agr III group (24, 34). To further determine the genetic backgrounds of our CA-MRSA isolates, we assessed their agr types by PCR, as described previously (31). Consistent with the findings of previous studies (30, 31), CA-MRSA strains of the MW2 clone (spa type 131) and those of spa types 35 and 194 belong to agr type III. Thus, agr type III appears to be linked to strains genetically related to MW2. Interestingly, all of the other strains of the different spa types were agr type I. Taken together, our data from the different genotyping methods (spa, PFGE, and agr typing) demonstrate that CA-MRSA has multiple genetic backgrounds and that the MW2 clone is genetically distinct from the other CA-MRSA strains.
Distribution and expression of PVL in CA-MRSA.Previous studies suggest that PVL is a significant virulence determinant in the pathogenesis of CA-MRSA. Therefore, we analyzed the distribution of PVL among the different CA-MRSA strains using PCR coupled with Southern blot analysis and a lukF-specific beacon. We used additional techniques to validate the PCR results because the genes encoding PVL share high homologies to other leucocidins, such as γ-hemolysin, and in some instances, the primers can amplify these homologous genes. Our results indicate that the PVL genes are present among all spa type 131 isolates (MW2 clone), an observation that supports the idea that these strains are clonal (Table 1). Furthermore, all spa type 194 strains analyzed possessed the PVL genes. On the other hand, none of the spa type 7 strains tested contained the genes encoding PVL. There was a differential distribution of PVL in spa types 1, 17, and 35; and only a subset of these strains contained the gene. Although not all CA-MRSA strains possessed PVL, we observed a significant increase in the number of PVL-positive strains (33 to 50%) among the CA-MRSA strains tested compared to that reported in the literature for other S. aureus strains (<5%) (21). This finding is in agreement with that of a recent study that demonstrated the increased prevalence of PVL among CA-MRSA strains compared to that among HA-MRSA strains (24).
Although a number of investigations have demonstrated an association of the PVL genes and CA-MRSA, their mRNA levels in different genetic backgrounds have not been analyzed. We first confirmed that CA-MRSA isolates containing the lukF and lukS genes expressed the corresponding messages. PVL transcripts were identified by real-time reverse transcription-PCR coupled with the use of a molecular beacon specific for lukF. CA-MRSA BK 9924, which lacks PVL genes, was used as a negative control. Notably, the PVL transcript was expressed in all strains harboring the PVL genes (Fig. 2). However, the levels of expression of genes encoding PVL varied from strain to strain, with isolate 9918 expressing nearly 10-fold more of the PVL message than any other isolate.
Differential expression of the lukF gene of Panton-Valentine leucocidin. The relative levels of lukF transcripts in CA-MRSA strains were determined by real-time PCR with a molecular beacon probe specific for the lukF gene. A molecular beacon directed against a Staphylococcus-specific region of 16S rRNA was used for normalization. †, Strain 9918 had greater than fivefold higher relative lukF transcript levels (1.29) than any of the other strains tested; *, the standard error for strain 11450 could not be reliably calculated due to the presence of an outlier.
PVL- and CA-MRSA-induced PMN lysis.Previous investigators have suggested that PVL facilitates the pathogenesis of S. aureus, presumably by altering human leukocyte responses to infection (10, 32). To assess the impact of PVL in the pathogenesis of CA-MRSA, we determined the cytotoxicity (lysis) for human PMNs during phagocytic interaction with PVL-positive and PVL-negative CA-MRSA strains (Fig. 3A). Each pair of PVL-positive and PVL-negative isolates comprised strains of closely related genetic backgrounds (Fig. 3B). Unexpectedly, there was no significant correlation between strains that expressed PVL and human PMN cytotoxicity. For example, in some cases the PVL-negative strain caused higher levels of PMN lysis compared to the levels of lysis caused by the genetically matched PVL-positive strain. Finally, in the spa type 35 genetic background, the cytotoxicity for PMNs was similar for both PVL-negative and PVL-positive strains (Fig. 3A). Although there is a higher incidence of PVL among CA-MRSA strains, our in vitro data suggest that other factors produced by CA-MRSA strains facilitate leukocidal activity.
PMN lysis during phagocytic interaction with PVL-negative (NEG) and PVL-positive (POS) S. aureus strains of closely related genetic backgrounds. (A) Percentage of PMN lysis over a period of 9 h. Ht648, heat-killed strain 648. (B) Genetic characteristics of the pairs used in the PMN assay.
CA-MRSA is an emerging pathogen that can cause severe and, in some cases, fatal infections in the community. CA-MRSA strains share certain properties such as the presence of a genomic SCCmec type IV element and increased susceptibilities to a variety of antibiotics (compared to those of HA-MRSA strains). We and others demonstrated that CA-MRSA strains have multiple genetic backgrounds, and we have extended this analysis in this study to show how the clonal MW2 CA-MRSA strains are differentiated from other strain families. This is particularly important to understanding of the relationship between genetic backgrounds and CA-MRSA disease.
The agr data indicate that CA-MRSA can be subdivided into two groups. The first group consists of the "true" CA-MRSA strains, i.e., MW2 and related strains (spa types 131 and 194). These strains tend to be susceptible to a wide array of non-beta-lactam antibiotics and have the agr III subtype (Table 1). The prevalence of agr type III among CA-MRSA strains has been reported previously (24, 34), and our results are in agreement with the findings from those studies. The second family consists of strains of spa types 1, 7, 17, and other miscellaneous spa types. In addition to the beta-lactams, this group is also resistant to erythromycin and, sometimes, fluoroquinolones. Unlike the MW2 clone, the second group belongs primarily to the agr I subtype. The grouping of CA-MRSA strains into two agr subgroups is in agreement with our previous study that classified CA-MRSA strains into two epidemiological groups, i.e., CA-MRSA strains with and without risk factors (29). Interestingly, CA-MRSA strains without risk factors correspond to the agr type III subgroup, and CA-MRSA strains with risk factors correspond to the agr type I subgroup.
Although the genes encoding PVL are rarely present in S. aureus strains (<5%) (21), they are highly represented among CA-MRSA strains. All spa type 131 and 194 strains tested contain lukF and lukS, consistent with their clonality, as determined by PFGE analysis (Fig. 1). On the other hand, only a subset of the other spa types has these genes, again, in concordance with their heterogeneous pulsotypes, as determined by PFGE. These results are in contrast to those of a recent study that reported on the presence of PVL in all CA-MRSA strains analyzed (34). Although the study by Vandenesch and colleagues (34) analyzed a collection of diverse CA-MRSA strains, they can be categorized as CA-MRSA strains without risk factors (34). In addition, 97% of the isolates tested were of agr type III. The discrepancy between the two studies may be explained by the difference in the two collections. For instance, the collection used by Vandenesch and colleagues (34) would exclude the agr type I CA-MRSA strains previously described in the Los Angeles population of men who have sex with men (6). As our collection is more inclusive (based on agr types) and consists of CA-MRSA strains both with and without risk factors, we report an unequal distribution of the genes encoding PVL among our strains.
PVL has been demonstrated to be cytotoxic to human PMNs, which are essential for the innate host defense against invading microorganisms (23, 32). Therefore, we used PMN lysis as a proxy for CA-MRSA virulence and pathogenesis. However, our results demonstrate that the presence (or absence) of PVL failed to correlate with the degree of PMN lysis, suggesting that additional factors are involved in leukotoxicity and the pathogenesis of CA-MRSA.
The conflicting results of the role of PVL in the pathogenesis of CA-MRSA are similar to the results reported on the role of the Salmonella plasmid virulence (spv) genes (9). One study demonstrated that spv genes are important virulence determinants, as an spvR mutant of Salmonella enterica serovar Dublin was attenuated in both enteric and systemic diseases. Conversely, another study reported that an spvR mutant of S. enterica serovar Typhimurium causes lethal enteric infection similar to that caused by the wild-type strain. Taken together, these results reiterate the multifactorial nature of bacterial pathogenesis. To better elucidate the role of PVL in the pathogenesis of CA-MRSA, comparative analyses with isogenic strains (PVL positive and negative) both in vitro and in vivo are needed. Deciphering of the pathogenesis of CA-MRSA strains will require multipronged strategies aimed at both the microbial level and the host level (2).
In summary, this study demonstrates that prototypic CA-MRSA strain MW2 and its related strains form a subclone different from other CA-MRSA strains at the genotypic and phenotypic levels. The presence and expression of the PVL-encoding genes are not uniform among CA-MRSA strains. Finally, these data do not support a clear correlation between the presence of PVL and the ability of the strain to cause PMN cytotoxicity.
We are grateful to Surbhi Leekha and Madhavi Koraboina for technical assistance with lukF beacon essays and Alex Ravikovitch for assistance with the figures.
Received 2 November 2004.
Returned for modification 23 December 2004.
Accepted 25 February 2005.
Baba, T., F. Takeuchi, M. Kuroda, H. Yuzawa, K. Aoki, A. Oguchi, Y. Nagai, N. Iwama, K. Asano, T. Naimi, H. Kuroda, L. Cui, K. Yamamoto, and K. Hiramatsu. 2002. Genome and virulence determinants of high virulence community-acquired MRSA. Lancet 359:1819-1827.
Blaser, M. J., and J. M. Musser. 2001. Bacterial polymorphisms and disease in humans. J. Clin. Investig. 107:391-392.
Boyum, A. 1968. Isolation of mononuclear cells and granulocytes from human blood. Isolation of mononuclear cells by one centrifugation, and of granulocytes by combining centrifugation and sedimentation at 1 g. Scand. J. Clin. Lab. Investig. Suppl. 97:77-89.
Centers for Disease Control and Prevention. 1999. Four pediatric deaths from community-acquired methicillin-resistant Staphylococcus aureus—Minnesota and North Dakota, 1997-1999. JAMA 282:1123-1125.
Centers for Disease Control and Prevention. 1999. National Nosocomial Infections Surveillance (NNIS) System report, data summary from January 1990-May 1999, issued June 1999. Am. J. Infect. Control 27:520-532.
Centers for Disease Control and Prevention. 2003. Outbreaks of community-associated methicillin-resistant Staphylococcus aureus skin infections—Los Angeles County, California, 2002-2003. Morb. Mortal. Wkly. Rep. 52:88.
Daum, R. S., T. Ito, K. Hiramatsu, F. Hussain, K. Mongkolrattanothai, M. Jamklang, and S. Boyle-Vavra. 2002. A novel methicillin-resistance cassette in community-acquired methicillin-resistant Staphylococcus aureus isolates of diverse genetic backgrounds. J. Infect. Dis. 186:1344-1347.
Fey, P. D., B. Said-Salim, M. E. Rupp, S. H. Hinrichs, D. J. Boxrud, C. C. Davis, B. N. Kreiswirth, and P. M. Schlievert. 2003. Comparative molecular analysis of community- or hospital-acquired methicillin-resistant Staphylococcus aureus. Antimicrob. Agents Chemother. 47:196-203.
Fierer, J., and D. G. Guiney. 2001. Diverse virulence traits underlying different clinical outcomes of Salmonella infection. J. Clin. Investig. 107:775-780.
Gillet, Y., B. Issartel, P. Vanhems, J. C. Fournet, G. Lina, M. Bes, F. Vandenesch, Y. Piemont, N. Brousse, D. Floret, and J. Etienne. 2002. Association between Staphylococcus aureus strains carrying gene for Panton-Valentine leukocidin and highly lethal necrotising pneumonia in young immunocompetent patients. Lancet 359:753-759.
Groom, A. V., D. H. Wolsey, T. S. Naimi, K. Smith, S. Johnson, D. Boxrud, K. A. Moore, and J. E. Cheek. 2001. Community-acquired methicillin-resistant Staphylococcus aureus in a rural American Indian community. JAMA 286:1201-1205.
Gross-Schulman, S., D. Dassey, L. Mascola, and C. Anaya. 1998. Community-acquired methicillin-resistant Staphylococcus aureus. JAMA 280:421-422.
Herold, B. C., L. C. Immergluck, M. C. Maranan, D. S. Lauderdale, R. E. Gaskin, S. Boyle-Vavra, C. D. Leitch, and R. S. Daum. 1998. Community-acquired methicillin-resistant Staphylococcus aureus in children with no identified predisposing risk. JAMA 279:593-598.
Hiramatsu, K., L. Cui, M. Kuroda, and T. Ito. 2001. The emergence and evolution of methicillin-resistant Staphylococcus aureus. Trends Microbiol. 9:486-493.
Hiramatsu, K., Y. Katayama, H. Yuzawa, and T. Ito. 2002. Molecular genetics of methicillin-resistant Staphylococcus aureus. Int. J. Med. Microbiol. 292:67-74.
Ito, T., Y. Katayama, K. Asada, N. Mori, K. Tsutsumimoto, C. Tiensasitorn, and K. Hiramatsu. 2001. Structural comparison of three types of staphylococcal cassette chromosome mec integrated in the chromosome in methicillin-resistant Staphylococcus aureus. Antimicrob. Agents Chemother. 45:1323-1336.
Jarraud, S., C. Mougel, J. Thioulouse, G. Lina, H. Meugnier, F. Forey, X. Nesme, J. Etienne, and F. Vandenesch. 2002. Relationships between Staphylococcus aureus genetic background, virulence factors, agr groups (alleles), and human disease. Infect. Immun. 70:631-641.
Jones, T. F., M. E. Kellum, S. S. Porter, M. Bell, and W. Schaffner. 2002. An outbreak of community-acquired foodborne illness caused by methicillin-resistant Staphylococcus aureus. Emerg. Infect. Dis. 8:82-84.
Koreen, L., S. V. Ramaswamy, E. A. Graviss, S. Naidich, J. M. Musser, and B. N. Kreiswirth. 2004. spa typing method for discriminating among Staphylococcus aureus isolates: implications for use of a single marker to detect genetic micro- and macrovariation. J. Clin. Microbiol. 42:792-799.
L'Heriteau, F., J. C. Lucet, A. Scanvic, and E. Bouvet. 1999. Community-acquired methicillin-resistant Staphylococcus aureus and familial transmission. JAMA 282:1038-1039.
Lina, G., Y. Piemont, F. Godail-Gamot, M. Bes, M. O. Peter, V. Gauduchon, F. Vandenesch, and J. Etienne. 1999. Involvement of Panton-Valentine leukocidin-producing Staphylococcus aureus in primary skin infections and pneumonia. Clin. Infect. Dis. 29:1128-1132.
Livermore, D. M. 2000. Antibiotic resistance in staphylococci. Int. J. Antimicrob. Agents 16(Suppl. 1):S3-S10.
Lowy, F. D. 1998. Staphylococcus aureus infections. N. Engl. J. Med. 339:520-532.
Naimi, T. S., K. H. LeDell, K. Como-Sabetti, S. M. Borchardt, D. J. Boxrud, J. Etienne, S. K. Johnson, F. Vandenesch, S. Fridkin, C. O'Boyle, R. N. Danila, and R. Lynfield. 2003. Comparison of community- and health care-associated methicillin-resistant Staphylococcus aureus infection. JAMA 290:2976-2984.
Okuma, K., K. Iwakawa, J. D. Turnidge, W. B. Grubb, J. M. Bell, F. G. O'Brien, G. W. Coombs, J. W. Pearman, F. C. Tenover, M. Kapi, C. Tiensasitorn, T. Ito, and K. Hiramatsu. 2002. Dissemination of new methicillin-resistant Staphylococcus aureus clones in the community. J. Clin. Microbiol. 40:4289-4294.
Oliveira, D. C., A. Tomasz, and H. de Lencastre. 2002. Secrets of success of a human pathogen: molecular evolution of pandemic clones of meticillin-resistant Staphylococcus aureus. Lancet Infect. Dis. 2:180-189.
Robinson, D. A., and M. C. Enright. 2003. Evolutionary models of the emergence of methicillin-resistant Staphylococcus aureus. Antimicrob. Agents Chemother. 47:3926-3934.
Said-Salim, B., P. M. Dunman, F. M. McAleese, D. Macapagal, E. Murphy, P. J. McNamara, S. Arvidson, T. J. Foster, S. J. Projan, and B. N. Kreiswirth. 2003. Global regulation of Staphylococcus aureus genes by Rot. J. Bacteriol. 185:610-619.
Said-Salim, B., B. Mathema, and B. N. Kreiswirth. 2003. Community-acquired methicillin-resistant Staphylococcus aureus: an emerging pathogen. Infect. Control Hosp. Epidemiol. 24:451-455.
Salmenlinna, S., O. Lyytikainen, and J. Vuopio-Varkila. 2002. Community-acquired methicillin-resistant Staphylococcus aureus, Finland. Emerg. Infect. Dis. 8:602-607.
Shopsin, B., B. Mathema, P. Alcabes, B. Said-Salim, G. Lina, A. Matsuka, J. Martinez, and B. N. Kreiswirth. 2003. Prevalence of agr specificity groups among Staphylococcus aureus strains colonizing children and their guardians. J. Clin. Microbiol. 41:456-459.
Staali, L., H. Monteil, and D. A. Colin. 1998. The staphylococcal pore-forming leukotoxins open Ca2+ channels in the membrane of human polymorphonuclear neutrophils. J. Membr. Biol. 162:209-216.
Tenover, F. C., R. D. Arbeit, R. V. Goering, P. A. Mickelsen, B. E. Murray, D. H. Persing, and B. Swaminathan. 1995. Interpreting chromosomal DNA restriction patterns produced by pulsed-field gel electrophoresis: criteria for bacterial strain typing. J. Clin. Microbiol. 33:2233-2239.
Vandenesch, F., T. Naimi, M. C. Enright, G. Lina, G. R. Nimmo, H. Heffernan, N. Liassine, M. Bes, T. Greenland, M. E. Reverdy, and J. Etienne. 2003. Community-acquired methicillin-resistant Staphylococcus aureus carrying Panton-Valentine leukocidin genes: worldwide emergence. Emerg. Infect. Dis. 9:978-984.
Journal of Clinical Microbiology Jul 2005, 43 (7) 3373-3379; DOI: 10.1128/JCM.43.7.3373-3379.2005
You are going to email the following Differential Distribution and Expression of Panton-Valentine Leucocidin among Community-Acquired Methicillin-Resistant Staphylococcus aureus Strains | CommonCrawl |
What rotates a door : torque or centripetal force?
I can realize that they are different, but can't understand what actually creates a rotational motion. Rotational motion doesn't mean just moving in a circle. There must be rotation of the position vector for rotational motion. For that, torque is required. Centripetal force,on the other hand,rotates the velocity vector. If this is the difference,what actually makes a door to rotate? Most book writes, it is torque, since the postition vector of the door's edge is rotating with respect to a hinge. But isn't there centripetal force,which is changing the direction of the velocity vector of the edge of the door to let it move in a circle?? What is actually happening? What is the difference between these two (apart from units!)??
rotational-dynamics
You need a torque/force to start or stop a door, because an external torque makes something slow down how fast it rotates or speed up how fast it rotates or change the direction of how it rotates.
But an external torque is not why the door keeps rotating. Imagine that you and your friend are out of the ice and you grasp each other hands facing each other. If a torque gets you spinning (say someone/something pushes your friend east and pushes you west) then you have hold on to each other to rotate in that symmetric non-expanding way. This is how rigid objects rotate, they are holding themselves together because they are stretching like a rubber band.
So let's now look at a top. At rest it is a certain size, the size that is a natural equilibrium based on the stuff it is made of of, how far the atoms can be and hold each other in place. But if you spin it, then it gets fatter in the middle, but that equilibrium position resists you making it bigger so it tries to pull itself back together, but the direction it pulls itself together is orthogonal to the velocity, so it doesn't change the speed, so it just keeps spinning. This force is a force that the object is exerting on itself. The parts on the outside are pulling the inner parts in an outward direction while those inner parts are pulling the outer parts in an inwards direction, but nobody goes any faster or slower because the directions they are pulling are orthogonal to the velocity.
TimaeusTimaeus
Centripetal force and torque are two very different things. Applying a torque to the door will cause angular acceleration while the centripetal force preserves rotation. The centripetal force is required to maintain circular motion and is provided by the bonds between the molecules in the door. A door is fairly rigid and so this force can be ignored as the "right amount" of centripetal force will be provided.
Your confusion is between torque and centripetal force. The torque produces the angular acceleration which changes the door's angular displacement, which causes the door to spin. The centripetal force, which is provided by the bonds between the molecules and in the hinge, keeps the door in its circular motion (otherwise the door would fly in a straight line).
curiousgeorgecuriousgeorge
First, let's grasp the idea that there is no force that is "the centripetal force." There are forces like gravity, Lorentz (EM), tension, and spring which can have components directed toward a center of curvature. A net force toward a center of curvature will have an associated centripetal acceleration, and the product of mass times this centripetal acceleration is commonly (and IMHO, confusingly) called centripetal force. But $m\frac{v^2}{r}$, while having force units, is not a force. It is mass ($m$) times acceleration ($\frac{v^2}{r}$).
The torque produced by a force involves the force component acting perpendicular to the line from a mass point to some defined point, usually a point on a fixed axis (but not always). This perpendicular component times the distance from the defined point to the application of the force is a torque. See the contrast between the perpendicular component used in the torque and it's product with the distance, and the radial (and usually) parallel component of force (and no distance multiplier) which contributes to the centripetal acceleration. Centripetally-directed force components product zero torque about the center of curvature.
Let's use a door swinging on hinges about a vertical axis to contrast the two. A net torque about the vertical axis colinear with the hinges will produce an angular acceleration about the hinges. This torque might be produced by someone's hand exerting a horizontal force perpendicular to the door.
As the door rotates, every part of the door has a changing velocity, changing in direction. This means (by Newton's First Law) that there is some force acting perpendicular to the velocity, and therefore, parallel to the door. This force is the electrostatic force holding the door together, and it acts toward the axis of rotation. It is producing a centripetal acceleration. It keeps the particles of the door from moving on straight lines. It didn't produce the rotation, but it holds the pieces of the door in a circular path.
Rotation is also a matter of perspective. Consider an airplane flying over a highway at constant speed and constant height. Let's ignore the slight curvature of the Earth for a moment. One might say the airplane is flying in a straight line, and that, to first order, is true. Now imagine a person sitting beside that highway, watching the plane through binoculars. That person will have to rotate the direction of the binoculars in order to watch the plane. One could then say the airplane is rotating around the person, which is also true.
Torques are always calculated about points, but there is no absolute point one must use. There are, however, some points which are more convenient to use. Centripetal accelerations, however, are always toward the instantaneous center of curvature. For circular paths, that happens to be the center of the circle.
Torques change angular momentum vectors and result in rotations of massive particles about points. Forces change momentum vectors and result in accelerations of massive objects. The rotation of a vector does not require a torque because the vector itself is not massive.
Bill NBill N
$\begingroup$ I didn't thank you for your answer when you posted at the mid-year. Sorry for that. Didn't look back again at the answer. However, +1 lately:) $\endgroup$ – user36790 Dec 21 '15 at 16:03
Finding force exerted on a turbine blade by water flow
Centripetal force of a rotating rigid body?
What sustains a rigid body's rotation at its constant angular(rotational) speed?
Optimal door opening
If direction of torque is upwards(or downwards), why does the body rotate perpendicular to the direction?
Torque for a door
What is really the force required to open a door? | CommonCrawl |
Only show content I have access to (77)
Over 3 years (417)
Physics And Astronomy (302)
Materials Research (282)
Statistics and Probability (13)
MRS Online Proceedings Library Archive (269)
Microscopy and Microanalysis (58)
Epidemiology & Infection (12)
Proceedings of the International Astronomical Union (10)
Symposium - International Astronomical Union (10)
European Journal of Anaesthesiology (4)
Bulletin of Entomological Research (3)
High Power Laser Science and Engineering (3)
International Astronomical Union Colloquium (3)
European Astronomical Society Publications Series (2)
Journal of Fluid Mechanics (2)
Journal of Plasma Physics (2)
Psychological Medicine (2)
The Journal of Anatomy (2)
Transactions of the International Astronomical Union (2)
Materials Research Society (277)
Brazilian Society for Microscopy and Microanalysis (SBMM) (58)
International Astronomical Union (25)
Ryan Test (2)
EDPS Sciences - Radioprotection (1)
MBA Online Only Members (1)
Mineralogical Society (1)
The Australian Society of Otolaryngology Head and Neck Surgery (1)
Weed Science Society of America (1)
World's Poultry Science Association (1)
Cambridge Contemporary Astrophysics (1)
International Hydrology Series (1)
Quantum electrodynamics experiments with colliding petawatt laser pulses
HPL_EP HEDP and High Power Laser 2018
I. C. E. Turcu, B. Shen, D. Neely, G. Sarri, K. A. Tanaka, P. McKenna, S. P. D. Mangles, T.-P. Yu, W. Luo, X.-L. Zhu, Y. Yin
Journal: High Power Laser Science and Engineering / Volume 7 / 2019
Published online by Cambridge University Press: 14 February 2019, e10
A new generation of high power laser facilities will provide laser pulses with extremely high powers of 10 petawatt (PW) and even 100 PW, capable of reaching intensities of $10^{23}~\text{W}/\text{cm}^{2}$ in the laser focus. These ultra-high intensities are nevertheless lower than the Schwinger intensity $I_{S}=2.3\times 10^{29}~\text{W}/\text{cm}^{2}$ at which the theory of quantum electrodynamics (QED) predicts that a large part of the energy of the laser photons will be transformed to hard Gamma-ray photons and even to matter, via electron–positron pair production. To enable the investigation of this physics at the intensities achievable with the next generation of high power laser facilities, an approach involving the interaction of two colliding PW laser pulses is being adopted. Theoretical simulations predict strong QED effects with colliding laser pulses of ${\geqslant}10~\text{PW}$ focused to intensities ${\geqslant}10^{22}~\text{W}/\text{cm}^{2}$ .
Research Highlights: Perovskites
Prachi Patel, Pabitra K. Nayak
Journal: MRS Bulletin / Volume 43 / Issue 9 / September 2018
Detection of modified measles and super-spreader using a real-time reverse transcription PCR in the largest measles outbreak, Yamagata, Japan, 2017 in its elimination era
J. Seto, T. Ikeda, S. Tanaka, K. Komabayashi, Y. Matoba, Y. Suzuki, S. Takeuchi, T. Yamauchi, K. Mizuta
Journal: Epidemiology & Infection / Volume 146 / Issue 13 / October 2018
We aimed to verify the effectiveness of real-time reverse transcription (rRT) polymerase chain reaction (PCR) for detecting cases of modified measles (M-Me) and for predicting super-spreader candidates through the experience of a measles outbreak dominated by M-Me in Yamagata, Japan, during March–April 2017. We applied rRT-PCR to specimens from 35 cases of M-Me, nine cases of typical measles (T-Me) and nine cases of prodromal stage of T-Me (P-Me). From rRT-PCR among the M-Me cases, peripheral blood mononuclear cells (PBMC) showed the highest positive rate (80.0%), followed by throat swab (48.6%), urine (33.3%) and serum (3.1%). The negative result of PBMC in M-Me cases was recovered by the result of a throat swab. In specimens of PBMC, throat swab and urine, M-Me group showed the significantly higher cycle of threshold (i.e., lower viral load) in the rRT-PCR than T-Me and P-Me groups, respectively. Furthermore, three super-spreaders in T-Me or P-Me showed an extremely low cycle of threshold in their throat swab specimens. rRT-PCR using PBMC and throat swab might be helpful for clinical management and measles control by certain detection of M-Me cases and by predicting super-spreading events resulting from measles cases with the high viral load.
Extraction of templates from phrases using Sequence Binary Decision Diagrams
D. HIRANO, K. TANAKA-ISHII, A. FINCH
Journal: Natural Language Engineering / Volume 24 / Issue 5 / September 2018
The extraction of templates such as 'regard X as Y' from a set of related phrases requires the identification of their internal structures. This paper presents an unsupervised approach for extracting templates on-the-fly from only tagged text by using a novel relaxed variant of the Sequence Binary Decision Diagram (SeqBDD). A SeqBDD can compress a set of sequences into a graphical structure equivalent to a minimal deterministic finite state automata, but more compact and better suited to the task of template extraction. The main contribution of this paper is a relaxed form of the SeqBDD construction algorithm that enables it to form general representations from a small amount of data. The process of compression of shared structures in the text during Relaxed SeqBDD construction, naturally induces the templates we wish to extract. Experiments show that the method is capable of high-quality extraction on tasks based on verb+preposition templates from corpora and phrasal templates from short messages from social media.
Diagnostic role of mean platelet volume in tonsillitis with and without peritonsillar abscess
Y Nakao, T Tanigawa, F Kano, H Tanaka, N Katahira, T Ogawa, K Murotani, T Nagata, R Shibata
Journal: The Journal of Laryngology & Otology / Volume 132 / Issue 7 / July 2018
To assess the diagnostic role of mean platelet volume in tonsillitis with and without peritonsillar abscess.
Mean platelet volume and other laboratory data were retrospectively investigated.
Mean platelet volume was significantly lower in the tonsillitis group (7.8 per cent ± 0.7 per cent) than in the control group (8.7 per cent ± 0.6 per cent; p < 0.0001), and it was significantly lower in the abscess group (7.5 per cent ± 0.6 per cent) than in the no abscess group (8.0 per cent ± 0.7 per cent; p = 0.0277). White blood cell counts and C-reactive protein levels were not significantly different between patients with an abscess and those without. The mean platelet volume cut-off values for the diagnosis of tonsillitis and peritonsillar abscess were 7.95 fl and 7.75 fl, respectively.
Our results suggest that a decreased mean platelet volume is associated with the development and severity of tonsillitis. This finding provides useful diagnostic information for physicians treating patients with tonsillitis.
Oxoammonium cation of 2,2,6,6-tetramethylpiperidin-1-oxyl: a very efficient dopant for hole-transporting triaryl amines in a perovskite solar cell
H. Maruo, S. Tanaka, M. Takamura, K. Oyaizu, H. Segawa, H. Nishide
Journal: MRS Communications / Volume 8 / Issue 1 / March 2018
Oxoammonium cation of 2,2,6,6-tetramethylpiperidin-1-oxyl (TEMPO) was used as an oxidizing dopant of triaryl amines to efficiently and almost quantitatively generate radical cations of the amines or a hole carrier. The doped-triaryl amines yielded an amorphous and homogeneous layer without any residual oxidant or neutral TEMPO molecule through its sublimination or warming the layer. The TEMPO cation-doped spiro-OMeTAD [tetrakis(dimethoxyphenylamine)spirobifluorene] produced a high hole mobility of 2 × 10−4 cm2/Vs. The perovskite solar cell fabricated with the TEMPO cation-doped or residual dopant-free spiro-OMeTAD as the hole-transporting layer displayed a photo-conversion efficiency of 20.1% with durability.
FARMERS' PERCEPTIONS ON MECHANICAL WEEDERS FOR RICE PRODUCTION IN SUB-SAHARAN AFRICA
JEAN-MARTIAL JOHNSON, JONNE RODENBURG, ATSUKO TANAKA, KALIMUTHU SENTHILKUMAR, KOKOU AHOUANTON, IBNOU DIENG, AGOSSOU KLOTOE, CYRIAQUE AKAKPO, ZACHARIE SEGDA, LOUIS P. YAMEOGO, HENRI GBAKATCHETCHE, GEORGE K. ACHEAMPONG, RALPH K. BAM, OLADELE S. BAKARE, ALAIN KALISA, ELIE R. GASORE, SÉKOU ANI, KOMLAN ABLEDE, KAZUKI SAITO
Journal: Experimental Agriculture / Volume 55 / Issue 1 / February 2019
Competition from weeds is one of the major biophysical constraints to rice (Oryza spp.) production in sub-Saharan Africa. Smallholder rice farmers require efficient, affordable and labour-saving weed management technologies. Mechanical weeders have shown to fit this profile. Several mechanical weeder types exist but little is known about locally specific differences in performance and farmer preference between these types. Three to six different weeder types were evaluated at 10 different sites across seven countries – i.e., Benin, Burkina Faso, Côte d'Ivoire, Ghana, Nigeria, Rwanda and Togo. A total of 310 farmers (173 male, 137 female) tested the weeders, scored them for their preference, and compared them with their own weed management practices. In a follow-up study, 186 farmers from Benin and Nigeria received the ring hoe, which was the most preferred in these two countries, to use it during the entire crop growing season. Farmers were surveyed on their experiences. The probability of the ring hoe having the highest score among the tested weeders was 71%. The probability of farmers' preference of the ring hoe over their usual practices – i.e., herbicide, traditional hoe and hand weeding – was 52, 95 and 91%, respectively. The preference of this weeder was not related to gender, years of experience with rice cultivation, rice field size, weed infestation level, water status or soil texture. In the follow-up study, 80% of farmers who used the ring hoe indicated that weeding time was reduced by at least 31%. Of the farmers testing the ring hoe in the follow-up study, 35% used it also for other crops such as vegetables, maize, sorghum, cassava and millet. These results suggest that the ring hoe offers a gender-neutral solution for reducing labour for weeding in rice as well as other crops and that it is compatible with a wide range of environments. The implications of our findings and challenges for out-scaling of mechanical weeders are discussed.
Long-term and highly frequent monitor of 6.7 GHz methanol masers to statistically research periodic flux variations around high-mass protostars using the Hitachi 32-m
Koichiro Sugiyama, Y. Yonekura, K. Motogi, Y. Saito, T. Yamaguchi, M. Momose, M. Honma, T. Hirota, M. Uchiyama, N. Matsumoto, K. Hachisuka, K. Inayoshi, K. E. I. Tanaka, T. Hosokawa, K. Fujisawa
Journal: Proceedings of the International Astronomical Union / Volume 13 / Issue S336 / September 2017
Published online by Cambridge University Press: 16 July 2018, pp. 45-48
We initiated a long-term and highly frequent monitoring project toward 442 methanol masers at 6.7 GHz (Dec >−30 deg) using the Hitachi 32-m radio telescope in December 2012. The observations have been carried out daily, monitoring a spectrum of each source with intervals of 9–10 days. In September 2015, the number of the target sources and intervals were redesigned into 143 and 4–5 days, respectively. This monitoring provides us complete information on how many sources show periodic flux variations in high-mass star-forming regions, which have been detected in 20 sources with periods of 29.5–668 days so far (e.g., Goedhart et al. 2004). We have already obtained new detections of periodic flux variations in 31 methanol sources with periods of 22–409 days. These periodic flux variations must be a unique tool to investigate high-mass protostars themselves and their circumstellar structure on a very tiny spatial scale of 0.1–1 au.
Soft X-ray Observation of Supernova Remnants in the Small Magellanic Cloud
H. Inoue, K. Koyama, Y. Tanaka
Journal: Symposium - International Astronomical Union / Volume 101 / 1983
The Small and Large Magellanic Clouds (SMC and LMC) are the nearest neighbouring galaxies. Their proximity enables us to investigate these galaxies in X-rays in fair detail.
Creation of High-Energy Electron Tails by the Lower-Hybrid Waves and its Relevance to Type II and III Bursts
Motohiko Tanaka, K. Papadopoulos
It is commonly anticipated that high-energy electrons play an important role for the wave emission in flare bursts. For instance, electrons with >100 KeV are considered to create microwave emissions through gyro-synchrotron process and hard x-rays may be due to bremstrahlung with >25 KeV electrons. However, electron acceleration mechanism itself is still in speculations.
Red Galaxies around a Quasar at z=1.1 and their Ages
I. Tanaka, T. Yamada, A. Aragón-Salamanca, T. Kodama, K. Ohta, N. Arimoto
Published online by Cambridge University Press: 25 May 2016, p. 164
We obtained near-infrared and new deep optical images of the field near the radio-loud quasar 1335.8+2834 at z=1.086 where excess of galaxy surface number density was reported by Huthings et al. [AJ, 106, 1324]. We found a clustering of objects with very red optical-NIR color, 4 < R–K < 6 and 3 < I–K < 5 near the quasar. The colors and magnitude of the reddest objects are consistent with those predicted for luminous (> 0.5L ∗) and old (2-4 Gyr old) passively evolving elliptical galaxies at z=1.1.
Clustering of Red Galaxies Near a Radio-Loud Quasar at z = 1.086
We investigated the environment of the radio-loud quasar 1335.8+2834 at z = 1.086 where an excess surface number density of galaxies was reported by Huthings et al. (1993). By obtaining near-infrared and new deep optical images of the field, we found a clustering of objects with very red optical-NIR color, 4 < R - K < 6 and 3 < I - K < 5 near the quasar. The colors and magnitude of the reddest objects are consistent with those predicted for old (2–4 Gyr) passively evolving elliptical galaxies at z = 1.1.
Quick Observations of the Fading X-Rays from Gamma-Ray Bursts with ASCA
T. Murakami, Y. Ueda, R. Fujimoto, M. Ishida, R. Shibata, S. Uno, F. Nagase, Isas Team, A. Yoshida, N. Kawai, F. Tokani, C. Otani, Riken Team, F.E. Marshall, R.H.D. Corbet, J.H. Swank, T. Takeshima, D.A. Smith, A. Levine, R.A. Remillard, R. Vanderspek, Rxte Team, C.R. Robinson, C. Kouveliotou, C. Meegan, V. Connaughton, R.M. Kippen, Batse Team, K. Hurley, UCB, S.D. Barthelmy, GCN, L. Piro, E. Costa, J. Heise, F. Fiore, SAX Team, J.V. Paradijs, Y. Tanaka, UOA, J. Greiner, AIP
Since the discovery of fading X-rays from Gamma-Ray Bursts (GRBs) with BeppoSAX (Piro et al. 1997, Costa et al. 1997), world-wide follow-up observations in optical band have achieved the fruitful results. The case of GRB 970228, there was an optical transient, coincides with the BeppoSAX position and faded (Paradijs et al. 1997, Sahu et al. 1997). These optical observations also confirmed the extended component, which was associated with the optical transient. The new transient are fading with a power-law function in time and the later observation of HST confirmed the extended emission is stable (Fruchter et al. 1997). This extended object seems to be a distant galaxy and strongly suggests to be the host.
X-Ray Determination of the Black-Hole Mass in Cygnus X-1
A. Kubota, K. Makishima, T. Dotani, H. Inoue, K. Mitsuda, F. Nagase, H. Negoro, Y. Ueda, K. Ebisawa, S. Kitamoto, Y. Tanaka
About 10 X-ray binaries in our Galaxy and LMC/SMC are considered to contain black hole candidates (BHCs). Among these objects, Cyg X-1 was identified as the first BHC, and it has led BHCs for more than 25 years(Oda 1977, Liang and Nolan 1984). It is a binary system composed of normal blue supergiant star and the X-ray emitting compact object. The orbital kinematics derived from optical observations indicates that the compact object is heavier than ~ 4.8 M⊙ (Herrero 1995), which well exceeds the upper limit mass for a neutron star(Kalogora 1996), where we assume the system consists of only two bodies. This has been the basis for BHC of Cyg X-1.
Chemical Abundances of Early Type Stars
S. Tanaka, S. Kitamoto, T. Suzuki, K. Torii, M.F. Corcoran, W. Waldron
X-rays from early-type stars are emitted by the corona or the stellar wind. The materials in the surface layer of early-type stars are not contaminated by nuclear reactions in the stellar inside. Therefore, abundance study of the early-type stars provides us an information of the abundances of the original gas. However, the X-ray observations indicate low-metallicity, which is about 0.3 times of cosmic abundances. This fact raises the problem on the cosmic abundances.
Energy Spectra of X 1636-536 Observed with ASCA
K. Asai, T. Dotani, K. Mitsuda, H. Inoue, Y. Tanaka, W. H. G. Lewin
Absorption line features were detected at 4.1 keV from X 1636-536 with the Tenma satellite in the spectra of X-ray bursts (Waki et al., 1984). Similar features were also detected from X 1608-52 and EXO 1747-214 during bursts (Nakamura et al., 1988; Magnier et al., 1989). These features at 4.1 keV may be interpreted as the redshifted Kα absorption line of helium-like iron atoms. However, such interpretation requires extremely soft equation of state for the nuclear matter, and confirmation with high resolution detectors is urged (Lewin et al., 1993). To investigate the line features, we observed X 1636-536 with ASCA for ~ 240 ksec.
IV. Results from Hinotori and P78-1
K. Tanaka
Journal: Transactions of the International Astronomical Union / Volume 19 / Issue 1 / 1985
Hinotori observed 720 flares through its operation February 1981-October 1981. General discussions of the results were given in two symposia: the Hinotori Symposium (ISAS 1982) and the U.S.-Japan Seminar (de Jager & Švestka 1983). The hard x-ray imaging made at the effective energy 20-35 keV showed a wide variety of morphology. Many flares (22 out of 30 events) showed single source structure, either compact (12) or extended (10) in the spatial resolution of 15 arc sec (Takakura et al. 1983a, Ohki et al. 1983, Takakura 1984). Evidences are given in some limb events that the main source is located in the high corona (1-4 x 104 km) (Takakura et al. 1983b). The extended single source could be the whole coronal loops which may include footpoints, but the maximum brightness is near the loop top. In some events (8 out of 30), weak subsource(s) which sould be identified with the footpoint(s) appear intermittently (Tsuneta et al. 1983). Takakura et al. (1984) found that the extended single source becomes compact and slightly shifts to higher altitudes in later phases of the impulsive burst (5 out of 10 events). Tanaka (1984), Takakura (1984), and Tanaka S Zirin (1984) argued that the hard x-ray morphology of the impulsive burst is consistent with the non-thermal electron beam model in high density corona. Sakurai (1983) investigated magnetic field structures of the hard x-ray sources based on the potential field calculations. Timing between the hard x-ray and microwave in the impulsive bursts was examined by Takakura et al. (1983c), who found correlated subsecond time structures and also by Takakura et al. (1983d), who found a long (5-10 s) delay of the peaks at 17 GHz and E > 300 keV to the peak at E < 100 keV. Kurokawa (1983) showed detailed coincidence between the hard x-ray spikes and Hα brightenings.
X-ray Observations of SN 1987A with Ginga
H. Inoue, K. Hayashida, M. Itoh, H. Kondo, K. Mitsuda, T. Takeshima, Y. Tanaka, K. Yoshida, Kazuo Makishima
Journal: Publications of the Astronomical Society of Australia / Volume 9 / Issue 1 / 1991
Published online by Cambridge University Press: 25 April 2016, p. 107
This paper summarises Ginga X-ray observations of SN 1987 A.
Commission 10. Solar Activity
E. Tandberg-Hanssen, V. Gaizauskas, M. Pick, A. Bhatnagar, V. Bumba, E. A. Gurtuvenko, M. Machado, D. B. Melrose, E. R. Priest, K. Tanaka, S. H. Ye, H. Zirin
In preparing the present report, which covers the period July 1, 1981, to June 30, 1984, close collaboration has taken place between Commissions 10 and 12, the two solar commissions, in order to avoid duplications and to insure that pertinent subjects are treated. The reader is referred to the report of Commission 12 for further solar topics. It is a pleasure to acknowledge the excellent work of the reviewers who wrote the different sections of this report, and all the members of the commission who provided information on research to be included.
Derivation of the Ionization Balance for Iron XIV/XXV and XXIII/XXIV Using Solar X-Ray Data
E. Antonucci, M.A. Dodero, A.H. Gabriel, K. Tanaka
Journal: International Astronomical Union Colloquium / Volume 86 / 1984
The relative concentrations of different ionization stages of iron are measured using the spectral emission of plasmas formed during solar flares. This is an extension of a study on the ionization balance of heavy elements, initiated with the analysis of calcium solar spectra (Antonucci et al., 1984). The data consist of a large set of iron spectra in the wavelength range from 1.84 to 1.88 Å, detected during the recent maximum of activity with the X-ray Polychromator Bent Crystal Spectrometer (BCS) on the NASA Solar Maximum Mission satellite and on the Soft X-ray Crystal Spectrometer (SOX) on the Hinotori satellite.
At the low densities typical of the solar corona, in the steady state the ionization balance of an element is a function of the plasma electron temperature. Hence, it can be measured for plasmas of known temperature and in slowly varying physical conditions, and in most cases, solar flare plasmas can be considered to be in such conditions. | CommonCrawl |
\begin{document}
\title[An inverse problem of determining the orders ... ]{An inverse problem of determining the orders of systems of fractional pseudo-differential equations}
\author{Ravshan Ashurov} \author{Sabir Umarov} \date{}
\address{ $^1$Institute of Mathematics of Academy of Sciences of Republic of Uzbekistan} \email{[email protected]}
\address{ $^2$University of New Haven, Department of Mathematics, 300 Boston Post Road, \\ West Haven, CT 06516, USA}
\email{[email protected]}
\begin{abstract} As it is known various dynamical processes can be modeled through the systems of time-fractional order pseudo-differential equations. In the modeling process one frequently faces with determining adequate orders of time-fractional derivatives in the sense of Riemann-Liouville or Caputo. This problem is qualified as an inverse problem. The right (vector) order can be found utilizing the available data. In this paper we consider this inverse problem for linear systems of fractional order pseudo-differential equations. We prove that the Fourier transform of the vector-solution $\widehat{U}(t, \xi)$ evaluated at a fixed time instance, which becomes possible due to the available data, recovers uniquely the unknown vector-order of the system of governing pseudo-differential equations.
\end{abstract}
\maketitle
\noindent {\it Keywords}: system of differential equations, fractional order differential equation, pseudo-differential operator, matrix symbol, inverse problem, determination of the fractional derivative's order
\section{Introduction}\label{sec:1}
In modern science and engineering researchers frequently use fractional order differential equations for modeling of dynamics of various complex stochastic processes arising in different fields; see, for example, \cite{Hilfer,Mainardi,MetzlerKlafter00,West} in physics, \cite{MachadoLopes,SGM} in finance, \cite{BMR} in hydrology, \cite{Magin} in cell biology, among others. In the last few decades several books, devoted to fractional order differential and pseudo-differential equations and their various applications, have been published (see e.g. \cite{SKM,Podlubny,KST,Handbook,Umarov,UHK_book_2018}).
In fractional order modeling, in contrast to integer order equations, orders of fractional order governing equations are often unknown, and requires to utilize available data to measure. Therefore, one of the key questions arising in the process of modeling is a proper determination of a fractional order of the governing equation. The problem of determining a correct order of an equation (or orders of equations if the model uses more than one governing equation) is classified as an inverse problem. Inverse problems naturally require additional conditions (or information) for a solution. For subdiffusion equations, in which the order is between zero and one, the inverse problem of determination of the order has been considered by a number of authors; see \cite{AA,AF,AU,Che,Jan,LiLiu,LiL,LiY} and references therein. For the survey paper, we refer the reader to \cite{LiLiu} by Li et. al. Note that in all the refereed works the subdiffusion equation was considered in a bounded domain $\Omega\subset \mathbb{R}^N$. In addition, it should be noted that in publications \cite{Che,Jan,LiL,LiY} the following relation was taken as an additional condition \begin{equation}\label{ex1} u(x_0,t)= h(t), \,\, 0<t<T, \end{equation} at a monitoring point $x_0\in \overline{\Omega}$. But this condition, as a rule (an exception is the work \cite{Jan} by J. Janno, where both the uniqueness and existence are proved), can ensure only the uniqueness of the solution of the inverse problem
\cite{Che,LiL,LiY}. In paper \cite{AU} authors obtained the existence and uniqueness result, considering as an additional condition, the value of the projection of the solution onto the first eigenfunction of the elliptic part of the subdiffusion equation. Note that the technique used in \cite{AU} is applicable only when the first eigenvalue is zero. In more general case the uniqueness and existence of a solution of the inverse problem of determination of an unknown order of the fractional derivative in the subdiffusion equation was proved in the recent work \cite{AA}. In this case, the additional condition is $ || u (x, t_0) ||^2 = d_0 $, and the boundary condition is not necessarily homogeneous. In paper \cite{AF} authors studied the inverse problem for the simultaneous determination of the order of the Riemann-Liouville time fractional derivative and the source function in the subdiffusion equations.
The purpose of this work is to investigate the inverse problem of determining the vector-order of the time-fractional derivatives of systems of pseudo-differential equations. We note that systems (linear and non-linear) of fractional ordinary equations and partial differential equations have rich applications and are used in modeling of various processes arising in modern science and engineering. For example, they are used in modeling of processes in biosystems \cite{DasGupta,Rihan,GuoFang}, ecology \cite{Khan,Rana}, epidemiology \cite{Zeb,Islam}, etc.
In this paper we consider the following system of linear homogenous time-fractional order pseudo-differential equations
\begin{equation} \label{system_0}
\begin{cases}
\mathcal{D}^{\beta_1}u_1(t,x) = A_{1, 1}(D) u_1(t,x) +\dots A_{1, m}(D) u_m(t,x), \\
\mathcal{D}^{\beta_2}u_2 (t,x)= A_{2, 1}(D) u_1 (t,x)+\dots A_{2, m} (D)u_m(t,x), \\
{\cdots } \\
\mathcal{D}^{\beta_m}u_m (t,x) = A_{m, 1}(D) u_1(t,x) +\dots A_{m, m}(D) u_m (t,x).
\end{cases}
\end{equation} where $\mathcal{B}=\langle \beta_1, \dots, \beta_m \rangle,$ $0<\beta_j \le 1, \ j=1,\dots, m,$ is an unknown vector-order to be determined, the operator $\mathcal{D}$ on the left hand side expresses either the Riemann-Liouville derivative $D_+$ or the Caputo derivative $D_{\ast}$, and $A_{j,k}(D)$ are pseudo-differential operators with (possibly singular) symbols depending only on dual variables (for simplicity) and described later. The initial conditions depend on the form of fractional derivatives.
As it follows from our main result, a predetermined value of the Fourier transform $\hat{U}(t,\xi) = \langle \hat{u}_1(t,\xi), \dots, \hat{u}_m(t,\xi) \rangle$ of the solution $U(t, x)= \langle u_1(t,x), \dots, u_m(t,x) \rangle$ of the initial value problem for system \eqref{system_0} at an appropriate fixed point $\xi_0\in \mathbb{R}^m$ satisfying some condition (see Eq. \eqref{xi0} ), that is \begin{equation}\label{3} \hat{u}_j (t_0, \xi^0)=d_j,\quad j=1,2, ..., m, \end{equation} where $t_0 \geq 1$ is an observation time, uniquely recovers the vector-order $\mathcal{B}=\langle \beta_1, \dots, \beta_m \rangle$ of the fractional derivatives.
In the particular case, the determining of a scalar order for one equation was considered in \cite{AU} and the forward problem for systems of pseudo-differential equations in \cite{UAY}. From this point of view the current paper is a logical continuation of these two papers.
We note that in \cite{AU} the additional condition is represented in the form
\[
\int\limits_{\Omega} u(t_0, x)v_1(x)dx=d \neq 0,
\] that is, in the form of a projection of the solution $u(t_0, x)$ onto the first eigenfunction $v_1(x)$ of the elliptic part of the equation considered in an arbitrary bounded domain $\Omega\subset \mathbb{R}^n$. Condition \eqref{3} can be considered as the projection of the solution $u(t_0, x)$ onto "the eigenfunction" $e^{-ix\xi_0}$:
\[
\int\limits_{\mathbb{R}^n} u(t_0, x)e^{-ix\xi_0} dx=d .
\]
To our best knowledge, the inverse problem for the system of equations \eqref{system_0} with the vector-order fractional derivatives under the additional condition \eqref{3} is considered for the first time.
\section{Main results}
\subsection{Notations.} We follow the notions and notations introduced in \cite{UAY}. For the reader's convenience, below we introduce the main notations used in the current paper; for details see \cite{Umarov,UAY}. Let $G \subseteq \mathbb{R}^n$ be an open set and $p \ge 1.$ The space ${\mathbf \Psi}_{G,p}(\mathbb{R}^n)$ comprises of functions $\psi\in L_p(\mathbb{R}^n)$, such that $\text{supp} \ \hat{\psi}\Subset G$, i.e. the Fourier transform \[ \hat{\psi}(\xi) = \int\limits_{\mathbb{R}^n} f(x) e^{-ix\xi} dx \] of $\psi$ has a compact support in $G$. This is a topological-vector space with respect to the following convergence: a sequence $\psi_n \to \psi$ if $\text{supp} \, \hat{\psi}_n \Subset G,$ and $\psi_n \to \psi$ in $L_p(\mathbb{R}^n).$ For relations of the spaces ${\mathbf \Psi}_{G,p}(\mathbb{R}^n)$ to Sobolev spaces and Schwartz distributions see \cite{Umarov}.
Let $A(\xi)$ be a continuous function in $G$. Outside of $G$ or on its boundary $A(\xi)$ may have singularities of arbitrary type. For a function $\varphi \in {\mathbf \Psi}_{G,p}(\mathbb{R}^n)$ the pseudo-differential operator $A(D)$ corresponding to the symbol $A(\xi)$ is defined by the formula
\begin{equation} \label{04} A(D)\varphi (x)= \frac{1}{(2 \pi)^n} \int_G A(\xi) \hat{\varphi} (\xi) e^{i x \xi} d \xi\, \quad x \in \mathbb{R}^n. \end{equation} For the systematic presentation of the theory of pseudo-differential operators being considered in this paper we refer the reader to \cite{Umarov}.
Let \[ \mathbb{A}(D)= \begin{bmatrix}
A_{1,1}(D) & \dots & A_{1, m} (D) \\
\dots & \dots & \dots \\
A_{m, 1}(D) & \dots & A_{m,m} (D)
\end{bmatrix}\] be the matrix pseudo-differential operator with constant (that is not depending on the variable $x$) matrix-symbol \begin{equation} \label{matrix_0} \mathcal{A}(\xi) = \begin{bmatrix}
A_{1,1}(\xi) & \dots & A_{1, m} (\xi) \\
\dots & \dots & \dots \\
A_{m, 1}(\xi) & \dots & A_{m,m} (\xi)
\end{bmatrix}, \end{equation} defined and continuous in $G$ in the sense of the matrix norm.
With the matrix form of the pseudo-differential operator we can represent system (\ref{system_0}) in the vector form: \begin{equation} \label{system_1} \mathcal{D}^{\mathcal{B}} {U}(t,x) = \mathbb{A}(D) {U} (t,x), \end{equation} where $ \mathcal{D}^{\mathcal{B}} {U}(t,x)=\langle \mathcal{D}^{{\beta_1}} {u}_1(t,x), \dots, \mathcal{D}^{{\beta_m}} {u}_m(t,x) \rangle. $
Below we will use two main forms of fractional derivatives, namely the Riemann-Liouville form and the Caputo form. Let $k$ be a natural number and $k-1 \le \beta <k.$ Then the fractional derivative of order $\beta$ of a measurable function $f$ in the sense of Riemann--Liouville is defined as \[ D^{\beta}_{+} f(t) = \frac{1}{\Gamma(k-\beta)}\frac{d^k}{dt^k} \int_0^t \frac{f(\tau)d\tau}{(t-\tau)^{\beta+1-k}}, \] provided the expression on the right exists. Here $\Gamma(t)$ is Euler's gamma function. If we replace differentiation and fractional integration in this definition, then we get the definition of a regularized derivative, that is, the definition of a fractional derivative in the sense of Caputo: \[ D^{\beta}_{\ast} f(t) = \frac{1}{\Gamma(k-\beta)} \int_0^t \frac{f^{(k)}(\tau)d\tau}{(t-\tau)^{\beta+1-k}}, \] provided the integral on the right exists.
We assume that the matrix-symbol is symmetric, $A_{k,j}(\xi)=A_{j,k}(\xi)$ for all $k,j=1,\dots,m,$ and $\xi\in G,$ and diagonalizable. Namely, there exists an invertible $(m\times m)$-matrix-function $M(\xi),$ such that \begin{equation} \label{matrix_01} \mathcal{A}(\xi) = M^{-1}(\xi) \Lambda(\xi) M(\xi), \quad \xi \in G, \end{equation} with a diagonal matrix
\begin{equation} \label{matrix_02} {\Lambda}(\xi) = \begin{bmatrix}
\lambda_1(\xi) & \dots & 0 \\
\dots & \dots & \dots \\
0 & \dots & \lambda_m (\xi)
\end{bmatrix}. \end{equation} We denote entries of matrices $M(\xi)$ and $M^{-1}(\xi)$ by $\mu_{j,k}(\xi), \ j,k=1,\dots,m,$ and $\nu_{j,k}(\xi)$, $j,k=1,\dots,m,$ respectively.
Since initial conditions depend on the form of the fractional derivative on the left hand side of equation (\ref{system_1}), we will consider the cases with the Caputo and Riemann-Liouville derivatives separately. We first formulate our main result in the case of Caputo fractional derivative. The case of Rieman-Liouville fractional derivative can be treated similarly.
\subsection{Forward problem} Let $\mathcal{B}$ be a known vector-order with $0<\beta_j\leq 1$, $j=1, \dots, m$. Consider the following Cauchy problem
\begin{align} \label{Cauchy_01_h} D_{\ast}^{\mathcal{B}} {U}(t,x) &= \mathbb{A}(D) {U} (t,x), \quad t>0, \ x \in \mathbb{R}^n, \\ U(0,x) & =\varPhi(x), \quad x \in \mathbb{R}^n, \label{Cauchy_02_h} \end{align} where $\varPhi(x) = \langle \varphi_1(x), \dots, \varphi_m(x) \rangle\in {\mathbf \Psi}_{G,p}(\mathbb{R}^n)$ and the fractional derivatives on the left are in the sense of Caputo.
We call the Cauchy problem (\ref{Cauchy_01_h})-(\ref{Cauchy_02_h}) \emph{the forward problem.}
A representation formula for the solution of the forward problem was obtained in \cite{UAY} and it has the form \[ u_j(t,x)= \frac{1}{(2\pi)^n} \int\limits_{\mathbb{R}^n}\sum\limits_{k=1}^m s_{j,k}(t,\xi)\hat{\varphi}_k(\xi)e^{ix\xi}d\xi, \quad j=1, \dots, m, \] where \[ s_{j,k}(t,\xi)=\sum\limits_{l=1}^m \mu_{j,l}(\xi)\nu_{l,k}(\xi)E_{\beta_l}(\lambda_l(\xi) t^{\beta_l}). \] Here we denoted by $E_{\beta_j}(z), j=1,\dots,m,$ the Mittag-Leffler functions of indices $\beta_1,\dots, \beta_m,$ respectively.
We rewrite function $u_j$ as \begin{align*} u_j(t,x) &= \frac{1}{(2\pi)^n} \int\limits_{\mathbb{R}^n}\sum\limits_{k=1}^m \sum\limits_{l=1}^m \mu_{j,l}(\xi)\nu_{l,k}(\xi)E_{\beta_l}(\lambda_l(\xi) t^{\beta_l})\hat{\varphi}_k(\xi)e^{ix\xi}d\xi \\ &= \frac{1}{(2\pi)^n} \int\limits_{\mathbb{R}^n}\sum\limits_{l=1}^m E_{\beta_l}(\lambda_l(\xi) t^{\beta_l})\bigg[\mu_{j,l}(\xi)\sum\limits_{k=1}^m\nu_{l,k}(\xi)\hat{\varphi}_k(\xi)\bigg]e^{ix\xi}d\xi \\ &= \frac{1}{(2\pi)^n} \int\limits_{\mathbb{R}^n}\sum\limits_{l=1}^m E_{\beta_l}(\lambda_l(\xi) t^{\beta_l})K_{j,l}\big(\xi, \widehat{\Phi}(\xi)\big)e^{ix\xi}d\xi, \end{align*} where \[ K_{j,l}\big(\xi, \widehat{\Phi}(\xi)\big)=\mu_{j,l}(\xi)\sum\limits_{k=1}^m\nu_{l,k}(\xi)\hat{\varphi}_k(\xi), \] and \[ \widehat{\Phi}(\xi)=\langle\hat{\varphi}_1(\xi),\hat{\varphi}_2(\xi),\cdot\cdot\cdot,\hat{\varphi}_m(\xi)\rangle. \]
For the Fourier transform of the solution we have \begin{equation}\label{1} \hat{u}_j(t,\xi)=\sum\limits_{l=1}^m E_{\beta_l}(\lambda_l(\xi) t^{\beta_l})K_{j,l}\big(\xi, \widehat{\Phi}(\xi)\big). \end{equation}
Note that under the above conditions on the matrix-symbol $A_{j,k}(\xi)$ and on the function $\Phi(x)$, this Fourier transform exists at each point $\xi\in \mathbb{R}^n$.
\subsection{Inverse problem.} Now let the parameter $\mathcal{B}$ be an unknown vector-order of the time derivative with $\beta_0\leq\beta_j< 1$, $j=1, \dots, m$, $\beta_0\in (0,1)$. The main purpose of this paper is to investigate the inverse problem of identifying of these parameters $\beta_j$. Since there are $m$ unknown parameters, we need to set $m$ conditions. We pass on to the determining of these additional conditions.
In what follows, we will assume that \begin{equation}\label{lambda}
|\arg \lambda_j(\xi)|> \frac{\pi}{2}, \quad \xi \in G,\quad j=1, \dots, m. \end{equation} Let $\xi^0=(\xi^0_1, \xi^0_2, ..., \xi^0_m)\in G\subset \mathbb{R}^n$ be a vector such that the determinant of the matrix \begin{equation} \label{matrixK} \mathcal{K}(\xi^0) \equiv \{K_{j,l}\big(\xi^0, \widehat{\Phi}(\xi^0)\big)\}, \ j, l =1,\dots,m, \end{equation}
satisfies the condition \begin{equation}\label{xi0}
\big|K_{j,l}\big(\xi^0, \widehat{\Phi}(\xi^0)\big)\big|\neq 0. \end{equation}
To find the unknown parameters $\beta_l, l=1,\dots, m$, we consider the following additional conditions \begin{equation}\label{ad1} f_j(\mathcal{B}, t_0, \xi^0)\equiv \hat{u}_j (t_0, \xi^0)=d_j,\quad j=1,\dots, m, \end{equation} where $d_j$ are given numbers and $t_0$ is defined later (see Lemmas \ref{ECaputo} and \ref{ERL}). We call problem (\ref{Cauchy_01_h})--(\ref{Cauchy_02_h}) together with the additional condition (\ref{ad1}) \emph{the inverse problem.}
It follows from (\ref{1}) and (\ref{ad1}) that for all $j=1,2, ... , m$ \begin{equation}\label{4} \sum\limits_{l=1}^m E_{\beta_l}(\lambda_l(\xi^0) t_0^{\beta_l})K_{j,l}\big(\xi^0, \widehat{\Phi}(\xi^0)\big)=d_j. \end{equation} These are in fact the system of equations to define the orders $\beta_l, l=1,2,..., m$.
Due to condition (\ref{xi0}) one can solve system (\ref{4}) with respect to the Mittag-Leffler functions $E_{\beta_l}$, i.e. \begin{equation}\label{ad2} E_{\beta_l}(\lambda_l(\xi^0) t_0^{\beta_l})= b_l, \quad l=1,2,..., m, \end{equation} where $b_l, \, l=1,2,..., m,$ are components of the vector $\mathcal{K}^{-1}(\xi^0) \textbf{d},$ with $\textbf{d}=\langle d_1,\dots,d_m \rangle$ and $\mathcal{K}^{-1}(\xi^0)$ is the inverse matrix to $\mathcal{K}(\xi^0),$ defined in \eqref{matrixK}. Thus to define each unknown parameter $\beta_l$ we obtained a separate equation (\ref{ad2}).
Let $R_{C, l}$ be the range of values of the function $e_{1, \lambda_l} (\beta_l)\equiv E_{\beta_l}(\lambda_l(\xi^0) t_0^{\beta_l})$ when $\beta_l$ runs over the half-interval $[\beta_0, 1)$, i.e. \[ e_{1, \lambda_l}: [\beta_0, 1)\rightarrow R_{C, l}\subset C, \] where $C$ is a complex plane, and the index $C$ emphasizes that we are considering the case of the Caputo derivatives.
Obviously, for equations (\ref{ad2}), in order to have solutions with respect to $\beta_l,$ the right-hand sides of these equations must lie within the values of the functions on the left-hand sides of these equations, i.e. \begin{equation}\label{Crange} b_l\in R_{C, l}, \quad l=1,2,..., m. \end{equation}
On the other hand, by virtue of Rolle's theorem, the strict monotonicity of either the real part or the imaginary part of the function $E_{\beta_l}(\lambda_l(\xi^0) t_0^{\beta_l})$ in the variable $\beta_l$ is sufficient for the uniqueness of the solution to equation (\ref{ad2}).
Let us introduce the following notation \[ R_C(\beta_l)=\Re(E_{\beta_l}(\lambda_l(\xi^0) t_0^{\beta_l})), \] where $\Re(z)$ is the real part of $z$ and the index $C$ again emphasizes that we are considering the case of the Caputo derivatives. The necessity of condition (\ref{lambda}) is that its fulfillment guarantees the strict monotonicity of the function $R_C(\beta_l)$ it the variable $\beta_l$ for each fixed $l.$
\subsection{Main results} \par The main results of this paper are stated in Theorems \ref{thm_Caputo} and \ref{thm_Riemann-Liouville}.
\begin{thm} \label{thm_Caputo} Let $\xi^0$ satisfy condition \eqref{xi0} and $t_0 > T_0,$ where $T_0$ is identified in Lemma \ref{ECaputo}. Let the numbers $d_l$ on the right hand side of equation (\ref{ad1}) be such that the components $b_l, \ l=1,\dots, m,$ of the vector $\mathcal{K}^{-1}(\xi^0) \textbf{d}$ satisfy the conditions (\ref{Crange}). Then for each $l$ there exists the unique number $\beta_l^\ast\in [\beta_0, 1]$ such that the Fourier transform of the solution $u_j(t, x), j=1,\dots,m,$ of the forward problem with $\beta_j=\beta_j^\ast, j=1,\dots,m,$ satisfies equation (\ref{ad1}). \end{thm}
The proof of this theorem follows from the existence and uniqueness theorem for the forward problem proved in \cite{UAY} (see Theorem 3.1) and Lemma \ref{ECaputo} below. Therefore, in order to prove the theorem we need to prove only this lemma. The proof of Lemma \ref{ECaputo} is given in Section \ref{sec_3}.
\begin{lem} \label{ECaputo}
Given $\beta_0$ in the interval $0<\beta_0< 1$ and $\xi_0$ satisfying condition (\ref{xi0}), there exists a number $T_0=T_0(\xi^0, \beta_0)>1$, such that for all $t_0\geq T_0$ the function $R_C(\beta_l)$ is positive and strictly monotonically decreasing with respect to $\beta_l\in [\beta_0, 1]$ and \begin{equation}\label{RICE} R_C(1) \leq R_C(\beta_l) \leq R_C(\beta_0),\,\, l=1, \dots, m. \end{equation}
\end{lem}
\begin{rem}\label{Uniqueness} Theorem \ref{thm_Caputo} defines the vector-order $\mathcal{B}^\ast=(\beta_1^\ast, \beta_2^\ast,..., \beta_m^\ast)$ uniquely from conditions (\ref{ad1}). Hence, if we define $f_j(\mathcal{B}, \cdot, \cdot)$ at another time instant $t_1$ and point $\xi^1$ and get a new $\mathcal{B^{\ast \ast}}$, i.e. $f_j(\mathcal{B}^{\ast \ast}, t_1, \xi^1) = d_j^1$, then from the equality $f_j(\mathcal{B}^{\ast \ast}, t_0, \xi^0) = d_j$, by virtue of the theorem, we obtain $\mathcal{B}^{\ast \ast}=\mathcal{B}^{\ast}$. \end{rem}
Now consider the following initial-value problem \begin{align} \label{Cauchy_10} D_{+}^{\mathcal{B}} {U}(t,x) &= \mathbb{A}(D) {U} (t,x), \quad t>0, \ x \in \mathbb{R}^n, \\ J^{1-\mathcal{B}}U(0,x) & =\varPhi(x), \quad x \in \mathbb{R}^n, \label{Cauchy_20} \end{align} where $\varPhi(x) = \langle \varphi_1(x), \dots, \varphi_m(x) \rangle\in {\mathbf \Psi}_{G,p}(\mathbb{R}^n)$ and the fractional derivatives on the left hand side of equation \eqref{Cauchy_10} are in the sense of Riemann-Liouville.
We call the Cauchy problem (\ref{Cauchy_10})-(\ref{Cauchy_20}) \emph{the second forward problem.}
A representation formula for the solution of the second forward problem was also obtained in \cite{UAY} and it has the form \[ u_j(t,x)= \frac{1}{(2\pi)^n} \int\limits_{\mathbb{R}^n}\sum\limits_{k=1}^m s^+_{j,k}(t,\xi)\hat{\varphi}_k(\xi)e^{ix\xi}d\xi, \quad j=1, \dots, m, \] where \[ s^+_{j,k}(t,\xi)=\sum\limits_{l=1}^m \mu_{j,l}(\xi)\nu_{l,k}(\xi)t^{\beta_l-1}E_{\beta_l, \beta_l}(\lambda_l(\xi) t^{\beta_l}). \] Here we denoted by $E_{\beta_j, \beta_j}(z), j=1,\dots,m,$ the two-parametric Mittag-Leffler functions.
For the Fourier transform of the solution we have \begin{equation}\label{Fu+} \hat{u}_j(t,\xi)=\sum\limits_{l=1}^m t^{\beta_l-1}E_{\beta_l, \beta_l}(\lambda_l(\xi) t^{\beta_l})K_{j,l}\big(\xi, \widehat{\Phi}(\xi)\big). \end{equation}
Suppose that condition (\ref{lambda}) is fulfilled and choose $\xi^0\in G$ so that inequality (\ref{xi0}) holds.
We call problem (\ref{Cauchy_10})-(\ref{Cauchy_20}) together with the additional condition (\ref{ad1}) \emph{the second inverse problem}.
Note that additional condition (\ref{ad1}) is in fact the equation to determine the unknown parameters $\beta_l$. Performing similar calculations as above, by virtue of condition (\ref{xi0}), we rewrite (\ref{ad1}) as \begin{equation}\label{ad3} t^{\beta_l-1}E_{\beta_l, \beta_l}(\lambda_l(\xi) t^{\beta_l})= b_l, \quad l=1,2,..., m, \end{equation} where $b_l, \, l=1,2,..., m,$ are the same numbers as above.
Let $R_{RL, l}$ be the range of values of the left-hand side of these equations when $\beta_l$ runs over the half-interval $[\beta_0, 1)$. Here the index $RL$ emphasizes that we are considering the case of the Riemann-Liouville derivatives.
Again, as in case of equations (\ref{ad2}), a necessary condition for the existence of solutions to equations (\ref{ad3}) is the inclusion \begin{equation}\label{RLrange} b_l\in R_{RL, l}, \quad l=1,2,..., m. \end{equation}
On the other hand, by virtue of Rolle's theorem, the strict monotonicity of the function $\Re(t^{\beta_l-1}E_{\beta_l, \beta_l}(\lambda_l(\xi) t^{\beta_l}))$ in the variable $\beta_l$ is sufficient for the uniqueness of the solution to equation (\ref{ad3}).
However, if $|\Re(\lambda_l(\xi^0))| = |\Im(\lambda_l(\xi^0))|$ (note, under the condition (\ref{lambda}) one has $\Re(\lambda_l(\xi^0))<0$), then the principal part of $\Re(t^{\beta_l-1}E_{\beta_l, \beta_l}(\lambda_l(\xi) t^{\beta_l}))$ vanishes, and in this case it is necessary to go to its next term in the asymptotic. Therefore, to simplify the presentation, we further assume that \begin{equation}\label{lambda1}
|\Re(\lambda_l(\xi^0))| \neq |\Im(\lambda_l(\xi^0))| \end{equation}
Let us introduce the following notation $$
R_{RL}(\beta_l)=sign(|\Re(\lambda_l(\xi^0))| -
|\Im(\lambda_l(\xi^0))|)\Re(t^{\beta_l-1}E_{\beta_l, \beta_l}(\lambda_l(\xi) t^{\beta_l})). $$ Here the index $RL$ again emphasizes that we are considering the case of the Riemann-Liouville derivatives.
\begin{lem}\label{ERL}
Given $\beta_0$ in the interval $0<\beta_0< 1$ and $\xi_0$ satisfying condition (\ref{xi0}),
there exists a number $T_1=T_1(\xi^0, \beta_0)>1$, such that for all $t_0\geq T_1$ the function $R_{RL}(\beta_l)$ is positive and strictly monotonically decreasing with respect to $\beta_l\in [\beta_0, 1]$ and \begin{equation}\label{RICE_1} R_{RL}(1) \leq R_{RL}(\beta_l) \leq R_{RL}(\beta_0),\,\, l=1, \dots, m. \end{equation}
\end{lem}
This lemma, similar to the Caputo derivative case, immediately implies the following main result of this paper in the case of the Riemann-Liouville derivatives. The existence and uniqueness theorem of the corresponding forward problem is proved in \cite{UAY} (see Theorem 3.4). The proof of Lemma \ref{ERL} is presented in the next section.
\begin{thm} \label{thm_Riemann-Liouville} Let $\xi^0$ satisfy condition \eqref{xi0} and $t_0>T_1,$ where $T_1$ is identified in Lemma \ref{ERL}. Let the numbers $d_l$ from condition (\ref{ad1}) be such that the corresponding numbers $b_l$ satisfy the conditions (\ref{RLrange}). Then for each $l$ there exists the unique number $\beta_l^\ast\in [\beta_0, 1]$ such that the Fourier transform of the solution $u_j(t, x)$ of the second forward problem with $\beta_j=\beta_j^\ast$ satisfies the equation (\ref{ad1}). \end{thm}
Similar to the Caputo derivative case, Theorem \ref{thm_Riemann-Liouville} defines the vector-order $\mathcal{B}^\ast=(\beta_1^\ast, \beta_2^\ast,..., \beta_m^\ast)$ uniquely from conditions (\ref{ad1}); see Remark \ref{Uniqueness}.
\section{Proofs of Lemmata \ref{ECaputo} and \ref{ERL}} \label{sec_3}
Let us denote by $\delta(1; \theta)$ a contour oriented by non-decreasing $\arg \zeta$ consisting of the following parts: the ray $\arg \zeta = -\theta$ with $|\zeta|\geq 1$, the arc
$-\theta\leq \arg \zeta \leq \theta$, $|\zeta|=1$, and the ray
$\arg \zeta = \theta$, $|\zeta|\geq 1$. If $0<\theta <\pi$, then the contour $\delta(1; \theta)$ divides the complex $\zeta$-plane into two unbounded parts, namely $G^{(-)}(1;\theta)$ to the left of $\delta(1; \theta)$ by orientation, and $G^{(+)}(1;\theta)$ to the right of it. The contour $\delta(1; \theta)$ is called the Hankel path.
In what follows, we fix $l$ out of $1,\dots, m$ and denote $\lambda=\lambda_l(\xi^0)$ and $\rho=\beta_l$. Let $\lambda=-\lambda_1+i\lambda_2$ and by virtue of condition (\ref{lambda}) one has $\lambda_1>0$.
Further let $\theta = (\frac{\pi}{2}+\varepsilon)\rho$, $\alpha = (\frac{\pi}{2}+2\varepsilon)\rho$, $\rho\in [\beta_0, 1)$ and $\varepsilon>0$ be such that
$\varepsilon\equiv\varepsilon(\xi^0)<\frac{1}{2}\min\{|\arg
\lambda(\xi^0)|-\pi/2, \pi/2\}$.
Then \[
\frac{\pi}{2}\rho<\theta<\alpha< \pi \rho, \quad \alpha <|\arg
\lambda|, \] and therefore $\lambda t_0^\rho\in G^{(-)}(1;\theta)$.
\
\subsection{Proof of Lemma \ref{ECaputo}} First we prove Lemma \ref{ECaputo}. By the definition of contour $\delta(1; \theta)$, we have (see \cite{Dzh66}, formula (2.29), p. 135) \begin{equation}\label{E1rho} e_{1,\lambda}(\rho)\equiv E_{\rho}(\lambda t_0^\rho)= -\frac{1}{\lambda t_0^\rho \Gamma (1-\rho)}+\frac{1}{2\pi i \rho\lambda t_0^\rho}\int\limits_{\delta(1;\theta)}\frac{e^{\zeta^{1/\rho}}\zeta}{\zeta+\lambda t_0^\rho} d\zeta = f_1(\rho)+f_2(\rho). \end{equation}
To prove the lemma, we need to determine the sign of the real part of the derivative $\frac{d}{d\rho} e_{1,\lambda} (\rho)$. It is not hard to estimate the derivative $f'_1(\rho)$. Indeed, let $\Psi(\rho)$ be the logarithmic derivative of the gamma function $\Gamma(\rho)$ (for the definition and properties of $\Psi$ see \cite{Bat}). Then $\Gamma'(\rho) = \Gamma (\rho) \Psi(\rho)$, and therefore, $$ f_1'(\rho)=\frac{\ln t_0 - \Psi (1-\rho)}{\lambda t_0^\rho \Gamma (1-\rho)}. $$ Since \[ \frac{1}{\Gamma(1-\rho)}=\frac{1-\rho}{\Gamma(2-\rho)}, \quad \Psi(1-\rho)=\Psi(2-\rho)-\frac{1}{1-\rho}, \] the function $f_1'(\rho)$ can be represented as follows \[ f_1'(\rho)=\frac{1}{\lambda t_0^\rho} \frac{(1-\rho)[\ln t_0 - \Psi (2-\rho)]+1}{ \Gamma (2-\rho)}. \]
If $\gamma\approx 0,57722$ is the Euler-Mascheroni constant, then $-\gamma <\Psi(2-\rho)< 1-\gamma$. By virtue of this estimate we may write \begin{equation}\label{Rf1} -\Re(f_1'(\rho))\geq
\frac{\lambda_1}{|\lambda|^2}\frac{(1-\rho)[\ln t_0 - (1-\gamma)]+1}{\Gamma(2-\rho) t_0^\rho}\geq
\frac{\lambda_1}{|\lambda|^2 t_0^{\rho}}, \end{equation} provided \begin{equation} \label{new_t0} \ln t_0 > 1-\gamma \quad \text{or} \quad t_0> T_0=e^{1-\gamma}>1. \end{equation}
To estimate the derivative $f'_2(\rho)$, we denote the integrand in (\ref{E1rho}) by $F(\zeta, \rho)$: \[ F(\zeta, \rho)=\frac{1}{2\pi i \rho\lambda t_0^\rho}\cdot \frac{e^{\zeta^{1/\rho}}\zeta}{\zeta+\lambda t_0^\rho}. \] Note, that the domain of integration $\delta(1; \theta)$ also depends on $\rho$. To take this circumstance into account when differentiating the function $f'_2(\rho)$, we rewrite the integral (\ref{E1rho}) in the form: \[ f_2(\rho)=f_{2+}(\rho)+f_{2-}(\rho)+f_{21}(\rho), \] where \[ f_{2\pm}(\rho)=e^{\pm i \theta}\int\limits_1^\infty F(s\,e^{\pm i \theta}, \rho)\, ds, \]
\[ f_{21}(\rho) = i \int\limits_{-\theta}^{\theta} F(e^{i y}, \rho)\, e^{iy} dy= i\theta \int\limits_{-1}^{1} F(e^{i \theta s}, \rho)\, e^{i\theta s} ds. \]
Let us consider the function $f_{2+}(\rho)$. Since $\theta = (\frac{\pi}{2}+\varepsilon)\rho$ and $\zeta= s\, e^{i\theta}$, then \[ e^{\zeta^{1/\rho}}=e^{-s^{\frac{1}{\rho}}(\varepsilon_1-i\varepsilon_2)}, \,\, \cos(\frac{\pi}{2}+\varepsilon) =-\varepsilon_1<0.\,\, \sin(\frac{\pi}{2}+\varepsilon) =\varepsilon_2>0. \] The derivative of the function $f_{2+}(\rho)$ has the form $$ f_{2+}'(\rho)=\frac{1}{2\pi i \rho\lambda t_0^\rho}\int\limits_1^\infty \frac{e^{-s^{\frac{1}{\rho}}(\varepsilon_1-i\varepsilon_2)}\,s\,e^{2ia\rho}\mathcal{M}(s)}{s\,e^{ia\rho}+\lambda t_0^\rho} ds, $$ where $a=\frac{\pi}{2}+\varepsilon,$ and \[ \mathcal{M}(s)=-\frac{\varepsilon_1-i\varepsilon_2}{\rho^2}s^{1/\rho}\ln s+2ia -\frac{1}{\rho}-\ln t_0-\frac{ia s\,e^{ia\rho}+\lambda t_0^\rho \ln t_0}{s\,e^{ia\rho}+\lambda t_0^\rho}. \] It is not hard to verify, that \[
|s\,e^{ia\rho}+\lambda t_0^\rho|\geq |\lambda| t_0^\rho \sin(\alpha - \theta)\geq
\frac{2}{\pi}|\lambda| t_0^\rho \varepsilon. \]
Therefor we arrive at $$
|f_{2+}'(\rho)|\leq \frac{C}{\rho(\varepsilon|\lambda| t_0^\rho)^2}\int\limits_1^\infty e^{-\varepsilon_1\,s^{1/\rho}}s\,\left[\frac{1}{\rho^2}s^{1/\rho}\ln s+\ln t_0\right] ds, $$ or $$
|f_{2+}'(\rho)|\leq \frac{C}{(|\lambda| t_0^\rho)^2}\left[\frac{1}{\rho}+\ln t_0\right], $$ where the constant $C$ depends only on $\varepsilon$ (and therefore only on $\xi^0$).
The function $f'_{2-}(\rho)$ has exactly the same estimate.
Now consider the function $f_{21}(\rho)$. It is not hard to verify that \[ f'_{21}(\rho)=\frac{a}{2\pi \lambda t_0^\rho}\int\limits_{-1}^1\frac{e^{e^{ias}}e^{2ia\rho s}\big[2ias-\ln t_0-\frac{ias e^{ia\rho s}+\lambda t_0^\rho \ln t_0}{e^{ia\rho s}+\lambda t_0^\rho}\big]}{e^{ia\rho s}+\lambda t_0^\rho} ds. \] Therefore, \[
|f'_{21}(\rho)|\leq C\frac{\ln t_0}{(|\lambda| t_0^\rho)^2}. \] Now we show that the real part of the derivative $\frac{d}{d\rho} e_{1,\lambda}(\rho)$ is negative. Taking into account estimate (\ref{Rf1}) and the estimates for $f'_{2\pm}$ and $f'_{21}$, we have \begin{equation}\label{der} \Re\bigg(\frac{d}{d\rho} e_{1,\lambda}(\rho)\bigg)<
-\frac{\lambda_1}{|\lambda|^2 t_0^{\rho}}+C\frac{1/\rho+\ln t_0}{(|\lambda| t_0^{\rho})^2}. \end{equation} In other words, this derivative is negative if \[ t_0^{\rho}>C\frac{1/\rho+ \ln t_0}{\lambda_1} \] for all $\rho\in [\beta_0, 1).$ Hence \begin{equation}\label{t01} t_0^{\beta_0}>C\frac{1/{\beta_0}+ \ln t_0}{\lambda_1}. \end{equation}
Thus, there exists a number $T_0 = T_0(\xi^0, \beta_0)>1$ (see \eqref{new_t0}) such, that for all $t_0\geq T_0$ we have the estimate \[ \Re\bigg(\frac{d}{d\rho} e_{1,\lambda}(\rho)\bigg)< 0\,\,\text{for}\,\,\text{all} \quad \rho\in [\beta_0,1]. \]
The positivity of $R_C(\beta_l)$ follows from the explicit form of the function $f_1(\rho)$.
Lemma \ref{ECaputo}, and therefore Theorem \ref{thm_Caputo} are completely proved.
\
\subsection{Proof of Lemma \ref{ERL}} We now turn to the proof of Lemma \ref{ERL}. By the definition of contour $\delta(1; \theta)$, we have for $e_{2,\lambda}(\rho)\equiv t_0^{\rho-1}E_{\rho, \rho}(\lambda t_0^\rho)$ the following equation (see \cite{Dzh66}, formula (2.29), p. 135) \begin{equation}\label{Erho} e_{2,\lambda}(\rho)= - \frac{1}{\lambda^2 t_0^{\rho+1} \Gamma (-\rho)}+\frac{1}{2\pi i \rho \lambda^2 t_0^{\rho+1}}\int\limits_{\delta(1;\theta)}\frac{e^{\zeta^{1/\rho}}\zeta^{\frac{1}{\rho}+1}}{\zeta+\lambda t_0^\rho} d\zeta = g_1(\rho)+g_2(\rho). \end{equation}
Since the positivity of $R_{RL}(1)$ is obvious, then in order to prove Lemma \ref{ERL} it suffices to show that the derivatives of $R_{RL}(\rho)$ is negative for all $\rho\in [\beta_0, 1)$.
For the derivative $g'_1(\rho)$ we have $$ g_1'(\rho)=\frac{\ln t_0 - \Psi (-\rho)}{\lambda^2 t_0^{\rho +1}\Gamma (-\rho)}. $$ To get rid of the singularity in the denominators, we use the equalities \begin{align*} \frac{1}{\Gamma(-\rho)} &= -\frac{\rho}{\Gamma(1-\rho)}=-\frac{\rho(1-\rho)}{\Gamma(2-\rho)}, \\ \Psi(-\rho) &=\Psi(1-\rho) +\frac{1}{\rho}=\Psi(2-\rho)+\frac{1}{\rho}-\frac{1}{1-\rho}. \end{align*} Then the function $g_1'(\rho)$ can be represented as follows \begin{equation}\label{f1} g_1'(\rho)=\frac{1}{\lambda^2 t_0^{\rho+1}} \frac{\rho(1-\rho)[\Psi (2-\rho)-\ln t_0]+1-2\rho}{ \Gamma (2-\rho)}=-\frac{g_{11}(\rho)}{\lambda^2 t_0^{\rho+1}\Gamma (2-\rho)}. \end{equation} Since $\Psi(2-\rho)< 1-\gamma$, then \[ g_{11}(\rho)>\rho (1-\rho)[\ln t_0 -(1-\gamma)])+2\rho-1. \] For $t_0=e^{1-\gamma} e^{2/\rho}$ one has $\rho (1-\rho)[\ln t_0 -(1-\gamma)])+2\rho-1=1$. Hence, $g_{11}(\rho)\geq 1$, provided $t_0\geq T_1,$ where \begin{equation}\label{t0} T_1= e^{1-\gamma} e^{2/\beta_0} > e^{3-\gamma}>1. \end{equation} Thus, by virtue of (\ref{f1}), for all such $t_0$ we arrive at \begin{equation}\label{f11}
sign(\lambda_1^2-\lambda_2^2)\Re(g_1'(\rho))\leq-\frac{|\lambda_1^2-\lambda_2^2|}{|\lambda|^4 t_0^{\rho+1}}. \end{equation}
To estimate the derivative $g'_2(\rho)$, we denote the integrand in (\ref{Erho}) by $G(\zeta, \rho)$: \[ G(\zeta, \rho)=\frac{1}{2\pi i \rho\lambda^2 t_0^{\rho+1}}\cdot \frac{e^{\zeta^{1/\rho}}\zeta^{1/\rho+1}}{\zeta+\lambda t_0^\rho}, \] and rewrite the integral (\ref{Erho}) in the form: \[ g_2(\rho)=g_{2+}(\rho)+g_{2-}(\rho)+g_{21}(\rho), \] where \[ g_{2\pm}(\rho)=e^{\pm i \theta}\int\limits_1^\infty G(s\,e^{\pm i \theta}, \rho)\, ds, \]
\[ g_{21}(\rho) = i \int\limits_{-\theta}^{\theta} F(e^{i y}, \rho)\, e^{iy} dy= i\theta \int\limits_{-1}^{1} G(e^{i \theta s}, \rho)\, e^{i\theta s} ds. \]
The derivative of the function $ g_{2+}(\rho)$ has the form $$ g_{2+}'(\rho)=I\cdot\int\limits_1^\infty \frac{e^{-s^{1/\rho}\, (\varepsilon_1-i\varepsilon_2)}s^{\frac{1}{\rho}+1}\,e^{2ia\rho} \mathcal{N}(s) }{s\,e^{ia\rho}+\lambda t_0^\rho} ds, $$ where $I=e^{ia}(2\pi i \rho\lambda^2 t_0^{\rho+1})^{-1},$ $a=\frac{3\pi}{4},$ and \[ \mathcal{N}(s) = -\frac{1}{\rho^2}((\varepsilon_1+i\varepsilon_2)s^{\frac{1}{\rho}}+1)\ln s+2ia -\frac{1}{\rho}-\ln t_0-\frac{ia s\,e^{ia\rho}+\lambda t_0^\rho \ln t_0}{s\,e^{ia\rho}+\lambda t_0^\rho}. \] By virtue of the inequality
$|s\,e^{ia\rho}+\lambda t_0^\rho|\geq \frac{2}{\pi}|\lambda| t_0^\rho \varepsilon$ we arrive at $$
|g_{2+}'(\rho)|\leq \frac{C}{\rho |\lambda|^3 t_0^{2\rho+1}}\int\limits_1^\infty e^{-\varepsilon_1\,s^{1/\rho}}s^{\frac{1}{\rho}+1}\,\big[\frac{1}{\rho^2}s^{1/\rho}\ln s+\ln t_0\big] ds, $$ or $$
|g_{2+}'(\rho)|\leq\frac{C}{|\lambda|^3 t_0^{2\rho+1}}\,\big[\frac{1}{\rho}+\ln t_0\big], $$ where the constant $C$ depends only on $\xi^0$.
The function $g'_{2-}(\rho)$ has exactly the same estimate.
Now consider the function $g_{21}(\rho)$. For its derivative we have \[ g'_{21}(\rho)=\frac{a}{2\pi i \lambda^2 t_0^{\rho+1}}\cdot\int\limits_{-1}^1\frac{e^{e^{ias}}\, e^{ias}\,e^{2ia\rho s}\big[2ias-\ln t_0-\frac{ias e^{ia\rho s}+\lambda t_0^\rho \ln t_0}{e^{ia\rho s}+\lambda t_0^\rho}\big]}{e^{ia\rho s}+\lambda t_0^\rho} ds. \] Therefore, \[
|g'_{21}(\rho)|\leq C\,\frac{\ln t_0}{|\lambda|^3 t_0^{2\rho+1}}. \] Taking into account estimate (\ref{f11}) and the estimates for $g'_{2\pm}$ and $g'_{21}$, we have \[ sign(\lambda_1^2-\lambda_2^2)\Re(e'_{2,\lambda}(\rho))\leq
-\frac{|\lambda_1^2-\lambda_2^2|}{|\lambda|^4 t_0^{\rho+1}}+C\frac{1/\rho+\ln t_0}{|\lambda|^3 t_0^{2\rho+1}}. \] In other words, the left hand side is negative if \[
t_0^{\beta_0}>\frac{C|\lambda|}{|\lambda_1^2-\lambda_2^2|}\big(\frac{1}{\beta_0}+ \ln t_0\big). \] Hence, there exists a number $T_1= T_1 (\xi^0, \beta_0)>1$ (see \eqref{t0}), such, that for all $t_0\geq T_1$ the function $sign(\lambda_1^2-\lambda_2^2)\Re(e'_{2,\lambda}(\rho))$ is negative.
Lemma \ref{ERL} and therefore Theorem \ref{thm_Riemann-Liouville} are proved.
\section{An example} To illustrate the theorems proved above consider the following Cauchy problem (see \cite{UAY})
\begin{align} \label{ex_01} D_{\ast}^{\beta_1} u_1(t,x) & = -D^2 u_1(t,x)-D u_2(t,x), \quad t>0, \ -\infty <x<\infty, \\\label{ex_02} D_{\ast}^{\beta_2} u_2(t,x) & = -D u_1(t,x)-D^2 u_2(t,x), \quad t>0, \ -\infty<x<\infty, \\ \label{ex_03} u_1(0,x) & =\varphi_1(x), \quad u_2(0,x)=\varphi_2(x), \quad -\infty<x<\infty. \end{align} It is not hard to see that the symbol of the operator on the right hand side of \eqref{ex_01}-\eqref{ex_02} is symmetric and has the representation
\begin{equation} \label{matrix_10} \mathcal{A}(\xi) = \begin{bmatrix}
-\xi^2 & -\xi \\
-\xi & -\xi^2
\end{bmatrix}
=
\begin{bmatrix}
1/2 & 1/2 \\
-1/2 & 1/2
\end{bmatrix}
\begin{bmatrix}
-\xi^2+\xi & 0 \\
0 & -\xi^2-\xi
\end{bmatrix}
\begin{bmatrix}
1 & -1 \\
1 & 1
\end{bmatrix} .
\end{equation} As is seen from \eqref{matrix_10} that $\lambda_1(\xi)=-\xi^2+\xi$ and $\lambda_2(\xi)=-\xi^2-\xi.$ The solution $U(t,x)= \langle u_1(t,x), u_2(t,x) \rangle$ to Cauchy problem \eqref{ex_01}-\eqref{ex_03} has the representation:
\begin{align*} u_{1} (t,x)&= \frac{1}{2\pi} \int\limits_{-\infty}^{\infty} \left[\frac{1}{2} E_{\beta_1}((-\xi^2+\xi)t^{\beta_1})+\frac{1}{2}E_{\beta_2}((-\xi^2-\xi)t^{\beta_2}) \right] \hat{\varphi}_1](\xi) d \xi \\ &+ \frac{1}{2\pi} \int\limits_{-\infty}^{\infty} \left[\frac{1}{2} E_{\beta_1}((-\xi^2+\xi)t^{\beta_1})-\frac{1}{2}E_{\beta_2}((-\xi^2-\xi)t^{\beta_2}) \right] \hat{\varphi}_2](\xi) d\xi; \\ u_{2} (t,x)&= \frac{1}{2\pi} \int\limits_{-\infty}^{\infty} \left[\frac{1}{2} E_{\beta_1}((-\xi^2+\xi)t^{\beta_1})-\frac{1}{2}E_{\beta_2}((-\xi^2-\xi)t^{\beta_2}) \right] \hat{\varphi}_1(\xi) d \xi \\ &+ \frac{1}{2 \pi} \int\limits_{-\infty}^{\infty} \left[\frac{1}{2} E_{\beta_1}((-\xi^2+\xi)t^{\beta_1})+\frac{1}{2}E_{\beta_2}((-\xi^2-\xi)t^{\beta_2}) \right] \hat{\varphi}_2(\xi) d \xi.
\end{align*} Moreover, obviously, $\lambda_k(\xi) \le 0, \ k=1,2,$ for all
$\xi$ satisfying the inequality $|\xi| \ge 1.$ It is not hard to verify, that \[ K_{1,1}(\xi, \hat{\Phi}(\xi))=K_{2,1}(\xi, \hat{\Phi}(\xi))=\frac{1}{2}\hat{\varphi}_1(\xi)+\frac{1}{2}\hat{\varphi}_2(\xi) \] and \[ K_{1,2}(\xi, \hat{\Phi}(\xi))=\frac{1}{2}\hat{\varphi}_1(\xi)-\frac{1}{2}\hat{\varphi}_2(\xi), \quad K_{2,2}(\xi, \hat{\Phi}(\xi))=-\frac{1}{2}\hat{\varphi}_1(\xi)+\frac{1}{2}\hat{\varphi}_2(\xi). \] Therefore for the corresponding determinant one has \[
\big|K_{j,l}\big(\xi^0,
\widehat{\Phi}(\xi^0)\big)\big|=\frac{1}{2} \big(\hat{\varphi}^2_2(\xi^0)-\hat{\varphi}^2_1(\xi^0)\big) \] and condition (\ref{xi0}) has the form \[ \hat{\varphi}^2_2(\xi^0)\neq\hat{\varphi}^2_1(\xi^0), \quad
\text{or} \quad |\hat{\varphi}_2(\xi^0)|\neq
|\hat{\varphi}_1(\xi^0)|. \] In this case the unknown orders $\beta_1$ and $\beta_2$ are the unique roots of the following equations \[ E_{\beta_1} \big(\lambda_1(\xi^0) t^{\beta_1}_0\big)= -\frac{d_1+d_2}{\hat{\varphi}_1(\xi^0)+\hat{\varphi}_2(\xi^0)}, \] \[ E_{\beta_2} \big(\lambda_2(\xi^0) t^{\beta_2}_0\big)= \frac{d_1-d_2}{\hat{\varphi}_2(\xi^0)-\hat{\varphi}_1(\xi^0)}, \] respectively.
\end{document}
\end{document} | arXiv |
Wireless Mobile Sensor Networks with Cognitive Radio Based FPGA for Disaster Management
G. A. Preethi Ananthachari*
G. A. Preethi Ananthachari*, Technology Studies, Endicott College of International Studies, Woosong University, Daejeon, South Korea, [email protected]
Received: May 18 2020
Revision received: November 7 2020
Accepted: March 5 2021
Abstract: The primary objective of this work was to discover a solution for the survival of people in an emergency flood. The geographical information was obtained from remote sensing techniques. Through helpline numbers, people who are in need request support. Although, it cannot be ensured that all the people will acquire the facility. A proper link is required to communicate with people who are at risk in affected areas. Mobile sensor networks with field-programmable gate array (FPGA) self-configurable radios were deployed in damaged areas for communication. Ad-hoc networks do not have a centralized structure. All the mobile nodes deploy a temporary structure and they act as a base station. The mobile nodes are involved in searching the spectrum for channel utilization for better communication. FPGA-based techniques ensure seamless communication for the sur-vivors. Timely help will increase the survival rate. The received signal strength is a vital factor for communi-cation. Cognitive radio ensures channel utilization in an effective manner which results in better signal strength reception. Frequency band selection was carried out with the help of the GRA-MADM method. In this study, an analysis of signal strength for different mobile sensor nodes was performed. FPGA-based implementation showed enhanced outcomes compared to software-based algorithms.
Keywords: Cognitive Radio , Disaster Management , FPGA , GRA , Sensors , Signal Strength
Our Earth has observed a large number of natural disasters recently, resulting in enormous fatalities for people and other living organisms, building infrastructure, and more or less the entire region involved. Catastrophes like cyclones, earthquakes, and floods, have constant effects in several places at irregular times, leading to the intensification of the destruction of life and public/individual property. Even though science and technology have made immense progress in many areas, scientists are still unable to precisely forecast the time and place of these disasters and the level of hazard that might occur. Some of the intense hurricane floods in recent times include Hurricane Dorian, which thrashed the Bahamas with its severe vicious winds on September 1, 2019. In 2015, north east monsoon generated heavy rainfall in South India especially in the region of Tamilnadu and Andhra, were there was a drastic flood that affected the daily life. In 2013, in the north Indian region Uttarkhand a flash flood, caused a huge catastrophic disaster in the form of a cloud burst and landslide. For all these, the focus has been on mitigating the harm that might be caused by natural adversity and to remain prepared for recovery management and reform operations. Calamity recovery and reconstruction operations have always been difficult tasks for the administration, confined authorities, and the Indian nation.
be caused by natural adversity and to remain prepared for recovery management and reform operations. Calamity recovery and reconstruction operations have always been difficult tasks for the administration, confined authorities, and the Indian nation.
Effective communication is inevitable in coordinating the rescue team and survivors of the disaster. To establish efficient and swift recovery, the rescue team, police officials, and individual volunteers involved need to be moving at a fast pace through diverse locations within the area to diminish the effects and to discover more survivors from the catastrophe. However, in many circumstances, the communication infrastructure breaks down and proper communication cannot be achieved in the area which leads to longer latencies in crisis operations and increased damage to humankind. Mobile sensor networks play an important role in risk management systems. Floods will affect the communication systems which can make it not possible to have regular communication. Thus a few infrastructure networks are implemented as a temporary arrangement. Since regions in the proximity also have the possibility of being affected by the flood. There may be disruption of signals, which are deployed using mobile sensor networks. At the destination end, signal reception will be interrupted, and result in noise signals. To prevent those compliances, FPGA is incorporated with cognitive radio and implemented. Cognitive radio embedded with FPGA substantially overcomes the interruptions incurred in communications. In turn, it paves way for the increased survival rate of people.
2. Related Work
A variety of wireless sensor studies have been conducted, starting from typical small-scale sensors to advanced mobile sensors (both wired and wireless types), temperature, underwater sensors, etc., in various research institutes such as in the Centre of Excellence in Wireless Technology, Chennai, India. A disaster detection and alerting system was presented by Kaur et al. [1]. The authors discussed the basic architecture of wireless sensor networks (WSNs) and how it will be helpful in disaster management schemes. The MANET or mobile ad hoc networks are capable of forming wireless networks with different topologies. The radios, which form the network use a specific band, data rate, and frequency. In [2], the authors focused on the implementation of an intelligent MANET (iMANET) in which wireless networks are established using reconfigurable radios so that intelligent networks can be formed. The nodes involved in the network formation are capable of detecting spectrum holes and are self-confi¬gurable to specific bandwidth and data rates based on the channel conditions. The cooperative sensing capability of the network helps to make accurate decisions. The entire iMANET is implemented in Xilinx Zynq, which is a GPU-FPGA system-on-a-chip (SoC).
Menon et al. [3] used the reliable routing technique (RRT) to ensure reliable data delivery at a destination device even when people with mobile devices are moving in the network, broadcasting property of the wireless network and a priority list of probable forwarding candidates at each device ensure increased throughput. Jayakumar and Gopinath [4] stated that the introduction of Bluetooth, 802.11, and hyper-LAN are helping commercial MANET deployments outside the military domain. To facilitate communication, routing protocols have been used to discover nodes nearby [4]. In [5], the author presented remote sensing application which plays a vital role in disaster management activities. It provides a database from which the evidence left behind disasters that have occurred before can be inferred. Hazard maps can be derived from remote sensing activities. Cyclone monitoring and warning, inundation mapping and damage assessment and drought management have been the major works carried out with the help of remote sensing techniques. Abba and Lee [6] suggested a low-power, low-cost, and highly accurate monitoring and control mechanism using autonomous sensor agents to dynamically reconfigure control of on-chip sensor networks by FPGAs. In [7], the authors proposed a "DistressNet," an ad-hoc wireless architecture that supports disaster response with distributed collaborative sensing, topology-aware routing using a multichannel protocol, and accurate resource localization.
However, designing a sensor network on FPGA is a tedious job to accomplish in a short period since designers are limited to extension work of the FPGA producer design requirement. Likewise, the amount of intricacy, as well as complexity of devise mechanization restricted in the FPGA, make it complicated for designers to twitch the FPGA circuitry to design and implement sensor networks. A cluster-based decentralized orchestration cooperative sensing scheme was proposed in [8], which was as entitled "Decentralized cooperative spectrum sensing for ad-hoc disaster relief network clusters" in another conference held in 2010 IEEE 71st Vehicular Technology Conference, Taipei, Taiwan. Choi et al. [9] proposed a distributed medium access control protocol that uses successive multiple collision detection phases for dense WSN environments by enhancing the typical carrier sense multiple access with collision resolution protocol that uses only a single a collision detection phase. Colliding stations are filtered so that only surviving stations are allowed to compete. The collision probability becomes considerably reduced. As a result, throughput increases.
Lotze et al. [10] developed a virtual architecture for hardware abstraction, an adaptive run-time system for managing cognition, and high-level design tools for cognitive radio development for embedding cognitive radio technology with a FPGA board. They proposed a framework for a receiver node that worked in two modes as discovery and communication. In [11], requirements for disaster management and innovative technology for an integrated disaster management communication and information system, addressing, in particular, network, configuration, scheduling, and data management issues during the response and recovery phases were discussed. The authors of [12] presented an experiment on ring oscillators in low voltage using thermal sensors. The sensibility of frequency increase was shown with the proposed system in which a quadratic polynomial function was utilized. A direct sensor FPGA interface was used by Oballe-Peinado et al. [13] to acquire data-parallel. FPGA does not have an analogue to digital converter. However it can be reprogrammed. Different capture modules have been analyzed.
In [14], the authors proposed FPGA-based on-chip sensors for network monitoring, which utilize autonomous reconfigurable sensors for controlling. It collects the signals of voltage and power; by self-aware capability, it reduces the power consumption of the FPGA. A 1,000 ms refresh time increases the on-chip sensor readings. Perera et al. [15] proposed a 1-wire protocol for on-chip sensor-based FPGA. Reprogrammable smart sensor nodes have been used with Zigbee. Processing and transducer func¬tionalities are given in a single core that increases the power speed due to inter-process communication between sensors. In [16], the authors presented an autonomous RFID sensor node using a single ISM band, which was used for both power transfer and data communication. A rectenna harvests the electromagnetic energy transported by the dedicated radiofrequency source for charging a few-mF super-capacitor. For super-capacitors of 7 mF, the proposed autonomous sensor nodes were able to wirelessly communicate with the reader at 868 MHz for 10 minutes without interruption for a tag-to-reader separation distance of 1 m. The authors obtained the result from effective radiated powers of 2 W during the super-capacitor charging of 100 mW during wireless data communication. In [17], the authors proposed a Markov decision process-based approach for seamless mobility of heterogeneous networks. The rest of the sections describe our work plan, methodology, signal strength implementation based on the hidden Markov model (HMM), the results and analysis, signal strength based on FPGA, conclusion.
3. Work Plan
Disaster management and recovery plans have become inevitable because of global warming. Mobile sensor networks play a crucial role in data transmission and communication. They can be very helpful in disaster-prone areas. Mobile sensor networks make use of the technologies which can warn and provide an alert message for the instantaneous rescue operation to activate, whenever a disaster. The aim was to assess the technological solutions for controlling a disaster using mobile sensor networks via a disaster discovery and preparedness system and search and rescue operations. Perhaps the focus was on the fundamental architecture of mobile sensor networks, which are very helpful for communicating with survivors during disaster management and for the FPGA models that can be employed for diverse disaster-prone locations. Disaster Relief Teams (DFT) were sent to all the devastated areas and coastal regions, which need quick aid during this study.
Voice communication may become interrupted in due course at an affected site. A suitable routing type should be used in which distress tolerant networks have been used. This overcomes the problem of disturbances in communication in congested network areas. In disaster-affected zones survivors and responders (people) move periodically for rehabilitation. Collective-channels medium access control usage enables continuous communication without commotion. Protocols, such as low rate wireless personal area networks, Zigbee, and Wi-Fi are used for communication. Problems, such as link breaks, deviation in networks, and an inability to access the protocol are significant challenges in networks. Congestion and collision occur while sensor nodes compete to send data through available channels. Optimized link state routing protocols will be used when it is a proactive protocol. It periodically updates the routing path and the time consumption is less compared to reactive protocols. Here mobile sensor networks were implemented using cognitive radios with effective channel selection, which was quite helpful in communicating in emergencies. A real execution is proposed for disaster management for observing consistent situations through a low pervasive sensing method. The network architecture and the interconnecting devices for consistent measurement of constraints by sensors and data communication through sensor networks were considered in this framework. At the base level, huge numbers of sensors were incorporated. A huge quantity of real-time device values was produced through the sensor. Fre¬quently, small amounts of data were often produced by the sensors.
Various sensors can be incorporated for measuring environmental conditions. Sensors connected to the FPGA board through a cognitive radio will get accurate data for further processing. The records are then distributed towards the system for serial monitoring. With the help of cognitive radio, sensor networks, and FPGA data analysis, the DFT can communicate rapidly and evacuate the people to a safer zone. FPGA sensor data is used to alert people for continuous automated monitoring of the affected regions.
This layer manages the infrastructure among all the sensor nodes. The interference of signals is mitigated through cognitive techniques of spectrum sensing. Cognitive radio technology increases channel utilization. The architecture design is shown in Fig. 1. In Algorithm 1, people from emergency areas will communicate through movement or by other means. Sensor data is collected through it and stored. Further processing is accomplished through cognitive ratio (CR) and FPGA analysis.
Architecture of the proposed cognitive radio based FPGA.
Channel selection by cognitive radio
Simulation: The FPGA board incorporated with sensors was used for the simulation. ISim is an FPGA simulator that helps program the board with various sensors and transducers. Having developed the On-chip sensor boards with FPGA [14], we used it as the basic concept for embedding the sensors in the simulation. CR incorporation with sensors will enhance channel utilization and prevent communication disconnection. CRs enable current WSNs to overcome the scarcity problem of the spectrum, which is shared with many other successful systems such as Wi-Fi and Bluetooth. It has been shown that the coexistence of such networks can significantly degrade a WSN's performance. Moreover, cognitive technology could provide access not only to the new spectrum but also to the spectrum with better propagation characteristics. Channel selection methodology using the CR is depicted in Fig. 2.
Experimental set-up: To verify the above methodology implemented simulation results, an FPGA board was used. Mobile sensors send the signals and those signals are captured through cognitive techniques, which helps with sensing the spectrum of frequency bands for channel utilization.
Channel selection methodology.
Study and comparison: The computer simulation and experimental results were compared for validation of the FPGA-based mobile networking model and working nature. The computer simulation was carried out based on HMM. HMM was implemented and compared with the FPGA-based signal strength obtained through cognitive sensing. Signals were selected through spectrum sensing. However, the available spectrum could not be guaranteed because a long duration call could be interrupted and disconnected during the congestion of channels. All these issues should be overcome by spectrum management, which involves spectrum detection (sensing), spectrum decision, and spectrum sharing. To ensure seamless spectrum availability, spectrum handover should be carried out successfully. Heterogeneous spectrum sharing involves many aspects, such as target channel, switching delay, channel latency, channel interference, and channel capacity.
Spectrum detection: Spectrum detection is sensing the unused spectrum. First, it should check for primary users and available free channels. Channels should be sensed without any interference. Many spectrum sensing schemes are available such as energy-based, radio identification-based schemes, and waveform-based schemes [18].
Spectrum decision: This involves selecting the spectrum from the available free spectrum. The spectrum hole is decided by the received signal strength, spectrum congestion, and the number of primary users sharing the spectrum. Spectrum selection is more challenging as there are many different routes available between the source and destination. Quality-of-service (QoS) parameter consideration will help to select the spectrum. Here, the multiple attribute decision making method was employed for spectrum selection.
Spectrum sharing: Here network maintains the QoS by avoiding collision with other CR's. This also involves resource allocation. As CR's require frequent movement from one band of frequency to another band, fairness in the handoff from one band to another band is mandatory. In this work, there were four different bands and for each band, there was its respective primary user. Whenever a primary user appears, the CR should leave the channel and search for a free channel. This will be accomplished by the heterogeneous handoff algorithm. Grey relational analysis (GRA) was used for spectrum selection [19].
5. Signal Strength Estimation based on the Hidden Markov Model
Normally, Markov chains are based on random selections. The basic Markov process involves states, transition probabilities, and actions. Actions are based on the transition probabilities, which can be calculated with the help of states. Transition probability represents moving from one state to another state. In contrast, HMM involves hidden states. If our Markov process represents signal strength from different nodes, we consider states as sensor nodes with RSS [TeX:] $$\left\{\mathrm{S}_{1}, \mathrm{~S}_{2} \ldots \mathrm{S}_{\mathrm{n}}\right\}$$. Data from one node to another node travel through the hops. If the source node to destination node is far apart, more hop counts will be added. The destination node has no idea about the traversal history of its data. This can be illustrated with the help of the HMM [20].
The Markov process property can be shown as
[TeX:] $$P\left(X_{t}=j \mid X_{1}=i_{1}, \ldots, X_{t-1}=i_{t-1}\right)=P\left(X_{t}=j \mid X_{t-1}=i_{t-1}\right)$$
The probability of moving from state i to state j.
States and transitions.
From Fig. 3, there are four states, the transition probability from [TeX:] $$\mathrm{S}_{1}$$ to [TeX:] $$\mathbf{S}_{2}$$ is given as 0.2 and from [TeX:] $$\mathrm{S}_{2}$$ to [TeX:] $$\mathrm{S}_{\underline{3}}$$, it is given as 0.3. [TeX:] $$\mathrm{S}_{3} \rightarrow \mathrm{S}_{1}$$ as 0.5. In matrix form,
[TeX:] $$p=\begin{array}{lll} 0.2 & 0.3 & 0.5 \\ 0.4 & 0.4 & 0.2 \\ 0.3 & 0.3 & 0.4 \end{array}$$
Likewise, note that the sum of each row is equal to 1. It represents weights. It shows the stochastic matrix. The i,j corresponds to [TeX:] $$p_{i, j}$$ which is the transition from state i to state j. Normally, probabilities are given at the initial stage as [TeX:] $$q=\left(q_{1}, \ldots, q_{m}\right)$$ to be at each state at time t=0. State i at time t will be equal to the [TeX:] $$i^{t h}$$ entry of the vector [TeX:] $$p^{m} q$$, which is the probability measure. Suppose that the signal [TeX:] $$\mathrm{S}_{1}, \mathrm{~S}_{2}$$, [TeX:] $$\mathbf{S}_{3}$$ has the transition probabilities, such as 0, 0.3 and 0.6 at some given time. The probability of the [TeX:] $$50^{\text {th }}$$ node receiving the data could be calculated as
[TeX:] $$p=\begin{array}{lll} 0.2 & 0.3 & 0.5^{50} \\ 0.4 & 0.4 & 0.2 \\ 0.3 & 0.3 & 0.4 \end{array} \cdot(0,0.3,0.6)^{t}$$
In the Hidden Markov Model, there is an invisible Markov chain that cannot be observed. Each state randomly generates one out of m observations that is visible. Dense deployment of sensor nodes produces signals at the same time and routes the signal through the hop count. For routing the data signal, sensors use some algorithms. During transmission of data, signals deteriorate, and not all the sensors data are received at the destination point. The transmissions of the data signal are depicted in terms of the Hidden Markov Model. As sensor node routes are unpredictable, the Hidden Markov Model suggests the best data reception method. Let us consider the Markov chain, with five states S1, S2, S3, S4, and S5. Suppose there are 2 observations, S1 and S2, in which,
[TeX:] $$P\left(S_{1} \mid S_{3}\right)=0, P\left(S_{2} \mid S_{3}\right)=1$$
[TeX:] $$P\left(S_{1} \mid S_{4}\right)=0.4, P\left(S_{2} \mid S_{4}\right)=0.6$$
By applying the Hidden Markov policy, consider that the data signal arrives from the sensor node S2. Suppose the data has been transmitted two times; then, the probability is 3x3, which has 9 options. The probability calculation is defined below in Eq (5);
[TeX:] $$\begin{aligned} P\left(\left(S_{2}, S_{2}\right),\left(S_{4}, S_{3}\right)\right) &\left.=P\left(\left(S_{2}, S_{2}\right) \mid\left(S_{4}, S_{3}\right)\right) \cdot P\left(S_{4}, S_{3}\right)\right) \\ &=P\left(S_{2} \mid S_{4}\right) \cdot P\left(S_{2} \mid S_{3}\right) \cdot P\left(S_{3} \mid S_{4}\right) \cdot P\left(S_{4}\right) \end{aligned}$$
5.1 Viterbi Algorithm: For Finding the Hidden States
Based on the observations, hidden states should be found. The straight forward approach will take multi-exponential time, like the traditional statistical approach that was discussed in an earlier section. A well-organized approach is the Viterbi algorithm [21]. The idea is as follows.
A series of observations are provided [TeX:] $$m_{1}, \ldots \ldots \ldots m_{t}$$. For each state [TeX:] $$s, t=1 \ldots \ldots T$$
[TeX:] $$\vartheta_{t}(s)=\max _{s_{1}, \ldots, s_{t-1}} P\left\{x_{1}=s_{1}, \ldots, x_{t-1}=x_{t-1}, x_{t}=s, m_{1}, \ldots, m_{t}\right\}$$
From the above Eq (6), the maximum probability of the path at state s that ends at time t is provided with the observations. By using the Markov property, the observation is the perceptive path the ends with state s at time t, which equals to [TeX:] $$\mathrm{s}^{*}$$ at time t-1. This confirms that [TeX:] $$\mathbf{S}^{*}$$ is the value of the final state of the perceptive path that ends at time t-1. The forward recursion is given by
[TeX:] $$\vartheta_{t}(s)=\max _{s} P_{s, n} \alpha_{n}\left(m_{t}\right) \vartheta_{t-1}(s)$$
[TeX:] $$\alpha_{n}\left(m_{t)}\right.$$ is the probability to acquire [TeX:] $$m_{t}$$ . provided that the hidden Markov state is j. Consider 15 different paths through which the sensor nodes send data. Here, 1 symbolizes successful data and 0 is more likely for unsuccessful data. By using the Viterbi algorithm, from the below code in Fig. 4, the simulated observations are shown.
Observed code and sequence.
The HMM matrix and observable signal transition probabilities are given in Fig. 5.
From the outcome shown in Fig. 6, s is the observed probability (Here we have 3 observations, such as s3, s4, and s5). Symbol "t" here represents different paths (hop) the signal traverses.
Nu[0,2] = 2.0 signifies that the sensor node s3 with path 2, emits a strong signal. Path results are shown in Fig. 7.
Hidden Markov Model matrix and observable matrix.
Signal emission through a different path.
Aggregation of the best path.
From the above result, 0 [TeX:] $$\longrightarrow$$ Noise, 1 [TeX:] $$\rightarrow$$ Signal, 2 [TeX:] $$\rightarrow$$ Strong Signal. Finally, it can be concluded that, from the above Table 1, the signal node [TeX:] $$\left(\mathrm{S}_{5}\right)$$ shows strong signal traversal and reaches the signal in [TeX:] $$\left(\mathrm{S}_{2}\right)$$. Signals are prone to deteriorate. On the account of such mitigation, the strong signal becomes less intense and reaches a normal signal. In path 2, the sensor node [TeX:] $$\left(\mathrm{S}_{4}\right)$$ sends an average signal to [TeX:] $$\mathbf{S}_{2}$$. Here average means a, signal near the threshold level. In path 3, [TeX:] $$\mathrm{S}_{3}$$ represents noise, which could be damaged data, and it results in signal [TeX:] $$\left(\mathrm{S}_{2}\right)$$. Here, the resultant signal would have lost some data packets. A large number of sensor nodes as well as, its signals could not be handled very easily. Therefore, FPGA usage paves the way for getting the manipulation of huge sensor signals. A huge number of sensor nodes are multiplexed and integrated to receive it as whole dataset. The advantage of multiplexing is to fault-tolerate some weak signal or no signal.
Observation of states and outcome
FPGA implementation using 40 sensor nodes.
Fig. 8 shows different input signals multiplexed through the OR gate to form integration and results as a single signal output. A 4:1 multiplexer type was used. Here the FPGA board uses 29 cells, 40 I/O ports, and 59 Nets. Nets represent the logical connection of more than a one pin symbol instance. Usually, mobile sensor communication is prone to attenuation. As it is deployed in an emergency where there are flood-affected areas, the sensor nodes deploy their own infrastructure and act as a base station at times. Fig. 9 shows the in-depth view of logic cells deployed in FPGA.
Zoom-in view of nodes.
6. Results and Analysis
As shown in Figs. 10–18, sensor nodes signal strength was measured and visualized. Received signal strength is represented in decibels (dB). Normally, the received signal strength depends on the transmitted power and received power. Overall, -60 dB to -70 dB is the signal strength range in mobile nodes. In sensor nodes, it might be less. As shown in Figs. 10–18, Node 1 measured a peak signal of 1.0; Node 2 showed its range from 20–60 –dBs; Node 3 signal range is from 0 to 5.5, which is a weak signal; Node 4 showed 0 to 10.5 at 500 ms and, -5 dBs at 1,250 ms; Node 5 displayed a peak of -3 dBs at 550 ms and fluctuated from -3 to +3 at 1,000 ms to 1,750 ms; Node 6 showed a low of 0.4 and a peak of -0.2; Node 7 showed -0.2 as a constant streaming signal; Node 8 showed a strong signal as -50 measured; Node 9 had a range from -30 to -70; Node 10 showed a range from -30 to -80; Nodes 11 and 12 consequently had their range as -20 to -80 and -30 to -70 dBs, respectively. Nodes 8 and 9 showed stronger signals compared to other nodes.
Signal strength (Node 1).
6.1 Signal Processing through FPGA
In Figs. 19–29, for Nodes 1–11, signal strengths which were captured via FPGA are shown: Node 1, showed a standard stream in which its signal is weak; Node 2 showed a peak of -2.5 at nearly 2,000 ms. Node 3 had an increase from 0 to 120 dBs, which could be noise or an interruption from other sensor nodes; Node 4 had its range from 7.75 to 9.5; Node 5 had a peak of -1; Node 6 had a streaming signal of 0.3 for the entire duration. Therefore, for Node 7, this also had a streaming signal. Node 8 had a stronger signal of -40 to -80 dBs; Nodes 9–11 are shown in Figs 27–29. These nodes exhibit stronger signals compared to previous simulation results. However, these particular nodes are executed for a short duration, the same as a computer simulation.
FPGA signal (Node 1).
FPGA signal (Node 10).
Comparison of simulation results.
As shown in the above Fig. 30, the FPGA-based implementation incorporates a proper sine wave, whereas simulated signals are prone to interruptions and cancel each other. This results in no available signal. The capabilities of proper signal waves are achieved by cognitive spectrum sensing. The Vivado Simulator has been used for simulating FPGA sources. It supports VHDL as well as Verilog code files. Parsers for VHDL are xvhdl and for Verilog files, they are xvlog, which stockpile the parsed files into an HDL library on the memory disk. The HDL elaborator is xelab and the linker command. For a specified high-level unit, xelab invokes all sub-level units, carries out static embellishment, and associates the produced executable code with the simulation core to generate an executable simulation scenario. The Vivado simulation command xsim loads a simulation scenario to achieve a batch mode simulation, a graphical user interface, or Tcl-based interactive simulation environment. Vivado Integrated Design Environment (IDE) is an "interactive design-editing environment that provides the simulator a user-interface and common waveform viewer."
CR channel selection could be statured as a multiple attribute decision-making problem (MADM). For a MADM problem, if there are 'm' alternatives and 'n' attributes, the i'th alternative could be expressed as [TeX:] $$Y_{i}=\left(y_{i 1}, y_{i 2}, \ldots y_{i n}\right)$$. Here alternatives are referred to as frequency bands and attributes are the QoS parameters. The comparability sequence is given by [TeX:] $$X_{i}=\left(x_{i 1}, x_{i 2}, \ldots x_{i n}\right)$$. The grey relational coefficient is calculated by [19],
[TeX:] $$\gamma\left(x_{0 j}, x_{i j}\right)=\frac{\Delta_{\min }+\varsigma \Delta_{\max }}{\Delta_{i j}+\varsigma \Delta_{\max }}$$
Here [TeX:] $$\Delta_{\min }$$ represents the minimum-better option. This applies to parameters, such as the delay in which should be at a minimum. [TeX:] $$\Delta_{\max }$$ represents the maximum better option. Channel capacity is considered to be maximum. [TeX:] $$\varsigma$$ represents the co-efficient value that is used for normalizing the values (0 to 1). [TeX:] $$\Delta_{i j}$$ is the difference between one attribute and another. The performance of GRA yields improved outcomes in relation to other MADM methods. The GRA method converges rapidly compared to other methods. From Figs. 31 and 32, shown that the computational complexity of GRA is less compared to the traditional MADM method. On the other hand, GRA converges at the optimal time, whereas MADM requires more iteration for channel selection. In Figs. 31 and 32, time is shown in terms of milliseconds.
Channel selection using GRA.
Channel selection using MADM.
FPGA mobile sensor nodes that are combined with cognitive radio amplify the communication channel in disaster regions. Cognitive radio-based FPGA sensor networks increase the scalability of nodes and interruptions in the communications are discarded extensively. This can be achieved with the help of spectrum sensing and effective channel utilization is established by cognitive radios. For the effective usage of CR's, the Handoff algorithm is used. Several algorithms are available for Handoff. MADM methods are characterized for decision-making. The GRA method illustrated significant results compared to the traditional method in terms of channel selection. As the versatility and consistency of FPGA enlarge, self-governing sensors can be embedded in FPGA to evaluate several dynamic parameters. This approach will considerably reduce power consumption and increase usability. It can be fully used for remote sensing applications. Sensor nodes will monitor the temperature, power, communication channel, etc. These mobile sensor nodes have better coverage capacity, enhanced energy efficiency, and outstanding channel capacity. The sensors connected to FPGA will detect the temperature and environmental conditions in close proximity. It will send an alert signal so that the evacuation process can proceed rapidly. This will help to save numerous lives. The planned architecture and layered work will support hardware engineers and system architects by supplying supple and proficient real-time observation and a control design for huge and complex embedded sensor networks and remote-sensing applications.
· FPGAs have a high amount of multipliers and internal memory. As such they are highly suitable for signal processing systems. Thus we can find them in hardware that performs signal conditioning and muxing/demuxing, e.g., wireless networking gear, such as base stations.
· The smallest logic element in an FPGA is called a logic block, which is an ALU+flip-flop at its minimum. Therefore, FPGAs are used extensively for computing problems that can benefit from SIMD types of architectures, e.g. cleaning up images being received from an image sensor, point, or local processing of image pixels, e.g. computing difference vectors in H.264 compression.
· Finally, ASIC emulation, or hardware/software in loop testing, etc. is another application. Logic design for FPGA shares the same flow and tools as ASIC design. As such, FPGAs are also used to verify some test cases during ASIC development where the interactions between hardware and software may be too complicated or time-consuming to model.
G. A. Preethi Ananthachari
She received her Ph.D. from Periyar University, Salem, TN, India. Currently, she is working as an assistant professor in the Technology Studies department, Endicott College of International Studies, Woosong University, Daejeon, South Korea. She has been awarded a distinction in her M.Phil. (Computer Science). She worked as an Assistant Professor at VIT Bhopal University, MP, India. She worked as a guest lecturer in the Department of Computer Science at Periyar University, Salem for a year from 2011-2012. She joined as a Project Fellow in the same department under the UGC Major Project in 2012. Her research interests are mobile computing, wireless technology, data analytics, machine learning, and robotics. She received the best paper award in an International Conference on Advances in Engineering and Technology (ICAET) held in E.G.S. Pillai Engineering College, Nagapattinam, Tamil Nadu, India.
1 H. Kaur, R. S. Sawhney, N. Komal, "Wireless sensor networks for disaster management," International Journal of Advanced Research in Computer Engineering & Technology, vol. 1, no. 5, pp. 2278-1323, 2012.custom:[[[-]]]
2 K. A. Kumar, "FPGA-ARM implementation of an intelligent mobile ad-Hoc network," in Proceedings of 2017 International Conference on Networks & Advances Computational Technologies (NetACT), Thiruvananthapuram, India, 2017;pp. 111-115. custom:[[[-]]]
3 V. G. Menon, J. P. Pathrose, J. Priya, "Ensuring reliable communication in disaster recovery operations with reliable routing technique," Mobile Information Systems, vol. 2016, no. 9141329, 2016.doi:[[[10.1155//9141329]]]
4 G. Jayakumar, G. Gopinath, "Ad hoc mobile wireless networks routing protocols: a review," Journal of Computer Science, vol. 3, no. 8, pp. 574-582, 2007.custom:[[[-]]]
5 B. Manikiam, "Remote sensing applications in disaster management," Mausam, vol. 54, no. 1, pp. 173-182, 2003.custom:[[[-]]]
6 S. Abba, J. A. Lee, "An autonomous self-aware and adaptive fault tolerant routing technique for wireless sensor networks," Sensors, vol. 15, no. 8, pp. 20316-20354, 2015.custom:[[[-]]]
7 S. M. George, W. Zhou, H. Chenji, M. Won, Y. O. Lee, A. Pazarloglou, R. Stoleru, P. Barooah, "DistressNet: a wireless ad hoc and sensor network architecture for situation management in disaster response," IEEE Communications Magazine, vol. 48, no. 3, pp. 128-136, 2010.custom:[[[-]]]
8 N. Pratas, N. Marchetti, N. R. Prasad, A. Rodrigues, R. Prasad, "Decentralized cooperative spectrum sensing for ad-hoc disaster relief network clusters," in Proceedings of the 71st IEEE V ehicular Technology Conference, Taipei, Taiwan, 2010;custom:[[[-]]]
9 H. H. Choi, H. Lee, S. Kim, J. R. Lee, I. H. Lee, "Distributed medium access control protocol based on successive collision detection for dense wireless sensor networks," International Journal of Distributed Sensor Networks, vol. 12, 2016.doi:[[[10.1177/1550147716664238]]]
10 J. Lotze, S. A. Fahmy, J. Noguera, B. Ozgul, L. Doyle, R. Esser, "Development framework for implementing FPGA-based cognitive network nodes," in Proceedings of 2009 IEEE Global Telecommu-nications Conference (GLOBECOM), Honolulu, HI, 2009;pp. 1-7. custom:[[[-]]]
11 A. Meissner, T. Luckenbach, T. Risse, T. Kirste, H. Kirchner, "Design challenges for an integrated disaster management communication and information system," in Proceedings of the 1st IEEE Workshop on Disaster Recovery Networks (DIREN), New York, NY, 2002;pp. 1-7. custom:[[[-]]]
12 J. J. L. Franco, E. Boemo, E. Castillo, L. Parrilla, "Ring oscillators as thermal sensors in FPGAs: experiments in low voltage," in Proceedings of 2010 VI Southern Programmable Logic Conference (SPL), Ipojuca, Brazil, 2010;pp. 133-137. custom:[[[-]]]
13 O. Oballe-Peinado, F. Vidal-V erdu, J. A. Sanchez-Duran, J. Castellanos-Ramos, A. Hidalgo-Lopez, "Smart capture modules for direct sensor-to-FPGA interfaces," Sensors, vol. 15, no. 12, pp. 31762-31780, 2015.custom:[[[-]]]
14 S. Abba, J. A. Lee, "FPGA-based design of an intelligent on-chip sensor network monitoring and control using dynamically reconfigurable autonomous sensor agents," International Journal of Distributed Sensor Networks, vol. 12, 2016.doi:[[[10.1155//4246596]]]
15 M. D. R. Perera, R. G. Meegama, M. K. Jayananda, "FPGA based single chip solution with 1-wire protocol for the design of smart sensor nodes," Journal of Sensors, vol. 2014, no. 125874, 2014.doi:[[[10.1155//125874]]]
16 A. Okba, D. Henry, A. Takacs, H. Aubert, "Autonomous RFID sensor node using a single ISM band for both wireless power transfer and data communication," Sensors2019, vol. 19, no. 15, 1915.doi:[[[10.3390/s3330]]]
17 G. A. Preethi, C. Chandrasekar, "Seamless mobility of heterogeneous networks based on Markov decision process," Journal of Information Processing Systems, vol. 11, no. 4, pp. 616-629, 2015.custom:[[[-]]]
18 K. Kumar, A. Prakash, R. Tripathi, "Spectrum handoff in cognitive radio networks: a classification and comprehensive survey," Journal of Network and Computer Applications, vol. 61, pp. 161-188, 2016.custom:[[[-]]]
19 U. Caydas, A. Hascalik, "Use of the grey relational analysis to determine optimum laser cutting parameters with multi-performance characteristics," Optics & Laser Technology, vol. 40, no. 7, pp. 987-994, 2008.custom:[[[-]]]
20 L. R. Rabiner, Proceedings of the IEEE, vol, 77, no. 2, pp. 257-286, 1989.custom:[[[-]]]
21 A. J. Viterbi, J. K. Omura, Principles of Digital Communications and Coding, NY: McGraw-Hill, New York, 1979.custom:[[[-]]]
Path (hop)
Observed state
0 Signal [TeX:] $$\mathrm{S}_{2}$$ Strong signal [TeX:] $$\mathrm{S}_{5}$$
2 Average signal [TeX:] $$\mathrm{S}_{1}$$ Signal [TeX:] $$\mathrm{S}_{4}$$
3 Signal [TeX:] $$\left(\mathrm{S}_{2}\right)$$ Noise [TeX:] $$\mathrm{S}_{3}$$
4 Average signal [TeX:] $$\left(\mathrm{S}_{1}\right)$$ Signal [TeX:] $$\left(\mathrm{S}_{4}\right)$$
6 Signal [TeX:] $$\left(\mathrm{S}_{2}\right)$$ Noise [TeX:] $$\left(\mathrm{S}_{3}\right)$$
9 Signal [TeX:] $$\left(\mathrm{S}_{2}\right)$$ Strong signal [TeX:] $$\mathrm{S}_{5}$$
10 Average signal [TeX:] $$\left(\mathrm{S}_{1}\right)$$ Signal [TeX:] $$\left(\mathrm{S}_{4}\right)$$
13 Signal [TeX:] $$\left(\mathrm{S}_{2}\right)$$ Noise [TeX:] $$\left(\mathrm{S}_{3}\right)$$ | CommonCrawl |
Pseudo-arc
In general topology, the pseudo-arc is the simplest nondegenerate hereditarily indecomposable continuum. The pseudo-arc is an arc-like homogeneous continuum, and played a central role in the classification of homogeneous planar continua. R. H. Bing proved that, in a certain well-defined sense, most continua in Rn, n ≥ 2, are homeomorphic to the pseudo-arc.
History
In 1920, Bronisław Knaster and Kazimierz Kuratowski asked whether a nondegenerate homogeneous continuum in the Euclidean plane R2 must be a Jordan curve. In 1921, Stefan Mazurkiewicz asked whether a nondegenerate continuum in R2 that is homeomorphic to each of its nondegenerate subcontinua must be an arc. In 1922, Knaster discovered the first example of a hereditarily indecomposable continuum K, later named the pseudo-arc, giving a negative answer to a Mazurkiewicz question. In 1948, R. H. Bing proved that Knaster's continuum is homogeneous, i.e. for any two of its points there is a homeomorphism taking one to the other. Yet also in 1948, Edwin Moise showed that Knaster's continuum is homeomorphic to each of its non-degenerate subcontinua. Due to its resemblance to the fundamental property of the arc, namely, being homeomorphic to all its nondegenerate subcontinua, Moise called his example M a pseudo-arc.[lower-alpha 1] Bing's construction is a modification of Moise's construction of M, which he had first heard described in a lecture. In 1951, Bing proved that all hereditarily indecomposable arc-like continua are homeomorphic — this implies that Knaster's K, Moise's M, and Bing's B are all homeomorphic. Bing also proved that the pseudo-arc is typical among the continua in a Euclidean space of dimension at least 2 or an infinite-dimensional separable Hilbert space.[lower-alpha 2] Bing and F. Burton Jones constructed a decomposable planar continuum that admits an open map onto the circle, with each point preimage homeomorphic to the pseudo-arc, called the circle of pseudo-arcs. Bing and Jones also showed that it is homogeneous. In 2016 Logan Hoehn and Lex Oversteegen classified all planar homogeneous continua, up to a homeomorphism, as the circle, pseudo-arc and circle of pseudo-arcs. In 2019 Hoehn and Oversteegen showed that the pseudo-arc is topologically the only, other than the arc, hereditarily equivalent planar continuum, thus providing a complete solution to the planar case of Mazurkiewicz's problem from 1921.
Construction
The following construction of the pseudo-arc follows (Wayne Lewis 1999) harv error: no target: CITEREFWayne_Lewis1999 (help).
Chains
At the heart of the definition of the pseudo-arc is the concept of a chain, which is defined as follows:
A chain is a finite collection of open sets ${\mathcal {C}}=\{C_{1},C_{2},\ldots ,C_{n}\}$ in a metric space such that $C_{i}\cap C_{j}\neq \emptyset $ if and only if $|i-j|\leq 1.$ The elements of a chain are called its links, and a chain is called an ε-chain if each of its links has diameter less than ε.
While being the simplest of the type of spaces listed above, the pseudo-arc is actually very complex. The concept of a chain being crooked (defined below) is what endows the pseudo-arc with its complexity. Informally, it requires a chain to follow a certain recursive zig-zag pattern in another chain. To 'move' from the mth link of the larger chain to the nth, the smaller chain must first move in a crooked manner from the mth link to the (n-1)th link, then in a crooked manner to the (m+1)th link, and then finally to the nth link.
More formally:
Let ${\mathcal {C}}$ and ${\mathcal {D}}$ be chains such that
1. each link of ${\mathcal {D}}$ is a subset of a link of ${\mathcal {C}}$, and
2. for any indices i, j, m, and n with $D_{i}\cap C_{m}\neq \emptyset $, $D_{j}\cap C_{n}\neq \emptyset $, and $m<n-2$, there exist indices $k$ and $\ell $ with $i<k<\ell <j$ (or $i>k>\ell >j$) and $D_{k}\subseteq C_{n-1}$ and $D_{\ell }\subseteq C_{m+1}.$
Then ${\mathcal {D}}$ is crooked in ${\mathcal {C}}.$
Pseudo-arc
For any collection C of sets, let $C^{*}$ denote the union of all of the elements of C. That is, let
$C^{*}=\bigcup _{S\in C}S.$
The pseudo-arc is defined as follows:
Let p and q be distinct points in the plane and $\left\{{\mathcal {C}}^{i}\right\}_{i\in \mathbb {N} }$ be a sequence of chains in the plane such that for each i,
1. the first link of ${\mathcal {C}}^{i}$ contains p and the last link contains q,
2. the chain ${\mathcal {C}}^{i}$ is a $1/2^{i}$-chain,
3. the closure of each link of ${\mathcal {C}}^{i+1}$ is a subset of some link of ${\mathcal {C}}^{i}$, and
4. the chain ${\mathcal {C}}^{i+1}$ is crooked in ${\mathcal {C}}^{i}$.
Let
$P=\bigcap _{i\in \mathbb {N} }\left({\mathcal {C}}^{i}\right)^{*}.$
Then P is a pseudo-arc.
Notes
1. Henderson (1960) later showed that a decomposable continuum homeomorphic to all its nondegenerate subcontinua must be an arc.
2. The history of the discovery of the pseudo-arc is described in Nadler (1992), pp. 228–229.
References
• Bing, R.H. (1948), "A homogeneous indecomposable plane continuum", Duke Mathematical Journal, 15 (3): 729–742, doi:10.1215/S0012-7094-48-01563-4
• Bing, R.H. (1951), "Concerning hereditarily indecomposable continua", Pacific Journal of Mathematics, 1: 43–51, doi:10.2140/pjm.1951.1.43
• Bing, R.H.; Jones, F. Burton (1959), "Another homogeneous plane continuum", Transactions of the American Mathematical Society, 90 (1): 171–192, doi:10.1090/S0002-9947-1959-0100823-3
• Henderson, George W. (1960), "Proof that every compact decomposable continuum which is topologically equivalent to each of its nondegenerate subcontinua is an arc", Annals of Mathematics, 2nd series, 72 (3): 421–428, doi:10.2307/1970224
• Hoehn, Logan C.; Oversteegen, Lex G. (2016), "A complete classification of homogeneous plane continua", Acta Mathematica, 216 (2): 177–216, doi:10.1007/s11511-016-0138-0
• Hoehn, Logan C.; Oversteegen, Lex G. (2020), "A complete classification of hereditarily equivalent plane continua", Advances in Mathematics, 368: 107131, arXiv:1812.08846, doi:10.1016/j.aim.2020.107131
• Irwin, Trevor; Solecki, Sławomir (2006), "Projective Fraïssé limits and the pseudo-arc", Transactions of the American Mathematical Society, 358 (7): 3077–3096, doi:10.1090/S0002-9947-06-03928-6
• Kawamura, Kazuhiro (2005), "On a conjecture of Wood", Glasgow Mathematical Journal, 47 (1): 1–5, doi:10.1017/S0017089504002186
• Knaster, Bronisław (1922), "Un continu dont tout sous-continu est indécomposable", Fundamenta Mathematicae, 3: 247–286, doi:10.4064/fm-3-1-247-286
• Lewis, Wayne (1999), "The Pseudo-Arc", Boletín de la Sociedad Matemática Mexicana, 5 (1): 25–77
• Lewis, Wayne; Minc, Piotr (2010), "Drawing the pseudo-arc" (PDF), Houston Journal of Mathematics, 36: 905–934
• Moise, Edwin (1948), "An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua", Transactions of the American Mathematical Society, 63 (3): 581–594, doi:10.1090/S0002-9947-1948-0025733-4
• Nadler, Sam B., Jr. (1992), Continuum theory. An introduction, Monographs and Textbooks in Pure and Applied Mathematics, vol. 158, Marcel Dekker, Inc., New York, ISBN 0-8247-8659-9{{citation}}: CS1 maint: multiple names: authors list (link)
• Rambla, Fernando (2006), "A counterexample to Wood's conjecture", Journal of Mathematical Analysis and Applications, 317 (2): 659–667, doi:10.1016/j.jmaa.2005.07.064
• Rempe-Gillen, Lasse (2016), Arc-like continua, Julia sets of entire functions, and Eremenko's Conjecture, arXiv:1610.06278
| Wikipedia |
Sharon M. Frechette
Asssociate Professor
[email protected]
Dept. of Mathematics & Computer Science
1 College Street
Haberlin 309
Ph.D., Dartmouth College, 1997.
Advisor: Thomas Shemanske.
Thesis: "Decomposition of Spaces of Half-Integral Weight Cusp Forms".
A.M.,Dartmouth College, 1994.
B.A., Boston University, 1988.
Senior Thesis Advisor: Paul Blanchard.
TEACHING EXPERIENCE AND INTEREST
I have taught courses at all levels of Holy Cross' undergraduate curriculum, including Calculus, Principles of Real Analysis, Algebraic Structures, Linear Algebra, Modern Algebra, Number Theory, Cryptography, and Combinatorics.
I have also developed specialty courses for the College's Montserrat Program, a first-year seminar program with an emphasis on writing and discussion, and the College Honors Program. Previous seminars have included some flavor of Mathematics of Art and Architecture. This year, I'm teaching a year-long sequence in Montserrat, on the history and mathematics of cryptology:
Ciphers and Heroes, Fall 2014:
How are secret codes constructed? What weaknesses allow many of them to be cracked by clever analysts? Welcome to cryptology, the scientific study of encoding and decoding secret messages. We will explore many cryptosystems, investigating their strengths and weaknesses, and surveying their historical developments, setbacks, and implications. This semester we focus on cryptosystems such as the shift ciphers used by Caesar, the Vigenere cipher used during the Victorian era, and most thrillingly, the ENIGMA cipher used during World War II. Along with the mathematics of these ciphers, we will discover fascinating facts about their creators and the clever analysts who crack the codes, including the Polish and British heroes who cracked the seemingly unbreakable ENIGMA.
Privacy in the Digital Age, Spring 2015:
How does Amazon.com keep your credit card information secure when you order online? What weaknesses can hackers exploit, in their quest to steal your identity online? Secure electronic communication is vital to today's society, and modern cryptosystems are at the heart of this enterprise. Most of these systems are based on the mathematics of elementary number theory, and the stunning development of public key cryptography, a revolutionary concept born in the computer revolution of the 1970s. This semester we focus on these modern cryptosystems, the visionaries who created them, and the advances in computing that have made them secure.
Number Theory: Specifically modular forms, multiple Dirichlet series, special values of L-functions, and finite-field hypergeometric functions as related to traces of Hecke operators.
"Gaussian Hypergeometric Functions and Rational Points on Calabi-Yau $n$-folds,'' (with Matthew Papanikolas, Jonathan Root, and Valentina Vega), in preparation.
Shalika Germs for $\mathfrak{sl}_n$ and $\mathfrak{sp}_{2n}$ are Motivic," (with Julia Gordon and Lance Robson), accepted for publication, Proceedings of the Women in Numbers--Europe Conference, October 2013m, CIRM. [arxiv version]
"Newly Irreducible Iterates in Families of Quadratic Polynomials," (with Katherine Chamberlin, Emma Colbert, Patrick Hefferman, Rafe Jones, and Sarah Orchard), Involve, 5 (no. 4), (2012), 481--495. [Preprint version]
" A Crystal Definition for Symplectic Multiple Dirichlet Series," (with Jennifer Beineke and Ben Brubaker), Multiple Dirichlet series, L-functions and automorphic forms, 37--63, Progr. Math., 300, Birkhauser/Springer, New York, 2012. [Preprint version]
" Weyl Group Multiple Dirichlet Series for Type $C$," (with Jennifer Beineke and Ben Brubaker), Pacific J. Math., 254 (1), (2011), 11--46. [Preprint version]
" An Interdisciplinary Course: 'On Beauty: Perspective, Proportion, and Rationalism in Western Culture,' (with Alison Fleming and Sarah Luria), refereed Conference Proceedings for Renaissance Banff: Mathematical Connections in Art, Music, and Science (July 2005), accepted for publication, 4 pages.
"Determinants Associated to Zeta Matrices of Posets," (with Cristina Ballantine and John Little), Linear Algebra Appl., 411C (2005), 364-370. [Preprint version]
"Nonvanishing Twists of $GL(2)$ Automorphic $L$-functions," (with Alina Bucur, Ben Brubaker, Gautam Chinta, and Jeffrey Hoffstein), Internat. Math. Res. Notices, 78 (2004), 4211-4239. [Preprint version]
"The Combinatorics of Traces of Hecke Operators," (with Ken Ono and Matthew Papanikolas), Proc. of the National Academy of Sciences, USA, 101 (2004), 17016-17020. [Preprint version]
"Gaussian Hypergeometric Functions and Traces of Hecke Operators," (with Ken Ono and Matthew Papanikolas), Internat. Math. Res. Notices, 60 (2004), 3233-3262. [Preprint version]
"A Classical Characterization of Newforms with Equivalent Eigenforms in S_{k + 1/2}(4N,\chi)," J. London Math. Soc., (3) 68 (2003), 563--578. [Preprint version]
"Nonvanishing of Special Values of $L$-functions for Quadratic Twists of Newforms," preprint.
"Hecke Structure of Spaces of Half-Integral Weight Cusp Forms," Nagoya Math. J., 159 (2000), 53--85. [Preprint version]
I was one of the co-organizers for the 18th Annual Workshop on Automorphic Forms and Related Topics, which was held March 21-24, 2004 at the University of California, Santa Barbara.
Back to Math Dept. page
Created by Sharon M. Frechette on Jan. 24, 2004
Last modified by Sharon M. Frechette on October 14, 2013 | CommonCrawl |
Combinatorial map
A combinatorial map is a combinatorial representation of a graph on an orientable surface. A combinatorial map may also be called a combinatorial embedding, a rotation system, an orientable ribbon graph, a fat graph, or a cyclic graph.[1] More generally, an $n$-dimensional combinatorial map is a combinatorial representation of a graph on an $n$-dimensional orientable manifold.
Combinatorial maps are used as efficient data structures in image representation and processing, in geometrical modeling. This model is related to simplicial complexes and to combinatorial topology. A combinatorial map is a boundary representation model; it represents object by its boundaries.
History
The concept of a combinatorial map was introduced informally by J. Edmonds for polyhedral surfaces[2] which are planar graphs. It was given its first definite formal expression under the name "Constellations" by A. Jacques[3][4] but the concept was already extensively used under the name "rotation" by Gerhard Ringel[5] and J.W.T. Youngs in their famous solution of the Heawood map-coloring problem. The term "constellation" was not retained and instead "combinatorial map" was favored.[6]
Combinatorial maps were later generalized to represent higher-dimensional orientable subdivided objects.
Motivation
Several applications require a data structure to represent the subdivision of an object. For example, a 2D object can be decomposed into vertices (0-cells), edges (1-cells), and faces (2-cells). More generally, an n-dimensional object is composed with cells of dimension 0 to n. Moreover, it is also often necessary to represent neighboring relations between these cells.
Thus, we want to describe all the cells of the subdivision, plus all the incidence and adjacency relations between these cells. When all the represented cells are simplexes, a simplicial complex may be used, but when we want to represent any type of cells, we need to use cellular topological models like combinatorial maps or generalized maps.
Definition
A combinatorial map is a triplet M = (D, σ, α) such that:
• D is a finite set of darts;
• σ is a permutation on D;
• α is an involution on D with no fixed point.
Intuitively, a combinatorial map corresponds to a graph where each edge is subdivided into two darts (sometimes also called half-edges). The permutation σ gives, for each dart, the next dart by turning around the vertex in the positive orientation; the other permutation α gives, for each dart, the other dart of the same edge.
α allows one to retrieve edges (alpha for arête in French), and σ allows one to retrieve vertices (sigma for sommet in French). We define φ = σ ∘ α which gives, for each dart, the next dart of the same face (phi for face also in French).
So, there are two ways to represent a combinatorial map depending if the permutation is σ or φ (see example below). These two representations are dual to each other: vertices and faces are exchanged.
Combinatorial maps example: a plane graph and the two combinatorial maps depending if we use the notation (D, σ, α) or (D, φ, α).
Higher-dimensional generalization
An n-dimensional combinatorial map (or n-map) is a (n + 1)-tuple M = (D, β1, ..., βn) such that:[7][8]
• D is a finite set of darts;
• β1 is a permutation on D;
• β2, ..., βn are involutions on D;
• βi ∘ βj is an involution if i + 2 ≤ j (i, j ∈ { 1, ,..., n }).
An n-dimensional combinatorial map represents the subdivision of a closed orientable n-dimensional space. The constraint on βi ∘ βj guarantees the topological validity of the map as a quasi-manifold subdivision. Two-dimensional combinatorial maps can be retrieved by fixing n = 2 and renaming σ by β1 and α by β2.
Spaces that are not necessarily closed or orientable may be represented using (n-dimensional) generalized maps.
See also
• Bollobás–Riordan polynomial
• Boundary representation
• Generalized maps
• Doubly connected edge list
• Quad-edge data structure
• Rotation system
• Simplicial complex
• Winged edge
References
1. Bollobás, Béla; Riordan, Oliver (2001). "A Polynomial Invariant of Graphs On Orientable Surfaces". Proceedings of the London Mathematical Society. Wiley. 83 (3): 513–531. doi:10.1112/plms/83.3.513. ISSN 0024-6115. S2CID 15895860.
2. Edmonds, J. (1960). "A Combinatorial Representation for Polyhedral Surfaces". Notices Amer. Math. Soc. 7. hdl:1903/24820.
3. Jacques, A. (1969). Constellations et propriétés algébriques des graphes topologiques (PhD). University of Paris.
4. Jacques, A. (1970). "Constellations et Graphes Topologiques". Colloque Math. Soc. János Bolyai: 657–672.
5. Ringel, G. (2012) [1974]. Map Color Theorem. Springer. ISBN 978-3-642-65759-7.
6. Cori, R. (1975). "Un code pour les graphes planaires et ses applications". Astérisque. 27. MR 0404045. Zbl 0313.05115.
7. Lienhardt, P. (1991). "Topological models for Boundary Representation : a comparison with n-dimensional generalized maps". Computer-Aided Design. 23 (1): 59–82. doi:10.1016/0010-4485(91)90082-8.
8. Lienhardt, P. (1994). "N-dimensional generalized combinatorial maps and cellular quasi-manifolds". International Journal of Computational Geometry and Applications. 4 (3): 275–324. doi:10.1142/S0218195994000173.
External links
• Combinatorial maps in CGAL, the Computational Geometry Algorithms Library:
• Damiand, Guillaume. "Combinatorial maps". Retrieved February 6, 2021.
• Combinatorial maps in CGoGN, Combinatorial and Geometric modeling with Generic N-dimensional Maps
• Combinatorial map at the nLab
| Wikipedia |
\begin{document}
\title{f Parabolic optimal control
problems with combinatorial switching constraints \ Part I: Convex relaxations hanks{This work has partially been supported by Deutsche Forschungsgemeinschaft (DFG) under grant no.~BU~2313/7-1 and ME~3281/10-1.} \renewcommand{\abstractname}{} \begin{abstract}
We consider optimal control problems for partial differential
equations where the controls take binary values but vary over the
time horizon, they can thus be seen as dynamic switches. The
switching patterns may be subject to combinatorial constraints such
as, e.g., an upper bound on the total number of switchings or a
lower bound on the time between two switchings. While such
combinatorial constraints are often seen as an additional
complication that is treated in a heuristic postprocessing, the core
of our approach is to investigate the convex hull of all feasible
switching patterns in order to define a tight convex relaxation of
the control problem. The convex relaxation is built by cutting
planes derived from finite-dimensional projections, which can be
studied by means of polyhedral combinatorics. A numerical example for the case of a bounded number of switchings shows
that our approach can significantly improve the dual bounds given by the straightforward continuous relaxation, which is obtained by relaxing binarity constraints.
\keywords{PDE-constrained optimization, switching time optimization, convex relaxations} \end{abstract}
\section{Introduction} \label{sec: intro}
Mixed-integer optimal control of a system governed by partial or ordinary differential equations became a hot research topic in the last decade, as a variety of applications leads to such control problems. In particular, the control often comes in form of a finite set of switches which can be operated within a given continuous time horizon, e.g., by shifting of gear-switches in automotive engineering~\cite{GER05,KSB10, SBF13} or by switching of valves or compressors in gas and water networks~\cite{FUEG09,HAN20}. Consequently, various approaches are discussed in the literature to address optimal control problems with discrete control variables, often known as mixed-integer optimal control problems (MIOCPs). Direct methods, based on the \textit{first-discretize-then-optimize}\/ paradigm, are widely used to tackle MIOCPs; see for instance~\cite{GER05} and~\cite{VSG20}. The control and, if desired, the state are discretized in time and space, in order to approximate the problem by a large, typically non-convex, finite-dimensional mixed-integer nonlinear programming problem (MINLP). The latter can be addressed by standard techniques; see \cite{Lee12} or~\cite{BKLL13} for surveys on algorithms for MINLPs. However, the size of the arising MINLPs easily becomes too large to solve them to proven optimality. In particular, direct methods are not promising for optimal control problems governed by partial differential equations~(PDEs)~\cite{GPRS19, SHL20}.
In contrast, arbitrary close approximations of MIOCPs can be computed efficiently by first replacing the set of discrete control values by its convex hull and then appropriately rounding the result. The most common approximation methods for systems governed by ordinary differential equations~(ODEs) are the \textit{Sum-Up Rounding} strategy~\cite{SA12,KLM20} and the \textit{Next Force Rounding} strategy~\cite{JUN14}. PDE-constrained problems can also be addressed with the Sum-Up Ronding strategy~\cite{HS13}. However, in the presence of additional combinatorial constraints, the latter may be violated \cite[Sect.~5.4]{MAN19}, and the heuristics used to obtain feasible solutions often do not perform well \cite[Example~3.2]{KML17}. Therefore, when aiming at globally optimal solutions, such approaches may only serve for computing primal bounds. To minimize the integrality error, the \textit{Combinatorial Integral Approximation
(CIA)}~\cite{SA05} tracks the average of a relaxed solution over a given rounding grid by a piecewise constant integer control and the discretized problem is solved by a tailored branch-and-bound algorithm~\cite{JRS15, KMS11}. The approach was again generalized to PDE-constrained problems~\cite{HKMS19}. To reduce the (undesired) chattering behavior of the rounded control the total variation is constrained~\cite{ZS20} or switching cost aware rounding algorithms are considered~\cite{BHKM20, BK20}.
Other approaches optimize the switching times, e.g., by controlling the switching times through a continuous time control function which scales the length of minor time intervals~\cite{GER06, ROL17} or by including a fixed number of transition times as decision variables into the MIOCP and solving the corresponding finite-dimensional non-convex problems by gradient descent techniques~\cite{SOG16,EWA06} or by second order methods~\cite{JM11,SOG17}. PDE-constrained optimal control problems can be addressed by the concept of switching time optimization as well~\cite{HR16}. Nevertheless, these methods have a limited applicability, since fairly restrictive assumptions on the objective and the state dynamics need to be made in order to guarantee differentiability in the discretized setting~\cite{FMO13}.
In the context of optimal control problems governed by PDEs, switching constraints are frequently imposed by penalty terms added to the objective functional~\cite{CIK16, CRKB16, CRK17}. The arising penalized problems are non-convex in general and are therefore convexified by means of the bi-conjugate functional associated with the penalty term. The desired switching structure of the optimal solutions of the convexified problems can however only be guaranteed under additional structural assumptions on the unknown solution. For the case of a switching between multiple constant control variables, a multi-bang approach might be favorable since optimal control problems subject to box constraints on the control may show a bang-bang behavior in the absence of a Tikhonov-type regularization term \cite{Troe79,DH12, CWW18, TW18}. However, the bang-bang structure of the optimal control cannot be guaranteed in general. In order to promote that the control attains the desired constant values, $L^0$-penalty terms or suitable indicator functionals are added to the objective and convex relaxations of the penalty terms based on the bi-conjugate functional are employed to make the problem amenable for optimization algorithms~\cite{CK14,CTW18}. Again, as in case of the penalization of the switching constraints mentioned above, the multi-bang structure of the optimal solutions of the convexified problems can only be ensured under additional assumptions that cannot be verified a priori. In~\cite{CK16}, the convexification of the $L^0$-penalty by means of the bi-conjugate functional is employed in the context of topology optimization, in~\cite{CKK18}, the $L^0$-penalty is enriched by the BV-seminorm. $L^0$-penalization techniques that go without regularization or convexification are for instance addressed in~\cite{CW20} from a theoretical perspective and in~\cite{Wac19} with regard to algorithms. However, to the best of our knowledge, additional combinatorial constraints on the switching structure have not yet been included in the penalization framework.
In summary, the design of global solvers for MIOCPs with dynamic switches and combinatorial switching constraints is an open field of research. The core of our new approach for addressing such problems is the computation of lower bounds by a tailored convexification of the set of feasible switching patterns in function space. A counterexample given in~Section~\ref{sec: convhull} shows that, even when the combinatorial constraint only consists in an upper bound on the total number of switchings, the naive approach of just relaxing the binarity constraint does not lead to the convex hull of the set of feasible switching patterns. Our aim is to determine tighter approximations of this convex hull by considering finite-dimensional projections that allow for the efficient computation of cutting planes. Based on the resulting outer description of the convex hull, in the companion paper~\cite{partII} we develop a tailored outer approximation algorithm which converges to a global minimizer of the convex relaxations. The resulting lower bounds could be used, e.g., in a branch-and-bound scheme to obtain globally optimal solutions of the control problems.
The remainder of this paper is organized as follows. In Section~\ref{sec:
opt}, we specify the prototypical optimal control problem as well as the class of combinatorial switching constraints considered in this work and show that the problem admits an optimal solution. In Section~\ref{sec: convhull}, we investigate the convex hull of feasible switching patterns and show that it can be fully described by cutting planes lifted from finite-dimensional projections. An example in Section~\ref{sec: qbounds} shows the strength of the lower bounds resulting from our tailored convexification.
\section{Optimal control problem} \label{sec: opt} For the sake of simplicity, throughout this paper, we restrict ourselves to a parabolic binary optimal control problem with switching constraints of the following form: \begin{equation}\tag{P}\label{eq:optprob} \left\{\quad \begin{aligned}
\text{min} \quad & J(y,u) = \tfrac{1}{2}\, \|y - y_{\textup{d}}\|_{L^2(Q)}^2 + \tfrac{\alpha}{2}\,\|u-\tfrac 12\|^2_{L^2(0,T; \mathbb{R}^n)}\\ \text{s.t.} \quad & \begin{aligned}[t] \partial_t y(t,x) - \Delta y(t,x) &= \sum_{j=1}^n u_j(t) \,\psi_j(x) & & \text{in } Q := \Omega \times (0,T),\\ y(t,x) &= 0 & & \text{on } \Gamma := \partial\Omega \times (0,T),\\ y(0,x) &= y_0(x) & & \text{in } \Omega, \end{aligned}\\ \text{and} \quad & u \in D. \end{aligned} \quad \right. \end{equation} Herein, $T > 0$ is a given final time and $\Omega\subset \mathbb{R}^d$, $d\in \mathbb{N}$, denotes a bounded domain, where a domain is an open and connected subset of a finite-dimensional vector space, with Lipschitz boundary $\partial \Omega$ in the sense of \cite[Def. 1.2.2.1]{GRIS85}. The form functions $\psi_j\in H^{-1}(\Omega)$, $j=1,\dots,n$, as well as the initial state~$y_0 \in L^2(\Omega)$ are given. Moreover, \[ D \subset \big\{ u \in BV(0,T;\mathbb{R}^n)\colon u(t) \in \{0,1\}^n \text{ f.a.a.\ } t \in (0,T)\big\} \] denotes the set of feasible \emph{switching controls}. Finally, $y_{\textup{d}} \in L^2(Q)$ is a given desired state and $\alpha \geq 0$ is a Tikhonov parameter weighting the mean deviation from~$\tfrac 12$. Note that the choice of~$\alpha$ does not have any impact on the set of optimal solutions of~\eqref{eq:optprob}, as $u \in \{0,1\}^n \text{ a.e.~in } (0,T)$ and hence the Tikhonov term is constant. However, the convex relaxations of~\eqref{eq:optprob} considered in this paper as well as their optimal values are influenced by~$\alpha$.
The particular challenge of our problem are the combinatorial switching constraints modeled by the set~$D$
of feasible controls. It is supposed to satisfy the two following assumptions: \begin{align} & \text{$D$ is a bounded set in $BV(0,T;\mathbb{R}^n)$,} \tag{D1}\label{eq:D1} \\ & \text{$D$ is closed in~$L^p(0,T;\mathbb{R}^n)$ for some fixed $p \in [1,\infty)$.} \tag{D2}\label{eq:D2} \end{align} Here, $BV(0,T;\mathbb{R}^n)$ denotes the set of all vector-valued functions with bounded variation, i.e., \[BV(0,T;\mathbb{R}^n):=\{u\in L^1(0,T;\mathbb{R}^n): u_i\in BV(0,T) \text{ for } i=1,\ldots,n\, \}\] equipped with the norm
\[\|u\|_{BV(0,T;\mathbb{R}^n)} := \|u\|_{L^1(0,T;\mathbb{R}^n)} + \sum_{j=1}^n |u_j|_{BV(0,T)}\;.\] For more details on the space of bounded variation functions, see, e.g.,
\cite[Chap.~10]{ATT14}. Note that, in our case, the BV-seminorm~$|u_j|_{BV(0,T)}$ agrees with the minimal number of switchings of any representative of~$u_j$ with values in~$\{0,1\}$.
A possible example for such a set is \begin{equation}\label{eq:Dex} \begin{aligned} D_{\max} := \big\{ u \in BV(0,T;\mathbb{R}^n)\colon \; & u(t) \in \{0,1\}^n \text{ f.a.a.\ } t \in (0,T),\\
& |u_j|_{BV(0,T)} \leq \sigma_{\max}\; \forall \,j = 1, \dots, n \big\}, \end{aligned} \end{equation}
where $|\cdot|_{BV(0,T)}$ denotes the BV-seminorm and $\sigma_{\max}\in \mathbb{N}$ is a given number. The set $D_{\max}$ meets the assumptions~\eqref{eq:D1} and~\eqref{eq:D2}, as we will show in Example~\ref{ex: boundedshift}. This choice of~$D$ is motivated by the following application-driven scenario: suppose $y$ is the temperature of a body covering the domain $\Omega$ and the aim of the optimization is to minimize the deviation of $y$ from a given desired state $y_{\textup{d}}$, by means of $n$ given heat sources modeled by the form functions~$\psi_j$, $j=1, \dots, n$. These heat sources can be switched on and off at arbitrary points in time, but we are only allowed to shift each switch for at most $\sigma_{\max}$ times. This leads to the set $D_{\max}$.
Various other practically relevant choices of~$D$ are conceivable. For instance, it could be required to bound the time interval between two shiftings of the same switch from below because of technical limitations; this kind of restriction is known as minimum dwell time constraints in the optimal control community and as min-up/min-down constraints in the unit commitment community. See Example~\ref{ex: combswpoint} for a discussion and generalization of this class of constraints. Another condition may be that certain switches are not allowed to be used (or switched on) at the same time.
Our previous assumptions guarantee that the PDE contained in~\eqref{eq:optprob} admits a unique weak solution $y\in W(0,T) := H^1(0,T;H^{-1}(\Omega)) \cap L^2(0,T; H^1_0(\Omega))$ for every $u\in D\subset~L^\infty(0,T;\mathbb{R}^n)$; see \cite[Chapter~3]{Troe10}. The associated solution operator~$S\colon L^2(0,T;\mathbb{R}^n) \to W(0,T)$ is affine and continuous. Using this solution operator, the problem~\eqref{eq:optprob} can be written as \begin{equation} \tag{P'} \label{eq:P'} \left\{\quad \begin{aligned} \mbox{min }~&f(u)=J(Su,u) \\ \mbox{s.t. }~ & u\in D\;. \end{aligned}\, \right. \end{equation}
Note that the objective function $f\colon L^2(0,T;\mathbb{R}^n) \to \mathbb{R}$ is weakly lower semi-continuous because both~$u \mapsto \|Su -
y_{\textup{d}}\|_{L^2(Q)}^2$ and~$u \mapsto \|u-\tfrac 12 \|_{L^2(0,T;\mathbb{R}^n)}^2$ are convex and lower semi-continuous, thus weakly lower semi-continuous, and the solution operator $S$ is affine and continuous, thus weakly continuous.
\begin{theorem}\label{thm:discr_ctrl}
Let $D\neq \emptyset$. Then Problem~\eqref{eq:P'} admits a global minimizer. \end{theorem} \begin{proof}
Since~$D\neq\emptyset$, we have ${f^\star}:=\inf_{u\in D}f(u)\in
\mathbb{R}\cup\{-\infty\}$. Let $\{u^k\}_{k\in \mathbb{N}}$ in $D$ be an
infimal sequence with
\[
\lim\limits_{k\to \infty} f(u^k) = {f^\star}\;.
\]
We know that $\{u^k\}_{k\in \mathbb{N}}$ is a bounded sequence
in~$BV(0,T;\mathbb{R}^n)$, since $D$ is a bounded set in~$BV(0,T;\mathbb{R}^n)$ by
assumption~\eqref{eq:D1}, i.e.,
\[\sup_{k\in \mathbb{N}}~\Vert u^k \Vert_{BV(0,T;\mathbb{R}^n)} = \sup_{k\in \mathbb{N}} \left(\|u^k\|_{L^1(0,T;\mathbb{R}^n)} + \textstyle\sum_{j=1}^n |u_j^k|_{BV(0,T)}\right) < \infty\;.\]
By Theorem 10.1.3 and Theorem 10.1.4 in~\cite{ATT14}, $BV(0,T;\mathbb{R}^n)$
is compactly embedded in~$L^p(0,T;\mathbb{R}^n)$, and hence there exists a
strongly convergent subsequence, which we again denote by
$\{u^{k}\}_{k\in \mathbb{N}}$, such that $ u^k \to {u^\star}\in
L^{p}(0,T;\mathbb{R}^n) \text{ for } k\to \infty.$
Since $D$ is closed in~$L^p(0,T;\mathbb{R}^n)$ by
condition~\eqref{eq:D2}, we deduce that ${u^\star}\in D$. The weak lower
semi-continuity of the objective function~$f$ leads to
\[f({u^\star})\leq \liminf\limits_{k\to\infty } f(u^k)={f^\star}\;.\]
This implies ${f^\star} > -\infty$ as well as the optimality of ${u^\star}$ for~\eqref{eq:P'}. \end{proof}
\section{Convex hull description} \label{sec: convhull}
The crucial ingredient of our approach is the outer description of the convex hull of the set~$D$ of feasible switching patterns by linear inequalities. In general, just replacing $\{0,1\}$ with $[0,1]$ in the definition of~$D$ does not lead to the convex hull of~$D$ in any $L^p$-space. This is true even in the case of just one switch that can be changed at most once on the entire time horizon, i.e., if the feasible switching control is required to belong to \begin{equation}\label{eq:counter} D:=\{ u \in BV(0,T)\colon \; u(t) \in \{0,1\} \text{ f.a.a.\ } t \in (0,T),
|u|_{BV(0,T)} \leq 1\}\;. \end{equation} Essentially, the naive approach does not consider the monotonicity of the switches in~$D$, as we will see in the following counterexample.
\begin{counterexample}
Let~$D$ be defined as in~\eqref{eq:counter} and consider the function
\[
u(t):=\begin{cases}
\tfrac{1}{2} & \text{if }t \in [\tfrac{1}{3}T,\tfrac{2}{3}T ] \\
0 & \text{otherwise}.
\end{cases}\]
Obviously, we have $u\in BV(0,T)$ with $u(t) \in [0,1]$ for
$t\in(0,T)$ and $|u|_{BV(0,T)}=1$. However, we claim that~$u$
does not belong to the closed convex hull of~$D$ in~$L^p(0,T)$ for
any~$p\in [1,\infty)$.
Assume on contrary that $u\in\overline{\operatorname{conv}(D)}^{L^p(0,T)}$
for some $p\in [1, \infty)$. Then there exists a sequence
$\{u^k\}_{k\in \mathbb{N}}\subset \operatorname{conv}(D)$ with $u^k \to u$ in~$L^p(0,T)$
for $k\to \infty$. In particular, $\{u^k\}_{k\in \mathbb{N}}$ converges
strongly to $u$ in $L^1(0,T)$ due to $L^p(0,T)\hookrightarrow
L^1(0,T)$, i.e.,
\[
\int_{0}^{T} |u^k-u|\ \text{d} t \to 0\mbox{ for } k\to \infty\;.
\]
Define $A^k:= \{t\in[\tfrac{1}{3}T,\tfrac{2}{3}T ]: u^k(t)\geq
\tfrac{2}{5}\}$. We claim that there exists $k_0\in \mathbb{N}$ such that
the sets~$A^k$, $k\geq k_0$, have a positive
Lebesgue-measure. Indeed, if such a $k_0\in \mathbb{N}$ did not exist, then
we could find a subsequence, which we denote by the same symbol
$\{A^k\}_{k\in\mathbb{N}}$ for simplicity, such that $\lambda(A^k)=0$ for
all $k\in \mathbb{N}$, where $\lambda(A^k)$ denotes the Lebesgue measure of
$A^k$. With $\lambda(A^k)=0$, it follows
\[ \int_{0}^{T} |u^k-u|\ \text{d} t \geq
\int_{\ttfrac{1}{3}T}^{\ttfrac{2}{3}T} |u^k-\tfrac{1}{2}|\ \text{d} t =
\int_{[\ttfrac{1}{3}T,\ttfrac{2}{3}T]\setminus A^k}
|u^k-\tfrac{1}{2}|\ \text{d} t >\tfrac{1}{30}T\;,
\]
where the last inequality holds due to $ |u^k-\tfrac{1}{2}|>
\tfrac{1}{10}$ for all $t \in [\tfrac{1}{3}T,\tfrac{2}{3}T
]\setminus A^k$ by definition of~$A^k$. This contradicts the strong
convergence of $u^k$ to $u$ in $L^1(0,T)$. Thus, a number $k_0\in
\mathbb{N}$ exists with $\lambda(A^k)>0$ for all $k\geq k_0$.
Now, let $k\geq k_0$ be arbitrary. We write $u^k\in \operatorname{conv}(D)$ as a convex combination
\[u^k=\sum_{l=1}^{m_k} \mu_l^k y_l^k\]
of functions in $D$. Let
$t_0\in A^k$ be a Lebesgue point of all functions $y_l^k\in D$,
$1\leq l\leq m_k$, which exists since the set of all non-Lebesgue
points of $y_l^k$ is a set of Lebesgue measure zero. Then, we know
\[
\tfrac{2}{5}\leq u^k(t_0)=\sum_{l=1}^{m_k} \mu_l^k y_l^k(t_0)\;.
\]
Set $I_k:=\{l\in\{1,\ldots,m_k\}: y_l^k(t_0)=1\}$. The
inequality then implies
\begin{equation}\label{eq:ineq}
\tfrac{2}{5}\leq \sum_{l\in I_k} \mu_l^k\;.
\end{equation}
Since $y_l^k(t_0)=1$ for $l\in I_k$ and $y_l^k$ might shift at most once due to $|y_l^k|_{BV(0,T)}\leq 1$, we deduce that either $y_l^k$ was first turned off and then turned on in $(0,t_0)$, such that $y_l^k(t)\equiv 1$ a.e. in $(t_0,T)$ holds, or $y_l^k$ was first turned on, i.e., $y_l^k(t)\equiv 1$ a.e. in $(0,t_0)$. Consequently, we get $y_l^k(t)\equiv 1$ a.e. in $(0,\tfrac{1}{3}T )$ or $(\tfrac{2}{3}T,T )$ for every $l\in I_k$. The latter, together with
~\eqref{eq:ineq}, yields
\[\int_{0}^{T} |u^k-u|\ \text{d} t \geq \int_{(0,\ttfrac{1}{3}T)\cup (\ttfrac{2}{3}T,T)} |u^k|\ \text{d} t
\geq\sum_{l\in I_k} \mu_l^k \int_{(0,\ttfrac{1}{3}T)\cup (\ttfrac{2}{3}T,T)}y_l^k\ \text{d} t
\geq \tfrac{2}{15} T\;,
\]
which contradicts the strong convergence of $u^k$ to $u$ in
$L^1(0,T)$. \end{counterexample}
This counterexample shows that we cannot expect to obtain a tight description of~$\operatorname{conv}(D)$ without a closer investigation of the specific switching constraint under consideration. Our basic idea is to reduce this investigation to a purely combinatorial task by projecting the set~$D$ to finite-dimensional spaces~$\mathbb{R}^M$, by means of~$M\in \mathbb{N}$ linear and continuous functionals~$\Phi_i \in L^p(0,T;\mathbb{R}^n)^*$, $i=1, \dots, M$. In the following, we restrict ourselves to local averaging operators of the form \begin{equation}\label{eq:localaveraging} \langle \Phi_{(j-1)N+i}, u
\rangle := \tfrac{1}{\lambda(I_{i})}\int_{I_{i}} u_{j}\,\text{d} t \end{equation} for $j=1,\ldots,n$ with suitably chosen subintervals $I_i\subset (0,T)$, $i=1,\ldots,N$, and $M:=n\,N$. The resulting projection then reads \begin{equation}\label{eq:pi} \Pi\colon BV(0,T;\mathbb{R}^n) \ni u \mapsto \big(\langle \Phi_l, u \rangle\big)^M_{l=1} \in \mathbb{R}^{M}\;. \end{equation} Note that~$\Pi$ is a linear mapping. The core result underlying our approach is that, for increasing~$N$, projections~$\Pi_N$ can be designed such that \begin{equation}\label{eq:convapprox} \overline{\operatorname{conv}(D)}^{L^p(0,T;\mathbb{R}^n)} = \bigcap_{N\in \mathbb{N}} \{ v \in L^p(0,T;\mathbb{R}^n)\colon \Pi_N(v) \in C_{D,\Pi_N}\} \end{equation} where \[ C_{D,\Pi} := \operatorname{conv}\{\Pi (u)\colon u \in D \}\subset \mathbb{R}^{M}\;. \] In other words, an outer description of all finite-dimensional convex hulls $C_{D,\Pi}$ also leads to an outer description of the convex hull of $D$ in function space.
We first observe that our general assumptions~\eqref{eq:D1} and ~\eqref{eq:D2} guarantee the closedness of the finite-dimensional set $C_{D,\Pi}$ in $\mathbb{R}^M$. \begin{lemma}\label{lem: finiteclosed}
For any\/ $\Pi$ as in~\eqref{eq:pi}, the set $C_{D,\Pi}$ is closed in $\mathbb{R}^M$. \end{lemma} \begin{proof}
Let $\{\Pi(u^k)\}_{k\in \mathbb{N}}\subset \mathbb{R}^M$ be a convergent sequence
in $\Pi (D)$, resulting from the projection of feasible switching
controls $u^k\in D$ for $k\in \mathbb{N}$, with $\Pi(u^k) \to \omega $ in
$\mathbb{R}^M$. The sequence~ $\{u^k\}_{k\in \mathbb{N}}\subset D$ is bounded in
$BV(0,T;\mathbb{R}^n)$ by~\eqref{eq:D1}. As in Theorem~\ref{thm:discr_ctrl}, the
compactness of the embedding $BV(0,T;\mathbb{R}^n)\hookrightarrow L^p(0,T;\mathbb{R}^n)$ by
Theorem~10.1.3 and Theorem~10.1.4 in~\cite{ATT14} implies the existence of a
strongly convergent subsequence, again denoted by
$\{u^{k}\}_{k\in \mathbb{N}}$, such that $ u^k \to u\in L^{p}(0,T;\mathbb{R}^n)
\text{ for } k\to \infty.$ Since $D$ is closed in~$L^p(0,T;\mathbb{R}^n)$
by~\eqref{eq:D2}, we deduce $u\in D$. By continuity of $\Pi$ in
$L^p(0,T;\mathbb{R}^n)$, we then have
\[
\omega = \lim\limits_{k\to \infty}\Pi (u^k) =\Pi(u)
\] so that $\omega$ lies in $\Pi(D).$ Hence, the set $\Pi(D)$ is closed in $\mathbb{R}^M$. It is also bounded, thus compact, such that $C_{D,\Pi}$ is closed as the convex hull of a compact set in $\mathbb{R}^M$. \end{proof}
As a consequence, we obtain that the subset of $L^p(0,T;\mathbb{R}^n)$ corresponding to the finite-dimensional projection $\Pi$ is convex and closed in~$L^p(0,T;\mathbb{R}^n)$. \begin{lemma}\label{lem: closed}
For any\/ $\Pi$ as in~\eqref{eq:pi}, the set $\{ v \in
L^p(0,T;\mathbb{R}^n)\colon\Pi(v) \in C_{D,\Pi}\}$ is convex and closed
in~$L^p(0,T;\mathbb{R}^n)$. \end{lemma} \begin{proof}
The convexity assertion follows from the convexity of $C_{D,\Pi}$
together with the linearity of~$\Pi$. Closedness follows from
Lemma~\ref{lem: finiteclosed} and the continuity of~$\Pi$
in~$L^p(0,T;\mathbb{R}^n)$. \end{proof}
By the following observation, each projection~$\Pi$ gives rise to a relaxation of the closed convex hull of~$D$ in~$L^p(0,T;\mathbb{R}^n)$. These relaxations can be used to derive outer approximations by linear inequalities. \begin{lemma}\label{lem: convD}
For any\/ $\Pi$ as in~\eqref{eq:pi}, we have
\[\overline{\operatorname{conv} (D)}^{ L^p(0,T;\mathbb{R}^n)}\subseteq \{ v \in
L^p(0,T;\mathbb{R}^n)\colon \Pi(v) \in C_{D,\Pi}\}=:V\;.\] \end{lemma} \begin{proof}
By construction of $C_{D,\Pi}$, every $u\in D$ satisfies $\Pi(u)
\in C_{D,\Pi}$. The linearity of~$\Pi$ leads to
$\operatorname{conv} (D)\subset V$,
using the convexity of~$V$ stated in Lemma~\ref{lem:
closed}. Again by Lemma~\ref{lem: closed}, the set~$V$ is closed
in~$L^p(0,T;\mathbb{R}^n)$, which shows the desired result. \end{proof}
The following result shows that the convex hull of the set of feasible switching controls can be fully described with the help of appropriate finite-dimensional sets $C_{D,\Pi}$. With a little abuse of notation, we slightly change the notation of the local averaging operators in the sense that the number of subintervals now differs from the dimension $M$ of the range of $\Pi$, see~\eqref{eq:Pik} below, in order to ease the proof of the following theorem.
\begin{theorem}\label{thm: convD}
For each~$k\in\mathbb{N}$, let~$I^k_1,\dots,I_{N_k}^k$, $N_k\in\mathbb{N}$, be
disjoint open intervals in~$(0,T)$ such that
\begin{itemize}
\item[(i)] $\bigcup_{i=1}^{N_k} \overline{I_i^k} = [0,T]$ for all $k\in\mathbb{N}$ and
\item[(ii)]$\max_{i=1,\dots,N_k}\lambda(I_i^k)\to 0\; \mbox{ for } k\to \infty$.
\end{itemize}
Set $M_k := n\, N_k$ and define projections~$\Pi_{k}\colon BV(0,T;\mathbb{R}^n) \to \mathbb{R}^{M_k}$,
for~$k\in\mathbb{N}$, by
\begin{equation}\label{eq:Pik}
\langle\Phi_{(j-1)N_k+i}^k,u\rangle :=
\tfrac{1}{\lambda (I_i^k)}\int_{I_i^k} u_j(t)\, \text{d} t\;
\end{equation}
for $j=1,\ldots,n$ and $i=1,\ldots,N_k$. Moreover, set
\[V_k:=\{ v \in L^p(0,T;\mathbb{R}^n)\colon \Pi_{k}(v) \in
C_{D,\Pi_k}\}\;.\]
Then
\begin{equation}\label{eq:convD}
\overline{\operatorname{conv} (D)}^{ L^p(0,T;\mathbb{R}^n)} = \bigcap_{k\in \mathbb{N}} V_k\;.
\end{equation}
\end{theorem}
\begin{proof}
The inclusion \grqq$\subseteq$\grqq\ in~\eqref{eq:convD} follows directly from
Lemma~\ref{lem: convD}, it thus remains to show \grqq$\supseteq$\grqq. For this,
let
\[u\in\bigcap_{k\in \mathbb{N}} V_k\;.
\]
By definition of $u$, we have $\Pi_{k}(u)\in C_{D,\Pi_{k}}$. Hence, there exist $v_l^k \in D$ for $l=1,\ldots,m$, where $m=m(k)\in\mathbb{N}$ may depend on $k$, as well coefficients $\mu^k_l \geq 0$ with $\sum_{l=1}^m \mu^k_l=1$ and
\[
\Pi_{k}(u) = \sum_{l=1}^m \mu^k_l\, \Pi_{k}(v_l^k)\;.
\] Set $u^k := \sum_{l=1}^m \mu_l^k v_l^k \in \operatorname{conv}(D)$. By construction and the linearity of the projection, we have $\Pi_k(u^k)=\Pi_k(u)$, i.e.,
\begin{equation}\label{eq:PiNeq}
\int_{I_i^k} (u^k - u)\,\text{d} t = 0\quad \forall\, i=1,\ldots,N_k, \;
k\in \mathbb{N}\;.
\end{equation} Let $k\in \mathbb{N}$ be fixed. Thanks to assumption (i), we conclude that for every $\ell\in \mathbb{N}$ it holds
\[
\lambda \Big(I_i^\ell \setminus \bigcup_{I_r^k\subset I_i^\ell} I_r^k\Big)\leq 2\max_{r=1,\dots,N_k} \lambda(I_r^k)\;.
\]
Set $E^l_i:=\bigcup_{I_r^k\subset I_i^\ell} I_r^k$ for all $\ell \in \mathbb{N}$ and $i=1,\ldots,N_\ell$.
Then~\eqref{eq:PiNeq} implies
\begin{equation*}
\begin{aligned}
\int_{I_i^\ell} (u^k - u)\, \text{d} t&= \int_{I_i^\ell\setminus E^l_i} (u^k - u)\, \text{d} t + \int_{E^l_i} (u^k - u)\, \text{d} t\ \\ &= \int_{I_i^\ell\setminus E^l_i} (u^k - u)\, \text{d} t
\end{aligned}
\end{equation*}
and thus
\begin{equation}\label{eq:PiLeq}
\begin{aligned}
\Big| \int_{I_i^\ell} (u^k - u)\, \text{d} t\Big|\leq\int_{I_i^\ell\setminus E^l_i} |u^k - &u|\, \text{d} t \leq \lambda(I_i^\ell\setminus E^l_i) \\&\leq 2 \max_{r=1,\dots,N_k} \lambda(I_r^k) \quad \forall i=1,\ldots,N_\ell,\ \ell \in \mathbb{N}
\end{aligned}
\end{equation}
Since $u^k(t)\in[0,1]^n$ holds almost everywhere in $(0,T)$, there
exists a weakly convergent subsequence, which we denote by the same
symbol for simplicity, with $u^k \rightharpoonup \tilde{u}$ in~$
L^p(0,T;\mathbb{R}^n)$. Together with~\eqref{eq:PiLeq} and $\max_{r=1,\dots,N_k}\lambda(I_r^k)\to 0\; \mbox{ for }
k\to \infty$, the weak convergence
of~$\{u^k\}_{k\in\mathbb{N}}$ to $\tilde{u} $ implies
\begin{equation}\label{eq:wutildeueq}
\int_{I_i^\ell} (\tilde u - u)\, \text{d} t = 0\quad \forall\, i=1,\ldots,N_\ell, \; \ell\in \mathbb{N}\;.
\end{equation} It is well known that the span of the characteristic functions $\chi_{I_i^l}$, $i=1,\ldots,N_\ell$, is dense in~$L^p(0,T)$, so that~\eqref{eq:wutildeueq} immediately yields $u=\tilde{u}$ in~$L^p(0,T;\mathbb{R}^n).$ We thus obtain $u^k
\rightharpoonup u$ in~$L^p(0,T;\mathbb{R}^n)$. The set $\overline{\operatorname{conv} D}^{L^p(0,T;\mathbb{R}^n)}$ is convex and closed, thus weakly closed, so that we deduce $u\in\overline{\operatorname{conv} D}^{L^p(0,T;\mathbb{R}^n)}$. \end{proof}
Our aim is to exploit the result of Theorem~\ref{thm: convD} in order to obtain outer descriptions of the convex hull of~$D$ in function space from outer descriptions of finite-dimensional sets of the form~$C_{D,\Pi}$. This approach is particularly appealing in case~$C_{D,\Pi}$ is a polyhedron. Before discussing some relevant classes of constraints where this holds true, we first show that polyhedricity cannot be guaranteed in general. In fact, the following construction shows that every closed convex set~$K\subseteq[0,1]^M$ can arise as~$C_{D,\Pi}$ for some feasible set~$D$. \begin{example}\label{ex: nonpoly}
Let~$M\in\mathbb{N}$ and~$K\subseteq[0,1]^M$ be a closed convex set. Define~$T=M$ and
\[
\begin{aligned}
D_{K} := \big\{ u \in BV(0,T)\colon \;
& u(t) \in \{0,1\} \text{ f.a.a.\ } t \in (0,T),\\
& |u|_{BV(0,T)} \le M,\; \textstyle(\int_{i-1}^{i}u\, \text{d} t)_{i=1}^M\in K \big\}\;.
\end{aligned}
\]
By definition, the set~$D_K$ satisfies Assumption~\eqref{eq:D1}. Also
Assumption~\eqref{eq:D2} is easy to verify for arbitrary
$p\in[1,\infty)$, using the closedness of $K$ and Proposition
10.1.1(i) in~\cite{ATT14}, which guarantees, for any
sequence~$\{u^k\}_{k\in \mathbb{N}}\subset D_K$ converging to some $u$
in~$L^p(0,T)\hookrightarrow L^1(0,T)$, that
\[
|u|_{BV(0,T)} \leq \liminf\limits_{k\to \infty}|u^k|_{BV(0,T)} \leq M\;.
\]
Defining the projection~$\Pi$ by local averaging on the
intervals~$(i-1,i)$, $i=1,\dots,M$, we obtain~$\Pi(D_K)=K$ and
hence, due to convexity of~$K$, we have~$K=C_{D_K,\Pi}$. \end{example}
In the following subsections, we discuss two of the practically most relevant classes of constraints~$D$ and investigate the associated sets~$C_{D,\Pi}$. The first class includes~$D_{\max}$ as defined in~\eqref{eq:Dex}, whereas the second class includes the minimum dwell time constraints mentioned in the introduction. For the remainder of this section, we always assume that the intervals defining the projection~$\Pi$ are pairwise disjoint.
\subsection{Pointwise combinatorial constraints}\label{ex: boundedshift}
By Assumption~\eqref{eq:D1}, the total number of shiftings of all switches is bounded by some~$\sigma\in\mathbb{N}$. A relevant class of constraints arises when the switches must additionally satisfy certain combinatorial conditions at any point in time. As an example, it might be required that two specific switches are never used at the same time, or that some switch can only be used when another switch is also used, e.g., because they are connected in series. More formally, we assume that a set~$U\subseteq\{0,1\}^n$ is given and consider the constraint \[ D_{\max}^\Sigma(U) := \Big\{ u \in BV(0,T;\mathbb{R}^n)\colon \; u(t) \in U \text{ f.a.a.\ } t \in (0,T),
\sum_{j=1}^n|u_j|_{BV(0,T)} \leq \sigma_{\max}\Big\}. \]
\begin{lemma}
The set $D_{\max}^\Sigma(U)$ satisfies Assumptions~\eqref{eq:D1} and~\eqref{eq:D2}. \end{lemma}
\begin{proof}
The set $D_{\max}^\Sigma(U)$ obviously satisfies
~\eqref{eq:D1}. Moreover, for any~$p\in[1,\infty)$,
Proposition~10.1.1(i) in~\cite{ATT14} again guarantees for any
sequence of controls $\{u^k\}_{k\in \mathbb{N}}\subset D_{\max}^\Sigma(U)$ in~$L^p(0,T;\mathbb{R}^n)\hookrightarrow
L^1(0,T;\mathbb{R}^n)$ that converges to some $u$ that
\[
|u_j|_{BV(0,T)} \leq \liminf\limits_{k\to \infty}|u_j^k|_{BV(0,T)} \leq \sigma_{\max}
\]
for $j=1,\ldots,n$, because of $\sup_{k\in \mathbb{N}} |u_j^k|_{BV(0,T)} \leq
\sigma_{\max}$. Furthermore, since convergence in~$L^p(0,T;\mathbb{R}^n)$ implies pointwise almost everywhere
convergence for a subsequence, the limit also satisfies $u(t) \in U$ f.a.a.\ $t\in (0,T)$.
It follows that $D_{\max}^\Sigma(U)$ is closed in
$L^p(0,T;\mathbb{R}^n)$ and thus fulfills~\eqref{eq:D2}. \end{proof}
We now show that the projections~$\Pi$ defined in~\eqref{eq:pi} not only lead to polytopes when applied to~$D_{\max}^\Sigma(U)$, but even yield integer polytopes, i.e., polytopes with integer vertices only. \begin{theorem}\label{thm: dmax01}
For any\/ $\Pi$ as in~\eqref{eq:pi}, the
set~$C_{D_{\max}^\Sigma(U),\Pi}$ is a 0/1-polytope in~$\mathbb{R}^M$. \end{theorem} \begin{proof}
We claim that~$C_{D_{\max}^\Sigma(U),\Pi}=\operatorname{conv}(K)$, where
\[
\begin{aligned}
K:=\{\Pi(u) \colon & u \in D_{\max}^\Sigma(U) \text{ and for all }i=1,\ldots,M \text{ there exists } w_i\in U \\
& \text{with } u(t)\equiv w_i \text{ f.a.a.\ } t\in I_i\}\;.
\end{aligned}
\]
From this, the result follows directly, as~$K\subseteq\{0,1\}^M$
holds by definition.
The direction \grqq$\supseteq$\grqq\ is trivial, since $K$ is a
subset of $\{\Pi(u) \colon u \in D_{\max}^\Sigma(U)\}$. It thus
remains to show \grqq$\subseteq$\grqq. For this, let $u\in
D_{\max}^\Sigma(U)$. We need to show that $\Pi(u)$ can be written as
a convex combination of vectors in~$K$. Let $m\in\{0,\dots,M\}$ denote the
number of intervals in which at least one of the switches is
shifted in~$u$. We prove the assertion by means of complete induction over
the number $m$. For $m=0$, we clearly have $\Pi(u)\in K\subseteq \operatorname{conv}(K)$.
So let the number of intervals in which at least one of the switches
is shifted be $m+1$. Additionally, let $\ell\in\{1,\ldots,M\}$ be an
index so that at least one switch is shifted in the interval
$I_\ell$. Since we have the upper bound $\sigma_{\max}$ on the total number
of shiftings, only finitely many shiftings can be in the interval
$I_\ell$. Hence, $I_\ell$ can be divided into disjoint subintervals
$I^1_\ell,\ldots,I^s_\ell$ such that
$\overline{I_\ell}=\bigcup_{k=1}^{s} \overline{I^k_\ell}$ and there
exist $w_k\in U$ with $u(t)=w_k$ f.a.a.\ $t\in I^k_\ell$, $1\leq k
\leq s$. Define functions $u^k$ for $k=1,\ldots,s$ as follows:
\[u^k(t):=\begin{cases}
w_k & \text{if }t\in I_\ell \\
u(t) & \text{otherwise}\;.
\end{cases}
\]
Due to $u\in U$ a.e.~in $(0,T)$ and $w_k\in U$, $u^k(t)$ is
a vector in~$U$ f.a.a.\ $t\in(0,T)$ and for~$k=1,\ldots,s$. Furthermore, $u^k$ has at most as many shiftings as
$u$ in total and we thus obtain $u^k \in D_{\max}^\Sigma(U)$. By
construction, we have
\[\tfrac{1}{\lambda(I_\ell)}\int_{I_\ell} u(t)\ \text{d} t =
\tfrac{1}{\lambda(I_\ell)} \sum_{k=1}^s \int_{I^k_\ell} w_k\ \text{d} t
=\sum_{k=1}^s \tfrac{\lambda(I^k_\ell)}{\lambda(I_\ell)} w_k\] with
$\nicefrac{\lambda(I^k_\ell)}{\lambda(I_\ell)}\geq 0$ for every
$k\in \{1,\ldots,s\}$ and $\sum_{k=1}^s
\nicefrac{\lambda(I^k_\ell)}{\lambda(I_\ell)}=1$. Since the control
is unchanged on the other intervals $I_i$, $i\neq \ell$, we obtain
$\Pi(u)=\sum_{k=1}^s
\nicefrac{\lambda(I^k_\ell)}{\lambda(I_\ell)}\Pi(u^k)$. The
functions $u^k$ have no shifting in $I_\ell$ so that the number of
intervals in which at least one of the switches is shifted is at
most~$m$. According to the induction hypothesis, the
vectors~$\Pi(u^k)$ can be written as a convex combinations of vectors in~$K$ and consequently, due to $\Pi(u)=\sum_{k=1}^s
\nicefrac{\lambda(I^k_\ell)}{\lambda(I_\ell)}\Pi(u^k)$, $\Pi(u)$ is
also a convex combination of vectors in~$K$. \end{proof} It is easy to see that Theorem~\ref{thm: dmax01} also extends to the constraint~$D_{\max}$ defined in~\eqref{eq:Dex}. Indeed, whenever the constraint set~$D$ is defined by switch-wise constraints as in~\eqref{eq:Dex}, polyhedricity and integrality can be verified for each switch individually, in which case~$D_{\max}$ reduces to~$D_{\max}^\Sigma(\{0,1\})$.
The fact that~$C_{D_{\max}^\Sigma(U),\Pi}$ is a polytope allows, in principle, to describe it by finitely many linear inequalities. However, the number of its facets may be exponential in~$n$ or~$M$, so that a separation algorithm will be needed for the outer approximation algorithm presented in the companion paper~\cite{partII}. It depends on the set~$U$ whether this separation problem can be performed efficiently. E.g., if~$U$ models arbitrary conflicts between switches that may not be used simultaneously, the separation problem turns out to be NP-hard, since~$U$ can model the independent set problem in this case.
Even for~$n=1$ and~$U=\{0,1\}$, the separation problem is non-trivial. In this case, the set~$K$ defined in Theorem~\ref{thm: dmax01} consists of all binary sequences~$v_1,\dots,v_M\in\{0,1\}$ such that~$v_{i-1}\neq v_i$ for at most~$\sigma_{\max}$ indices~$i\in\{2,\dots,M\}$. For the slightly different setting where $v_1$ is fixed to zero, it is shown in~\cite{buchheim22} that the separation problem for~$\operatorname{conv}(K)$ and hence for~$C_{D_{\max},\Pi}$ can be solved in polynomial time. More precisely, a complete linear description of~$C_{D_{\max},\Pi}$ is given by~$v\in[0,1]^M$, $v_1=0$, and inequalities of the form \begin{equation}\label{eq:alt} \sum_{j=1}^m (-1)^{j+1} v_{i_j} \leq \Big\lfloor \frac{\sigma_{\max}}{2} \Big \rfloor\;, \end{equation} where~$i_1,\dots,i_m\in \{2,\dots,M\}$ is an increasing sequence of indices with~$m-\sigma_{\max}$ odd and~$m>\sigma_{\max}$. For given~$\bar v\in[0,1]^M$, a most violated inequality of the form~\eqref{eq:alt} is obtained by choosing~$\{i_1,i_3,\dots\}$ as the local maximizers of~$\bar v$ and~$\{i_2,i_4,\dots\}$ as the local minimizers of~$\bar v$ (excluding~$1$); such an inequality can thus be computed in~$O(M)$ time. This separation algorithm is used in Section~\ref{sec: qbounds} to investigate the strength of our convex relaxation.
\subsection{Switching point constraints}\label{ex: combswpoint}
In this section, we focus on the case~$n=1$. It is well known that a function $u \in BV(0,T)$ admits a right-continuous representative given by $\hat u(t) = c+\mu([0,t])$, $t\in (0,T)$, where $\mu$ is the regular Borel measure on $[0,T]$ associated with the distributional derivative of $u$ and $c\in \mathbb{R}$ a constant. Note that $\hat u$ is unique on~$(0,T)$. Given $u \in BV(0,T)$ with its right-continuous representative $\hat u$, we denote the essential jump set of $u$ by \[ J_u := \Big\{ t\in (0,T) \colon\; \lim_{\tau \nearrow t} \hat u(\tau) \neq \lim_{\tau \searrow t}\hat{u}(\tau) \Big\}. \]
In the following, we assume that $u\in BV(0,T)$ always starts with zero. More formally, if $\lim_{\tau \searrow 0}\hat{u}(\tau)=1$, we already count this as one switching from zero to one and add a switching point $t=0$ to $J_u$. If $J_u$ is a finite set, we denote its cardinality by $|J_u|$. For the rest of this section, let~$\sigma \in \mathbb{N}$ be given.
\begin{definition}
Let $0 \leq t_1\leq \ldots \leq t_\sigma < \infty$ be given and set
\[
\begin{aligned}[t]
\eta_\le : \mathbb{R} \to \{0, \ldots, \sigma\}, & &
\eta_\le(t) & := |\{i \in \{1, \ldots, \sigma\} \colon \, t_i \le t \}| \\
\eta_= : \mathbb{R} \to \{0, \ldots, \sigma\}, & &
\eta_=(t) & := |\{i \in \{1, \ldots, \sigma\} \colon \, t_i = t \}|
\end{aligned}
\]
with the usual convention $|\emptyset| = 0$.
Then we define the function $u_{t_1,\dots,t_\sigma}$ by
\begin{equation}\label{eq:urepr_def}
\begin{aligned}
& u_{t_1,\dots,t_\sigma}\colon [0, T] \to \{0,1\} ,\\
&u_{t_1,\dots,t_\sigma}(t) :=
\begin{cases}
0, & \text{if \,$\eta_\le(t)$ is even},\\
1 , & \text{if\, $\eta_\le(t)$ is odd}.
\end{cases}
\end{aligned}
\end{equation} \end{definition}
\begin{lemma}\label{lem:urepr}
Let $u \in BV(0,T;\{0,1\})$ be given. Then $u_{t_1,\dots,t_\sigma}$ is a representative of $u$,
i.e., $u_{t_1,\dots,t_\sigma} \in [u]$, if and only if $\{t_1, \ldots, t_\sigma\}$ fulfill
the following conditions:
\begin{itemize}
\item[\textup{(0)}] $0 \leq t_1 \leq \ldots \leq t_\sigma < \infty$
\item[\textup{(1)}] $J_u \subseteq \{t_1, \ldots, t_\sigma\}$
\item[\textup{(2)}] If $i\in \{1, \ldots, \sigma\}$ is such that $t_i\in J_u$, then $\eta_=(t_i)$ is odd.
If $t_i \notin J_u$ and $t_i < T$, then $\eta_=(t_i)$ is even.
\end{itemize} \end{lemma}
\begin{proof}
It is easy to verify that $u_{t_1,\dots,t_\sigma} $ agrees with
the right-continuous representative~$\hat u$ on~$(0,T)$ if and only
if (0)--(2) are fulfilled, which gives the assertion. \end{proof}
Now, given any polytope~$P\subseteq \mathbb{R}_+^\sigma$, we define the \emph{set of switching point constraints} by
\[ \begin{aligned}
D_P := \{ u \in BV(0,T;\{0,1\}) \colon \;
& \exists\, 0\leq t_1\leq \cdots\leq t_\sigma < \infty\\ & \text{ s.t.\ } (t_1, \ldots, t_\sigma) \in P, \ u_{t_1,\dots,t_\sigma} \in [u] \}\;.
\end{aligned}
\]
\begin{lemma}
The set $D_P$ satisfies the assumptions in~\eqref{eq:D1} and~\eqref{eq:D2}. \end{lemma}
\begin{proof}
Since $u\in \{0,1\}$ a.e.~in $(0,T)$ and $|J_u| \leq \sigma$ by Lemma~\ref{lem:urepr}(1)
for all $u\in D_P$, every $u\in D_P$ satisfies
$|u|_{BV(0,T)} \leq \sigma$ such that~\eqref{eq:D1} is fulfilled.
To verify~\eqref{eq:D2}, consider a sequence $\{u^k\}\subset D_P$ with $u^k \to u$ in~$L^p(0,T)$.
From~\eqref{eq:D1} and \cite[10.1.1(i)]{ATT14}, we deduce $u \in BV(0,T)$.
Moreover, there is a subsequence, denoted by the same symbol for convenience,
such that the sequence of representatives~$\{u_{t^k_1,\dots,t^k_\sigma}\}$ converges pointwise
almost everywhere in $(0,T)$ to $u$. This yields $u\in \{0,1\}$ a.e.~in $(0,T)$.
Furthermore, as a polytope, $P$ is compact by definition, so that
there is yet another subsequence such that
$t^k := (t_1^k, \ldots, t_\sigma^k)$ converges to $\bar t\in \mathbb{R}^\sigma$
with $0 \leq \bar t_1 \leq \ldots \leq \bar t_\sigma < \infty$ and $\bar t \in P$.
The pointwise almost everywhere convergence of $\{u_{t^k_1,\dots,t^k_\sigma}\}$
implies that $u$ is constant a.e.~in $(\bar t_i, \bar t_{i+1}) \cap (0,T)$ for all $i=1, \ldots, \sigma-1$
and a.e.~in $(0,\bar t_1)$ and $(\bar t_\sigma, T)$.
Therefore, the essential jump set of the limit $u$ is contained in $\{\bar t_1, \ldots, \bar t_\sigma\}$
as required in Lemma~\ref{lem:urepr}(1).
To show Lemma~\ref{lem:urepr}(2) assume on contrary that there is an index $i\in \{1, \ldots, \sigma\}$ such that
$t_i \in J_u$ and $\eta := \eta_=(t_i)$ is even. Let $\bar t_j, \ldots, \bar t_{j + \eta}$ be those elements of $\bar t$ that
equal $t_i$. Then, due to $|J_u| \leq \sigma$ and $t^k \to \bar t$,
there exists an $\varepsilon > 0$ such that $[t_i - \varepsilon, t_i + \varepsilon] \cap J_u = \{t_i\}$
and, for $k\in \mathbb{N}$
sufficiently large, $t_j^k, \ldots, t_{j + \eta}^k \in [t_i - \varepsilon, t_i + \varepsilon]$, while
$t_\ell^k \notin [t_i - \varepsilon, t_i + \varepsilon]$ for all $\ell \neq j, \ldots j+\eta$.
Because of $t_i \in J_u$, $t_j^k, \ldots, t_{j+\eta}^k \to t_i$, $u^k \to u$ in $L^1(0,T)$, and the construction of
$u_{\bar t_1,\dots,\bar t_\sigma}$ in~\eqref{eq:urepr_def}, we then obtain
\[
\begin{aligned}
\varepsilon =
\int_{t_i - \varepsilon}^{t_i+ \varepsilon} u(t) \,\text{d} t
&= \lim_{k\to\infty}
\begin{aligned}[t]
& \Big(\int_{t_i - \varepsilon}^{t_{j}^k} u_{t^k_1,\dots,t^k_\sigma}(t) \, \text{d} t \\
& + \sum_{m=j}^{j+\eta-1} \int_{t_m^k}^{t_{m+1}^k} u_{t^k_1,\dots,t^k_\sigma}(t) \, \text{d} t \\
& + \int_{t_{j+\eta}^k}^{t_i + \varepsilon} u_{t^k_1,\dots,t^k_\sigma}(t) \, \text{d} t\Big) \end{aligned} \\
& \in \Big\{ \int_{t_i - \varepsilon}^{t_i+ \varepsilon} 0 \,\text{d} t,
\int_{t_i - \varepsilon}^{t_i+ \varepsilon} 1 \,\text{d} t \Big\}
= \{0, 2\varepsilon\},
\end{aligned}
\]
which is the desired contradiction. Analogously, one shows that, if $t_i \notin J_u$ and $t_i < T$, then
$\eta_=(t_i)$ is even.
In summary, we have shown that the limit $\bar t\in \mathbb{R}^\sigma$ satisfies the
conditions~(0)--(2) in Lemma~\ref{lem:urepr} with the essential jump set $J_u$ corresponding to the limit function $u$.
Thus, the associated function $u_{\bar t_1,\dots,\bar t_\sigma}$ is a representative of $[u]$ and, thanks to
$\bar t \in P$, this implies $u \in D_P$, which finishes the proof. \end{proof}
\begin{theorem}\label{thm: combswpoint}
For any\/ $\Pi$ as in~\eqref{eq:pi}, the
set~$C_{D_P,\Pi}$ is a polytope in~$\mathbb{R}^M$. \end{theorem} \begin{proof}
Let~$0=s_0< s_1< \dots< s_{r-1}<s_r=\infty$ include all end points
of the intervals~$I_{1}, \ldots, I_{M}$ defining~$\Pi$. Let~$\Phi$ be the set of all
maps~$\varphi\colon\{1,\dots,\sigma\}\rightarrow\{1,\dots,r\}$. Then we have
\begin{equation}\label{eq:decomp}
\big\{ (t_1,\dots,t_\sigma)\in P\colon t_1\le\dots\le
t_\sigma\big\} =\bigcup_{\varphi\in\Phi} P_\varphi
\end{equation}
with
\[P_\varphi:=\big\{ (t_1,\dots,t_\sigma)\in P\colon t_1\le\dots\le
t_\sigma,\;s_{\varphi(i)-1}\le t_i\le
s_{\varphi(i)}\;\forall i=1,\dots,\sigma \big\}\;.\] Now each
set~$P_\varphi$ is a (potentially empty) polytope. Moreover, by
construction, the function~$P_\varphi\ni
(t_1,\dots,t_\sigma)\mapsto\Pi(u_{t_1,\dots,t_\sigma})\in\mathbb{R}^M$ is
linear, since
\[
\Pi(u_{t_1,\dots,t_\sigma})_j
= \tfrac{1}{\lambda(I_j)}\int_{I_j}u_{t_1,\dots,t_\sigma}(t)\,\text{d} t\\
= \tfrac{1}{\lambda(I_j)}\sum_{\tiny\substack{i\in\{1,\dots,\sigma+1\}\\\text{ even}}}\int_{I_j} \chi_{[t_i-t_{i-1}]}\,\text{d} t
\] for~$j=1,\dots,M$, where we set~$t_0:=0,t_{\sigma+1}:=\infty$,
and $\int_{I_j} \chi_{[t_i-t_{i-1}]}\,\text{d} t$ is linear
in~$t_i$ and~$t_{i-1}$ for a fixed assignment~$\varphi$. It
follows from~\eqref{eq:decomp} that~$\Pi(D_P)$ is a finite union of
polytopes and hence its convex hull~$C_{D_P,\Pi}$ is a polytope
again. \end{proof}
An important class of constraints of type~$D_P$ are the minimum dwell-time constraints. For a given minimum dwell time~$s>0$, it is required that the time elapsed between two switchings is at least~$s$. This implies, in particular, that the number of such switchings is bounded by~$\sigma:=\lceil T/s\rceil$. We thus consider the constraint \[ \begin{aligned} D_s:=\big\{ u\in BV(0,T)\colon &\exists \ t_1,\ldots,t_\sigma\geq 0 \\ &\text{ s.t.\ } t_{j}-t_{j-1}\ge s\; \forall \,j = 2, \dots,\sigma, \;u_{t_1,\dots,t_\sigma} \in [u] \big\}. \end{aligned} \] By Theorem~\ref{thm: combswpoint}, the set~$C_{D_s,\Pi}$ is a polytope in~$\mathbb{R}^M$. However, it is not a 0/1-polytope in general. As an example, consider the time horizon~$[0,3]$ with intervals~$I_j:=[j-1,j]$ for each~$j=1,2,3$, and let~$s=\tfrac 32$. Then it is easy to verify that~$C_{D_s,\Pi}$ has several fractional vertices, e.g., the vector~$(0,1,\tfrac 12)^\top$, being the unique optimal solution when minimizing~$(1,-1,\tfrac 12)^\top x$ over~$x\in C_{D_s,\Pi}$. Nevertheless, the separation problem for~$D_s$ can be solved efficiently, as we will show in the following. Our approach is thus well-suited to deal with minimum dwell time constraints as well.
In order to show tractability, we first argue that it is enough to consider as switching points the finitely many points in the set \[ S:=[0,T]\cap \Big(\mathbb{Z} s+\big(\{0,T\}\cup\{a_i,b_i\colon i=1,\dots,M\}\big)\Big) \] where~$I_i=[a_i,b_i]$ for~$i=1,\dots,M$. The set~$S$ thus contains all end points of the intervals~$I_1,\dots,I_M$ and $[0,T]$ shifted by arbitrary integer multiples of~$s$, as long as they are included in~$[0,T]$. Clearly, we can compute~$S$
in~$O(M\sigma)$ time. Let~$\tau_1\dots,\tau_{|S|}$ be the elements of~$S$ sorted in ascending order. \begin{lemma}\label{lem:dwell}
Let~$v$ be a vertex of~$C_{D_s,\Pi}$. Then there exists~$u\in D_s$ with~$\Pi(u)=v$ such that~$u$ switches only in~$S$. \end{lemma} \begin{proof}
Choose~$c\in\mathbb{R}^M$ such that~$v$ is the unique minimizer of~$c^\top
v$ with~$v\in C_{D_s,\Pi}$. Moreover, choose any~$u\in D_s$
with~$\Pi(u)=v$ and let~$t_1,\dots,t_\sigma$ be the switching points of~$u$, i.e., let $0\le t_1\le\dots\le t_\sigma<\infty$ such that $u_{t_1,\dots,t_\sigma}\in [u]$. For the following, define
$$S'_j:=\{t_\ell\mid
\ell\in\{1,\dots,\sigma\},\ t_\ell-t_j=s(\ell-j)\}$$
for~$j=1,\dots,\sigma$.
Assume
first that $t_j\in(a_i,b_i)\setminus S$ for
some~$i\in\{1,\dots,M\}$ and some~$j\in\{1,\dots,\sigma\}$. By
definition of~$S$, all switching points having minimal distance
to~$t_j$ do not belong to~$S$ as
well, i.e., $S'_j\cap S=\emptyset$. Hence all points in $S_j'$ can be shifted simultaneously by some
small enough~$\varepsilon>0$, in both directions, maintaining
feasibility with respect to~$D_s$ and without any of these points
leaving or entering any of the intervals~$I_1,\dots,I_M$ and
$[0,T]$. This shifting thus changes the value of~$c^\top\Pi(u)$
linearly, as seen in the proof of~Theorem~\ref{thm: combswpoint}, which is
a contradiction to unique optimality of~$v$.
We have thus shown that any~$u\in D_s$ with~$\Pi(u)=v$ must have all
switching points either in~$S$ or outside of any interval~$I_i$. So
consider some~$u\in D_s$ with~$\Pi(u)=v$, defined by switching
points~$t_1,\dots,t_\sigma$ as above, and let~$t_j\not\in
S$ be any switching point of~$u$ not belonging to any
interval~$I_i$. By shifting all switching points in~$S'_j$ simultaneously
to the left until~$S_j'\cap S\neq\emptyset$, taking into account that the set~$S_j'$ may increase when~$t_j$ decreases, we obtain another
function~$u'\in D_s$. By construction of~$S$, no shifting point is
moved beyond the next point in~$S$ to the left of its original
position. In particular, none of the shifting points being moved
enters any of the intervals~$I_i$, so that we
derive~$\Pi(u')=\Pi(u)=v$, but~$u'$ has strictly less switching
points outside of~$S$ than~$u$. By repeatedly applying the same
modification, we eventually obtain a function projecting to~$v$ with
switching points only in~$S$. \end{proof} \begin{theorem}\label{thm:dwell}
One can optimize over~$C_{D_s,\Pi}$ (and hence also separate
from~$C_{D_s,\Pi}$) in time polynomial in~$M$ and~$\sigma$. \end{theorem} \begin{proof}
By Lemma~\ref{lem:dwell}, it suffices to optimize over the
projections of all~$u\in D_s$ with switchings only in~$S$. This
can be done by a simple dynamic programming approach:
given~$c\in\mathbb{R}^M$, we can compute the optimal value
\[c^*(t,b):=\min\;c^\top \Pi(u\cdot\chi_{[0,t]})\; \text{
s.t. }u\in D_s,\;\lim\limits_{\tau\searrow t}\hat{u}(\tau)=b\text{ if }t<T\] for~$b\in\{0,1\}$
recursively for all~$t\in S$.
Starting with~$c^*(\tau_1,b)=0$, we obtain
\[
c^*(\tau_j,b)=\min\begin{cases}
\begin{array}{ll}
c^*(\tau_{j-1},b)+c^\top\Pi(b\chi_{[\tau_{j-1},\tau_{j}]})\\
c^*(\tau_{j}-s,1-b)+c^\top\Pi((1-b)\chi_{[\tau_{j}-s,\tau_{j}]}), & \text{if }\tau_j\ge s\\
c^\top\Pi((1-b)\chi_{[0,\tau_j]}), & \text{if }\tau_j< s
\end{array}
\end{cases}
\]
for~$j=1,\dots,|S|$. The desired optimal value
is~$\min\{c^*(T,0),c^*(T,1)\}$ then, and a corresponding optimal
solution can be derived easily. \end{proof} Note that~$\sigma$ is not polynomial in the input size in general, but only pseudopolynomial, if~$T$ and~$s$ are considered part of the input.
In practice, it is necessary to design an explicit separation algorithm for~$C_{D_s,\Pi}$ instead of using the theoretical equivalence between separation and optimization. This might be possible by generalizing the results presented in~\cite{lee04}. In fact, in the special case that~$[0,T]$ is subdivided into intervals~$I_1,\dots,I_M$ of the same size and this size is a divisor of~$s$, it follows from Lemma~\ref{lem:dwell} that~$C_{D_s,\Pi}$ agrees with the min-up/min-down polytope investigated in~\cite{lee04}. In this case,~$C_{D_s,\Pi}$ is a 0/1-polytope and a full linear description, together with an exact and efficient separation algorithm, is given in~\cite{lee04}. It might be possible to obtain similar polyhedral results for~$C_{D_s,\Pi}$ also in the general case. We leave this as future work.
To conclude this section, we note that the latter results can easily be transferred to a situation where the minimum dwell time after switching up is different from the minimum dwell time after switching down, which is often considered in the literature. More generally, we may consider any~$\bar s\in\mathbb{R}_+^\sigma$ and define
\[
\begin{aligned}
D_{\bar s}:=\big\{ u\in BV(0,T)\colon &\exists \ t_1,\ldots,t_\sigma\geq 0 \\ &\text{ s.t.\ } t_1\ge \bar s_1,\;t_{j}-t_{j-1}\ge \bar s_j\; \forall \,j = 2, \dots,\sigma,\;u_{t_1,\dots,t_\sigma} \in [u] \big\}.
\end{aligned}
\]
In order to generalize the results obtained for~$D_s$, it suffices to replace the set~$S$ used above by the set
$$\bar S:=[0,T]\cap \Big(\{0\}\cup\big\{\pm\textstyle\sum_{j=\ell_1}^{\ell_2}\bar s_j\mid 1\le \ell_1\le \ell_2\le \sigma\big\}+\big(\{0,T\}\cup\{a_i,b_i\colon
i=1,\dots,M\}\big)\Big)\;,$$
which can be computed in~$O(M\sigma^2)$ time. Using~$\bar S$ in place of~$S$ and following the same reasoning, both Lemma~\ref{lem:dwell} and Theorem~\ref{thm:dwell} also hold for~$D_{\bar s}$.
\section{Numerical evaluation of bounds} \label{sec: qbounds}
In this section, we test the quality of our outer description of the convex hull and, in particular, the strength of the resulting lower bounds. For this, we concentrate on the case of a single switch with an upper bound~$\sigma_{\max}$ on the number of switchings, i.e., we consider \[
D := \big\{ u \in BV(0,T)\colon \; u(t) \in \{0,1\} \text{ f.a.a.\ } t \in (0,T),\; |u|_{BV(0,T)} \leq \sigma_{\max} \big\}. \] However, we assume that $u$ is fixed to zero before the time horizon, so that we count it as a shift if $u$ is $1$ at the beginning. Moreover, we consider exemplarily a square domain $\Omega=[0,1]^2$, the end time $T=2$, the upper bound $\sigma_{\max}=2$ on the number of switchings and the form function $\psi$ as well the desired state $y_\textup{d}$ given as \[ \begin{aligned} \psi(x)&:=12\pi^2 \exp(x_1+x_2)\sin(\pi\,x_1)\sin(\pi\,x_2) \\ y_\textup{d}(t,x)&:=2\pi^2\,\max(\cos(2\pi\, t),0)\sin(\pi\,x_1)\sin(\pi\,x_2). \\ \end{aligned} \] We always choose $\alpha=0$, so that the computed bounds are not deteriorated by the Tikhonov term.
For the discretization of the optimal control problem, we use the \textsc{DUNE}-library~\cite{SAN21}. To obtain exact optimal solutions for comparison, we use the MINLP solver \textsc{Gurobi~9.1.2}~\cite{gurobi} for solving the discretized problem of~\eqref{eq:optprob}. The source code is part of the implementation at~\url{www.mathematik.tu-dortmund.de/lsv/codes/switch.tar.gz}. The spatial discretization uses a standard Galerkin method with continuous and piecewise linear functionals. For the state $y$ and the desired temperature $y_\textup{d}$ we also use continuous and piecewise linear functionals in time, while the temporal discretization for the controls chooses piecewise constant functionals. The BV-seminorm condition then simplifies to \begin{equation}\label{eq:semidisc}
u_0+\sum_{i=1}^{N_t-1} |u_i-u_{i-1}|\leq\sigma_{\max}\;, \end{equation}
where the term~$u_0$ is added in order to count a shift if $u_0=1$. We linearize~\eqref{eq:semidisc} by introducing $N_t-1$ additional real variables $z_i$ expressing the absolute values $|u_i-u_{i-1}|$. More precisely, we require~$z_i\ge u_i-u_{i-1}$ and~$z_i\ge u_{i-1}-u_i$ and use the linear constraint $u_0+\sum_{i=1}^{N_t}z_i\le \sigma_{\max}$ instead of~\eqref{eq:semidisc}. The naive convex relaxation now replaces the binarity constraint~$u_i\in\{0,1\}$ with~$u_i\in[0,1]$ for~$i=0,\dots,N_t-1$. For the tailored convexification presented in this paper, we instead omit the constraint~\eqref{eq:semidisc} and iteratively add a most violated cutting plane for~$C_{D,\Pi}$, where the intervals $I_1,\ldots,I_M$ for the projection are the ones given by the discretization in time, until the relative change of the bound is less than $0.1\%$ in three successive iterations. To the best of our knowledge, there is no standard procedure for solving the convexified control problems with additional linear control constraints arising at this point, we thus also use \textsc{Gurobi~9.1.2}~\cite{gurobi} for this.
We investigate the bounds for a sequence of discretizations with various numbers~$N_t$ of time intervals and uniform spatial triangulations of~$\Omega$ with $N_x\times N_x$ nodes. \textsc{Gurobi} is run with default settings except that the parallel mode is switched off for better comparison and the dual simplex method is used due to better performance. All computations have been performed on a 64bit Linux system with an Intel Xeon E5-2640 CPU @ 2.5 GHz and $32$ GB RAM.
\bgroup \def1.5{1.5} \begin{table}[htb]
\centering
\begin{scriptsize}
\begin{tabular}{rrrrrrrrrrr}
\hline
$N_x$ & $N_t$ & \multicolumn{2}{c}{MINLP} & \multicolumn{2}{c}{naive rel.} & \multicolumn{4}{c}{tailored convexification} &\\
\cmidrule(lr){3-4}\cmidrule(lr){5-6}\cmidrule(lr){7-11} & & Obj & Time (s) & Obj & Gap & Obj & \#Cuts & \#Ex & Gap & Filled gap \\
\hline
10& 20 & 13.69 & 4.38 & 8.41 & 38.60 \% & 9.67 & 21 & 5&29.39 \% & 23.85 \% \\
& 40 & 12.76 & 39.51 & 7.39 & 42.09 \% & 9.03 & 56 & 16&29.22 \% & 30.59 \% \\
& 60 & 12.51 & 152.44 & 7.29 & 41.71 \% & 8.86 & 108 & 30&29.19 \% & 30.01 \% \\
& 80 & 12.50 & 465.19 & 7.28 & 41.75 \% & 8.53 & 143 & 67&31.74 \% & 23.98 \% \\
& 100 & 12.54 & 663.20 & 7.25 & 42.18 \% & 7.95 & 136 & 88&36.62 \% & 13.18 \% \\
\hline
15& 20 & 13.69 & 32.07 & 8.38 & 38.79 \% & 9.67 & 21 & 5&29.40 \% & 24.20 \% \\
& 40 & 12.76 & 272.97 & 7.38 & 42.13 \% & 9.08 & 52 & 16&28.83 \% & 31.57 \% \\
& 60 & 12.51 & 1183.68 & 7.29 & 41.75 \% & 8.71 & 91 & 28&30.35 \% & 27.30 \% \\
& 80 & 12.50 & 2837.12 & 7.28 & 41.78 \% & 8.35 & 117 & 60&33.19 \% & 20.57 \% \\
& 100 & 12.54 & 4686.54 & 7.25 & 42.22 \% & 7.79 & 131 & 93&37.91 \% & 10.20 \% \\
\hline
20& 20 & 13.69 & 109.08 & 8.37 & 38.85 \% & 9.69 & 23 & 5&29.24 \% & 24.73 \% \\
& 40 & 12.76 & 1305.88 & 7.38 & 42.14 \% & 8.96 & 59 & 20&29.75 \% & 29.40 \% \\
& 60 & 12.51 & 5147.66 & 7.29 & 41.76 \% & 8.57 & 86 & 35&31.52 \% & 24.53 \% \\
& 80 & 12.50 & 15185.22 & 7.28 & 41.78 \% & 8.30 & 123 & 62&33.62 \% & 19.53 \% \\
& 100 & 12.54 & 19550.01 & 7.25 & 42.23 \% & 8.00 & 153 & 91 &36.20 \% & 14.27 \% \\
\hline
\end{tabular}
\end{scriptsize}
\caption{Comparison of naive and tailored convexification.}\label{tab:gurobi} \end{table} \egroup
The results are presented in Table~\ref{tab:gurobi}. For given choices of~$N_t$ and~$N_x$, we report the objective values (Obj) obtained by the exact approach and the two relaxations. We emphasize that, for a given optimal solution of the respective problem, we recalculate the objective value with a much finer discretization, choosing $N_t=200$ and $N_x=100$. In particular, the bounds do not necessarily behave monotonously. It can be seen from the results that the new bounds are clearly stronger than the naive bounds. In the last column (Filled gap), we state how much of the gap left open by the naive relaxation is closed by the new relaxation. We also state how many cutting planes are computed altogether (\#Cuts) and how many of them are needed to obtain at least the same bound as the naive relaxation (\#Ex). The main message of Table~\ref{tab:gurobi} is that our new approach yields better bounds than the naive approach even after adding relatively few cutting planes. Additionally, the naive relaxation includes inequality constraints involving the BV-seminorm, such that its solution is very challenging in practice.
For the exact approach, we also state the time (in seconds) needed for the solution of the problem (Time). It is obvious from the results that only very coarse discretizations can be considered when using a straightforward MINLP-based approach. In the companion paper~\cite{partII}, we thus develop a tailored outer approximation algorithm based on the the convex hull description in~\eqref{eq:convapprox} in order to compute the dual bounds of our convex relaxation of~\eqref{eq:optprob} more efficiently.
\fontsize{9}{10.5}\selectfont
\end{document} | arXiv |
Modeling the two- and three-dimensional displacement field in Lorca, Spain, subsidence and the global implications
Jose Fernandez ORCID: orcid.org/0000-0001-5745-35271,
Juan F. Prieto ORCID: orcid.org/0000-0002-7235-52952,
Joaquin Escayo ORCID: orcid.org/0000-0002-4394-50181,
Antonio G. Camacho1,
Francisco Luzón3,
Kristy F. Tiampo4,
Mimmo Palano ORCID: orcid.org/0000-0001-7254-78555,
Tamara Abajo1,
Enrique Pérez6,
Jesus Velasco2,
Tomas Herrero6,
Guadalupe Bru1,
Iñigo Molina2,
Juan López6,
Gema Rodríguez-Velasco7,
Israel Gómez1 &
Jordi J. Mallorquí8
Land subsidence associated with overexploitation of aquifers is a hazard that commonly affects large areas worldwide. The Lorca area, located in southeast Spain, has undergone one of the highest subsidence rates in Europe as a direct consequence of long-term aquifer exploitation. Previous studies carried out on the region assumed that the ground deformation retrieved from satellite radar interferometry corresponds only to vertical displacement. Here we report, for the first time, the two- and three-dimensional displacement field over the study area using synthetic aperture radar (SAR) data from Sentinel-1A images and Global Navigation Satellite System (GNSS) observations. By modeling this displacement, we provide new insights on the spatial and temporal evolution of the subsidence processes and on the main governing mechanisms. Additionally, we also demonstrate the importance of knowing both the vertical and horizontal components of the displacement to properly characterize similar hazards. Based on these results, we propose some general guidelines for the sustainable management and monitoring of land subsidence related to anthropogenic activities.
Land subsidence, ranging from local collapse to the broad regional lowering of Earth's surface, represents the main geomechanical effect related to the removal of subsurface support. Subsidence can occur as a result of (i) natural factors (e.g., tectonic activity, self-consolidation of recent sedimentary deposits, oxidation and shrinkage of organic soils)1,2 and (ii) anthropogenic processes (e.g., groundwater pumping3,4,5,6, urban development7, hydrocarbon or mining exploitation8,9).
In this study, we focus on land subsidence related to groundwater pumping because it represents a hazard commonly affecting large areas worldwide, usually associated with the increasing demand upon groundwater resources due to expanding metropolitan and agricultural areas in semiarid and arid regions4. The surface ground deformation thus constitutes a signature of the processes in the reservoir and can provide information about those subsurface processes10. A number of recent studies have focused on this topic3,4,5,11,12,13,14,15,16,17,18,19,20,21,22.
Frequently, land settlement goes unnoticed, only to be discovered later, after severe damage has occurred or in the framework of broader scientific or technical studies4,7,12. Recently, awareness on the damage threat posed by anthropogenic subsidence has increased significantly at both the political and public levels, thus contributing to lowering of the alarm threshold12. As a result, recent plans for subsurface resource management, including the study of the related environmental impact, have incorporated numerical predictions of the anticipated subsidence in the specific area of interest. Also in this context, the issue of anthropogenic land subsidence was included as one of the most urgent threats to sustainable development in the UNESCO International Hydrological Program VIII (2014–2020)12,23.
The modelling of surface deformation patterns can provide significant insights into the temporal changes of pore pressure as well as the 3D geometry of a reservoir in response to its exploitation over the time16,21. A number of different techniques have been developed in recent decades to estimate the surface deformation pattern related to volume changes in elastic and poroelastic media6,21,24,25,26,27,28,29. Inverse modeling is required to achieve success in such an endeavor10,30. A proper understanding of the subsidence mechanism is essential to calibrate protocols and best practices for monitoring natural and anthropogenic phenomena, with the aim to reduce vulnerability and risk for infrastructures, economies, natural environments and human life.
Given the limitations on the type and/or number of observation data, and on the geophysical and geological information for the study area, analytical models are used to estimate the amplitude and pattern of surface deformation based on assumptions about the media and perturbation source (e.g., using elastic or poroelastic theory)29,31,32. They provide a relatively simple method to model surface deformation for reservoirs of any geometric shape. Furthermore, given that these techniques assume that most of the surface deformation is explained by the poroelastic expansion or contraction of the reservoir, less in situ geological data is required than that needed for numerical models29.
One method of computing surface deformation is Geerstma's nucleus of strain model in a half space24,25, in which pressure change occurs within many small prisms in the reservoir. Surface deformation can be computed by adding the influence of these depleting prisms. Given that Geertsma's models are linear and the entire subsurface is assumed to be isotropic, superposition is allowable. Using this assumption, these linear equations permit the computation of surface deformation based on the superposition of many prismatic blocks within a compacting reservoir of any geometric shape24. See the Methods section for more details on the forward model and on the inversion technique30,32,33 used in this work.
The Lorca region, located in the Alto Guadalentín Basin of southeastern Spain (Fig. 1), is affected by subsidence rates of up to 10 cm/yr as a direct consequence of long-term aquifer exploitation4,5 (Fig. 2). This region is characterized by semi-arid climate conditions, with average precipitation rates of 150 mm/yr and an average annual temperature34 of ~18 °C. The basin is infilled with Quaternary alluvial fan systems overlapping Tertiary sediments transported by the Guadalentín River along the depression located in the eastern part of the Betic Mountain Range (an ENE-WSW oriented alpine orogenic belt resulting from the Nubia-Iberia ongoing convergence35,36,37).
Geographical location of the study area. Location of the Alto Guadalentín Basin, the Bajo Guadalentín Basin and the Guadalentín River that formed the two basins. Black lines depict main faults in the area. The locations and names of the main cities in the area are shown. The topography has been obtained from MDT05 2015 CC-BY 4.0 digital elevation model74. This figure was generated using Arc Map 10.3 (http://desktop.argis.com/es/arcmap/).
Subsidence area and location of the GNSS stations. (a) Subsidence area detected in previous studies31 by means of InSAR techniques along the Alto Guadalentín Basin. Subsidence rates have a maximum of 16 cm/yr for the period 2006–2011 located ~4 km south-west the city of Lorca. The black stars are damage locations due to the M = 5.1 May 2008 Lorca earthquake. Red lines are main faults (AMF, Alhama de Murcia Fault). The contour lines indicate 2 cm/yr InSAR subsidence due to groundwater pumping. (b) Location of the monitoring GNSS control stations deployed in the area of Alto Guadalentín. The network consists of 33 monitoring stations (blue circles show their location) and covers an area of about 70 km2. The network is designed to allow high accuracy GNSS surveys and also includes two existing continuous GNSS stations. Main population centers are depicted with white stars. GMT software was used to create this figure75.
The Guadalentín Basin aquifer is composed of two contiguous sub-basins: Alto and Bajo Guadalentín (Fig. 1). From a hydrogeological point of view, the basement beneath the aquifer is composed of several relatively impermeable Paleozoic metamorphic complexes overlain by permeable Miocene conglomerate and/or calcarenite series. The top of the succession comprises Pliocene-Quaternary, low-permeability, compressible conglomerates, sand, silt, and clays4,38. The Alto Guadalentín aquifer covers an area of approximately 277 km2. Historically, piezometric levels were located closely to the land surface, allowing the development of a number of artesian wells and permanent lagoons38. Since the 1960s–1970s, the Guadalentín Basin aquifer has reflected gradually increased overdraft and contamination (e.g., high electrical conductivity, CO2 positive thermal anomaly), and was legally declared provisionally overexploited39 in 1987. Pumping has occurred in ~1000 wells at rates of 24 (in 1973), 69 (in 1987), and 86 hm3/yr (in 2006)38,39, which led to a spatially variable continuous piezometric level decrease (at rates within the 0.5–10 m/yr range). Available piezometric information consists of water-level time series for a few points, from the 1970s to present. A long drought period from 1990 to 1995 (also in 1999–2000 and 2005–2007) reduced natural recharge and increased pumping in the Guadalentín Basin, which led to an increased resources deficit. All this information indicates a long-term trend in the consumption of groundwater resources.
Interferometric synthetic aperture radar (InSAR) studies, while detecting the high subsidence rates affecting the Alto Guadalentín Basin, also identified a delayed transient of nonlinear compaction of the Alto Guadalentín aquifer due to the 1990–1995 drought period4. This suggested a relationship between local crustal unloading and stress change on active faults bordering the basin31. Later work5 extended those studies using advanced differential InSAR (A-DInSAR) techniques to process ALOS PALSAR (2007–2010) and COSMO-SkyMed (2011–2012) radar images. The combination of multi-sensor SAR images with different resolutions allowed for a longer monitoring time span of 20 years (1992–2012) over the Alto Guadalentín Basin. Additionally, the satellite measurements provided locally comparable results with measurements acquired by two continuous GNSS stations located in the study area. Furthermore, the work presented a new soft soil thickness map and collected historical piezometric data, in order to assess aquifer system compressibility and groundwater level changes in the past 50 years. From the analysis of these data with A-DInSAR displacement measurements, the authors concluded that the governing mechanism of the Alto Guadalentín aquifer system is an inelastic, unrecoverable and delayed compaction process between water level depletion and ground surface displacement, related to the presence of very thick (>100 m) unconsolidated sediments (clay and silts).
Despite the aforementioned achievements, the previous studies focusing on the deformation in the area are based on InSAR analysis using ascending and/or descending acquisitions, without any combination of the datasets to estimate both vertical and horizontal (E-W) components4,5. Therefore, only the line-of-sight (LOS) displacement field is known in the Alto Guadalentín area at a regional level and it was assumed to correspond completely to vertical displacement. Although this is a common procedure in subsidence studies using InSAR measurements40,41,42,43,44,45,46, the main consequences are i) the neglecting of possible horizontal displacement components and ii) the likely overestimation of vertical displacement.
Here, while we afforded the problem on the decomposition of LOS measurements in the E-W and vertical components over the investigated area, we can provide additional constraints on the spatial and temporal evolution of the subsidence process as well as on the main governing mechanisms (e.g. temporal changes of pore pressure, geometry of the reservoir). With this primary aim, we established a GNSS network consisting of 33 stations in 2015, which densely covers the Alto Guadalentín basin (Fig. 2). This network has been observed in survey, or campaign, mode. Here we analyzed the measurements carried out in November 2015, June-July 2016 and February 2017. GNSS raw data have been processed by adopting standard processing strategies for this type of network and referred to a local reference frame in order to estimate the 3D deformation field (see Methods Section and Supplementary Information). Despite the limited time interval covered by the surveys, we estimated, for the first time, a significant 3D deformation field which is primarily related to the local exploitation of the aquifer. SAR data from the Sentinel-1 Copernicus constellation, acquired in ascending and descending orbits for the same time period, also were processed to obtain the respective LOS displacements. Using the GNSS and SAR-based deformation fields, we estimated both the vertical and horizontal components of the displacement over the entire area. In the following sections, the main results are described, compared and interpreted using the forward model and inversion technique previously mentioned and described in the Methods section.
Global Navigation Satellite System (GNSS) results
Three geodetic campaigns have been carried out in November 2015, June-July 2016 and February 2017. These surveys were conducted using 10 dual frequency Topcon GPS + GLONASS receivers and choke ring antennas on a four hour session basis (see Supplementary Information for details about the monuments and antennae setting characteristics). All stations were measured at least twice during the 2015 campaign and at least three times in the 2016 and 2017 campaigns with a 1 Hz sampling interval data recording (see Methods section for the description of the GNSS data processing). The 3D velocity field results are shown in Fig. 3 for both the vertical and horizontal components determined by comparing the coordinates obtained for the time spans of the three surveys. Time series for selected stations are shown in Supplementary Fig. S2.
Displacement rates determined from GNSS observations. Results corresponding to the period November 2015–February 2017. (a) Annual vertical displacement rates, subsidence, measured with standard confidence bars. (b) Average annual horizontal displacements with standard confidence regions. Additional results are shown in the Supplementary Information. This figure was generated using GMT software75.
The maximum vertical subsidence rate (9.0 ± 0.5 cm/yr) is of the same order of magnitude as that previously detected by earlier InSAR studies4,5. The maximum horizontal displacement rate detected is 2.5 ± 0.3 cm/yr (about 28% of the vertical displacement rate), a non-negligible amplitude. In the area showing the highest subsidence rate, again previously detected by InSAR techniques, a characteristic pattern of horizontal deformation appears (Fig. 3). These deformations, as theoretically expected, show a centripetal pattern towards the zone of maximum subsidence, located in the central part of the monitored area.
Also in the southern area, where there is a relative maximum in the LOS displacement detected by InSAR, significant horizontal motions are detected (see Fig. 3b) associated with GNSS stations 23 and 28. After comparison with A-DInSAR results and field inspection, we conclude that these are produced by very local movements related to monument instabilities (see Supplementary Fig. S3).
Advanced Differential Satellite Synthetic Aperture Radar (A-DInSAR) results
The low revisit time (12 days per satellite, 6 days as a constellation) of the Sentinel-1 satellites, the total coverage of the European Plate, and the free availability of these products, make them an optimal choice for this study. We used the Interferometric Wide Swath (IW) mode to perform A-DInSAR processing of the Lorca area Sentinel-1A images (see Methods section for a description of the advanced processing of the satellite radar images).
Both orbits, ascending and descending (tracks 103 and 8 respectively) from Sentinel-1A, were used to decompose the measured LOS movement/mean velocities into horizontal (E-W) and vertical components47,48,49 over the studied area.
Our A-DInSAR study covers the same time interval spanned by the GNSS campaigns (November 2015 – February 2017). Radar data were processed using the Coherent Pixel Technique (CPT)50 (see Methods Section). The total area covered by the GNSS network is approximately 70 km2. For the A-DInSAR we processed an extended region with a total area of 170 km2. In both geometries, ascending and descending, the study area is covered by three bursts of the same swath. We have used a total of 42 Interferometric Wide Swath (IW) SLC images from the Sentinel-1A satellite, which results in 185 interferograms (22 images and 137 interferograms for ascending data, 20 and 48 respectively for descending; see Supplementary Tables S2 to S4). The results are shown in Fig. 4, while some selected time series are shown in the Supplementary Fig. S5 for descending LOS. This is the first InSAR study of the Alto Guadalentin Basin using two different geometries for the same time period.
Results obtained from the A-DInSAR processing using CPT technique. Both geometries, ascending and descending, have been processed using a multilook window of 3 × 13 pixels (azimuth × range) which generates a square pixel of about 60 × 60 meters in ground resolution. Coherence method has been used for pixel selection coherence method. Results are shown for the period November 2015–February 2017. (a) Line of Sight (LOS) velocity values obtained for the ascending orbit. (b) LOS velocity values for the descending orbit. Black dots locate the GNSS stations. GMT software was used to create this figure75.
Because SAR is sensitive in the perpendicular direction to its azimuth and describes an almost polar orbit, it is assumed that the detected displacement is caused by vertical and E-W motion, and the N-S motion is neglected. To obtain the vertical and E-W components of the displacement from the ascending and descending LOS motions, the following equation system must be solved10,51.
$$[\begin{array}{c}{u}_{z}\\ {u}_{ew}\end{array}]=[\begin{array}{cc}-\cos ({\theta }_{asc}) & \sin ({\theta }_{asc})\,\cos ({\alpha }_{asc})\\ -\cos ({\theta }_{dsc}) & \sin ({\theta }_{dsc})\,\cos ({\alpha }_{dsc})\end{array}]\cdot [\begin{array}{c}{u}_{los}^{asc}\\ {u}_{los}^{dsc}\end{array}],$$
where ulos is the displacement detected for each geometry (considered positive when it is away from the satellite and negative when it is towards the satellite), αasc and αdsc are the heading angles of the satellite and θasc and θdsc are the incidence angle of the SAR beam which are determined for each pixel. An additional minor correction due to the squint angle of the SAR beam can be made51. However, in order to apply this last correction, the coordinates of each pixel over the original SAR image are necessary and SUBSIDENCE-GUI (the software implementation of CPT) currently is unable to produce this information, so we were unable to apply this correction to our results.
The decomposition into vertical and E-W displacements also introduces the need for interpolation52,53,54 because the pixels identified for ascending and descending satellite orbits are not identical in most cases. Such a decomposition is allowable only when the deformation signal is sufficiently smooth and well-sampled. Interpolation can be avoided by using the LOS data directly in the parameter estimation procedure10,55,56. E-W and vertical components of the displacement fields in the area obtained using ascending and descending LOS results are shown in Fig. 5.
East-West and Vertical displacements obtained by A-DInSAR. (a) Horizontal (East-West) and (b) vertical (Up-Down) displacement rates estimations obtained by decomposition of the LOS detected velocity using ascending and descending orbits. GNSS displacements are also plotted with arrows to compare. Results are shown for the period November 2015 - February 2017. GMT software was used to create this figure75.
For this case, the decomposition above does not produce particularly good results due to the previously mentioned methodological aspects and to the fact that the magnitude of the E-W motion is at sub-centimeter levels in many of the coherent pixels, i.e., the same order of the A-DInSAR uncertainty. In Supplementary Table S5 we compare GNSS and A-DInSAR results for those stations which have coherent pixels from the ascending and descending time series within 100 meters. The comparison allows us to estimate the error of the A-DInSAR processing, relative to GNSS, as ~0.7 cm in the vertical velocity component (good agreement comparable to the GNSS precision in this component) and ~1.0 cm for the E-W velocity component (worse agreement but consistent with the previously described limitations). If we project the three components of the measured displacement rates from the GNSS into the LOS (see Supplementary Table S6) and compare with the measured LOS (A-DInSAR), we obtain better results: ~0.7 cm for both ascending and descending orbits.
Other methodologies that can be used to obtain the North-South component of the displacement, such as Pixel Offset Tracking or Multiple Aperture Interferometry, were considered but discarded since the magnitude of the displacement in this component is not enough to obtain a reliable result with those or other techniques57,58,59,60,61. Multi-platform and multi-angle InSAR-driven combination methods, such as Multidimensional Small Baseline Subset (MSBAS)62, could increase the temporal span in the InSAR time-series but there are no GNSS data available for comparison during those time periods.
As previously noted, the establishment and observation (spanning November 2015 - February 2017) of a local GNSS network allows, for the first time, for measurement of the 3D displacement field in the Alto Guadaletin area, associated with exploitation of the local aquifer. Also, for the first time, A-DInSAR results have been obtained using both ascending and descending radar images from the Sentinel-1A Copernicus radar satellite, allowing estimation of both vertical and horizontal (E-W) displacement components, a 2D displacement field, at higher spatial resolution than GNSS. See Figs 3 to 5.
Our results highlight how the ad hoc establishment of survey mode GNSS networks improves the spatio-temporal monitoring of the 3D displacement field of areas subjected to extensive groundwater extraction, therefore representing a valuable monitoring technique. Moreover, GNSS observation provides complementary information to A-DInSAR results, allowing for their validation and scaling. In addition, at a local level, it is observed that the GNSS network does not completely cover the current displacement area, in particular along the SW region (see Figs 4 and 5), because the network was defined based on displacements obtained prior to 2012 (see Fig. 1). But our Sentinel-1 A-DInSAR results show that the deformation has extended in the SW direction, which today is the region of the most significant water extraction63. Therefore, the GNSS network needs to be extended over that region with additional GNSS stations.
Our results for the studied area highlight that: (i) simultaneous GNSS and A-DInSAR results are consistent with each other (Fig. 5 and Supplementary Tables S5 and S6); (ii) the results obtained for rates and pattern of the displacement are consistent with previous DInSAR results4,5; however, (iii) the horizontal displacement rate has a maximum amplitude of 2–3 cm/year (Figs 3 and 4) and it is a significant component of the observed deformation field. Therefore, the horizontal displacement cannot be neglected, as in the discussion and interpretation sections of previous studies4,5.
Because the results here demonstrate that the horizontal displacements represent a significant component of the deformation field of the studied area, we also performed sensitivity tests by neglecting/including this horizontal motion in order to assess the variability (or bias percentage) on the determination of the aquifer characteristics and their temporal evolution using deformation modeling. To do this, we employed our GNSS and A-DInSAR results. Also, taking into account the linear time behavior of the displacement field (see Supplementary Figs S2 and S5), we considered displacement rates in our study.
We employed four different data sets (Cases) of surface displacement covering the period November 2015 to February 2017, and we carried out the inversion using the described forward model and inversion methodology (see introduction and Methods section). In the Supplementary Information (pages 13–17) we describe a complementary study carried out considering ten Cases, which have been obtained by combining the available and different data sets. Here it is clearer to show only the most representative ones, to demonstrate the main consequences of neglecting horizontal displacements on the resulting interpretation. Subsequently we evaluate the consequences and implications for operative monitoring at a global scale (see the Supplementary Information study for additional details).
The cases described here are the following:
LOS A-DInSAR results obtained for descending orbit images, assuming 100% as vertical displacement.
Purely LOS A-DInSAR results obtained for descending orbit images.
Purely LOS A-DInSAR results obtained for ascending and descending orbit images.
Purely LOS A-DInSAR results obtained for ascending and descending orbit images together with the 3D displacements determined using the GNSS surveys results.
Case A is one-dimensional (1D), B and C are 2D (indirectly by combining Up-Down and E-W in the measured LOS), and D is a combination of 2D and 3D data (2D + 3D data).
We invert each case and estimate the volume changes of the water table (volume and geometry) assuming a given pressure change value. Moreover, based on hydrogeological observations, we impose the criteria that sources are shallower than one kilometer. A summary of the results is provided in Table 1 and Fig. 6. In Fig. 6, the blue colors indicate negative pressure values, while white colors indicate positive pressure change cells. The former are related to the loss of pore pressure due to aquifer overdrawing, the latter are related to modelling of measurement errors and/or effects related to other deformation sources (e.g., of tectonic origin). Noting this, it is interesting to observe how the cells with positive pressure changes tend to accumulate along directions of faults existing in the area36 (see Figs 1 and 6). This potentially indicates some relation with the thickness of the compacting material across the fault. While this is outside the scope of this study, these results suggest that additional research in these aspects should be carried out in the future, and that next models of the observed deformation should introduce additional sources.
Table 1 Numerical summary of the inversion results obtained for selected cases.
Representation of the inversion results obtained for the 1D, 2D and 2D + 3D considered data sets. (a) Obtained source for Case A; (b) for Case B; (c) for Case C; and (d) for Case D. Blue color indicates negative pressure value cells, produced by water extraction. White color indicates positive pressure change cells. These positive pressure sources adjust the errors and the effects of other deformation sources, different from water extraction (e.g., of tectonic origin). This figure was created using Surfer 8.02 Surface Mapping System (www.goldensoftware.com/products/surfer) and Paint, Microsoft Windows 10.
Inversion results include the volume and geometry of the active part of the aquifer which has produced the measured displacements. Here this is quantified by the intensity, which is equal to the product of volume by pressure change; it is impossible to determine both quantities separately. If we increase pressure, we decrease volume and vice versa. Here, in order to determine a general geometry, we have constrained the value of the pressure change30,31,32. We consider a pressure value of −3 MPa, after a trial analysis, selecting the value that gives us a source geometry most consistent with the characteristics of the aquifer.
Note that the inversion results obtained from these data sets can be organized into two subsets: (i) Case A (1D, vertical displacement) and (ii) Cases B–D (2D and 2D + 3D). Results for group (ii) are internally very consistent, with scattering on the order of 3% (Table 1).
The results of (i), 1D results, are ~24% greater in intensity/volume than those of (ii) (2D and 2D + 3D results), indicating that using only one component of the displacement field and assuming that displacements are only vertical significantly overestimates the volume of water extracted during the study period (on the order of tens of hm3). This can have an important effect in predictions of future volume variations and surface displacements.
Another important result is that we do not observe significant differences (3–4%, at the level of error or lower, see Table 1 and Supplementary Information) between using just one LOS (ascending or descending), both LOS (ascending and descending) displacement data sets together, or both combined with the GNSS results. The estimation of source characteristics is very similar for all cases, just slightly changing the misfit of the minimum values for Cases B–D.
In summary, contrary to previous studies in the Lorca area, the measurement and use of the horizontal and vertical displacements at the surface is important for the prediction of future volume variations and surface displacements. These differences can have important effects on the design of monitoring systems, help in the decision-making process related to the sustainable management of the aquifer resources, and improve the assessment of potential hazards related to the aquifer exploitation.
Our main conclusions, summarized above, do not include any local assumptions which could condition our interpretation methodology, but they have general applicability worldwide. The Lorca case can be considered an extreme case, taking into account that it has significant E-W horizontal deformation, but only in geographically limited areas of maximum deformation. In other regions, where significant horizontal E-W deformation may be more scattered and cover more extended areas (see the synthetic test case in the Supplementary Information, text, Figs S8 and S9, and Table S8) the effect of considering vertical deformation alone could be even more dramatic.
We have shown, using inversion results from different data sets, that the operational monitoring of the aquifer can be done using A-DInSAR with ascending and/or descending satellite radar images. Considering the inversion results described previously and in the Supplementary Information, the most effective method is to carry out a joint inversion of LOS measurements determined using ascending and descending radar images. GNSS, using continuous or survey observation mode, can be used for validation and scaling purposes. GNSS stations should be installed in those locations that will best constrain the A-DInSAR results in areas of zero deformation for reference, maximum deformation areas for scaling and study of the time variation, or in low coherence areas so that both techniques complement each other. Continuous GNSS observations are preferable, if possible. The proposed methodology will potentially reduce the cost of the geodetic monitoring system in a very important way.
In addition, this effective A-DInSAR monitoring can be accomplished with the freely-available Copernicus Sentinel-1A and -1B satellite data, considering their global coverage and repeatability, ensuring their effective use for monitoring on a global scale.
Global Navigation Satellite System (GNSS)
Raw GNSS observations collected on the episodic geodetic network were processed using GAMIT/GLOBK 10.6 software64. To improve the overall configuration of the network and tie the local measurements to a regional reference frame, data coming from more than 20 continuous stations belonging to regional (REGAM and MERISTEMUM), wide-scale (IGNE and EPN) networks were introduced in the processing (see Supplementary Information for additional details). In a first step, we used daily double-differenced GNSS phase observations, on a 30-sec sampling basis; the observations were weighted according to the elevation angle, for which a cut-off angle of 10° was chosen. In addition, we used the latest absolute receiver antenna models by the IGS and we adopted atmospheric zenith delay models65, coupled with the Global Mapping Functions for the neutral atmosphere. The results of this processing step are daily estimates of loosely constrained station coordinates, and other parameters, along with the associated variance-covariance matrices. In a successive step, the loosely constrained daily solutions were used as quasi observations in a Kalman filter (GLOBK) in order to estimate a consistent set of daily coordinates (i.e. time series) for all sites involved. Each time series was analyzed for linear velocities and antenna jumps; in order to obtain clean time series, any position estimate whose uncertainty was greater than 20 mm or whose value differed by more than 10 mm from the best-fitting linear trend was removed. In a final step, all loosely constrained daily solutions and their full covariance matrices were combined to compute a set of coordinates and velocities related to ITRF2008 geodetic reference frame66. In this step, to account for correlated errors, we added a random walk component62 of 1.5 and 2.5 mm yr−0.5 to the assumed error in horizontal and vertical positions, respectively. To adequately show the crustal deformation pattern over the studied area, i.e. to isolate the local deformation field from the regional tectonic pattern, we rotated our estimated GNSS velocities to a local reference system defined by the minimization of the long-term velocities67 of ALAC, ALBA and ALME continuous stations from EPN68.
Advanced Differential Synthetics Aperture Radar Interferometry (A-DInSAR)
The dataset used in this study is composed of 37 Sentinel-1A Single Look Complex (SLC) images (19 ascending from track 103 and 18 descending from track 8) that were processed using SUBSIDENCE-GUI interferometric software. Prior to the interferometric generation, all images for each orbit were registered to a common master image. The master image was selected to minimize perpendicular and temporal baselines across the dataset in order to avoid registration errors.
Interferogram pairs were generated by a double minimum criteria, avoiding those with high temporal and perpendicular baselines. This selection mode proved to generate better velocity estimation as well as a lower sensitivity to Digital Elevation Model (DEM) errors. A total of 185 interferograms (137 for ascending orbit and 48 for descending), have been generated (see Supplementary Table S3 for the entire list). To remove the topographic phase from the interferograms, an external high-resolution DEM have been used. Since there are missing acquisitions from some orbits, our study covers a temporal span from October 2015 to February 2017 which covers the period of the GNSS surveys.
To obtain the surface displacements we use SUBSIDENCE-GUI, the software implementation of the Coherent Pixel Technique algorithm, CPT50. This method works with distributed scatters at low-resolution over the multi-looked interferograms, similar to the wide-used Small Baselines Subset (SBAS)69. Because of the characteristics of the ground surface in Lorca (mostly agricultural and bare soil), this kind of analysis is more suitable than a full-resolution approach like Point Scatters (PS) method. For the two geometries (ascending and descending ones) the principal parameters of the processing have been preserved. A mean coherence map has been processed to establish a pixel selection by means of coherence, using a multilook window of fifteen samples in range and three lines in azimuth. This multilooking results in low resolution pixels obtained from an average of 45 pixels from the original interferogram, which correspond to a square pixel in a ground resolution of about 60 m × 60 m.
To select those pixels with enough phase quality to obtain surface deformation, a coherence criterion has been chosen. A threshold of a medium coherence of 0.4 corresponding to a phase standard deviation of 18° degrees has been used70, this value provides good spatial coverage and enough phase quality to obtain a convergent solution. A Delaunay triangulation between pixels is used, and to reduce the atmospheric artifacts in the lineal processing a limit of 800 m among pixels is also used. To estimate the linear velocity CPT needs velocity and DEM error seeds, points with known velocity and known altitude for the entire studied period50. For velocity seeds, several points outside of the main deformation area have been selected and for DEM seeds, large human-made flat zones were used, such as roads or parking lots (see Supplementary Fig. S4).
For the non-linear velocity estimation, the atmospheric contribution to the phase must be calculated. To filter the atmospheric perturbations, two filters were applied: a spatial low pass filtering with a 1-km correlation window and a high pass temporal filtering with a window of 60 days and a minimum of 4 samples. After this processing, the non-linear displacement can be calculated71.
Direct Modeling and Inverse Technique
Considering the linear theory of poroelasticity72, the horizontal and vertical components (du, dv, dw) of the movements at a point (X, Y, Z) of the free surface, due to a differential nucleus, located at (x, y, z), with sides dx, dy, dz, corresponding to the reservoir with local overpressure Δp are24:
$$(\begin{array}{c}du\\ dv\\ dw\end{array})={\rm{\Delta }}p\frac{1-\nu }{\pi }{c}_{m}\,(\begin{array}{c}X-x\\ Y-y\\ Z-z\end{array})\,\frac{dxdydz}{{({(X-x)}^{2}+{(Y-y)}^{2}+{(Z-z)}^{2})}^{3/2}}$$
where ν denotes Poisson's ratio (≈0.25), cm the uniaxial compaction coefficient. Assuming that displacements at the surface happen to be almost directly proportional to the thickness Δz of the reservoir, the volume integrations for a parallelepiped cell of sides Δx, Δy, Δz and overpressure Δp in equation (2) can be simplified to integration in the horizontal plane only given rise to24:
$$(\begin{array}{c}du\\ dv\\ dw\end{array})={\rm{\Delta }}p\,\frac{1-\nu }{\pi }\,{c}_{m}I\,{\rm{\Delta }}z$$
$$\begin{array}{rcl}I & = & {I}_{i}(X-(x+{\Delta }x/2),Y-(y+{\Delta }y/2),Z-z)\\ & & -\,{I}_{i}(X-(x+{\Delta }x/2),Y-(y-{\Delta }y/2),Z-z)\\ & & -\,{I}_{i}(X-(x-{\Delta }x/2),Y-(y+{\Delta }y/2),Z-z)\\ & & +\,{I}_{i}(X-(x-{\Delta }x/2),Y-(y-{\Delta }y/2),Z-z)\end{array}$$
The integrals Ii for the displacements in the i-direction are:
$${I}_{z}(p,q,r)=\frac{1}{2}\frac{p}{|p|}\{arcsin\,\frac{{p}^{2}{q}^{2}-{r}^{2}({p}^{2}+{q}^{2}+{r}^{2})}{({p}^{2}+{r}^{2})\,({q}^{2}+{r}^{2})}+\frac{\pi }{2}\}$$
$${I}_{x}(p,q,r)=arcsinh\frac{p}{\sqrt{{q}^{2}+{r}^{2}}}$$
$${I}_{y}(p,q,r)=arcsinh\frac{q}{\sqrt{{p}^{2}+{r}^{2}}}$$
This formulation provides the direct calculation of the surface effect of a single parallelepiped cell. The total effect of an anomalous structure described as aggregation of m small parallelepiped cells is obtained, according24, as addition of the partial effects. This direct formulation can be used to carry out the inverse approach in order to determine the pressure 3D source structure responsible of the observed surface deformations.
Camacho30 presented an original methodology for simultaneous inversion of three dimensional displacement data, LOS, or any combination of terrestrial and space displacement data, by means of 3D extended bodies with free geometry for anomalous pressure. The approach determines a general geometrical configuration of pressurized sources corresponding to prescribed values of anomalous pressure. These sources are described as a 3D aggregate of (thousands of) pressure elemental sources, and they fit the entire data set within some regularity conditions. The approach works in a step-by-step growth process that allows us to build very general geometrical configurations.
The observation equations are:
$${\boldsymbol{ds}}={\boldsymbol{d}}{{\boldsymbol{s}}}^{{\boldsymbol{c}}}+{\boldsymbol{v}}$$
where ds, dsc represent the vector of observed and calculated three component (3D) deformations, and v is the vector for residual values coming from inaccuracies in the observation process and also from insufficient model fit. In that methodology surface deformation, dsc, due to a buried over pressure structure is computed as the aggregated effect for several point sources, as due to the deformation effects from the incremental pressure pk and expansion radius within the elastic semi space, originally formulated as a Mogi model73. In this work we substitute this source with the poroelastic expressions24 for 3D reservoirs according the preceding formulation for parallelepiped cells.
The inversion equations (8) are solved by means of adding a regularization misfit conditions
$${{\boldsymbol{v}}}^{T}{{\boldsymbol{Q}}}_{D}^{-1}{\boldsymbol{v}}+\lambda \,{{\boldsymbol{m}}}^{T}{{\boldsymbol{Q}}}_{M}^{-1}{\boldsymbol{m}}=\,{\rm{\min }}\,.$$
where model vector m is constituted by the values of pressure × volume, mk = pk∆xk∆yk∆zk, k = 1, …, m, for the m cells of the model, QD is a covariance matrix for the data, QM is a suitable covariance matrix corresponding to the physical configuration and λ is a smoothing factor for selected balance between fitness and smoothness of the model. The inversion approach is a non-linear problem.
The anomalous source is determined as a free aggregation of a large number of small sources with anomalous pressure. We carry out a step-by-step process of growth of the 3D models, using an exploratory technique to find each new cell to be filled with anomalous pressure values and aggregated to the models.
Sentinel-1 data can be downloaded through the Copernicus Open Data Hub. The datasets generated during and analysed during the current study are available from the corresponding author on reasonable request.
Galloway, D., Jones, D. R. & Ingebritsen, S. Land subsidence in the United States. (US Geological Survey Circular, 1999).
Dokka, R. K. Modern-day tectonic subsidence in coastal Louisiana. Geology 34, 281 (2006).
Amelung, F., Galloway, D. L., Bell, J. W., Zebker, H. A. & Laczniak, R. J. Sensing the ups and downs of Las Vegas: InSAR reveals structural control of land subsidence and aquifer-system deformation. Geology 27, 483 (1999).
González, P. J. & Fernández, J. Drought-driven transient aquifer compaction imaged using multitemporal satellite radar interferometry. Geology 39, 551–554 (2011).
Bonì, R. et al. Twenty-year advanced DInSAR analysis of severe land subsidence: The Alto Guadalentín Basin (Spain) case study. Eng. Geol. 198, 40–52 (2015).
Samsonov, S. V., Tiampo, K. F. & Feng, W. Fast subsidence in downtown of Seattle observed with satellite radar. Remote Sens. Appl. Soc. Environ. 4, 179–187 (2016).
Bru, G. et al. A-DInSAR Monitoring of Landslide and Subsidence Activity: A Case of Urban Damage in Arcos de la Frontera, Spain. Remote Sens. 9, 787 (2017).
Gourmelen, N., Amelung, F., Casu, F., Manzo, M. & Lanari, R. Mining-related ground deformation in Crescent Valley, Nevada: Implications for sparse GPS networks. Geophys. Res. Lett. 34 (2007).
Ma, C. et al. Investigation on Mining Subsidence Based on Multi-Temporal InSAR and Time-Series Analysis of the Small Baseline Subset—Case Study of Working Faces 22201-1/2 in Bu'ertai Mine, Shendong Coalfield, China. Remote Sens. 8, 951 (2016).
Fokker, P. A., Wassing, B. B. T., van Leijen, F. J., Hanssen, R. F. & Nieuwland, D. A. Application of an ensemble smoother with multiple data assimilation to the Bergermeer gas field, using PS-InSAR. Geomech. Energy Environ. 5, 16–28 (2016).
Bakr, M. Influence of Groundwater Management on Land Subsidence in Deltas. Water Resour. Manag. 29, 1541–1555 (2015).
Gambolati, G. & Teatini, P. Geomechanics of subsurface water withdrawal and injection. Water Resour. Res. 51, 3922–3955 (2015).
Chen, M. et al. Imaging Land Subsidence Induced by Groundwater Extraction in Beijing (China) Using Satellite Radar Interferometry. Remote Sens. 8, 468 (2016).
de Luna, R. M. R., Garnés, S. JdosA., Cabral, J. JdaS. P. & dos Santos, S. M. Groundwater overexploitation and soil subsidence monitoring on Recife plain (Brazil). Nat. Hazards 86, 1363–1376 (2017).
Motagh, M. et al. Quantifying groundwater exploitation induced subsidence in the Rafsanjan plain, southeastern Iran, using InSAR time-series and in situ measurements. Eng. Geol. 218, 134–151 (2017).
Vasco, D. W., Wicks, C., Karasaki, K. & Marques, O. Geodetic imaging: reservoir monitoring using satellite interferometry. Geophys. J. Int. 149, 555–571 (2002).
Stramondo, S. et al. Surface movements in Bologna (Po Plain — Italy) detected by multitemporal DInSAR. Remote Sens. Environ. 110, 304–316 (2007).
Motagh, M. et al. Land subsidence in Iran caused by widespread water reservoir overexploitation. Geophys. Res. Lett. 35, L16403 (2008).
Gutiérrez-Yurrita, P. J. Land subsidence, associated hazards and the role of natural resources development. In Land Subsidence, Associated Hazards and the Role of Natural Resources Development (Proceedings of EISOLS 2010) 396–401 (2010).
Sahu, P. & Sikdar, P. K. Threat of land subsidence in and around Kolkata City and East Kolkata Wetlands, West Bengal, India. J. Earth Syst. Sci. 120, 435–446 (2011).
Tiampo, K. F. et al. An Elliptical Model for Deformation Due to Groundwater Fluctuations. Pure Appl. Geophys. 169, 1443–1456 (2012).
Tosi, L., Teatini, P. & Strozzi, T. Natural versus anthropogenic subsidence of Venice. Sci. Rep. 3, 2710 (2013).
UNESCO. International Hydrological Program VIII (2014–2020). Available at: https://en.unesco.org/themes/water-security/hydrology (Accessed: 15th January 2018) (2017).
Geertsma, J. Land Subsidence Above Compacting Oil and Gas Reservoirs. J. Pet. Technol. 25, 734–744 (1973).
Geertsma, J. & Van Opstal, G. A numerical technique for predicting subsidence above compacting reservoirs based on the nucleus of strain concept. Verh. Kon. Ned. Geol. Mijnbouwk 28, 63–78 (1975).
Segall, P. Stress and subsidence resulting from subsurface fluid withdrawal in the epicentral region of the 1983 Coalinga Earthquake. J. Geophys. Res. 90, 6801 (1985).
Vasco, D. W., Karasaki, K. & Doughty, C. Using surface deformation to image reservoir dynamics. Geophysics 65, 132–147 (2000).
Walsh, J. B. Subsidence above a planar reservoir. J. Geophys. Res. Solid Earth 107, ETG 6-1–ETG 6-9 (2002).
Brown, N. J., Woods, A. W., Neufeld, J. A. & Richardson, C. Constraining Surface Deformation Predictions Resulting from Coal Seam Gas Extraction, https://doi.org/10.11636/Record.2014.044 (2014).
Camacho, A. G., González, P. J., Fernández, J. & Berrino, G. Simultaneous inversion of surface deformation and gravity changes by means of extended bodies with a free geometry: Application to deforming calderas. J. Geophys. Res. 116, B10401 (2011).
González, P. J., Tiampo, K. F., Palano, M., Cannavó, F. & Fernández, J. The 2011 Lorca earthquake slip distribution controlled by groundwater crustal unloading. Nat. Geosci. 5, 821–825 (2012).
Cannavò, F. et al. Real Time Tracking of Magmatic Intrusions by means of Ground Deformation Modeling during Volcanic Crises. Sci. Rep. 5, 10970 (2015).
Camacho, A. G., Fernández, J. & Cannavò, F. PAF: A software tool to estimate free-geometry extended bodies of anomalous pressure from surface deformation data. Comput. Geosci. 111, 235–243 (2018).
AEMET. Agencia Estatal de Meteorología. Available at: www.aemet.es/es/datos_abiertos/AEMET_OpenData (Accessed: 15th January 2018) (2017).
Bourgois, J., Mauffret, A., Ammar, A. & Demnati, A. Multichannel seismic data imaging of inversion tectonics of the Alboran Ridge (western Mediterranean Sea). Geo-Marine Lett. 12, 117–122 (1992).
Martı́nez-Dı́az, J. J. Stress field variation related to fault interaction in a reverse oblique-slip fault: the Alhama de Murcia fault, Betic Cordillera, Spain. Tectonophysics 356, 291–305 (2002).
Palano, M., González, P. J. & Fernández, J. Strain and stress fields along the Gibraltar Orogenic Arc: Constraints on active geodynamics. Gondwana Res. 23, 1071–1088 (2013).
Cerón, J. C. & Pulido-Bosch, A. Groundwater problems resulting from CO 2 pollution and overexploitation in Alto Guadalentín aquifer (Murcia, Spain). Environ. Geol. 28, 223–228 (1996).
Confederación Hidrográfica del Segura. Plan Especial de Actuación en Situaciones de Alerta y Eventual Sequía. Tech. Rep. 298 (2006).
Tomas, R. et al. Study of the land subsidence in Orihuela City (SE Spain) using PSI data: Distribution, evolution and correlation with conditioning and triggering factors. Eng. Geol. 115, 105–121 (2010).
Herrera, G. et al. Analysis of subsidence using TerraSAR-X data: Murcia case study. Eng. Geol. 116, 284–295 (2010).
Zhu, L. et al. Land subsidence due to groundwater withdrawal in the northern Beijing plain, China. Eng. Geol. 193, 243–255 (2015).
Pacheco-Martínez, J. et al. Application of InSAR and Gravimetry for Land Subsidence Hazard Zoning in Aguascalientes, Mexico. Remote Sens. 7, 17035–17050 (2015).
Solari, L., Ciampalini, A., Raspini, F., Bianchini, S. & Moretti, S. PSInSAR Analysis in the Pisa Urban Area (Italy): A Case Study of Subsidence Related to Stratigraphical Factors and Urbanization. Remote Sens. 8, 120 (2016).
Chaussard, E. et al. Remote Sensing of Ground Deformation for Monitoring Groundwater Management Practices: Application to the Santa Clara Valley During the 2012–2015 California Drought. J. Geophys. Res. Solid Earth 122, 8566–8582 (2017).
Chen, G. et al. Detection of Land Subsidence Associated with Land Creation and Rapid Urbanization in the Chinese Loess Plateau Using Time Series InSAR: A Case Study of Lanzhou New District. Remote Sens. 10, 270 (2018).
Hanssen, R. F. Radar Interferometry: Data Interpretation and Error Analysis, https://doi.org/10.1007/0-306-47633-9 (2001).
Samieie-esfahany, S., Hanssen, R. F., Thienen-visser, K. Van, Muntendam-bos, A. & Systems, S. On the Effect of Horizontal Deformation on InSAR Subsidence Estimates. 2009 (2010).
Mora, O., Ordoqui, P., Iglesias, R. & Blanco, P. Earthquake Rapid Mapping Using Ascending and Descending Sentinel-1 TOPSAR Interferograms. Procedia Comput. Sci. 100, 1135–1140 (2016).
Blanco-Sánchez, P., Mallorquí, J. J., Duque, S. & Monells, D. The Coherent Pixels Technique (CPT): An Advanced DInSAR Technique for Nonlinear Deformation Monitoring. Pure Appl. Geophys. 165, 1167–1193 (2008).
ADS Article MATH Google Scholar
González, P. J. et al. The 2014–2015 eruption of Fogo volcano: Geodetic modeling of Sentinel-1 TOPS interferometry. Geophys. Res. Lett. 42, 9239–9246 (2015).
Rucci, A., Vasco, D. W. & Novali, F. Monitoring the geologic storage of carbon dioxide using multicomponent SAR interferometry. Geophys. J. Int. 193, 197–208 (2013).
Klemm, H., Quseimi, I., Novali, F., Ferretti, A. & Tamburini, A. Monitoring horizontal and vertical surface deformation over a hydrocarbon reservoir by PSInSAR. First Break 28 (2010).
Janna, C., Castelletto, N., Ferronato, M., Gambolati, G. & Teatini, P. A geomechanical transversely isotropic model of the Po River basin using PSInSAR derived horizontal displacement. Int. J. Rock Mech. Min. Sci. 51, 105–118 (2012).
Jónsson, S., Zebker, H., Segall, P. & Amelung, F. Fault Slip Distribution of the 1999 Mw 7.1 Hector Mine, California, Earthquake, Estimated from Satellite Radar and GPS Measurements. Bull. Seismol. Soc. Am. 92, 1377–1389 (2002).
Anderssohn, J. et al. Surface deformation time series and source modeling for a volcanic complex system based on satellite wide swath and image mode interferometry: The Lazufre system, central Andes. Remote Sens. Environ. 113, 2062–2075 (2009).
Jun, H., Wei, L. Z., Jun, Z. J., Chong, R. X. & XiaoLi, D. Inferring three-dimensional surface displacement field by combining SAR interferometric phase and amplitude information of ascending and descending orbits. Sci. China Earth Sci. 53, 550–560 (2010).
Hu, J. et al. Kalman-filterbased approach for multisensor, multitrack, and multitemporal InSAR. IEEE Trans. Geosci. Remote Sens. 51, 4226–4239 (2013).
Shirzaei, M. A seamless multitrack multitemporal InSAR algorithm. Geochem. Geophys. Geosyst. 16, 1656–1669 (2015).
Casu, F. & Manconi, A. Four-dimensional surface evolution of active rifting from spaceborne SAR data. Geosphere 12, 697–705 (2016).
Pepe, A., Solaro, G., Calò, F. & Dema, C. A Minimum Acceleration Approach for the Retrieval of Multi-Platform InSAR Deformation Time-Series. IEEE J. Sel. Appl. Earth Obs. 9, 3883–3898 (2016).
Samsonov, S. & D'Oreye, N. Multidimensional time-series analysis of ground deformation from multiple InSAR data sets applied to Virunga Volcanic Province. Geophys. J. Int. 191, 1095–1108 (2012).
ADS Google Scholar
Ezquerro, P. et al. Groundwater and Subsidence Modeling Combining Geological and Multi-Satellite SAR Data over the Alto Guadalentín Aquifer (SE Spain). Geofluids 2017, 1–17 (2017).
Herring, T. A., King, R. W., Floyd, M. A. & McClusky, S. C. GAMIT Reference Manual. GPS Analysis at MIT GLOBK, Release 10.6. (Massachusetts Institute of Technology, 2015).
Saastamoinen, J. Atmospheric Correction for the Troposphere and Stratosphere in Radio Ranging Satellites, https://doi.org/10.1029/GM015p0247 (2013).
Altamimi, Z., Métivier, L. & Collilieux, X. ITRF2008 plate motion model. J. Geophys. Res. Solid Earth 117 (2012).
Palano, M., González, P. J. & Fernández, J. The Diffuse Plate boundary of Nubia and Iberia in the Western Mediterranean: Crustal deformation evidence for viscous coupling and fragmented lithosphere. Earth Planet. Sci. Lett. 430, 439–447 (2015).
Bruyninx, C. et al. Enhancement of the EUREF Permanent Network Services and Products. In 27–34, https://doi.org/10.1007/978-3-642-20338-1_4 (2012).
Lanari, R. et al. An Overview of the Small BAseline Subset Algorithm: a DInSAR Technique for Surface Deformation Analysis. Pure Appl. Geophys. 164, 637–661 (2007).
Hanssen, R. F. Radar Interferometry, https://doi.org/10.1007/0-306-47633-9 (Springer Netherlands, 2001).
Mora, O., Mallorqui, J. J. & Broquetas, A. Linear and nonlinear terrain deformation maps from a reduced set of interferometric sar images. IEEE Trans. Geosci. Remote Sens 41, 2243–2253 (2003).
Biot, M. A. General Theory of Three‐Dimensional Consolidation. J. Appl. Phys. 12, 155–164 (1941).
Mogi, K. Relations between the eruptions of various volcanoes and the deformations of the ground surfaces around them. B. Earthq. Res. Inst. Univ. Tokyo 36, 99–134 (1958).
IGN. Instituto Geográfico Nacional., Available at: http://www.ign.es/ (2018).
Wessel, P., Smith, W. H. F., Scharroo, R., Luis, J. & Wobbe, F. Generic Mapping Tools: Improved Version Released. Eos, Trans. Am. Geophys. Union 94, 409–410 (2013).
The research has been supported by the Spanish Ministry of Economy and Competitiveness research projects ESP2013-47780-557 C2-1-R and TEC2017-85244-C2-2-P. It is a contribution to the Moncloa Campus of International Excellence. GMT (Wessel, 2013) was used for the production of cartographic charts. This work contains modified Copernicus Sentinel data from years 2015 to 2017, processed by ESA and downloaded through Copernicus Open Data Hub. Digital Elevation Model used in this work was from MDT25 data from Instituto Geográfico Nacional (IGN). We thank the support given by the Comunidad de Regantes of Lorca and the City Council of Lorca. We thank F. Tornos by comments in previous versions of this manuscript. CommSensLab is Unidad de Excelencia Maria de Maeztu MDM-2016-0600.
Instituto de Geociencias (CSIC, UCM), Calle del Doctor Severo Ochoa, no 7, Facultad de Medicina (Edificio Entrepabellones 7 y 8, 4a planta), Ciudad Universitaria, 28040, Madrid, Spain
Jose Fernandez, Joaquin Escayo, Antonio G. Camacho, Tamara Abajo, Guadalupe Bru & Israel Gómez
ETSI Topografía, Geodesia y Cartografía, Universidad Politécnica de Madrid, Ctra. Valencia km 7, 28031, Madrid, Spain
Juan F. Prieto, Jesus Velasco & Iñigo Molina
Dpto de Química y Física, Universidad de Almería, Edificio CITE-IIA, Cañada de San Urbano s/n, 04120, Almería, Spain
Francisco Luzón
Cooperative Institute for Research in Environmental Sciences (CIRES), 216UCB, University of Colorado at Boulder, Boulder, CO, 80309, USA
Kristy F. Tiampo
Istituto Nazionale di Geofisica e Vulcanologia, Osservatorio Etneo, 95125, Catania, Italy
Mimmo Palano
Dpto. Ingeniería Agroforestal, ETSI Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, Avda. Puerta de Hierro, no 2 – 4, 28040, Madrid, Spain
Enrique Pérez, Tomas Herrero & Juan López
Dpto. Física de la Tierra y Astrofísica, Unidad Departamental Astronomía y Geodesia, Universidad Complutense de Madrid, Fac. C. Matemáticas, Plaza de Ciencias, 3, 28040, Madrid, Spain
Gema Rodríguez-Velasco
CommSensLab, Dep. Signal Theory and Communications, Universitat Politècnica de Catalunya (UPC), D3-Campus Nord-UPC, C. Jordi Girona 1-3, 08034, Barcelona, Spain
Jordi J. Mallorquí
Juan F. Prieto
Joaquin Escayo
Antonio G. Camacho
Tamara Abajo
Jesus Velasco
Tomas Herrero
Guadalupe Bru
Iñigo Molina
Israel Gómez
J.F. conceived the study, coordinated the preparation of the manuscript and wrote it with the collaboration of all the coauthors. J.F.P. and J.F. designed the GNSS network and observation methodology. M.P. and J.F.P. carried out the GNSS data processing. J.F.P. coordinates the GNSS surveys. J.E., J.F. and J.J.M. designed the Sentinel-1 data download from ESA and their processing. J.E. carried out, with the support from J.J.M., the A-DInSAR processing. A.G.C. with F.L. and help from J.F. and K.F.T. developed the direct model and inversion software. A.G.C. carried out the inversion of the different data sets used in this study. J.F., J.F.P., J.E., A.G.C., T.A., E.P., J.V., T.H., G.B., I.M., J.C.L., G.R.-V. and I.G. carried out the GNSS observation. J.J.M. provided the A-DInSAR processing software used.
Correspondence to Jose Fernandez.
Fernandez, J., Prieto, J.F., Escayo, J. et al. Modeling the two- and three-dimensional displacement field in Lorca, Spain, subsidence and the global implications. Sci Rep 8, 14782 (2018). https://doi.org/10.1038/s41598-018-33128-0
Three-dimensional Displacement Field
GNSS Network
Interferometric Synthetic Aperture Radar (InSAR)
GNSS Stations
Modeling historical subsidence due to groundwater withdrawal in the Alto Guadalentín aquifer-system (Spain)
J.A. Fernández-Merodo
, P. Ezquerro
, D. Manzanal
, M. Bejar-Pizarro
, R.M. Mateos
, C. Guardiola-Albert
, J.C. García-Davalillo
, J. López-Vinielles
, R. Sarro
, G. Bru
, J. Mulas
, R. Aragon
, C. Reyes-Carmona
, P. Mira
, M. Pastor
& G. Herrera
Engineering Geology (2021)
Quantifying Ground Subsidence Associated with Aquifer Overexploitation Using Space-Borne Radar Interferometry in Kabul, Afghanistan
Gauhar Meldebekova
, Chen Yu
, Zhenhong Li
& Chuang Song
Remote Sensing (2020)
3D multi-source model of elastic volcanic ground deformation
, José Fernández
, Sergey V. Samsonov
, Kristy F. Tiampo
& Mimmo Palano
Earth and Planetary Science Letters (2020)
Geodetic Study of the 2006–2010 Ground Deformation in La Palma (Canary Islands): Observational Results
Joaquín Escayo
, Juan F. Prieto
, Antonio G. Camacho
, Mimmo Palano
, Alfredo Aparicio
, Gema Rodríguez-Velasco
& Eumenio Ancochea
Extreme subsidence in a populated city (Mashhad) detected by PSInSAR considering groundwater withdrawal and geotechnical properties
Mohammad Khorrami
, Saeed Abrishami
, Yasser Maghsoudi
, Babak Alizadeh
& Daniele Perissin
Scientific Reports (2020) | CommonCrawl |
\begin{document}
\selectlanguage{english} \begin{abstract} In this article, for degree $d\geq 1$, we construct an embedding $\Phi_d $ of the connectedness locus $\mathcal{M}_{d+1}$ of the polynomials $z^{d+1}+c$ into the connectedness locus of degree $2d+1$ bicritical odd polynomials. \end{abstract} \maketitle
\section{Introduction} Relationships between different families of rational maps have been studied in various contexts in complex dynamics. In rational dynamics, quadratic polynomials of the form $z^2+c$ are the fundamental objects of study, and much of the field involves the study of the Mandelbrot set pioneered by Douady and Hubbard, in \cite{Hubbarditer}, \cite{MR728980},\cite{MR812271} and \cite{MR1215974}, and developed by Milnor (\cite{MR1755445}), Lyubich and Dudko (\cite{https://doi.org/10.48550/arxiv.1808.10425}) and several others. In general, polynomials with a single critical point, normalized as $z^d+c$, and their connectedness loci $\mathcal{M}_d$ - that is, the set of parameters $c$ for which the filled Julia set is connected, commonly referred to as the Multibrot sets, have also been studied in \cite{Schleicher98onfibers}, \cite{MR3444240}, etc., and the properties of these sets have been used to conjecture and prove several results in both rational and transcendental dynamics.
For example, the theory of matings and the work of Tan Lei, Rees, Shishikura and others (see \cite{Rees1990}, \cite{Shishikura2000OnAT}, \cite{lei_1992}) created a link between polynomials and rational maps, by combining two polynomials to create a rational map. This made it easier to study certain hyperbolic components in rational parameter spaces, as well as the structure of Julia sets of rational maps that arise as matings.
\begin{figure}
\caption{$\mathcal{CBO}_1$}
\label{fig:odd1}
\caption{$\mathcal{CBO}_2$}
\label{fig:odd2_1}
\caption{$\mathcal{CBO}_3$}
\label{fig:odd3}
\caption{$\mathcal{CBO}_4$}
\label{fig:odd4}
\caption{$\mathcal{CBO}_5$}
\label{fig:odd5}
\caption{The families $\mathcal{CBO}_d$ for $d=1,2,3,4,5$}
\label{fig:odds}
\end{figure}
Renormalization is a powerful tool in dynamics. It restricts certain holomorphic functions to smaller domains in which they ``look'' like some $z^d+c$ to make them easier to study. In his study of renormalizable maps in \cite{McMullen19972TM}, McMullen proved that unicriticals are universal in the sense that there are small copies of the Multibrot sets found in any holomorphic family of maps.
Branner and Douady constructed a continuous map from the basilica limb of the Mandelbrot set $\mathcal{M}_2$ to the rabbit limb (see \cite{10.1007/BFb0081395}). This was later extended by Branner and Fagella in \cite{MR1757453} into homeomorphisms between various limbs of the Mandelbrot set, and in a different spirit, by Dudko and Schleicher in \cite{MR2888182}. In \cite{riedl}, Riedl and Schleicher also construct a homeomorphism from a subset of any $\frac{p}{nq}$-limb of the Mandelbrot set to the $\frac{p}{q}$-limb.
We establish another such relationship between two holomorphic families - unicritical polynomials and the family of symmetric polynomials which we introduce below and describe in detail in Section~\ref{section:prelim}. We take symmetric polynomials to mean polynomials that commute with some affine map $M$ satisfying $M^{\circ 2} = \text{Id}$. Symmetric cubic polynomials are encountered, for example, in the study of core entropy (see \cite{gao2019core}). We focus on the more specific family of symmetric polynomials with exactly two critical points. As we will show in Section~\ref{section:prelim}, such a polynomial is affine conjugate to
$$p_{a,d}(z) = a\int_0^z\Big(1-\frac{w^2}{d}\Big)^ddw$$ for some $a \in \mathbb{C}^*$ and $d\geq 2$. For each $a$,
$p_{a,d}$ is an odd polynomial- that is, it commutes with $z \mapsto -z$. For a fixed $d$, we let $p_a = p_{a,d}$. We let $\mathcal{CBO}_d$ denote the set of $a \in \mathbb{C}^*$ such that $p_a$ has connected filled Julia set. This is a closed, compact connected subset of $\mathbb{C}$. In this article we shall prove the following.
\begin{thm}\label{thm:maintheorem}
For $d\geq 1$, there exists a continuous map $\Phi_d: \mathcal{M}_{d+1} \longrightarrow \mathcal{CBO}_d$ that is a homeomorphism onto its image.
\end{thm}Our proof is along the lines of Douady and Branner's use of quasiconformal surgery in \cite{10.1007/BFb0081395}. We shall perform a quasiconformal surgery along a $\beta-$ fixed point and its pre-images. The map $\Phi_d$ is natural in the sense that its inverse can be described by a renormalization operator on a subset of $\mathcal{CBO}_d$. We shall also give a complete description of the image under $\Phi_d$ (see Section~\ref{section:imgdefn}).
Although we do not provide details here, our construction holds in the following generality:
\begin{thm}
For any integer $k\geq 2$, there exists a continuous map from $\mathcal{M}_{d+1}$ to the collection of $a \in \mathbb{C}^*$ such that $a {\displaystyle \int_{0}^{z} } \Big(1-\frac{w^k}{d}\Big)^ddw$ has connected Julia set, that is a homeomorphism onto its image.
\end{thm}The family $p_{a,d}$ is interesting in its own right: as $d\longrightarrow \infty$, $p_{a,d}(z) \longrightarrow a{\displaystyle \int_{0}^{z}} e^{-w^2}dw$ locally uniformly on $\mathbb{C}$. The limit function is entire, odd, has two asymptotic values $\pm \frac{a\sqrt{\pi}}{2}$ and no critical points. It is called an ``error'' function (see \cite{nevanlinna1970analytic} for an introduction). Error functions belong to the larger Speiser class- the family of entire functions with finitely many critical and asymptotic values. This family is studied in \cite{goldberg_keen_1986}, \cite{AIF_1992__42_4_989_0} and several others.
The simplest of the Speiser class is the family of exponential functions. A lot of the analysis of exponential functions is a direct application of the tools used in the analysis of unicritical polynomials, normalized as $\lambda(1+\frac{z}{d})^d$ and using the fact that they converge to $\lambda \exp{z}$ as $d \longrightarrow \infty$. This is a theme that is explored in \cite{MR1785056}. Our work in progress aims to carry out a similar analysis for the error functions $a{\displaystyle \int_0^z}e^{-w^2}dw$. This paper presents a structural similarity between the polynomials approximating exponential functions, and the polynomials $p_{a,d}$ that approximate error functions, and prompts us to make the following conjecture:
\begin{conj}
Let $E_c(z) = \exp z+c$, and $\mathscr{E}_a(z) = a {\displaystyle \int_{0}^{z}} e^{-w^2}dw$.
There exists a continuous map from $\big\{c \in \mathbb{C}| \{E_c^{\circ n}(c)\}_{n \geq 0} \text{ is bounded}\big\}$ to the set $\big\{a \in \mathbb{C}^*| \{\mathscr{E}^{\circ n}_a(a)\}_{n\geq 0} \text{ is bounded}\big\}$ that is a homeomorphism onto its image.
\end{conj}
There is some evidence to show that this is reasonable; work in progress indicates that it may be possible to embed postsingularly finite exponential functions into the collection of postsingularly finite error functions in a combinatorially meaningful manner. We do not, however, address error functions in this article.
The paper is organized as follows. In Section~\ref{section:prelim}, we introduce symmetric polynomials and establish some of their basic properties, provide motivation for Theorem~\ref{thm:maintheorem} while laying out our proof strategy, and describe the image of $\Phi_d$. In Sections~\ref{section:defn} and ~\ref{section:continuity} respectively, we define $\Phi_d$ and prove that it is continuous. We end in Section~\ref{section:injectivity} by constructing a continuous inverse for $\Phi_d$ on its image.
\subsection*{Acknowledgements.}The author is indebted to John Hubbard and Sarah Koch for their continued guidance throughout this study, and to Dierk Schleicher for helping widen the horizons of the author's research perspective. Special thanks to Jack Burkart, Alex Kapiamba, Leticia Pardo-Sim\'{o}n and Vasiliki Evdoridou for helpful conversations and resources, and to Lukas Geyer, N\'{u}ria Fagella, Lasse Rempe, Laurent Bartholdi, Joanna Furno, Giulio Tiozzo, Eriko Hironaka, David Mart\'{i}-Pete, Mikhail Hlushchanka, Nikolai Prochorov and others for their time and insightful comments. The author also thanks Daniel Stoll for an introduction to the Mandel software package.
This material is based upon work supported by the National Science Foundation under Grant No.\ DMS-1928930 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2022 semester. \begin{figure}
\caption{$\mathcal{M}_3$}
\label{fig:m3}
\caption{$\mathcal{CBO}_2$}
\label{fig:odd2_2}
\caption{A portion of $\mathcal{CBO}_2$. The cut point on the mid-left is $a=1$. Note the resemblance to $\mathcal{M}_3$}
\label{fig:odd2_right}
\caption{The figure on the top left is the unicritical locus $\mathcal{M}_3$. The figure on the top right is the locus $\mathcal{CBO}_2$ of odd bicritical polynomials of degree $5$. The figure at the bottom zooms in on the right of the figure on the top right- we will show that this region contains a copy of $\mathcal{M}_3$ }
\label{fig:fig2}
\end{figure} \section{Preliminaries}\label{section:prelim} For an introduction to polynomial dynamics and the Multibrot sets, see \cite{MR1755445}, \cite{MR2193309}, \cite{MR3675959} and \cite{MR3444240}. \subsection{Introduction} We call a polynomial $f$ of degree $>1$ symmetric if it commutes with an affine map $M$ that satisfies $M^{\circ 2}(z) = z$. $M$ is of the form $M(z) = -z+b$ for some $b \in \mathbb{C}$, and we may conjugate $M$ by a translation $\tau$ so that $\tau^{-1}\circ M \circ \tau(z) = -z$. We have \begin{align*}
\tau^{-1}\circ f \circ \tau(z) & = \tau^{-1}\circ (M \circ f \circ M^{-1} )\circ \tau(z) \\& = (\tau^{-1}\circ M \circ \tau) \circ (\tau^{-1} \circ f \circ \tau) \circ (\tau^{-1} \circ M^{-1}\circ \tau)(z) \\& = -(\tau^{-1} \circ f \circ \tau)(-z) \end{align*} That is, $\tau^{-1} \circ f \circ \tau$ is an odd polynomial. Therefore, every symmetric polynomial contains an odd polynomial in its affine conjugacy class. We recall that an odd polynomial has only odd degree terms. \subsection{Bicritical odd polynomials} $f$ is bicritical if it has, upto multiplicity, exactly two critical points on the plane. Let $f$ be a bicritical odd polynomial of degree $2d+1$, with $d\geq 1$. The Riemann Hurwitz formula shows that $f$ has local degree $d+1$ at both critical points. Furthermore, the critical points are of the form $\pm x$ for some $x \in \mathbb{C}^*$. Let $\phi(z) = kz$ be such that $\phi(\{x,-x\}) = \{\sqrt{d},-\sqrt{d}\}$. Then there exists a constant $a \in \mathbb{C}^*$ such that \begin{align*} (\phi \circ f \circ \phi^{-1})'(z) & = a\Big(1-\frac{z^2}{d}\Big)^d \end{align*} Therefore, \begin{align*}
\phi \circ f \circ \phi^{-1}(z) & = a\int_0^z\Big(1-\frac{w^2}{d}\Big)^ddw \end{align*} \noindent For $a \in \mathbb{C}^*$, let \begin{align*}
p_a(z) & = a\int_0^z \Big(1-\frac{w^2}{d}\Big)^ddw \end{align*}
It is evident that $p_{a}$ is affine conjugate to $p_{a'}$ if and only if $a = a'$. Therefore, the space of bicritical odd polynomials modulo conjugation by scaling (or, the space of symmetric bicritical polynomials modulo affine conjugation) is the family $a \mapsto p_a$ over $\mathbb{C}^*$. We shall denote this family $\mathcal{BO}_d$, and let \begin{align*}
\mathcal{CBO}_d = \{a: K_{p_a} \text{ is connected}\} \end{align*} Figure~\ref{fig:odds} illustrates $\mathcal{CBO}_d$ for $d=1,2,3,4,5,19$.
We consider the part of $\mathcal{CBO}_2$ illustrated in Figure~\ref{fig:odd2_right}, and present some of the Hubbard trees of postcritically finite polynomials in this region, in Figure~\ref{fig:trees}. The pictures indicate a relationship between $\mathcal{M}_{3}$ and $\mathcal{CBO}_2$, and in general, between $\mathcal{M}_{d+1}$ and $\mathcal{CBO}_d$. \begin{figure}
\caption{A section of $\mathcal{CBO}_1$}
\label{fig:odd1_right}
\caption{The $(+,-)$ type $z\mapsto z^2$}
\label{fig:maincardioid}
\caption{The $(+,-)$ type basilica}
\label{fig:basilica}
\caption{Postcritically finite polynomials along with their Hubbard trees in the family $p_a(z)=a{\displaystyle \int_0^z}(1-w^2)dw$. $x_0^- = -1$, $x_0^+=1$ are the two critical points, with $x_i^{\pm} = p_a^{\circ i}(x_0^\pm)$. Terminology:`$(+,-)$ type polynomial $p$' refers to the polynomial in $\mathcal{CBO}_1$ that looks like a pair of copies of the polynomial $p = f_c$, where $c \in \mathcal{M}_2$}
\label{fig:trees}
\end{figure} \begin{figure}
\caption{The $(+,-)$ type airplane}
\label{fig:airplane}
\caption{The $(+,-)$ type rabbit}
\label{fig:rabbit}
\caption{The $(+,-)$ type $z\mapsto z^2 -0.10003 + 0.95227i $}
\label{fig:trip_point}
\caption{More examples of Hubbard trees in $\mathcal{CBO}_1$}
\label{fig:trees2}
\end{figure} \subsection{Monic representatives of polynomials in $\mathcal{BO}_d$}\label{section:imgdefn} Any polynomial $p_a$ for $a \in \mathcal{BO}_d$ has leading coefficient $T(a):=\frac{(-1)^da}{d^d(2d+1)}$ attached to $z^{2d+1}$. Let $w(z) = \frac{z}{s}$. We note that $P_s(z)=w^{-1} \circ p_a \circ w$ is monic if and only if $s^{2d} = T(a)$.
The polynomial $P_s$ admits a unique B\"{o}ttcher chart $\varphi_s$ in a neighborhood of $\infty$ that satisfies $\lim_{z \rightarrow \infty}\frac{\varphi_s(z)}{z} = 1$, and if $s_1^{2d} = s_2^{2d}$, then $\varphi_{s_1}(z) = \omega \varphi_{s_2}(z)$ where $\omega = \frac{s_2}{s_1}$ is a $2d-$th root of unity. Let $\mathcal{R}_\theta(s)$ denote the ray at angle $\theta$ in the dynamical plane of $P_s$. Then it is easy to see that if $\frac{s_2}{s_1} = e^{\frac{2\pi ij}{2d}}$ for some integer $j$, then for all $\theta \in \mathbb{R}/\mathbb{Z}$, \begin{align*}
\mathcal{R}_\theta(s_2) & = \mathcal{R}_{\theta+\frac{j}{2d}}(s_1) \end{align*} Additionally, since $P_s$ is odd, $\varphi_s(z) = \lim _{n \rightarrow \infty} P_s^{\circ n}(z) ^{\frac{1}{(2d+1)^n}}$ satisfies \begin{align*}
\varphi_s(-z) & = -\varphi_s(z)\\
\implies \mathcal{R}_{\theta+\frac{1}{2}}(s) & = -\mathcal{R}_\theta(s) \end{align*}
For any $s$ such that $0 \in J_{P_s}$, there exists a subset $\Theta$ of $\big\{0,1,...,\frac{2d-1}{2d}\big\}$ satisfying $\Theta+\frac{1}{2}=\Theta$ such that the dynamical rays landing at $0$ are exactly those with angles in $\Theta$. Moreover, if $s'=e^{\frac{2 \pi i j}{2d}} s$, then the set of angles that land at $0$ in the dynamical plane of $P_{s'}$ is $\frac{j}{2d}+\Theta$.
This shows that for any $a \in \mathcal{CBO}_d$ such that $0 \in J_{p_a}$, there exists a monic representative $P_s$ of $p_a$ so that $0$ is the landing point of $\mathcal{R}_0(s)$ and $\mathcal{R}_\frac{1}{2}(s)$. The union of the rays $\mathcal{R}_0(s)$ and $\mathcal{R}_\frac{1}{2}(s)$ separates the plane into two connected components $F^L_s$ and $F^R_{s}$, named so that $F^L_{s}$ is the component that contains the critical point $-s\sqrt{d}$, and $F^R_s$ contains $s\sqrt{d}$ ($L$ and $R$ stand for left and right). \subsubsection{The Image under $\Phi_d$} The set of $a \in \mathcal{CBO}_d$ such that $0 \in J_{p_a}$ is exactly $\mathcal{CBO}_d \setminus \mathbb{D}$. Let $H$ be the component of $\mathcal{CBO}_d \setminus \mathbb{D}$ that intersects the right half plane. By the previous paragraph, on $H$, there exists a branch of $a \mapsto (T(a))^\frac{1}{2d}$, which we shall denote $s(a)$, so that $0$ is the landing point of $\mathcal{R}_0(s(a))$ and $\mathcal{R}_\frac{1}{2}(s(a))$.
We note that the critical points of $P_{s}$ are $\pm s \sqrt{d}$. Let $\mathscr{O}^R_s$ and $\mathscr{O}^L_s$ be the orbits under $P_s$ of $s\sqrt{d}$ and $-s\sqrt{d}$ respectively. \begin{align*}
\Phi_d(\mathcal{M}_{d+1})& = \big\{a\big| \mathscr{O}^L_{s(a)} \subset F^L_{s(a)} \cup \{0\} \text{ and }\mathscr{O}_{s(a)}^R \subset F^R_{s(a)} \cup \{0\}\big \} \end{align*} That is, the image is the set of polynomials where the dynamical rays at angles $0,\frac{1}{2}$ separate the orbits of the two distinct critical points. It is easy to see that the latter is a closed set in $\mathcal{CBO}_d$. We shall henceforth denote this image as $\mathcal{CBO}^{(+,-)}_d$. This is also a proper subset of $\mathcal{CBO}_d$: for example, $a=-1$ is not in this set. We have described in detail the dynamics of polynomials in $\mathcal{CBO}_d^{(+,-)}$ in Section~\ref{section:imgdynamics}.
Figure~\ref{fig:odd1_right} illustrates the way $\Phi_d$ maps $\mathcal{M}_{2}$ by pointing out the position of the images of well-known polynomials like the rabbit, co-rabbit, airplane, etc. \subsection{Quotienting by $z^2$} Given $p_a \in \mathcal{BO}_d$, there exists a unique polynomial $\mathcal{P}_a:\hat{\mathbb{C}} \longrightarrow \hat{\mathbb{C}}$ so that the following diagram commutes.\\ \[ \begin{tikzcd} \hat{\mathbb{C}} \arrow[d,"z\mapsto z^2"] \arrow[r,"p_a"] & \hat{\mathbb{C}} \arrow[d,"z\mapsto z^2"]\\ \hat{\mathbb{C}} \arrow[r,"\mathcal{P}_a"] & \hat{\mathbb{C}} \end{tikzcd} \] The critical points of $\mathcal{P}_a$ are $d$ and $\big\{x_\ell^2\big\}_{\ell=1}^d$, where $\pm x_\ell$, $\ell = 1,2,...,d$, are the pre-images of $0$ that are not equal to $0$.
When $d=1$, the family $\mathcal{P}_a$ corresponds to the collection of cubic polynomials where one critical point is a pre-image of a $\beta-$fixed point (that is, the landing point of a dynamical ray at angle $0$ or $\frac{1}{2}$) and the other is free. This is isomorphic to the collection $\mathcal{F} = \big\{(a,b) | Q_{a,b}(a) = -2a\big\}$, where $Q_{a,b}(z) = z^3 - 3a^2z+b$ discussed in \cite[Chapters~I,II]{10.1007/BFb0081395} in the following way: letting $\widetilde{a} = 9a^2$, we have \begin{align*}
\mathcal{P}_{-a} &=\mathcal{P}_a \simeq Q_{\widetilde{a},2\widetilde{a}^3-2\widetilde{a}} \end{align*} where $\simeq$ refers to affine conjugacy.
Let $F_+ \subset \mathcal{F}$ be the collection of polynomials $Q_{a,b}$ for which the critical point $-a$ maps to the landing point of the dynamical ray at angle $0$, and the other critical point $a$ is in the filled Julia set. Douady and Branner show that there exists a homeomorphism $\Phi_B$ from the basilica limb of the Mandelbrot set to $F_+$. The relationship between $\mathcal{CBO}_1^{(+,-)}$ and $F_+$ is as follows: \begin{align*}
\big\{\big(\widetilde{a},2{\widetilde{a}}^3 - 2\widetilde{a}\big) \big|a \in \mathcal{CBO}_1^{(+,-)}, \widetilde{a} = 9a^2\big\} \subsetneq F_+ \end{align*} For $d=1$, the map $\Phi_d$ we construct in this paper exhibits different behaviour from the $\Phi_B$ that the authors construct in \cite[Chapter~II]{10.1007/BFb0081395}. Firstly, it is defined on the whole of the Mandelbrot set, and not just the basilica limb. Secondly, generally, given $c$ in the basilica limb, if $Q_{\widetilde{a},2\widetilde{a}^3 - 2\widetilde{a}}$ is the polynomial corresponding to $\Phi_B(c)$, $\widetilde{a}$ does not equal $9\Phi_d(c)^2$. Thirdly, it is evident that our map does not change the combinatorics of critical portraits, whereas $\Phi_B$ does. \subsection{Properties of $\mathcal{CBO}_d$}Let $\mathcal{MBO}_d$ denote the set of $s$ such that $P_s$ has connected filled Julia set. Using the methods used by Douady and Hubbard in their proof that the Mandelbrot set is connected (see, for example, \cite[Chapter~10]{MR3675959}), we find that the map \begin{align*}
\hat{\mathbb{C}} \setminus \mathcal{MBO}_d & \longrightarrow \hat{\mathbb{C}} \setminus \overline{\mathbb{D}}\\
c &\mapsto \varphi_c(P_s(-s\sqrt{d}))\\
\infty &\mapsto \infty \end{align*} is a proper map of degree $2d$ ramified at $\infty$. This implies $\mathcal{MBO}_d$, and consequently $\mathcal{CBO}_d$, is connected and compact. \subsection{Proof Strategy and Tools} We will use all the theorems listed in this section. Their statements are borrowed from \cite{Douady1985}. \begin{thm}(The Measurable Riemann Mapping Theorem) Let $\mu_0$ be the standard complex structure on $\mathbb{C}$. If $\mu$ is a complex structure on a simply connected domain $U \subset \mathbb{C}$ that has bounded dilitation with respect to $\mu_0$, then there exists a quasiconformal map $f:U \longrightarrow V\subset \hat{\mathbb{C}}$ satisfying \begin{align*} f^*\mu_0 = \mu \end{align*} unique up to post composition by a M\"{o}bius transformation. \begin{enumerate}
\item Let $\mu_n $ be a sequence of Beltrami forms on a bounded domain $U \subset \mathbb{C}$ such that $||\mu_n||_\infty \leq m<1$ and $\mu_n \longrightarrow \mu$ in the $L^1$ norm, where $\mu$ is a Beltrami form on $U$ with $||\mu||_\infty \leq m$. There exists a sequence of integrating maps $\phi_n$ for $\mu_n$ and an integrating map $\phi$ for $\mu$ such that $\phi_n\longrightarrow \phi$ uniformly on $U$.
\item
Let $\Lambda$ be an open set in $\mathbb{C}^{n}$ and $(\mu_\lambda)_{\lambda \in \Lambda}$ be a family of Beltrami forms on $U$. Suppose $\lambda \mapsto \mu_\lambda(z)$ is holomorphic for almost every $z \in U$ , and that there exists a constant $m<1$ such that $||\mu_\lambda||_\infty<m$ for each $\lambda$. For each $\lambda$, extend $\mu_\lambda$ to $ \mathbb{C}$ by $\mu_\lambda=0$ on $\mathbb{C}\setminus U$, and let $f_\lambda: \mathbb{C}\ \longrightarrow \mathbb{C}$ be the unique quasi-conformal homeomorphism such that $f_\lambda^*\mu_0 = \mu_\lambda$, and $\frac{f_\lambda(z)}{z}\longrightarrow 1$ when $|z| \longrightarrow \infty$. Then $(\lambda,z) \mapsto (\lambda,f_\lambda(z))$ is a homeomorphism of $\Lambda \times \mathbb{C}$ onto itself, and for each $z \in \mathbb{C}$ the map $\lambda \mapsto f_\lambda(z)$ is holomorphic. \end{enumerate} \end{thm} \begin{defn}[Polynomial-like maps] Given Jordan domains $U,V \subset \mathbb{C}$ with $\overline{U} \subset V$, a polynomial-like map $f: U\longrightarrow V$ is an analytic proper map of finite degree $d$.
The filled Julia set of $f$ is the set \begin{align*}
K_f & = \bigcap _{n\geq 0}f^{\circ n}(U) \end{align*}
\end{defn}
Given a polynomial $p:\hat{\mathbb{C}} \longrightarrow \hat{\mathbb{C}}$ of degree $d$, we can always find suitable domains $U,V$ such that $\overline{U} \subset V$ and $p\big |_U: U\longrightarrow V$ is polynomial-like of degree $d$. \begin{defn}[Hybrid Equivalence] Given two polynomial like maps $f: U \longrightarrow V$ and $g: U' \longrightarrow V'$, we say that $f,g$ are hybrid equivalent if there exists a quasiconformal homeomorphism $\psi: (V,U) \longrightarrow (V',U')$ satisfying $g\circ \psi = \psi \circ f $, with zero dilitation on $K_f$. \end{defn} The following theorem is due to Douady and Hubbard, and we shall be using it several times. \begin{thm}[The Straightening Theorem for polynomial-like maps] Every polynomial-like map of degree $d$ is hybrid equivalent to a polynomial of degree $d$. \end{thm}
Our strategy for constructing $\Phi_d$ follows the general layout in \cite{10.1007/BFb0081395}. Given $c \in \mathcal{M}_d$, we will perform a topological surgery in the dynamical plane using the dynamics around one of the $\beta-$ fixed points. At the end of this surgery, we will construct a quasiregular map $g_c$ from a simply-connected Riemann surface $X_1$ to a simply connected Riemann surface $X$ with $\overline{X}_1 \subset X$.
Next, we will show that $g_c$ has an invariant complex structure, and is therefore equivalent to a polynomial-like map of degree $2d+1$. We will finally show that this map is hybrid equivalent to an odd polynomial $p_a$ with $a \in \mathcal{CBO}_d$. The strategy for constructing an inverse for $\Phi_d$ is similar.
All the figures in this paper are illustrations of the case $d=2$. \begin{figure}
\caption{The dynamical plane of $z^3+c$}
\label{fig:dynplaneinit}
\end{figure}
\begin{figure}
\caption{Cutting the dynamical plane of $z^3+c$.}
\label{fig:my_label}
\end{figure} \section{Construction of $\Phi_d$}\label{section:defn} As mentioned before, we proceed along the lines of holomorphic surgery as outlined in \cite{10.1007/BFb0081395}. Let $f_c(z) = z^{d+1}+c$, and $\varphi_c$ be a B\"{o}ttcher chart at $\infty$ that satisfies $\lim_{z \rightarrow \infty} \frac{\varphi_c(z)}{z} = 1$. In the absence of the latter condition, $\varphi_c$ is unique only upto multiplication by a $d$th root of unity. By including the condition, we fix a choice of $\varphi_c$ for every $c$ that makes it continuous in the following sense: given $c \in \mathbb{C}$ and $z \in \mathbb{C}$ such that $\varphi_{\tilde{c}}(z)$ is well-defined for $\tilde{c}$ in a neighborhood of $c$, $\tilde{c} \mapsto \varphi_c(z)$ is continuous in $\tilde{c}$.
Now fix $c \in \mathcal{M}_{d+1}$. The dynamical ray $\mathcal{R}_0$ at angle $0$ lands at a fixed point $\beta$ on the dynamical plane of $c$. For a fixed $r>0$, choose $q,\eta$ such that $q\eta < r$. We will explain how to choose $r$ in the following passages. Let $G_c$ denote the Green's escape rate function.\\ Let \begin{align*}
W_0 & = \{z: G_c(z) < \eta\} \end{align*} and $W_i = f_c^{-1}(W_0)$. We note that \begin{align*}
W_i & = \Biggl\{z: G_c(z) < \frac{\eta}{(d+1)^i}\Biggl\} \end{align*} Also define \begin{align*}
\widetilde{S}(0) & = \{\varphi_c^{-1}(e^{s+2\pi i t}): s \in (0,\eta), |t| < qs\} \end{align*} We call $\widetilde{S}(0)$ a ``sector'' based at $\beta$. It is invariant under $f_c$, and its inverse image under $f_c$ is a union of similar sectors, each based at a pre-image of $\beta$. More precisely, for $\ell \in \{1,2,...,d\}$, let \begin{align*}
\widetilde{S}\Big(\frac{\ell}{d+1}\Big) & = \Big\{\varphi_c^{-1}(e^{s+2\pi i t}): s \in (0,\eta), \Big|t-\frac{\ell}{d+1}\Big| < qs\Big\} \subset W_0\\ \end{align*} Then \begin{align*}
f_c^{-1}(\widetilde{S}(0)) & = \bigcup_{\ell=0}^d \Big(W_1 \cap \widetilde{S}\Big(\frac{\ell}{d+1}\Big) \Big) \end{align*} \begin{figure}
\caption{The Riemann Surface $\widetilde{X}$ }
\label{fig:dynplanefinal}
\end{figure} When imposing the condition $q\eta < r$, we choose $r$ small enough so that the sectors $\widetilde{S}(\frac{\ell}{d+1})$, $\ell=0,1,...,d$, are pairwise disjoint (see Figure~\ref{fig:dynplaneinit}).
Additonally, form open subsets $\widetilde{S_i}(0) = W_i \cap \widetilde{S}(0)$ of $\widetilde{S}(0)$. All points in $\widetilde{S}_i(0)$ have escape rates in the interval $\Big(0,\frac{\eta}{(d+1)^{i}}\Big)$, and $f_c$ maps $\widetilde{S}_i(0)$ conformally onto $\widetilde{S}_{i-1}(0)$. By definition, there is a branch of $\log$ that satisfies $$\log \circ \varphi_c(\widetilde{S}(0)) = \{z: Re(z) >0 \text{ and } |Im(z)| < 2\pi q Re(z) \}$$ \subsection{Steps in the definition of $\Phi_d(c)$}\label{sec:defnsteps} As in \cite[Chapter~II]{10.1007/BFb0081395}, we shall follow this sequence of steps: \begin{enumerate}
\item First, we cut along $\mathcal{R}_0$ and glue together two copies of $W_0$, one rotated by $180^\circ$, to get a quotient Riemann surface $\widetilde{X}$
\item We then construct $f$ on an open subset of $\widetilde{X}$ that is
\begin{itemize}
\item analytic and acts like $f_c$ away from the sectors $\widetilde{S}\big(\frac{\ell}{d+1}\big)$, $\ell \in \{0,1,2,...,d\}$ on both copies of $W_0$
\item has lines of discontinuities at the two copies of $\mathcal{R}_{\frac{\ell}{d+1}}$ for $\ell \in \{1,2,...,d\}$
\end{itemize}
\item We show that by changing $f$ in sectors around these rays, and by modifying the boundary of these sectors, we may construct a quasiregular map $g :X_1 \longrightarrow X$ between simply connected Riemann surfaces with $\overline{X_1} \subset X$, and an almost complex structure $\sigma$ on $X$ that is $g$ invariant. Under the measurable Riemann mapping theorem, there exists a quasi-conformal map $\psi$ such that $\psi \circ g \circ \psi^{-1}$ is analytic.
\item Finally, we will apply the straightening theorem to obtain a unique polynomial $p_{a}$ hybrid equivalent to $\psi \circ g \circ \psi^{-1}$.\\ \end{enumerate}
We will now implement these steps one by one.\\ \subsubsection{Cutting along $\mathcal{R}_0$} Let us cut along $\mathcal{R}_0$. In this slit disk, $\widetilde{S}(0)$ is now split into two components $\widetilde{S}_1$ and $\widetilde{S}_2$; we will call the copy of $\mathcal{R}_0$ bounding $\widetilde{S}_1$ $\mathcal{R}_0^{(A)}$ and the one bounding $\widetilde{S}_2$, $\mathcal{R}_0^{(B)}$. Every $x \in \mathcal{R}_0$ now has two copies $x^{(A)}$ and $x^{(B)}$.
Consider a second copy of this slit $W_0$, and rotate it by $\pi$. We will accent all objects in this (slit) second copy with a $^-$ superscript), and all objects in the original copy with a $^+$ superscript. Glue the slit copies $W_0^+, W_0^-$ together using the following rule: \begin{align*}
\forall x \in \mathcal{R}_0, \hspace{10pt}
x^{(A^+)} &\sim x^{(B^-)}\\
x^{(B^+)} &\sim x^{(A^-)} \end{align*} This gives a quotient map \begin{align*}
\pi : W^+_0 \sqcup W_0^- \longrightarrow W^+_0 \sqcup W_0^- / \sim \end{align*} \noindent This quotient surface can be endowed with a Riemann surface structure that makes $\pi$ analytic away from $\beta^\pm$. We can think of $\widetilde{X}$ as an open subset of the branched cover over $\mathbb{C}$ corresponding to $w \mapsto w^2+\beta$, and $\pi$ as a branch of $\sqrt{z-\beta}$ on each of the slit copies $W_0^+$ and $W_0^-$. \begin{figure}
\caption{The set $\Delta_q \subset \log \circ \varphi_c(\widetilde{S}_1(0))$. We will eventually define a map that is conformal on the white and lightly shaded regions, and quasiconformal on the darkly shaded region}
\label{fig:triangle}
\end{figure} We name this Riemann surface $\widetilde{X}$, and note that $\widetilde{X}$ has smooth boundary. Let $\widetilde{X_1} = \pi(W^+_1 \sqcup W_1^-)$. Then $\widetilde{X_1}$ is an open subset of $X$. We also define the ``sectors'' $\widetilde{A}$ and $\widetilde{B}$ as follows: \begin{align*}
\widetilde{A} &= \pi(\widetilde{S}_1^- \cup \widetilde{S}_2^+)\\
\widetilde{B} &= \pi(\widetilde{S}_2^- \cup \widetilde{S}_1^+) \end{align*} See Figure~\ref{fig:dynplanefinal} for an illustration. \begin{rem}\label{rem:whichbeta} We could have performed our cut and paste surgery by cutting along $\mathcal{R}_{\frac{j}{d}}$ for any $i \in \{0,1,...,d-1\}$ (the landing points of these rays are precisely the $\beta$ fixed points of $f_c$). To get a continuous embedding $\Phi_d$ of $\mathcal{M}_{d+1}$, however, we will use the same $j$ for all $c \in \mathcal{M}_{d+1}$. \end{rem} \subsubsection{Constructing a map $f$ on a subset of $\widetilde{X}_1$}\label{sec:fdefn} For $z \in \pi(W_1^{\pm})$, define \begin{align*}
f(\pi(z)) & =
\begin{cases}
\pi(f_c(z)) & z \not \in \mathcal{R}_0^{A^\pm} \cup \mathcal{R}_0^{B^\pm}\\
\pi(f_c(z^{(A^+)})) = \pi(f_c(z^{(B^-)}))& z \in (\mathcal{R}_0^{(A^+)} \cap W_1^+) \cup (\mathcal{R}_0^{(B^-)} \cap W_1^-)\\
\pi(f_c(z^{(B^+)})) = \pi(f_c(z^{(A^-)}))& z \in (\mathcal{R}_0^{(B^+)} \cap W_1) \cup (\mathcal{R}_0^{(A^-)} \cap W_1^-)\\
\pi(\beta^+) = \pi(\beta^-) & z \in \{\beta,\beta^-\}
\end{cases} \end{align*}
For $\ell \in \{1,2,...,d\}$, $f$ is not well defined on $\pi(\mathcal{R}^\pm(\frac{\ell}{d+1}))$ and we cannot extend it over any of these rays continuously since one component of the complement of such a ray in $\pi(\widetilde{S}^\pm_{i-1}(\frac{\ell}{d+1}))$ is mapped to $\widetilde{A}$, and the other is mapped to $\widetilde{B}$. \begin{figure}
\caption{The Riemann map $\widetilde{h}$ maps the lightly shaded region $T_1(0)$ to the region $V_0$ (the non-white region at the bottom). The darkly shaded region in the bottom figure is $f_c(Y_1(0))$}
\label{fig:riemannmap}
\end{figure} However, on the complement in $\widetilde{X}_1$ of the rays above , $f$ is analytic. \subsubsection{A new map on some sectors} By our definition of sectors, note that $\widetilde{S}\Big(\frac{\ell}{d+1}\Big) = \omega^\ell \tilde{S}(0)$.
Our strategy will be to produce a quasiregular map $g$ that agrees with $f$ on the complement of the sets $\pi\Big(\widetilde{S}_1^{\pm}\big(\frac{\ell}{d+1}\big)\Big)$ for $\ell=1,2,...,d$. We let \begin{align*}
\widetilde{S}_i\Big(\frac{\ell}{d+1}\Big) & = \widetilde{S}\Big(\frac{\ell}{d+1}\Big) \cap W_i \end{align*} $f_c$ maps $\widetilde{S}_i(\frac{\ell}{d+1})$ conformally to $\widetilde{S}_{i-1}(0)$.
Choose $p,q'$ such that $0<p<q'<q$, and consider the set $\Delta_q$ in $\log$ B\"{o}ttcher coordinates as illustrated in Figure~\ref{fig:triangle}. Its boundary is defined so that it is smooth away from the points $0, \log {(\frac{\eta}{d+1})}(1 \pm 2 \pi q i)$, and such that it coincides with an arc of the circle $x^2+y^2=\frac{\eta^2}{(d+1)^2}$ on the two connected regions region bounded by $y=\pm 2 \pi px$ and $y=\pm 2\pi q'x$. We will also require the boundary of $\Delta_q$ to be symmetric about the $x$ - axis in Figure~\ref{fig:triangle}. Additionally, let \begin{align*}
\Delta_{q'} & = \Delta_q \cap \{|y| < 2\pi q'x\}\\
\Delta_{p} & = \Delta_q \cap \{|y| < 2\pi px\} \end{align*} For $\ell \in \{0,1,...,d\}$, define \begin{align}
S_1\Big(\frac{\ell}{d+1}\Big) &= \omega^\ell \varphi_c^{-1} \circ \exp (\Delta_{q}) \label{eqn:s1defn}\\
Q_1\Big(\frac{\ell}{d+1}\Big)&= \omega^\ell\varphi_c^{-1} \circ \exp (\Delta_{q'}) \label{eqn:q1defn}\\
T_1\Big(\frac{\ell}{d+1}\Big) &=\omega^\ell \varphi_c^{-1} \circ \exp (\Delta_{p}) \label{eqn:t1defn}\\
Y_1\Big(\frac{\ell}{d+1}\Big)&=\omega^\ell\varphi_c^{-1} \circ \exp (\Delta_{q}\setminus \overline{\Delta_{q'}}) = S_1\Big(\frac{\ell}{d+1}\Big) \setminus \overline{Q_1\Big(\frac{\ell}{d+1}\Big)}\label{eqn:y1defn} \end{align} Clearly, \begin{align*}
T_1\big(\frac{\ell}{d+1}\big) \subset Q_1\big(\frac{\ell}{d+1}\big) \subset S_1\big(\frac{\ell}{d+1}\big) \subset \widetilde{S}_1\big(\frac{\ell}{d+1}\big) \end{align*} On the slit disk $W_0 \setminus \mathcal{R}_0$, we define $V_0$ as follows: \begin{align}
V_0 & = f_c\Big(W_1 \setminus \bigcup_{\ell = 0}^d S_1\Big(\frac{\ell}{d+1}\Big)\Big) \cup f_c(Y_1(0)) \label{eqn:vodefn} \end{align} See Figure~\ref{fig:riemannmap} for details.
\begin{figure}\label{fig:Delta}
\end{figure} By the Riemann mapping theorem, there exist analytic maps $\widetilde{k}: \mathbb{D} \longrightarrow V_0$ and $\widetilde{m}: T_1(0) \longrightarrow \mathbb{D}$ such that \begin{align*}
\widetilde{k}(\widetilde{m}( \beta))&= \beta\\
\widetilde{k}(\widetilde{m}(z_1)) &= \widetilde{z}_1\\
\widetilde{k}(\widetilde{m}(z_2))&=\widetilde{z}_2 \end{align*} Let $\widetilde{h} = \widetilde{k}\circ \widetilde{m}$. The map $\widetilde{h}:T_1(0) \longrightarrow V_0$ is the unique analytic function that sends the triple $(\beta,z_1,z_2)$ to $(\beta, \widetilde{z}_1,\widetilde{z}_2)$ (see Figure~\ref{fig:riemannmap}). $\partial V_0, \partial T_1(0)$ are quasi-circles. Therefore, $\widetilde{k},\widetilde{m}$ extend to quasisymmetric maps on the boundaries of their respective domains. \noindent Furthermore, by \cite[Chapter~3.4, Exercise 1]{pommerenke}, \begin{align*}
\widetilde{k}(z) &= \beta + a_k(z-1)^{2-4q'} + O(|z-1|^{2-4q' + (2-4q')\gamma_k})\\
\widetilde{m}^{-1}(z) &= \beta + a_m(z-1)^{4p} + O(|z-1|^{4p + 4p\gamma_k}) \end{align*} for some $\gamma_k, \gamma_m \in (0,1), a_k, a_m \in \mathbb{C}$.
It then follows that \begin{align*}
\widetilde{h}(z) = \beta + a_h\big(z-\beta\big)^{\frac{1-2q'}{2p}} + O\Big(\big|z-\beta\big|^{\frac{1-2q'}{2p} + \big(\frac{1-2q'}{2p}\big) \gamma_h}\Big) \end{align*} for some $\gamma_h \in (0,1), a_h \in \mathbb{C}$. Since $\varphi_c$ does not distort angles, conjugating $\tilde{h}$ by $\varphi_c$ should not change this equation, and we have the following: \begin{prop}\label{boundarybeh} For all $z \in \exp(\Delta_p) = \varphi_c(T_1(0))$, \begin{align}
\varphi_c \circ \widetilde{h} \circ \varphi_c^{-1}(z) = 1 + a'\big(z-1\big)^{\frac{1-2q'}{2p}} + O\Big(\big|z-1\big|^{\frac{1-2q'}{2p}+ (\frac{1-2q'}{2p})\gamma}\Big) \label{eqn1} \end{align} for some $\gamma \in (0,1), a' \in \mathbb{C} \setminus \{0\}$. \end{prop} \begin{comment} \begin{proof} This is because $\varphi_c$ does not distort angles at $\beta$.\\
We will also remark here that this shows that \begin{align*}
|\varphi_c \circ \widetilde{h} \circ \varphi_c^{-1}(z)| \geq |C|z-1|^{\frac{1-2q'}{p}}-1| \end{align*} for some constant $C \neq 0$. \end{proof} \end{comment} Let $G$ be a connected component of $\Delta_{q'} \setminus \overline{\Delta_p}$, say the one bounded by $y=2\pi q'x$ and $y=2\pi p x$. The polynomial $f_c$ induces the map $\mu_{d+1}(z)= (d+1)z$ on the part of its boundary where $y = 2\pi q'x$. The map $\hat{h} = (\log \circ \varphi_c) \circ \widetilde{h} \circ ( \varphi_c^{-1} \circ \exp)$ extends to a continuous map on the part of the boundary where $y= 2\pi px$.
Let $\Delta$ be the set $\Delta_{q'} \cap\{ x^2+y^2< \eta^2\}$(see Figure~\ref{fig:Delta} for an illustration of $\Delta$). $S(0):=\varphi_c^{-1} \circ \exp(\Delta)$ is an open subset of $\widetilde{S}(0)$. For $\ell \in \{1,2,...,d\}$, let $S(\frac{\ell}{d+1}) = \omega^\ell S(0)$. The sector $S\big(\frac{\ell}{d+1}\big)$ contains $S_1\big(\frac{\ell}{d+1}\big)$.\\
The following is a crucial lemma. \begin{figure}
\caption{The map $H$ is defined by mapping vertical lines to lines joining the images of their endpoints}
\label{fig:maponstrips}
\end{figure} \begin{lemma}\label{lemma:quasiexists} There exists a quasiconformal map from $G$ to $\Delta$ that restricts to $\mu_{d+1}$ on one boundary, and to $\hat{h}$ on the other boundary. \end{lemma} \begin{proof} $G$ has a positive angle at the vertex $0$. In $\log$ coordinates, $G' = \log G$ is a half-infinite horizontal strip with $\hat{\mu}_{d+1}(z) = z+\log (d+1)$ induced by $\mu_{d+1}$ on the part of the boundary where $y = \arctan(2 \pi q')$, and $H(z) = \log \hat{h}(e^z)$ on the part of the boundary where $y = \arctan(2 \pi p)$.
We will interpolate between $\hat{\mu}_{d+1}$ and $H$ by mapping vertical lines in $G'$ to lines joined by the images of the endpoints. If we can show that these image lines have uniformly bounded slope, the resulting map will be quasiconformal. We explain this in detail below.
Set $\theta_p = \arctan(2 \pi p), \theta_{q'} = \arctan(2\pi q').$ We will define $H$ on $\overline{G'}$ by extending along vertical lines: \begin{align*}
H\Big(x+i\big((1-t)\theta_p +t \theta_{q'}\big)\Big) & = (1-t)H\big(x + i\theta_p\big) + t\hat{\mu}_{d+1}\big(x+i \theta_{q'}\big) \end{align*}
\begin{prop}\label{boundedlines} There exists $R > 0$ such that for all $z \in \{Im(z) = \arctan(2\pi p)\} \cap \overline{G'}$, \begin{align}
|H(z) - z| \leq R \label{eqn2} \end{align} \end{prop} \begin{proof} We will prove this by showing that both $z \mapsto z - H(z)$ and $z \mapsto H(z)-z$ are bounded above.
Suppose $z-H(z)$ is not bounded above, then for each natural number $n$, there exists $z_n$ such that \begin{align*}
z_n - H(z_n) &> n \end{align*} and upto a subsequence, the $z_n$ tend to $ -\infty$. But this implies that \begin{align*}
|\hat{h}(u_n)|&< \frac{|u_n|}{e^n} \end{align*} where $u_n = \exp z_n$. Furthermore, \begin{align*}
Re(\hat{h}(u_n)) &\leq |\hat{h}(u_n)| \\& \leq \frac{|u_n|}{e^n} = \frac{1}{e^n}\sqrt{Re(u_n)^2(1+4\pi^2 p^2)} \\& \leq C\frac{|Re(u_n)|}{e^n} =C\frac{Re(u_n)}{e^n} \end{align*} for some constant $C>0$. \\
Set $w_n = \exp {u_n}$, and note that $\exp \hat{h}(u_n) = \varphi_c \circ \widetilde{h}\circ \varphi_c^{-1}(w_n)$. Thus \begin{align}
|\varphi_c \circ \widetilde{h}\circ \varphi_c^{-1}(w_n)| < b|w_n|^{\frac{1}{e^n}} \label{eqn3} \end{align} for some $b>0$.
But Equation~\ref{eqn3} implies that $|\varphi \circ \widetilde{h}\circ \varphi_c^{-1}(w_n)|$ converges much faster to $1$ than allowed by Equation~\ref{eqn1}, and forms a contradiction. This proves that $z - H(z)$ is bounded above.
Similarly, suppose $H(z)-z$ is not bounded above as $z \rightarrow -\infty$, there exists a sequence $z_n \rightarrow -\infty$ such that \begin{align*}
H(z_n) - z_n &> n \end{align*} Thus \begin{align*}
|\hat{h}(u_n)| &> |u_n|e^n \end{align*} Consequently, we have \begin{align*}
Re\hat{h}(u_n) &= \frac{|\hat{h}(u_n)| }{\sqrt{1+4\pi^2q'^2}} \\
&> \frac{|u_n|e^n}{\sqrt{1+4\pi^2q'^2}} \\
&= \frac{\sqrt{1+4\pi^2p^2}}{\sqrt{1+4\pi^2q'^2}}Re(u_n)e^n \end{align*} But this gives us \begin{align*}
|\varphi_c \circ \widetilde{h}\circ \varphi_c^{-1}(w_n)| > \iota |w_n|^{e^n} \end{align*} for some constant $\iota >0$, which also contradicts Equation~\ref{eqn1}.
This proves Proposition~\ref{boundedlines}. \end{proof} \begin{figure}
\caption{An illustration of $h$ on $ \pi\big(S_1\big(\frac{1}{3}\big)^+\big)$; the dark components above are quasiconformally mapped to the dark components below, the lightly shaded region maps by the Riemann map $\omega^{-1}\widetilde{h}$, and the white region maps by $f$ }
\label{fig:quasimap}
\end{figure}
It is clear that $H$ interpolates between the maps on the two horizontal boundaries, and that $H(G' \cap \{Re(z) = \log(\frac{\eta}{3})\}) = \{\log \eta + iy : |y| < \arctan(2 \pi q')\}$. Furthermore, $H$ is a quasiconformal map whose dilitation is bounded above by some $M\geq 1$ (see Figure~\ref{fig:maponstrips}): this is because vertical lines in the domain are mapped by $H$ to lines whose slopes are bounded below by some uniform constant, by Equation~\ref{eqn2}. We conjugate $H$ by the exponential map to obtain a quasiconformal map $\hat{h}$ from $G$ that satisfies the properties in the statement of Lemma~\ref{lemma:quasiexists}. \end{proof}
$G$ could be taken to be either connected component of $\Delta_{q'} \setminus \overline{\Delta_{p}}$. On the dynamical plane, it corresponds to a component $\mathcal{G}$ of $Q_1(0) \setminus \overline{T_1(0)}$. We shall henceforth denote the copy of $S(0)$ in $\widetilde{A}$ as $A$, and the copy in $\widetilde{B}$ as $B$. With this in mind, we will take $h_{\mathcal{G},A}$ to mean the map $ (\omega^\ell \circ \varphi_c^{-1} \circ \exp) \circ \hat{h} \circ (\log \circ \varphi_c \circ \omega^{-\ell})$ from the component $\mathcal{G}$ of $Q_1\big(\frac{\ell}{d+1}\big) \setminus T_1\big(\frac{\ell}{d+1}\big)$ to $A$, and $h_{\mathcal{G},B}$ to mean the same map, but from $\mathcal{G}$ to $B$. We will use the same names for the extended maps from $\overline{\mathcal{G}}$. \subsubsection{Constructing a quasiregular map $g$} Let $S$ be a sector of the form $\pi\Big(S^+_1\big(\frac{\ell}{d+1}\big)\Big)$, where $\ell \in \{1,2,...,d+1\}$. The map $f$ has a line of discontinuities in $S$ along the ray $\pi\Big(\mathcal{R}^+_{\frac{\ell}{d+1}}\Big)$- on one side of this ray, $f$ maps into $\widetilde{A}$ and approaches $\pi(\mathcal{R}_0^{\widetilde{A}})$, whereas on the other side, $f$ maps into $\widetilde{B}$ and approaches $\pi(\mathcal{R}_0^{\widetilde{B}})$. Define a map $h$ on $\overline{S}$ as follows: \begin{itemize}
\item On $\pi\Big(\overline{T^+_1\big(\frac{\ell}{d+1}\big)}\Big)$, let $h(\pi(z^+)) = \pi(\widetilde{h}(\omega^{-\ell}z)^-) \in \pi(V_0^-)$.
\item On $\pi\Big(\overline{Y_1^+\big(\frac{\ell}{d+1}\big)}\Big)$, let $h(\pi(z^+)) = f(\pi(z^+))$, where $f$ is defined as in Section~\ref{sec:fdefn}.
\item On the connected component $\mathcal{G}$ of $\pi\Big(Q^+_1\big(\frac{\ell}{d+1}\big) \setminus T^+_1\big(\frac{\ell}{d+1}\big)\Big)$ with $\ell=1,2,...,d$ part of whose boundary $f$ maps into $\partial \widetilde{A}$, let $h(\pi(z^+)) = h_{\mathcal{G},A}(z)$.
\item On the connected component $\mathcal{G}$ of $\pi\Big(Q^+_1\big(\frac{\ell}{d+1}\big) \setminus T^+_1\big(\frac{\ell}{d+1}\big)\Big)$ with $\ell=1,2,...,d$ part of whose boundary $f$ maps into $\partial \widetilde{B}$, let $h(\pi(z^+)) = h_{\mathcal{G},B}(z)$. \end{itemize} \begin{figure}
\caption{The Riemann surfaces $X_1$, $X$}
\label{fig:dynplanemodified1}
\caption{The quasiregular map $g$}
\label{fig:dynplanemodified2}
\caption{The dynamics of a quasiregular model for $\Phi_d(c)$}
\label{fig:dynplanemodified}
\end{figure}
The map $h$ so defined is a quasiconformal homeomorphism from $S$ to $\pi(V_0^-) \cup A \cup B$ (see Figure~\ref{fig:quasimap} for an illustration of $h$ on $\pi\Big(S\big(\frac{1}{3}\big)^+\Big)$ when $d=2$), and restricts to an analytic map on $\pi\Big(T_1\Big(\frac{\ell}{d+1}\Big)\Big)$.
Furthermore, the latter set has smooth boundary at the points $\pi(\widetilde{z}_1^-)$ and $\pi(\widetilde{z_2}^-)$ in Figure~\ref{fig:quasimap}: consider $\pi(\widetilde{z}_1^-)$ for instance. In B\"{o}ttcher coordinates, the boundary in a neighborhood of $\widetilde{z}_1$ looks like $f_c(\varphi_c^{-1} \circ \exp(\gamma))$, where $\gamma$ is a neighborhood of the boundary of $\Delta_{q}$ at the point $\log \circ \varphi_c(z_1)$; $\gamma$ is clearly smooth.
On $\pi\Big(S_1^{-}\big(\frac{\ell}{d+1}\big)\Big)$, we define $h$ the same way, except with the following change: on $\pi\Big(\overline{T_1^-\big(\frac{\ell}{d+1}\big)}\Big)$, let $h(\pi(z^-)) = \pi(\widetilde{h}(\omega^{-\ell}z)^+) \in \pi(V_0^+)$. This $h$ is a quasiconformal hoemoemorphism from $S$ to $\pi(V_0^+) \cup A \cup B$. Finally, we construct a quasiregular map on newly defined subsets of $\widetilde{X}_1, \widetilde{X}$. \\ Let \begin{align*}
X & = \pi(V_0^+) \cup A \cup B \cup \pi(V_0^-) \end{align*} Also let $X_1$ be the subset of $\widetilde{X_1}$ where all sectors of the form $\pi\Big(\widetilde{S}^\pm_1\big(\frac{\ell}{d+1}\big)\Big)$, $\pi\Big(\widetilde{S}^-_1\big(\frac{\ell}{d+1}\big)\Big)$ are replaced by $\pi\Big(S_1^\pm\big(\frac{\ell}{d+1}\big)\Big)$ for $\ell=1,2,..,d$, and let $X$ be the open subset of $\widetilde{X}$ where $\widetilde{A}$ and $\widetilde{B}$ are replaced by $A$, $B$ respectively. See Figure~\ref{fig:dynplanemodified1} for details. \noindent Clearly, $X_1$ is an open subset of the Riemann surface $X$ compactly contained in $X$. We define \begin{align*}
g&: X_1 \longrightarrow X\\
g(\pi(z)) & = \begin{cases}
f(\pi(z)) & z \in \Big(W^+_1 \setminus \bigcup _\ell S^+_1\big(\frac{\ell}{d+1}\big)\Big) \cup \Big(W^-_1 \setminus \bigcup_\ell S^-_1\big(\frac{\ell}{d+1}\big)\Big),\ell=1,2,...,d \\
h(\pi(z)) & z \in S^\pm_1\big(\frac{\ell}{d+1}\big), \ell=1,2,...,d\\
\end{cases} \end{align*} See Figure~\ref{fig:dynplanemodified2} for an illustration of $g$.\\
$g$ is quasiregular. Furthermore, any $g$ orbit visits $\pi\Big(Q^+_1\big(\frac{\ell}{d+1}\big) \setminus T^+_1\big(\frac{\ell}{d+1}\big)\Big)$ or $\pi\Big(Q^-_1\big(\frac{\ell}{d+1}\big) \setminus T^-_1\big(\frac{\ell}{d+1}\big)\Big)$ at most once, and these are the only regions where $g$ is not analytic. We will use this fact to define a new complex structure $\sigma$ (given by an ellipse field $E_x$ for $x \in X$) by setting \begin{itemize}
\item $E_x = \mathbb{S}^1$ if $x \in X \setminus X_1$ or if the orbit of $x$ never visits $\pi\Big(Q^+_1\big(\frac{\ell}{d+1}\big)\setminus T^+_1\big(\frac{\ell}{d+1}\big)\Big)$ or $\pi\Big(Q^-_1\big(\frac{\ell}{d+1}\big) \setminus T^-_1(\frac{\ell}{d+1}\big)\Big)$ \item $E_x = (T_xg)^{-1}(\mathbb{S}^1)$ for $x \in \pi\Big(Q^+_1\big(\frac{\ell}{d+1}\big)\setminus T^+_1\big(\frac{\ell}{d+1}\big)\Big)$ or $x \in \pi\Big(Q^-_1\big(\frac{\ell}{d+1}\big) \setminus T^-_1\big(\frac{\ell}{d+1}\big)\Big)$ for some $\ell \in \{1,2,...,d\}$
\item $E_x = (T_x g^{n})^{-1}(E_{g^n(x)})$ if $g^n(x)$ is the first point in the $g-$orbit of $x$ that is in one of the regions above. \end{itemize} The complex structure $\sigma$ thus defined has bounded dilitation, and $g^*\sigma = \sigma$. \subsubsection{Obtaining a polynomial} Define the map $\tau: X \longrightarrow X$ by sending $\pi(z^+)$ to $\pi(z^-)$. $\tau$ satisfies $\tau^*\sigma = \sigma$, $\tau(X_1) = X_1$, and $\tau^{\circ 2} = id$. We note that \begin{align*}
g \circ \tau = \tau \circ g \end{align*}
We find an integrating map $\psi$ for $\sigma$ sending $\pi(\beta^\pm)$ to $0$, and satisfying $\frac{\psi(z)}{z} \longrightarrow 1$ as $z \longrightarrow \infty$.
The map $G= \psi \circ g\circ \psi^{-1} : U' \longrightarrow U$ is polynomial-like, and has two critical points with local degree $d+1$. $\kappa = \psi \circ \tau \circ \psi^{-1}$ an analytic involution of $U$, and commutes with $G$ on $U$. We can further conjugate by a Riemann map taking the pair $(U,0)$ to $(\mathbb{D},0)$. By the Schwarz lemma, we can assume without loss of generality that $\kappa\big |_U(z) = -z$; in particular, $\kappa$ has a global extension.
$G$ is hybrid equivalent to a degree $2d+1$ polynomial $p$ with two critical points by the straightening theorem. We may choose this hybrid equivalence $h:\mathbb{C} \longrightarrow \mathbb{C}$ such that $h(0) = 0$. Then $\delta = h \circ \kappa\circ h^{-1}$ is an affine map of $\mathbb{C}$ with $\delta(0) = 0, \delta^{\circ 2} = id, \delta \neq id$. Therefore, $\delta(z) = -z$. $p$ commutes with $\delta$, and can now be normalized to the form $a{\displaystyle \int_{0}^{z} } \Big(1-\frac{w^2}{d}\Big)^ddw$ for a unique $a \in \mathbb{C}^*$. The choice of $a$ does not depend on our initial choice of $q,\eta,\psi$ ($h$ is determined by $c,p,q,q',\eta$) - we can show that different choices give rise to hybrid equivalent polynomials. \subsection{The image of $\Phi_d$} Clearly, $\Phi_d(\mathcal{M}_{d+1}) \subset \mathcal{CBO}_d$. Let $a=\Phi_d(c)$. By our construction, $0$ is a fixed point of $p_{a}$ belonging to the Julia set, and it disconnects the Julia set into two components.
Under our surgery, the original dynamical ray $\mathcal{R}_0$ landing at $\beta$ gets transformed into an arc $\Gamma$ from $0$ to $\infty$ in the dynamical plane of $p_{\Phi_d(c)}$ whose interior is contained in the escaping set. In the monic representation $P_{s(a)}$ of $p_a$, $\Gamma$ has the same access as $\mathcal{R}_{s(a)}(0)$. The union $\Gamma \cup -\Gamma$, and indeed $\mathcal{R}_0(s(a)) \cup \mathcal{R}_\frac{1}{2}(s(a))$, separates the orbits of the two critical points of $P_{s(a)}$. That is, $\Phi_d(\mathcal{M}_{d+1}) \subset \mathcal{CBO}^{(+,-)}_d$. We will show in Section~\ref{section:injectivity} that the image under $\Phi_d$ is equal to this set. \section{Continuity of $\Phi_d$}\label{section:continuity} To show continuity of $\Phi_d$, we will follow the strategy laid out in \cite[Chapter~II.8]{10.1007/BFb0081395}, and show it separately when $c$ is on the boundary, or in the interior of $\mathcal{M}_{d+1}$. Throughout this section, we shall index all sets and functions in Section~\ref{section:defn} in constructing $\Phi_d(c)$ by the subscript $c$. For example, the projecton $\pi$ is referred to as $\pi_c$, the quasiregular map $g$ as $g_c$, the domain of $g_c$ as $(X_1)_c$ and so on. \begin{lemma}\label{lemm:hybridimpliesaffine} If $p_a,p_{a'}$ with $a,a' \in \mathcal{CBO}_d$ are hybrid equivalent, then they are affine conjugate. \end{lemma} \begin{proof} This follows from \cite[Chapter~I.6, Corollary~2]{Douady1985}. \end{proof} \subsection{The interior case}
If $c \in \mathcal{M}_{d+1}^\circ$, then proof is based on the proof of \cite[Chapter~II.5, Proposition~12]{Douady1985}. \begin{defn}\label{defn:analyticfamily}
Given
$f_\lambda : U_\lambda' \longrightarrow U_\lambda$, for $\lambda \in \Lambda$, let
\begin{align*}
\mathcal{U}' & = \{(\lambda, z)| z \in U_\lambda'\}\\
\mathcal{U} & = \{(\lambda, z)| z \in U_\lambda\}
\end{align*}
and define $f: \mathcal{U}' \longrightarrow \mathcal{U}$ as $f(\lambda, z) = f_\lambda(z)$. If
\begin{enumerate}
\item $\mathcal{U}',\mathcal{U}$ are homeomorphic over $\Lambda$ to $\Lambda \times \mathbb{D}$.
\item projection of $\overline{\mathcal{U}'}$ in $\mathcal{U}$ to $\Lambda$ is proper
\item $f$ is holomorphic and proper
\end{enumerate}
then $f_\lambda$ is called an analytic family.
\end{defn}
Let us go back to the construction of $\Phi_d(c)$ from $f_c$. We first construct a quasiregular map $g_{c}: (X_1)_{c} \longrightarrow X_{c}$. This map is built from $f_c$ away from certain escaping sectors, and from the Riemann map $h_{c}$ on other sectors. Then we find an invariant complex structure $\sigma_c$ for $g_{c}$ and find integrating maps $\psi_c$. This gives us the polynomial-like family $G_{c}: U'_{c} \longrightarrow U_{c}$.
\begin{prop}\label{prop:analyticfam}
On a connected component $\Lambda$ of $\mathcal{M}^\circ_{d+1}$, $(c,z) \mapsto (c,G_{c}(z))$ is an analytic family of structurally stable polynomials.
\end{prop}
\begin{proof} We show that $G_{c}$ satisfies the three properties of Definition~\ref{defn:analyticfamily}.
\begin{enumerate}
\item $U'_{c}, U_{c}$ are homeomorphic to $\mathbb{D}$ and $c' \mapsto {U}'_{c}, c \mapsto U_{c}$ are both continuous maps in the Hausdorff topology
\item Let $\Pi$ be this projection. Given any compact set $K$ in $\Lambda$, and a sequence $(c_n, z_n) \in \Pi^{-1}(K)$, upto a subsequence, $c_n \longrightarrow c \in K$. We note that $z_n \in \overline{U'_{c_n}}$. $\overline{U'}_{c_n} \longrightarrow \overline{U'}_c$, and hence, there exists a sequence $\widetilde{z}_n \in \overline{U'}_c$ such that $|z_n - \widetilde{z}_n| \longrightarrow 0$. Since $\overline{U'}_c$ is compact, upto a subsequence, $\widetilde{z}_n \longrightarrow z \in \overline{U'}_c$. So $z_n \longrightarrow z$ upto the same subsequence. This shows that $\Pi^{-1}(K)$ is compact.
\item By \cite{mane1982dynamics}, every parameter $c \in \Lambda$ is structurally stable. More particularly, given $c \in \Lambda$, there exists a holomorphic motion $L: \Lambda \times \hat{\mathbb{C}} \longrightarrow \hat{\mathbb{C}}$ such that $L_c = id$, and for all $\widetilde{c} \in \Lambda$, $L_{\widetilde{c}}$ is quasiconformal and satisfies $L_{\widetilde{c}}\circ f_c\circ L_{\widetilde{c}}^{-1} = f_{\widetilde{c}}$.
\begin{figure}
\caption{Dynamics of $P_{s(a)}$ for $a \in \mathcal{CBO}_d^{(+,-)}$}
\label{fig:inversedefn}
\end{figure}
But this also means that $h_{\widetilde{c}} =L_{\widetilde{c}}\circ h_c\circ L_{\widetilde{c}}^{-1}$ on $\varphi^{-1}_{\widetilde{c}} \circ \exp (\Delta_p)$, and by definition of the quasiconformal extension, $h_{\widetilde{c}} =L_{\widetilde{c}}\circ h_c\circ L_{\widetilde{c}}^{-1}$ on the sector $S_1\big(\frac{\ell}{d+1}\big)$ in the dynamical plane of $f_{\widetilde{c}}$.
Therefore, $g_{c}$, and consequently, $\sigma_c$ depend analytically on $c$. By the measurable Riemann mapping theorem, the integrating maps $\psi_c$ depend holomorphically on $c$.
For a fixed $z \in U'_c$, when $\widetilde{c}$ is close to $c$, $G_{\widetilde{c}}(z)$ is well-defined, and $\widetilde{c} \mapsto G_{\widetilde{c}}(z) = \psi_{\widetilde{c}} \circ g_{\widetilde{c}}\circ \psi_{\widetilde{c}}^{-1}(z)$ is holomorphic in $\widetilde{c}$.
For a fixed $c$, $z \mapsto G_c(z)$ is holomorphic by definition. Thus $G_c(z)$ is holomorphic in both $c$ and $z$; by Hartog's theorem, it is holomorphic as a function of $(c,z)$. Proof that $G_c(z)$ is proper is similar to Point~2 above.
\end{enumerate}
\end{proof} In Proposition~\ref{prop:analyticfam}, we showed that $(c,z) \mapsto G_c(z)$ is an analytic family over every connected component $\Lambda$ of $\mathcal{M}^{\circ}_{d+1}$. Given $c \in \Lambda$, let us pick the hybrid equivalence $k_{{c}}$ conjugating $G_c(z)$ to a polynomial in such a way that it fixes $0$ and satisfies $k_{{c}}(z)/z \longrightarrow 1$. Then, by \cite[Chapter~II.5, Proposition~12]{Douady1985}, the polynomials $k_{{c}}\circ G_{{c}}\circ k_{{c}}^{-1}$ form a continuous family over $\Lambda$. As proved in Section~\ref{section:defn}, these are affine conjugate to bicritical odd polynomials, and their critical points vary continuously with respect to ${c}$. Hence there exists a continuous family of scaling maps $M_{{c}}$ that map these critical points to $\pm \sqrt{d}$. But then
\begin{align*}
p_{\Phi_d(\widetilde{c})} = M_{\widetilde{c}}\circ k_{\widetilde{c}}\circ G_{\widetilde{c}}\circ k_{\widetilde{c}}^{-1}\circ M_{\widetilde{c}}^{-1},
\end{align*} is clearly continuous in $\widetilde{c}$. \subsection{The Boundary case} The following lemma and its proof are similar to \cite[Chapter~II.8, Lemma~3]{10.1007/BFb0081395}. \begin{lemma}\label{lemm:quasi} If $p_a$ and $p_{a'}$ are quasiconformally equivalent via $\psi$, with $a \in \partial \mathcal{CBO}_d$, such that $\psi$ satisfies the conditions below: \begin{align*}
\psi(0) &= 0\\
\psi(\sqrt{d}) & =\sqrt{d}\\ \lim_{z \longrightarrow \infty}\frac{\psi(z)}{z} &=1 \end{align*} then $a=a'$. \end{lemma} \begin{proof} We first note that any $\psi$ as above also satisfies $\psi(-\sqrt{d}) = -\sqrt{d}$. If $K_{p_a}$ has measure $0$, then $\psi$ is a hybrid equivalence and the result follows.
Otherwise, our strategy is to build a hybrid equivalence between the two polynomials, similar to \cite[Chapter~I.6, Corollary~2]{Douady1985} and use Lemma~\ref{lemm:hybridimpliesaffine}. Consider the Beltrami form $\mu = \frac{ \overline{\partial} \psi}{\partial \psi}$ and let $\mu_0$ be the form that agrees with $\mu$ on $K_{p_a}$ and equals $0$ on $\mathbb{C} \setminus K_{p_a}$. Set $k = ||\mu_0||_\infty$.
We note that $k<1$. By the measurable Riemann mapping theorem, for every $t \in \mathbb{D}_{\frac{1}{k}}$, there exists a unique quasiconformal homeomorphism $\psi_t : \mathbb{C} \longrightarrow \mathbb{C}$ such that \begin{align*}
\frac{\overline{\partial }\psi_t}{\partial \psi_t} & = t\mu_0\\
\psi_t(0) &= 0\\
\lim_{z \rightarrow \infty}\frac{\psi_t(z)}{z}& = 1 \end{align*}
We note that $\kappa_t (z)= \psi_t(-\psi_t^{-1}(z))$ is a family of affine maps that satisfy \begin{align*} \kappa_t(0) & = 0\\
\lim_{z \mapsto \infty} \frac{\kappa_t(z)}{z} &= -1 \end{align*} Therefore, $ \kappa_t(z) = -z $.
$\psi_t \circ p_a \circ \psi_t^{-1}$ is a polynomial with exactly two critical points that commutes with $\kappa_t$. It has the form $\widetilde{a}(t){\displaystyle \int_{0}^{z}} \Big(1-\frac{w}{x(t)}\Big)^d\Big(1+\frac{w}{x(t)}\Big)^ddw$. The functions $\widetilde{a}, x:\mathbb{D}_{\frac{1}{k}} \longrightarrow \mathbb{C}$ are holomorphic, with $x(0) = \sqrt{d}$, $\widetilde{a}(0)=a$. These polynomials are odd, therefore by conjugating them by $h_t(z) = \frac{z\sqrt{d}}{x(t)}$, we obtain polynomials of the form $a(t){\displaystyle \int_0^z}\Big(1-\frac{w^2}{d}\Big)^ddw$, where $a: \mathbb{D}_{\frac{1}{k}} \longrightarrow \mathbb{C}$ is holomorphic, with $a(t) \in \mathcal{CBO}_d$, and $a(0) = a \in \partial \mathcal{CBO}_d$. But this implies that $a(t)$ is a constant function, and $\psi \circ \psi_1^{-1} \circ h_1^{-1}$ is a hybrid equivalence between $p_a$ and $p_{a'}$. Lemma~\ref{lemm:quasi} implies $a=a'$.\\ \end{proof}
\noindent To show continuity of $\Phi_d$ at $c \in \partial \mathcal{M}_{d+1}$, it suffices to show that its graph is closed, that is, if $c_n \in \mathcal{M}_{d+1}$ converge to $c$ and $a_n=\Phi_d(c_n) \longrightarrow \widetilde{a}$, then $\widetilde{a} = \Phi_d(c)$.
Let \begin{align*} a&=\Phi_d(c) &f_n & = f_{c_n}\\ g&=g_c & g_n&=g_{c_n}\\ \sigma & = \sigma_c &\sigma_n&=\sigma_{c_n}\\ \psi & = \psi_c & \psi_n & = \psi_{c_n}\\ G&=G_c = \psi \circ g \circ \psi^{-1} &G_n& = G_{c_n} = \psi_n \circ g_n \circ \psi_n^{-1}\\
\varphi &=\varphi_c &\varphi_{n} & =\varphi_{c_n}\\
V_n& = (V_0)_{c_n} & V & = (V_0)_c\\
Q_n(\ell) & = \Big(Q_1\Big(\frac{\ell}{d+1}\Big)\Big)_{c_n} & Q(\ell) & = \Big(Q_1\Big(\frac{\ell}{d+1}\Big)\Big) _c\\
T_n(\ell) & = \Big(T_1\Big(\frac{\ell}{d+1}\Big)\Big)_{c_n} & T(\ell) & = \Big(T_1\Big(\frac{\ell}{d+1}\Big)\Big) _c \end{align*}
\begin{prop}
The sequence of quasiregular maps $g_n$ converge to $g$.
\end{prop}
\begin{proof}
On both the $+$ and $-$ copies of $(W_0)_{c_n}$, $g_n$ coincides with $f_n$ away from the sectors $Q_1(\ell)$, for $\ell=1,2,...,d$. On each of these sectors, $g_n$ has as its components a conformal map $\widetilde{h}_n: T_n(\ell) \longrightarrow V_n$, chosen uniquely so that the triple $((z_1)_n , (z_2)_n, \omega^\ell\beta_n)$ is mapped to the triple $((\widetilde{z}_1)_n, (\widetilde{z}_2)_n), \beta_n)$, and a quasiconformal extension to $Q_n(\ell) \setminus \overline{T_n(\ell)}$. Similarly, on $T(\ell)$, $g$ agrees with an analytic map $\widetilde{h}: T(\ell) \longrightarrow V$ chosen so that the triple $(z_1,z_2,\omega^\ell\beta)$ is sent to $(\widetilde{z}_1, \widetilde{z}_2, \beta)$. Fix an $\ell \in \{1,2,...,d\}$. We will first show that the $\widetilde{h}_n$ converge to $\widetilde{h}$.
Let $\rho_n$ be the Riemann map that sends $\mathbb{D}$ to $V_n$, with $\rho_n(0) = 0$ and $\rho_n'(0) >0$. Observe that $V_n$ converges to $V$ with respect to the point $0$ in the sense of kernel convergence (see \cite[Section~1.4]{pommerenke}). By Carath\'{e}odory's kernel convergence theorem (\cite[Theorem~1.8]{pommerenke}), $\rho_n \longrightarrow \rho$ uniformly in $\mathbb{D}$, where $\rho :\mathbb{D} \longrightarrow V$ is a conformal map that sends $0$ to $0$ and satisfies $\rho'(0)>0$. Since the boundaries of $V_n, V$ are quasicircles, the $\rho_n$ extend to $\partial \mathbb{D}$ and these boundary maps converge uniformly to the boundary extension of $\rho$. Thus, the triples $(s_n, t_n, w_n)$ in $\mathbb{S}^1$ that map under $\rho_n$ to $((\widetilde{z}_1)_n, (\widetilde{z}_2)_n), \beta_n)$ converge to the triple $(s,t,w)$ in $\mathbb{S}^1$ that maps under $\rho$ to $(\widetilde{z}_1, \widetilde{z}_2, \beta)$.
Let $M_n: \mathbb{D} \rightarrow\mathbb{D} $ be a sequence of automorphisms that send $(1,i,-1)$ to $(s_n,t_n, w_n)$, and let $M$ be the automorphism of $\mathbb{D}$ that sends $(1,i,-1)$ to $(s,t,w)$. Then $M_n \longrightarrow M$ on $\overline{\mathbb{D}}$.
Lastly, for a given $\ell$, note that $\varphi_n(T_n(\ell))$ is the same domain $D = \exp(\Delta_q) = \varphi(T(\ell))$ for each $n$, and furthermore, $(\varphi_n((z_1)_n), \varphi_n((z_2)_n), \varphi_n(\omega^\ell\beta_n)) = (\varphi(z_1), \varphi(z_2), \varphi(\omega^\ell\beta))$ (we note that $\omega^\ell\beta_n$ and $\omega^\ell \beta$ are tips, ie. a unique dynamical ray lands at each of these points, so evaluating the B\"{o}ttcher chart at these points makes sense). Let $e: D \longrightarrow \mathbb{D}$ be the Riemann map that takes the triple $(\varphi(z_1), \varphi(z_2), \varphi(\omega^\ell\beta))$ in $\partial D$ to $(1,i,-1)$. Then \begin{align*}
\widetilde{h}_n & = \rho_n \circ M_n\circ e \circ \varphi_n\\
\widetilde{h} & = \rho \circ M \circ e \circ \varphi \end{align*} It is clear by our discussion that $\widetilde{h}_n \longrightarrow \widetilde{h}$.
Therefore, on the sectors $T_n(\ell)$, the sequence $g_n$ converges to $g$. But note that the quasiconformal extension to $Q_n(\ell)$ is done in the same way for each $n$. Therefore, $g_n \longrightarrow g$.
By definition of $\sigma_n$ and $\sigma$, we must have $\sigma_n \longrightarrow \sigma$, and consequently, by the Measurable Riemann Mapping Theorem, $\psi_n\longrightarrow \psi$.
\end{proof} \noindent This discussion tells us that \begin{align*}
G_n&= \psi_n \circ g_n \circ \psi_n^{-1} \longrightarrow \psi \circ g \circ \psi^{-1} =G
\end{align*}
Now consider the hybrid equivalences $k_n$ that conjugate $G_n$ to $p_{a_n}$. These have bounded dilitation ratio and map $0$ to $0$, and hence form an equicontinuous family. Upto a subsequence, $k_n$ converge to a quasiconformal map $\widetilde{k}$. Thus, $k_n \circ G_n \circ k_n^{-1} \longrightarrow \widetilde{k} \circ G \circ \widetilde{k}^{-1}$. We will call the latter map $\widetilde{G}$.
Using \cite[Chapter~II.7, Lemma, p.313]{Douady1985}, $\widetilde{G}$ is quasiconformally equivalent to $p_{\widetilde{a}}$ (not necessarily hybrid equivalent), but it is also quasiconformally equivalent to $k \circ G \circ k^{-1}$, which in turn is hybrid equivalent to $p_a$.
This shows that $p_{\widetilde{a}}$ is quasiconformally equivalent to $p_a$. We can choose the equivalence so that the conditions of Lemma~\ref{lemm:quasi} are satisfied. But in order to use this lemma, we also need to show that $a \in \partial \mathcal{CBO}_d$.
Consider a sequence $c_n^*$ of Misiurewicz parameters tending to $c$, and let $a_n^* = \Phi_d(c_n^*)$. Then $a_n^*$ is Misiurewicz, and there exists a subsequence $a_n^* \longrightarrow a^* \in \partial \mathcal{CBO}_d$. By the paragraphs above, $a^* = a$, and hence, $a \in \partial \mathcal{CBO}_d$. Now we apply Lemma~\ref{lemm:quasi} again to get $a = a'$.
\section{Injectivity of $\Phi_d$}\label{section:injectivity}
In this section will construct an inverse $\Psi_d:\mathcal{CBO}_d^{(+,-)} \longrightarrow \mathcal{M}_{d+1}$ of $\Phi_d$. \subsection{Dynamics of maps in $\mathcal{CBO}_d^{(+,-)}$}\label{section:imgdynamics} Given $a \in \mathcal{CBO}_d^{(+,-)}$, let $P_{s(a)}$ be the monic representative of $p_a$ as defined in Section~$\ref{section:prelim}$. As in the construction of $\Phi_d$, for $\theta=0,\frac{1}{2}$, let $S_\theta$ be invariant sectors at $0$ with same slope. That is, \begin{align*}
S_0 & =\{\varphi_{s(a)}^{-1}(e^{s+2\pi i t}): s \in (0,\eta), |t| < qs\} \\
S_{\frac{1}{2}} & = \{\varphi_{s(a)}^{-1}(-e^{s+2\pi i t}): s \in (0,\eta), |t| < qs\} \end{align*} Note that $S_{\frac{1}{2}} = -S_0$.
We choose $q$ to be small enough so that $S_0 \cap S_{\frac{1}{2}} = \{0\}$, and the inverse image of each $S_\theta$ under $P_{s(a)}$ consists of exactly $2d+1$ components. The point $0$ has pre-images $\{0=x_0, x_1,x_2,...,x_{2d}\}$ under $P_{s(a)}$, of which $d$ - say $x_1,x_2,...,x_d$, are in $F^L_{s(a)}$, and $d$ are in $F^R_{s(a)}$. Let $S_\theta(x_\ell)$ be the inverse image of $S_\theta$ based at $x_\ell$ for $\ell \neq 0$.
Let $W$ be the region bounded by an equipotential $\{z| G_s(z)=\eta\}$ and define $W_i = P_{s(a)}^{-\circ i}(W)$. For a given $\ell \in \{1,2,...,2d\}$, let $S$ be the connected component of $W_1 \setminus (S_0(x_\ell) \cup S_{\frac{1}{2}}(x_\ell))$ that does not contain $0$. Then $P_{s(a)}$ maps $S$ to $F^L_{s(a)}$ if $S\subset F^R_{s(a)}$, and to $F^R_{s(a)}$ if $S \subset F^L_{s(a)}$.
We have illustrated this in Figure~\ref{fig:inversedefn}.
\subsection{Definition of $\Psi_d$}\label{section:inversedefn} With $a$ as above, \begin{figure}
\caption{Cut and paste surgery on $P_{s(a)}$}
\label{fig:odd_to_uni_quasiregular}
\end{figure} construct the Riemann surface $Y$ as follows: let $Y_0 = W \cap F^L_{s(a)}$, and identify the boundaries $Y_0\cap \mathcal{R}_0(s(a))$ and $Y_0 \cap \mathcal{R}_0(s(a))$ by identifying points on either ray with same speed of escape. Additionally, if necessary, smoothe the boundary of $Y_0$ at the point $w$ as shown in Figure~\ref{fig:odd_to_uni_quasiregular}. $S_0 \cap F^L_{s(a)}$ and $S_{\frac{1}{2}} \cap F^L_{s(a)}$ with this boundary identification become a single sector which we shall call $\widetilde{S}$. We let $Y_1 = P_{s(a)}^{-1}(Y_0)$ with this boundary identification. Clearly, $\overline{Y_1}\subset Y_0$.
Given $\ell \in \{1,2,...,d\}$, let $S$ be the connected component of $Y_1 \setminus (S_0(x_\ell) \cup S_{\frac{1}{2}}(x_\ell))$ that does not contain $0$, and let $S'$ be the component that does. Let $S_\ell = S \cup S_0(x_\ell) \cup S_{\frac{1}{2}}(x_\ell)$ (see Figure~\ref{fig:odd_to_uni_quasiregular}). Pick a quasiconformal homeomorphism $e_\ell: S_\ell \mapsto \widetilde{S}$ that extends to a homeomorphism from $\partial S_\ell$ to $\partial \widetilde{S}$, and coincides with $P_{s(a)}$ on $\partial S_\ell \cap \partial S'$. For example, this can be constructed in a manner similar to $g_c \Big| _{\pi_c\big(S_1^\pm\big(\frac{\ell}{d+1}\big)\big)}$ in Section~\ref{section:defn}. Define \begin{align*}
F: Y_1 &\longrightarrow Y_0\\
F(z) & = \begin{cases}
P_{s(a)}(z) & z \in Y_1 \setminus \bigcup_{\ell=1}^d S_\ell \\
e_\ell(z) & z \in S_\ell \text{ for some }\ell \in \{1,2,...,d\}
\end{cases} \end{align*} $F$ is clearly a quasiregular map of degree $d+1$ with a single critical point. We define an $F-$ invariant complex structure $\sigma$ on $Y_0$ as\\ \begin{itemize}
\item $E_z=\mathbb{S}^1$ if $z \in Y_0 \setminus Y_1$ or the $F-$ orbit of $z$ does not intersect $S_\ell$ for any $\theta,\ell$ \\
\item $E_z = (DF^{n})^{-1}(\mathbb{S}^1)$ if $F^{n}(z)$ is the first point in the orbit of $z$ that is in $S_\ell$\\ \end{itemize} \begin{figure}
\caption{Alternative construction of $\Psi_d(a)$ by renormalization; the
`$*$' marks the critical point $-\sqrt{d}s(a)$ of $P_{s(a)}$}
\label{fig:odd_to_uni_renorm}
\end{figure} Every $F-$ orbit visits $S_\theta(x_i)$ at most once. So, $\sigma$ has bounded dilitation. Note that $F^*\sigma = \sigma$, and thus, $F$ is quasiconformally equivalent to a polynomial-like map $y: V_1 \longrightarrow V$ with degree $d+1$ and a single critical point. The map $y$ is hybrid equivalent to a polynomial of the form $f_c(z) = z^{d+1}+c$. Note that $y$ only determines the affine equivalence class of $f_c$, and thus $c$ is not unique if $d>1$; however, we impose the condition that the identified rays $\mathcal{R}_0(s(a))$ and $\mathcal{R}_{\frac{1}{2}}(s(a))$ are eventually mapped to the same access as the dynamical ray at angle $0$ to $f_c$ (with respect to the B\"{o}ttcher chart where $\frac{\varphi_c(z)}{z} \rightarrow 1$ as $z \rightarrow \infty$). This determines $c$ uniquely. It is clear that $c \in \mathcal{M}_{d+1}$; we therefore define $\Psi_d(a) = c$. \begin{rem} We may also construct $\Psi_d(a)$ by choosing a renormalization of $P_{s(a)}$.
Choose a neighborhood $\mathcal{E}$ of $0$, in which $P_{s(a)}$ is conjugate to $z \mapsto rz$ for some $r \in \mathbb{C}$ with $|r|>1$, small enough so that $\overline{\mathcal{E}}$ does not contain any critical points, and satisfying \begin{align*}
\mathcal{E} \cap S_0 &= S_0 \cap W_i\\
\mathcal{E} \cap S_{\frac{1}{2}} &= S_\frac{1}{2} \cap W_i \end{align*}
Let $\mathcal{V}$ be an open set defined the union of $W_i \cap F^L_{s(a)}$ and $\mathcal{E}$. Then, there exists a connected component $\mathcal{V}'$ of $P_{s(a)}^{-1}(\mathcal{V})$ such that $\overline{\mathcal{V}'} \subset \mathcal{V}$, and $P_{s(a)}\big |_{\mathcal{V}'} : \mathcal{V}' \longrightarrow \mathcal{V}$ is polynomial-like of degree $d+1$ (see Figure~\ref{fig:odd_to_uni_renorm}). This polynomial-like map has a unique critical point at $-\sqrt{d}$, and by the straightening theorem, it is hybrid equivalent to a unicritical degree $d+1$ polynomial.
We can show for an appropriate choice of domains, the map $F$ defined above in the first definition of $\Psi_d(a)$ and $P_{s(a)} \big |_{\mathcal{V}'}$ are hybrid equivalent. \end{rem}
We may use the same methods as in Section~\ref{section:continuity} to show that $\Psi_d$ is continuous. \subsection{$\Psi_d$ is the inverse of $\Phi_d$} Given $c \in \mathcal{M}_{d+1}$, let ${c'} = \Psi_d \circ \Phi_d(c)$. We will follow the construction to show that $f_{c'}$ and $f_c$ are hybrid equivalent, and thus, $c'=c$.
Let $a=\Phi_d(c)$. The construction $a \mapsto \Psi_d(a)$ involves picking the sectors $S_0$ and $S_\frac{1}{2}$ in the dynamical plane of $P_{s(a)}$, constructing a Riemann surface $ Y$, a quasiregular map $F_{s(a)}$, and lastly, a polynomial like map $y_{s(a)}$.
On the other hand, the construction $c \mapsto \Phi_d(c)$ goes through the steps $f_c \mapsto g_c \mapsto G_c \mapsto P_{s(a)}$. We will only be working with the `$-$' copies of $S(\frac{\ell}{d+1}), K_{f_c}$, etc., and so we shall drop the `$-$' superscript. The first step in the construction of $\Phi_d(c)$ uses the quotient map $\pi_c$, and we have \begin{align*}
g_c (\pi_c(z))&= \pi_c(f_c(z)) \text{ away from sectors }\pi_c\Big(S\Big(\frac{\ell}{d+1}\Big)\Big)\\
G_c& = \psi_c \circ g_c \circ \psi_c^{-1}\\
P_{s(a)} & = k_c \circ G_c \circ k_c^{-1} \end{align*} where $\psi_c$ is quasiconformal and $k_c$ is a hybrid equivalence.
\begin{figure}
\caption{Building a conjugacy between $f_c$ and $F_{s(a)}$. The wavily shaded region in the top figure is $S(0)$; it is contained in $\mathcal{S}$ and its two copies map under $k_c \circ \psi_c \circ \pi_c$ to the sectors $S_0$ and $S_1$ respectively, which we cut to make $\tilde{S}$. We define $\hat{\phi}$ on the darkly shaded region on the top to the darkly shaded region at the bottom.}
\label{fig:inverse}
\end{figure} \\In the dynamical plane of $P_{s(a)}$, for $\ell \in \{1,2,...,d\}$, define \begin{align*}
\widetilde{S}_0(x_\ell) & =\Big\{\varphi_{s(a)}^{-1}(e^{r+2\pi i t}): s \in (0,\eta), \Big|t-\frac{\ell}{2d}\Big| < qs\Big\} \\
\widetilde{S}_{\frac{1}{2}}(x_\ell) & =\Big\{\varphi_{s(a)}^{-1}(-e^{s+2\pi i t }): s \in (0,\eta), \Big|t-\frac{\ell}{2d}\Big| < qs\Big\} \end{align*} and let $\widetilde{S}_\ell$ be the union of $\widetilde{S}_0(x_\ell), \tilde{S}_{\frac{1}{2}}(x_\ell)$ and the connected component of $Y_0 \setminus $ $\widetilde{S}_0(x_\ell) \cup \tilde{S}_{\frac{1}{2}}(x_\ell)$ that contains $S_\ell$, as defined in Section~\ref{section:inversedefn}. See Figure~\ref{fig:odd_to_uni_quasiregular} for an illustration of $\widetilde{S}_\ell$. Let $\widetilde{\phi} = k_c \circ \psi_c \circ \pi_c$.
In the dynamical plane of $f_c$, let $S_1\big(\frac{\ell}{d+1}\big)$ , $\ell = 0,1,...,d$, be as defined in Equations~\ref{eqn:s1defn} to \ref{eqn:y1defn} (the equipotential $\eta$ and the slope factor $q$ may be different from the ones used for $P_{s(a)}$). There are two copies of $S(0) = f_c\big(S_1(0)\big)$ in the dynamical plane of $G_c$, but we will pick the copy that eventually gets mapped to a sector that intersects $S_0$. More generally, for a suitable choice of equipotential and slope factor in the $f_c$ - plane, we may assume that the open sets $S\big( \frac{\ell}{d+1}\big) = \omega^\ell S(0)$ are eventually mapped into $\widetilde{S}_\ell$, and that $S(0)$ is eventually mapped to $S_0$ (or to $S_{\frac{1}{2}}$). That is, \begin{align*} S_\ell(c) &=\widetilde{\phi} \Big(S\Big( \frac{\ell}{d+1}\Big)\Big) \subset \widetilde{S}_\ell \text{ for } \ell = 0,1,...,d\\ \tilde{\phi}(V_0) &= Y_0 \setminus S_0(c) \end{align*} where the domain $V_0$ is as defined in Equation~\ref{eqn:vodefn}.
Our strategy will be to set up a quasiconformal map $\phi: V_0 \cup S(0) \longrightarrow Y_0$ that has agrees with $\widetilde{\phi}$ away from certain sectors, and conjugates $f_c$ and $F_{s(a)}$. \\ Let \begin{align*}
V& = \widetilde{\phi}^{-1}(Y_0 \setminus \widetilde{S})\\
V_1 & = f_c^{-1}(V)\\
\mathcal{S} & = V_0 \cup S(0) \setminus V\\
\mathcal{S}\Big(\frac{\ell}{d+1}\Big) & = \widetilde{\phi}^{-1}(\tilde{S}_\ell) \text{ for }\ell = 1,2,...,d \end{align*} \noindent See Figure~\ref{fig:inverse} for details. For all $z \in V_1 \setminus \Big(\mathcal{S} \cup \bigcup_\ell \mathcal{S}\big(\frac{\ell}{d+1}\big)\Big)$, \begin{align*}
F_{s(a)} \circ \widetilde{\phi}(z) &= \widetilde{\phi} \circ f_c(z) \end{align*} Furthermore, with degree one, \begin{align*}
f_c\Big(\mathcal{S}\Big(\frac{\ell}{d+1}\Big) \cap V_1\Big) &= \mathcal{S} \text{ for }\ell = 1,2,...,d\\
f_c(\mathcal{S} \cap V_1) &=\mathcal{S} \end{align*}
For $z \in \mathcal{S}$, $z = f_c(w)$ for $d$ distinct $w \in V_1$. We can assume that $F_{s(a)} \circ \widetilde{\phi}(w)$ does not depend on the choice of preimage $w$, since $\widetilde{\phi}(w) \in S_\ell$ and $F_{s(a)}\big|_{S_\ell}$ depends on the homeomorphisms $e_\ell$ defined as in Section~\ref{section:inversedefn}, which we have freedom in choosing.\\ So we set \begin{align*}
\hat{\phi}(z) & = F_{s(a)} \circ \widetilde{\phi}(w) \end{align*} Define \begin{align*}
\phi&: V_0 \cup S(0) \longrightarrow Y_0\\
\phi(z) & = \begin{cases}
\widetilde{\phi}(z) & z \not \in \mathcal{S}\\
\hat{\phi}(z) & z \in \mathcal{S}
\end{cases} \end{align*} By the discussion above, for all $z \in f_c^{-1}(V_0 \cup S(0))$, \begin{align*}
F_{s(a)} \circ \phi(z) & = \phi \circ f_c(z) \end{align*}
We note that $\pi_c$ changes the angle at $\beta_c$ from $2\pi$ to $\pi$, and has zero dilitation on $K_{f_c} \setminus \{\beta_c\}$. Also note that $\psi_c$ has zero dilitation on $\pi_c(K_{f_c})$.
On the other hand, the cutting procedure in Section~\ref{section:inversedefn} changes the angle $\pi$ made by the boundary of $F^L_{s(a)}$ at $0$ to the angle $2 \pi$ in the plane of $F_{s(a)}$. Lastly, note that $k_c$ has zero dilitation on $\psi_c\circ \pi_c(K_{f_c}) \setminus \{0\}$.
Combined, this information tells us that we have constructed a quasiconformal map $\phi: V_0 \cup S(0) \longrightarrow Y$ that has zero dilitation on $K_c$, and conjugates $f_c$ to $F_{s(a)}$.
Now, if $z \in \phi(K_c)$, a point $F^{\circ n}_{s(a)}$ in the orbit of $z$ cannot be in the interior of $S_\ell$ for any $\ell$. Therefore, the quasiconformal map that conjugates $F_{s(a)}$ to $y_{s(a)}$ has zero dilitation on $\phi(K_c)$. That is, $y_{s(a)}$ and $f_c$ are hybrid equivalent. Thus, $f_{c'}$ and $f_c$ are hybrid equivalent, implying $c=c'$.
In a similar manner, we can show that $p_{\widetilde{a}}$, where $\widetilde{a} = \Phi_d \circ \Psi_d(a)$, is hybrid equivalent to $p_a$. That is, $\Phi_d \circ \Psi_d(a) = a$.
This finishes the proof of Theorem~\ref{thm:maintheorem}.
We will end with a discussion of how the image under $\Phi_d$ fits inside $\mathcal{CBO}_d$. \begin{lemma} $\mathcal{CBO}^{(+,-)}_d$ disconnects $\mathcal{CBO}_d$ into infinitely many components. \end{lemma} \begin{proof} Let $f_c(z) = z^{d+1}+c$ be a polynomial where the orbit of $c$ contains the $\beta-$fixed point where the dynamical ray at angle $0$ lands. There are infinitely many values of $c$ in $\mathcal{M}_{d+1}$ that satisfy this condition - these are precisely the landing points of parameter rays at angles $\frac{i}{d^n}$ for $n\geq 1$ and $0 < i < d^n$. These are included in the set of ``tips'' of $\mathcal{M}_{d+1}$.
Given such a $c$, let $a=\Phi_d(c)$. Then the orbit of both critical points $\pm \sqrt{d}$ of $p_a$ eventually lands on $0$ - that is, there exists $k$ such that $p_a^{\circ k}(\pm \sqrt{d}) = 0$.
In the dynamical plane of the monic representative $P_{s(a)}$, the dynamical rays at angles $0,\frac{1}{2}$ land at $0$. Thus there exist two angles $\theta_1,\theta_2$ such that $(2d+1)^{k-1} \theta_1 \equiv 0 $ and $(2d+1)^{k-1} \theta_2 \equiv \frac{1}{2}$, which both land at the critical value $P_{s(a)}\big(-s(a)\sqrt{d}\big)$. In the parameter plane of $\mathcal{MBO}_d$, the parameter rays at angle $\theta_1,\theta_2$ both land at $s(a)$- which means that $s(a)$ is a cut-point of $\mathcal{MBO}_d$, which is equivalent to saying that $a$ is a cut-point of $\mathcal{CBO}_d$.
Another way to show this is to see that exists $a' \in \mathcal{CBO}_d$ close to $a$ such that $P_{s(a')}^{ \circ k}(\sqrt{d}) \in F^L_{s(a')}$ and $P_{s(a')}^{ \circ k}(-\sqrt{d}) \in F^R_{s(a')}$. That is, the orbits of both critical points eventually ``cross over'' to the other side. So $a' \not \in \mathcal{CBO}_d^{(+,-)}$. \end{proof}
\end{document} | arXiv |
Find the value of $a_2+a_4+a_6+a_8+\dots+a_{98}$ if $a_1, a_2, a_3, \ldots$ is an arithmetic progression with common difference $1$ and \[a_1+a_2+a_3+\dots+a_{98}=137.\]
Let $S = a_1 + a_3 + \dots + a_{97}$ and $T = a_2 + a_4 + \dots + a_{98}$. Then the given equation states that $S + T = 137$, and we want to find $T$.
We can build another equation relating $S$ and $T$: note that \[\begin{aligned} T-S &= (a_2-a_1) + (a_4-a_3) + \dots + (a_{98}-a_{97}) \\ &= \underbrace{1 + 1 + \dots + 1}_{49 \text{ times }} \\ &= 49 \end{aligned}\]since $(a_n)$ has common difference $1$. Then, adding the two equations $S+T=137$ and $T-S=49$, we get $2T=137+49=186$, so $T = \tfrac{186}{2} = \boxed{93}$. | Math Dataset |
Dunce hat (topology)
In topology, the dunce hat is a compact topological space formed by taking a solid triangle and gluing all three sides together, with the orientation of one side reversed. Simply gluing two sides oriented in the opposite direction would yield a cone much like the dunce cap, but the gluing of the third side results in identifying the base of the cap with a line joining the base to the point.
Name
The name is due to E. C. Zeeman, who observed that any contractible 2-complex (such as the dunce hat) after taking the Cartesian product with the closed unit interval seemed to be collapsible. This observation became known as the Zeeman conjecture and was shown by Zeeman to imply the Poincaré conjecture.
Properties
The dunce hat is contractible, but not collapsible. Contractibility can be easily seen by noting that the dunce hat embeds in the 3-ball and the 3-ball deformation retracts onto the dunce hat. Alternatively, note that the dunce hat is the CW-complex obtained by gluing the boundary of a 2-cell onto the circle. The gluing map is homotopic to the identity map on the circle and so the complex is homotopy equivalent to the disc. By contrast, it is not collapsible because it does not have a free face.
See also
• House with two rooms
• List of topologies
References
• Zeeman, E. C. (1964). "On the dunce hat". Topology. 2 (4): 341–358. doi:10.1016/0040-9383(63)90014-4.
| Wikipedia |
Extremal orders of an arithmetic function
In mathematics, specifically in number theory, the extremal orders of an arithmetic function are best possible bounds of the given arithmetic function. Specifically, if f(n) is an arithmetic function and m(n) is a non-decreasing function that is ultimately positive and
$\liminf _{n\to \infty }{\frac {f(n)}{m(n)}}=1$
we say that m is a minimal order for f. Similarly if M(n) is a non-decreasing function that is ultimately positive and
$\limsup _{n\to \infty }{\frac {f(n)}{M(n)}}=1$
we say that M is a maximal order for f.[1]: 80 Here, $\liminf _{n\to \infty }$ and $\limsup _{n\to \infty }$ denote the limit inferior and limit superior, respectively.
The subject was first studied systematically by Ramanujan starting in 1915.[1]: 87
Examples
• For the sum-of-divisors function σ(n) we have the trivial result
$\liminf _{n\to \infty }{\frac {\sigma (n)}{n}}=1$
because always σ(n) ≥ n and for primes σ(p) = p + 1. We also have
$\limsup _{n\to \infty }{\frac {\sigma (n)}{n\ln \ln n}}=e^{\gamma },$
proved by Gronwall in 1913.[1]: 86 [2]: Theorem 323 [3] Therefore n is a minimal order and e−γ n ln ln n is a maximal order for σ(n).
• For the Euler totient φ(n) we have the trivial result
$\limsup _{n\to \infty }{\frac {\phi (n)}{n}}=1$
because always φ(n) ≤ n and for primes φ(p) = p − 1. We also have
$\liminf _{n\to \infty }{\frac {\phi (n)\ln \ln n}{n}}=e^{-\gamma },$
proven by Landau in 1903.[1]: 84 [2]: Theorem 328
• For the number of divisors function d(n) we have the trivial lower bound 2 ≤ d(n), in which equality occurs when n is prime, so 2 is a minimal order. For ln d(n) we have a maximal order ln 2 ln n / ln ln n, proved by Wigert in 1907.[1]: 82 [2]: Theorem 317
• For the number of distinct prime factors ω(n) we have a trivial lower bound 1 ≤ ω(n), in which equality occurs when n is a prime power. A maximal order for ω(n) is ln n / ln ln n.[1]: 83
• For the number of prime factors counted with multiplicity Ω(n) we have a trivial lower bound 1 ≤ Ω(n), in which equality occurs when n is prime. A maximal order for Ω(n) is ln n / ln 2[1]: 83
• It is conjectured that the Mertens function, or summatory function of the Möbius function, satisfies $\limsup _{n\to \infty }{\frac {|M(x)|}{\sqrt {x}}}=+\infty ,$ though to date this limit superior has only been shown to be larger than a small constant. This statement is compared with the disproof of Mertens conjecture given by Odlyzko and te Riele in their several decades old breakthrough paper Disproof of the Mertens Conjecture. In contrast, we note that while extensive computational evidence suggests that the above conjecture is true, i.e., along some increasing sequence of $\{x_{n}\}_{n\geq 1}$ tending to infinity the average order of ${\sqrt {x_{n}}}|M(x_{n})|$ grows unbounded, that the Riemann hypothesis is equivalent to the limit $\lim _{x\to \infty }M(x)/x^{{\frac {1}{2}}+\varepsilon }=0$ being true for all (sufficiently small) $\varepsilon >0$.
See also
• Average order of an arithmetic function
• Normal order of an arithmetic function
Notes
1. Tenenbaum, Gérald (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge studies in advanced mathematics. Vol. 46. Cambridge University Press. ISBN 0-521-41261-7.
2. Hardy, G. H.; Wright, E. M. (1979). An Introduction to the Theory of Numbers (5th ed.). Oxford: Clarendon Press. ISBN 0-19-853171-0.
3. Gronwall, T. H. (1913). "Some asymptotic expressions in the theory of numbers". Transactions of the American Mathematical Society. 14 (4): 113–122. doi:10.1090/s0002-9947-1913-1500940-6.
Further reading
• Nicolas, J.-L. (1988). "On Highly Composite Numbers". In Andrews, G. E.; Askey, R. A.; Berndt, B. C.; Ramanathan, K. G. (eds.). Ramanujan Revisited. Academic Press. pp. 215–244. ISBN 978-0-12-058560-1. A survey of extremal orders, with an extensive bibliography.
| Wikipedia |
5.5 More About Heat Engines
[ "article:topic", "authorname:crowellb", "license:ccbysa", "showtoc:no" ]
\(\require{cancel}\)
Physical Constants
Units and Conversions
College Physics
Book: Conceptual Physics (Crowell)
5: Thermodynamics
Contributed by Benjamin Crowell
Professor (Physics) at Fullerton College
So far, the only heat engine we've discussed in any detail has been a fictitious Carnot engine, with a monoatomic ideal gas as its working gas. As a more realistic example, figure l shows one full cycle of a cylinder in a standard gas-burning automobile engine. This four-stroke cycle is called the Otto cycle, after its inventor, German engineer Nikolaus Otto. The Otto cycle is more complicated than a Carnot cycle, in a number of ways:
The working gas is physically pumped in and out of the cylinder through valves, rather than being sealed and reused indefinitely as in the Carnot engine.
The cylinders are not perfectly insulated from the engine block, so heat energy is lost from each cylinder by conduction. This makes the engine less efficient that a Carnot engine, because heat is being discharged at a temperature that is not as cool as the environment.
Rather than being heated by contact with an external heat reservoir, the air-gas mixture inside each cylinder is heated by internal combusion: a spark from a spark plug burns the gasoline, releasing heat.
The working gas is not monoatomic. Air consists of diatomic molecules (\(\text{N}_2\) and \(\text{O}_2\)), and gasoline of polyatomic molecules such as octane (\(\text{C}_8\text{H}_{18}\)).
The working gas is not ideal. An ideal gas is one in which the molecules never interact with one another, but only with the walls of the vessel, when they collide with it. In a car engine, the molecules are interacting very dramatically with one another when the air-gas mixture explodes (and less dramatically at other times as well, since, for example, the gasoline may be in the form of microscopic droplets rather than individual molecules).
This is all extremely complicated, and it would be nice to have some way of understanding and visualizing the important properties of such a heat engine without trying to handle every detail at once. A good method of doing this is a type of graph known as a P-V diagram. As proved in homework problem 2, the equation \(dW=Fdx\) for mechanical work can be rewritten as \(dW=PdV\) in the case of work done by a piston. Here \(P\) represents the pressure of the working gas, and \(V\) its volume. Thus, on a graph of \(P\) versus \(V\), the area under the curve represents the work done. When the gas expands, \(dx\) is positive, and the gas does positive work. When the gas is being compressed, \(dx\) is negative, and the gas does negative work, i.e., it absorbs energy.
a / P-V diagrams for a Carnot engine and an Otto engine.
Notice how, in the diagram of the Carnot engine in the top panel of figure a, the cycle goes clockwise around the curve, and therefore the part of the curve in which negative work is being done (arrowheads pointing to the left) are below the ones in which positive work is being done. This means that over all, the engine does a positive amount of work. This net work equals the area under the top part of the curve, minus the area under the bottom part of the curve, which is simply the area enclosed by the curve. Although the diagram for the Otto engine is more complicated, we can at least compare it on the same footing with the Carnot engine. The curve forms a figure-eight, because it cuts across itself. The top loop goes clockwise, so as in the case of the Carnot engine, it represents positive work. The bottom loop goes counterclockwise, so it represents a net negative contribution to the work. This is because more work is expended in forcing out the exhaust than is generated in the intake stroke.
To make an engine as efficient as possible, we would like to make the loop have as much area as possible. What is it that determines the actual shape of the curve? First let's consider the constant-temperature expansion stroke that forms the top of the Carnot engine's P-V plot. This is analogous to the power stroke of an Otto engine. Heat is being sucked in from the hot reservoir, and since the working gas is always in thermal equilibrium with the hot reservoir, its temperature is constant. Regardless of the type of gas, we therefore have \(PV=nkT\) with \(T\) held constant, and thus \(P\propto V^{-1}\) is the mathematical shape of this curve --- a \(y=1/x\) graph, which is a hyperbola. This is all true regardless of whether the working gas is monoatomic, diatomic, or polyatomic. (The bottom of the loop is likewise of the form \(P\propto V^{-1}\), but with a smaller constant of proportionality due to the lower temperature.)
Now consider the insulated expansion stroke that forms the right side of the curve for the Carnot engine. As shown on page 324, the relationship between pressure and temperature in an insulated compression or expansion is \(T \propto P^b\), with \(b=2/5\), 2/7, or 1/4, respectively, for a monoatomic, diatomic, or polyatomic gas. For \(P\) as a function of \(V\) at constant \(T\), the ideal gas law gives \(P\propto T/V\), so \(P\propto V^{-\gamma}\), where \(\gamma=1/(1-b)\) takes on the values 5/3, 7/5, and 4/3. The number \(\gamma\) can be interpreted as the ratio \(C_P/C_V\), where \(C_P\), the heat capacity at constant pressure, is the amount of heat required to raise the temperature of the gas by one degree while keeping its pressure constant, and \(C_V\) is the corresponding quantity under conditions of constant volume.
Example 22: The compression ratio
Operating along a constant-temperature stroke, the amount of mechanical work done by a heat engine can be calculated as follows:
\[\begin{align*} PV &= nkT \\ \text{Setting $c=nkT$ to simplify the writing,} P &= cV^{-1} \\ W &= \int_{V_i}^{V_f} P dV \\ &= c \int_{V_i}^{V_f} V^{-1} dV \\ &= c \ln V_f - c \ln V_i \\ &= c \ln (V_f/V_i) \end{align*}\]
The ratio \(V_f/V_i\) is called the compression ratio of the engine, and higher values result in more power along this stroke. Along an insulated stroke, we have \(P\propto V^{-\gamma}\), with \(\gamma\ne1\), so the result for the work no longer has this perfect mathematical property of depending only on the ratio \(V_f/V_i\). Nevertheless, the compression ratio is still a good figure of merit for predicting the performance of any heat engine, including an internal combustion engine. High compression ratios tend to make the working gas of an internal combustion engine heat up so much that it spontaneously explodes. When this happens in an Otto-cycle engine, it can cause ignition before the sparkplug fires, an undesirable effect known as pinging. For this reason, the compression ratio of an Otto-cycle automobile engine cannot normally exceed about 10. In a diesel engine, however, this effect is used intentionally, as an alternative to sparkplugs, and compression ratios can be 20 or more.
Example 23: Sound
b / Example 23.
Figure b shows a P-V plot for a sound wave. As the pressure oscillates up and down, the air is heated and cooled by its compression and expansion. Heat conduction is a relatively slow process, so typically there is not enough time over each cycle for any significant amount of heat to flow from the hot areas to the cold areas. (This is analogous to insulated compression or expansion of a heat engine; in general, a compression or expansion of this type, with no transfer of heat, is called adiabatic.) The pressure and volume of a particular little piece of the air are therefore related according to \(P\propto V^{-\gamma}\). The cycle of oscillation consists of motion back and forth along a single curve in the P-V plane, and since this curve encloses zero volume, no mechanical work is being done: the wave (under the assumed ideal conditions) propagates without any loss of energy due to friction.
The speed of sound is also related to \(\gamma\). See example 13 on p. 375.
Example 24: Measuring \(\gamma\) using the "spring of air"
c / Example 24.
Figure c shows an experiment that can be used to measure the \(\gamma\) of a gas. When the mass \(m\) is inserted into bottle's neck, which has cross-sectional area \(A\), the mass drops until it compresses the air enough so that the pressure is enough to support its weight. The observed frequency \(\omega\) of oscillations about this equilibrium position \(y_\text{o}\) can be used to extract the \(\gamma\) of the gas.
\[\begin{align*} \omega^2 &= \frac{k}{m} \\ &= \left.-\frac{1}{m}\:\frac{dF}{dy}\right|_{y_\text{o}} \\ &= \left.-\frac{A}{m}\:\frac{dP}{dy}\right|_{y_\text{o}} \\ &= \left.-\frac{A^2}{m}\:\frac{dP}{dV}\right|_{V_\text{o}} \end{align*}\]
We make the bottle big enough so that its large surface-to-volume ratio prevents the conduction of any significant amount of heat through its walls during one cycle, so \(P\propto V^{-\gamma}\), and \(dP/dV=-\gamma P/V\). Thus,
\[\begin{align*} \omega^2 &= \gamma\frac{A^2}{m}\:\frac{P_\text{o}}{V_\text{o}} \end{align*}\]
Example 25: The Helmholtz resonator
When you blow over the top of a beer bottle, you produce a pure tone. As you drink more of the beer, the pitch goes down. This is similar to example 24, except that instead of a solid mass \(m\) sitting inside the neck of the bottle, the moving mass is the air itself. As air rushes in and out of the bottle, its velocity is highest at the bottleneck, and since kinetic energy is proportional to the square of the velocity, essentially all of the kinetic energy is that of the air that's in the neck. In other words, we can replace \(m\) with \(AL\rho\), where \(L\) is the length of the neck, and \(\rho\) is the density of the air. Substituting into the earlier result, we find that the resonant frequency is
\[\begin{align*} \omega^2 &= \gamma\frac{P_\text{o}}{\rho}\:\frac{A}{LV_\text{o}} . \end{align*}\]
This is known as a Helmholtz resonator. As shown in figure d, a violin or an acoustic guitar has a Helmholtz resonance, since air can move in and out through the f-holes. Problem 10 is a more quantitative exploration of this.
d / The resonance curve of a 1713 Stradivarius violin, measured by Carleen Hutchins. There are a number of different resonance peaks, some strong and some weak; the ones near 200 and 400 Hz are vibrations of the wood, but the one near 300 Hz is a resonance of the air moving in and out through those holes shaped like the letter F. The white lines show the frequencies of the four strings.
We have already seen, based on the microscopic nature of entropy, that any Carnot engine has the same efficiency, and the argument only employed the assumption that the engine met the definition of a Carnot cycle: two insulated strokes, and two constant-temperature strokes. Since we didn't have to make any assumptions about the nature of the working gas being used, the result is evidently true for diatomic or polyatomic molecules, or for a gas that is not ideal. This result is surprisingly simple and general, and a little mysterious --- it even applies to possibilities that we have not even considered, such as a Carnot engine designed so that the working "gas" actually consists of a mixture of liquid droplets and vapor, as in a steam engine. How can it always turn out so simple, given the kind of mathematical complications that were swept under the rug in example 22? A better way to understand this result is by switching from P-V diagrams to a diagram of temperature versus entropy, as shown in figure e.
e / A T-S diagram for a Carnot engine.
An infinitesimal transfer of heat \(dQ\) gives rise to a change in entropy \(dS=dQ/T\), so the area under the curve on a T-S plot gives the amount of heat transferred. The area under the top edge of the box in figure e, extending all the way down to the axis, represents the amount of heat absorbed from the hot reservoir, while the smaller area under the bottom edge represents the heat wasted into the cold reservoir. By conservation of energy, the area enclosed by the box therefore represents the amount of mechanical work being done, as for a P-V diagram. We can now see why the efficiency of a Carnot engine is independent of any of the physical details: the definition of a Carnot engine guarantees that the T-S diagram will be a rectangular box, and the efficiency depends only on the relative heights of the top and bottom of the box.
Benjamin Crowell (Fullerton College). Conceptual Physics is copyrighted with a CC-BY-SA license.
5.4 Entropy As a Microscopic Quantity
5.E: Thermodynamics (Exercises)
Section or Page
Ben Crowell
© Copyright 2019 Physics LibreTexts | CommonCrawl |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.